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An integer valued Hausdorff-like metric

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An integer valued Hausdorff-like metric
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Digitized images ( jstor )
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Hamming distances ( jstor )
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Thesis (Ph. D.)--University of Florida, 1980.
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Bibliography: leaves 64-65.
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Typescript.
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Vita.
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by Carolyn Roche Johnson.

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AN INTEGER VALUED HAUSDORFF-LIKE METRIC


BY

CAROLYN ROCHE JOHNSON













A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY





UNIVERSITY OF FLORIDA


1980



































TO MY FAMILY
AND ESPECIALLY TO KARL















ACKNOWLEDGEMENTS


The author would like to express her gratitude to all those who contributed, directly or indirectly, to the completion of this work.


Above all, her sincere thanks go to the chairman of her supervisory committee, Dr. A. R. Bednarek, for his guidance throughout her graduate studies. This study was made possible by his many suggestions and intuitions.


The author wishes to acknowledge the remainder of her supervisory committee: Dr. B. B. Baird, Dr. M. P. Hale, Dr. S. Y. Su, and Dr. S. M. Ulam for their contributions to her academic training. She also wishes to acknowledge Dr. J. E. Keesling and Dr. R. E. Osteen for their comments and suggestions.


Finally, she wishes to thank her husband, Karl, whose encouragement and love have endured her throughout her graduate studies.


iii















TABLE OF CONTENTS


PAGE

ACKNOWLEDGEMENTS ...... ............................... iii

ABSTRACT ...... ....................................... V

INTRODUCTION ......................................... 1

CHAPTER

I BASIC DEFINITIONS AND ELEMENTARY PROPERTIES
OF THE HAUSDORFF METRIC ........................ 4

II A DISCRETE ANALOGUE ............................ 10

III TOPOLOGICAL CONNECTION ......................... 18

IV APPLICATION TO DIGITIZED GREY LEVEL IMAGES ..... 35

V A SHARPENING TRANSFORMATION FOR GRADED
PATTERNS ....................................... 42

VI SOME COMPUTATIONAL EXPERIMENTS ................. 50

VII POSSIBLE APPLICATIONS AND PROBLEMS ................ 59

REFERENCES ........................................... 64

APPENDIX ............................................. 67

BIOGRAPHICAL SKETCH .................................. 80


iv















Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


AN INTEGER VALUED HAUSDORFF-LIKE METRIC By

Carolyn Roche Johnson

August 1980


Chairman: Dr. A. R. Bednarek Major Department: Mathematics


Motivated by the need for a non-Euclidean metric between sets of objects and using the Hausdorff metric as a prototype, an integer valued discrete analogue of the Hausdorff metric is presented. Basic properties common to both metrics are examined, and a topological relationship is established. The Hausdorff space of nonempty subsets of the unit square is shown to be homeomorphic to the completion of the direct limit of a sequence of finite spaces having the discrete Hausdorff distance as a metric.

The potential usefulness of the discrete Hausdorff

metric in pattern recognition as applied to graded patterns, or digitized grey level images, is examined and an algorithm for computing the distance between two graded patterns is presented. Also, a nonlinear neighborhood dependent


v











sharpening transformation particularly suited to graded patterns is presented and an alternative proof of convergence of the sharpening procedure is provided. Several computations studying the interaction of the discrete Hausdorff metric with the sharpening transformation are reported.


vi














INTRODUCTION


The evolution of the discipline of mathematical taxonomy

7 1 has increased the need for sensitive measures of similarity or distances between objects or classes of objects. The objects of interest may be biological sequences, digitized grey level images, individuals of a population, or elements of abstract point sets. Traditionally classification involved associating the objects to be classified with some n-dimensional vector in a Euclidean space, employing Euclidean distance as the measure of similarity and then clustering. The need for a non-Euclidean metric between sets of objects (clusters) was expressed in Jardine and Sibson [7 ].


The concept of distances between subsets of a metric

space was first examined systematically by F. Hausdorff [6 1. In particular, for nonempty closed subsets A and B of a compact metric space (X,d) the Hausdorff distance p H between A and B is defined to be the maximum of the two values max min d(x,y) and max min d(x,y). Using the Hausdorff xEA yEB xEB yEA

distance as a prototype, and motivated by the need for a nonEuclidean intertaxal distance, Bednarek and Smith [1 1 introduced an integer valued discrete analogue of the


1






2




Hausdorff metric. The present work is devoted to a more detailed examination of this discrete Hausdorff metric and, in particular, to an examination of its potential for application in pattern recognition problems.


In Chapters I and II we discuss the original Hausdorff metric, the discrete analogue and their basic properties.


Chapter III contains one of the main results, namely, that the Hausdorff space of nonempty closed subsets of the unit square is the completion of the direct limit of a sequence of finite metric spaces having the discrete Hausdorff distance as a metric. This, in part, justifies the terminology and the use of the discrete metric for the study of digitized pictures.


A particularization, first suggested by Bednarek and Ulam [2 ], to the case of graded patterns (digitized grey level images) is examined more closely in Chapter IV. In particular, we provide an algorithm for the computation of the distance between two graded patterns and illustrate its application in the Appendix.


In [9 1, Kramer and Bruchner introduced a nonlinear neighborhood dependent sharpening transformation particularly applicable to the patterns considered in Chapter IV. In Chapter V we provide an alternate proof of the convergence of this sharpening procedure.





3


We feel that the relationships between our metric and the Kramer-Bruchner sharpening transformation, both being highly neighborhood dependent and both being potentially applicable to the processing and recognition of digitized images, were worth examining. Toward this end some computations were carried out and their results are reported in Chapter VI. While not definitive, these studies suggest some possible applications and additional investigations. Some of these are delineated in Chapter VII.


This work was supported in part by NSF Grant No. MSC 75-21130.














CHAPTER I

BASIC DEFINITIONS AND ELEMENTARY PROPERTIES OF THE HAUSDORFF METRIC


Let X be a set and let R+ denote the nonnegative reals.


Definition 1.1. A function d from the cartesian product X x X into R+ satisfying the following conditions for all elements x, y, and z of X is called a metric:

(1) d(x,y) = 0 implies that x = y

(2) d(x,x) = 0

(3) d(x,y) = d(y,x)

(4) d(x,z) d(x,y) + d(y,z).

The pair (X,d) is called a metric space.


Let (X,d) be a compact metric space, and let 2X denote

the family of all nonempty closed subsets of X. For elements
X
A and B of 2X, defining


PH(A,B) max {max min d(x,y),max min d(x,y)} xEA yEB XEB yEA


we obtain the well-known metric on 2X introduced by Hausdorff [ 6 ]. A topologically equivalent definition of the Hausdorff metric is given by pH', where


PH'(A,B) = max min d(x,y) + max min d(x,y).
xEA yEB xEB yeA


4






5


When it is necessary to identify the metric d underlying the Hausdorff metric we will write pH instead of pH'


Let B be an element of 2 X, and x an element of X, where (X,d) is a compact metric space. Definition 1.2. The distance from x to B, d(x,B) is given by d(x,B) = inf fd(x,b): b E B}. For any positive number r, define U(B,r) = fx E X: d(x,B) < r}.


If we let N r(b) denote the usual neighborhood about b of radius r, that is, N r(b) = fx E X: d(x,b) < r}, then we observe that U(B,r) = ufN r(b): b E B).


An equivalent and perhaps more intuitive definition of PH (A,B) is given by


PH(AB) = inf fr: A c U(B,r) and B c U(A,r)}.

Note that the value of the infimum is not always realized. That is, if pH(A,B) = r, then it is possible that A is not a subset of U(B,r) or B is not a subset of U(A,r), as these are open sets. See Example 1.7 for an illustration of this situation.


Intuitively, the Hausdorff distance between sets A and B can be viewed as the smallest number c such that an "e-expansion" of the sets A and B will lead to mutual absorption, as shown in Figure I-1.






6


/
/



/ A


I



B


Figure I-1


Proposition 1.3. If A and B are elements of 2X and A c B, then U(A,r) c U(B,r).


Proof. This follows from the fact that if A c B, then inf {d(x,b): b E B} inf fd(x,a): a E A}. Proposition 1.4. For A, B, C, and D elements of 2 P H(A u B,C u D) max {pH(A,C),pH(A,D),pH(B,C),pH(B,D)}. Proof. Let p H (A,C) = q, p H (A,D) = r, p H (B,C) = s, pH(B,D) = t, and let m = max {q,r,s,t}. Let c be a positive number. By definition, A c U(C,q + e) c U(C,m + c) c U(C u D,m + c) and B c U(C,s + e) c U(C,m + c) c U(C u D,m + c), so that A u B c U(C u D,m + c). Analogously, C u D c U(A u B,m + c).


I





7


Then pH(A u B,C u D) m + E for every positive number c, which implies that pH(A u B,C u D) 5 m.


The following example shows that strict inequality is possible.


Example 1.5. Let X = [0,1] and d(x,y) = Ix - yI for elements x and y of X. Let A = [0,1/4], B = [1/2,3/4], C = [1/4,1/2], and D = [3/4,1]. Then pH(A,C) = 1/4, pH(A,D) = 3/4, PH(B,C) = 1/4, pH(B,D) = 1/4, and pH(A U B,C U D) = 1/4 < max { H(A,C),pH(AD),pH(BC),pH(BD)) = 3/4. Corollary 1.6. For elements A and B of 2X, PH(A u B,A)

PH(A,B).


Example 1.7. An example of strict inequality is given by letting X = [0,1] with the usual metric; i.e., d(x,y) Ix - yj. Let A = {l/21 u'{l/2 + (1/2)k: k = 3,4,...} {l/2,5/8,9/16,17/32,...} and B = {l}. Then clearly PH(A,B) = 1/2, while pH(A u B,A) = 3/8 because d(l,A) = inf {d(l,a): a E Al = inf {l/2,3/8,7/16,15/32,...} = 3/8. Corollary 1.8. For elements A, B, and C of 2 PH (A u B,C) max {p H(AC),p H(B,C)}. Proposition 1.9. For elements A, B, and C of 2X PH(A u B,C) p H(A,B) + pH(B,C).


Proof. Let pH(A,B) = r, pH(B,C) = s, and c > 0.


Then






8


A c U(B,r + 6), B c U(C,s + 6) c U(C,r + s + 2c), and C c U(B,s + c) c U(A u B,r + s + 2c). This implies that for every a E A there exists a b E B such that d(a,b) < r + c, and for every b E B there exists a c E C such that d(b,c) < s + e. So d(a,c) d(a,b) + d(b,c) < r + s + 2E, which implies that A u B c U(C,r + s + 2E). Thus, PH(A u B,C) r + s + 2E, but c was arbitrary, so that PH (A u B,C) r + s.

Definition 1.10. If X and Y are topological spaces, then a continuous map f from X onto Y which is one-to-one and such that f~ is also continuous is called a homeomorphism. Definition 1.11. Let (X,d1) and (X2'd2) be metric spaces. An isometry is a map f from X onto X2 such that d1(x,y) = d2(f(x),f(y)) for all elements x and y of X .


Note that every isometry is a homeomorphism. In particular, if (X,d) is a compact metric space, then the map f: x - {xl is an isometry of X onto a subspace of 2


An interesting property of the Hausdorff metric, though somewhat disappointing from a topological standpoint, is that the topology of the underlying space (X,d) does not determine the topology of (2 ,pH ). That is, two metrics d and d2
d
can generate the same topological spaces (X,d1) and (X,d2) while the spaces (2Xp'Hd ) and (2XPH d) can be different. For an example of this, see Kelley [ 8 ].






9




Borsuk [ 3 1 also points out that pH does not measure the difference in topological structure of sets A and B. Thus, p H (A,B) can be arbitrarily small although the topological structures of A and B are quite different. For a broader discussion of the topological properties of 2 , the reader is referred to [11].














CHAPTER II

A DISCRETE ANALOGUE


In this chapter we introduce an integer valued metric

which is analogous to the Hausdorff metric. Basic properties and similarities between it and the Hausdorff metric are examined. Several examples of the metric are included. This metric was first introduced in a paper by Bednarek and Smith [ 1], and the description which follows is found in [1].


Let X be a set with cardinality lxi equal to n. For most purposes, we only consider the case in which X is finite. We assume that with every x in X there is associated a unique nonempty subset N(x) of X called the neighborhood of x. The only restriction that we place on N(x) is that every point x is contained in its neighborhood, x E N(x).


For subsets A of X, define E(A) to be the set

E(A) = ufN(a): a E A}. Recursively, define E (A) = E(E(A)), and in general, E k+l(A) = E(E k(A)) for any positive integer k. Define E 0(A) to be just A itself, E 0(A) = A, and note that A = E 0(A) c E(A) by definition. Moreover,


E0(A) c E1(A) c E (A) c ...cEk A) cE k+l(A)c.... We observe that the operator E is monotone, that is, if


10






11


A c B then E(A) c E(B). We also note that E is additive, that is, E(A U B) = E(A) u E(B).


We now define an integral metric on the nonempty

subsets of X. Let A and B be two nonempty subsets of X and define the distance between A and B, p(A,B), by min {k: A c Ek(B) and B c E k(A)} p(A,B) = {
IX! otherwise


and note that p(A,B) = JXI if and only if there is no positive integer k such that A c Ek(B) and B c E k(A).


Theorem 2.1. If X is finite, then p is a metric on the nonempty subsets of X.


Proof. By definition p(A,B) is a nonnegative integer. If p(A,B) = 0 then A c E 0(B) = B and B c E 0(A) = A, which implies that A = B. Conversely, if A = B then A c B = E 0(B) and B c A = E 0(A), so that p(A,B) = 0. The symmetry of p, p(A,B) = p(B,A), follows directly from the definition. In order to prove the triangle inequality for p, let A, B, and C be nonempty subsets of X. We wish to show that p(A,B)

p(A,C) + p(C,B). If either p(A,C) = n or p(C,B) = n,

where IXI = n, the result is true. Suppose that p(A,C) =j, P(C,B) = k, and that neither j nor k is equal to n. Then A c E (C) and C C Ek (B) so that A c Ej(Ek (B)) = Ej+k (B). Also, B c Ek (C) and C c E (A) so that B c E k(E(A)) = Ej+k(A).






12


By definition of p this implies that p(A,B) 5 j + k = p(A,C) + p(C,B), and triangularity is established.


For the case in which X is infinite, we obtain an

extended integer valued distance between sets A and B which fails to be a proper metric only in the sense that it can take on the value infinity.


Corollary 2.2. If X is infinite, p is an extended integer valued metric on the nonempty subsets of X.


We shall refer to this metric as either p or the

discrete Hausdorff metric throughout further discussions. We now give several particular examples of the metric p. Example 2.3. A graph G is a nonempty finite set of points (or vertices), V, together with a set E of unordered pairs of distinct points of V, called edges. We say two points v, w are edged if (v,w) e E. We write G = (V,E).


Suppose G = (V,E) is a graph and let X = V. For points x in X, define the neighborhood of x, N(x), to be the set consisting of x and all those points edged with x; that is, N(x) = {x} u {y: (x,y) e El. Then p is a metric on graphs. In particular, consider the example given in Figure II-i. In Figure II-1, B c E (A) and A c E (B) = X, so p(A,B) = 4.






13


Figure II-i




Example 2.4. Let N and N2 be positive integers and let X be the cells of an N1 x N2 grid. Formally, X = {(i,j): 1 i : N ,1 ! j ! N2}. For each cell x let N(x) be a cruciform neighborhood consisting of those cells to the left, right, above, and below x, as shown in Figure 11-2.




N(x) =




Figure 11-2



N(x) can be defined formally by N(x) = {(h,k) E X: Ih - il + Ik - jf 11. An example of the expansion of a set A by the operator E is shown in Figure 11-3. The points of A are indicated by * and the points of Ek (A) - Ek-l (A) are indicated by the number k for k = 1,2,3,... ,7. Note that E14 (A) = X for all nonempty sets A of X if N = N2 =8.






14


5 4 3 2 1 2 3 4
4 3 2 1 * 1 2 3
3 2 1 * * 12 3
3 2 1 * * 12 3
4 3 2 1 * * 1 2
5 4 3 2 1 1 2 3
6 5 4 3 2 2 3 4
7 6 5 4 3 3 4 5


Figure 11-3



Many properties of the Hausdorff metric also hold for the discrete Hausdorff metric. The following sequel is a partial list of these analogous properties. Because the proofs for the case in which X is infinite are the same as those for the finite case, we shall only consider sets X with cardinality JlX = n where n is finite.


Proposition 2.5. Let A, B, C, and D be nonempty subsets of X. Then p(A u B,C u D) max fp(A,C),p(A,D),p(B,C),p(B,D)}.


Proof. Let p(A,C) = h, p(A,D) = i, p(B,C) = j, and p(B,D) = k. If h, i, j, or k equals n, then the result follows as p(A u B,C u D) n by definition. Suppose h, i, j, and k are all less than n. Let m = max {h,i,j,k}. Then A c E h(C) c Em (C) c Em (C u D) and B c E (C) c Em(C) c Em (C u D), so that A u B m C u D). Similarly, C c E m(A)

c Em (A u B) and D c Em (A) c Em(A u B), so C u D c Em (A u B). Thus p(A u B,C u D) m.





15


By letting C = D = A, we obtain the following corollary.


Corollary 2.6. Let A and B be nonempty subsets of X. Then p(A u B,A) p(A,B).


Example 2.7. A simple example showing that strict inequality can hold in Corollary 2.6 is shown in Figure 11-4, where the cells of set A are indicated by a and the cells of B by b. Cruciform neighborhoods (see Example 2.4) are used in this example to compute p(A,B). In this example, p(A u B,A) = 1 < 4 = p(A,B).




b a a
a a a




Figure 11-4



Proposition 2.8. Let A, B, and C be nonempty subsets of X. Then p(A u B,C) p(A,B) + p(B,C). Proof. Let p(A,B) = j and p(B,C) k. If either j = n or k = n, then the result follows, so assume j and k are not equal to n. Then A c E (B) and B k C) so that

A u B c E (B) c E-(Ek C)) = j+k C). Also, C c Ek B) and B c E (A) so C c Ek (E(A)) = Ej+k (A) c Ej+k(A u B). Thus p(A u B,C) : j + k.






16


At this point it would seem that increasing the cardinality of sets decreases the resulting metric value. This is shown to be false in Figure 11-5 where A c B but p(B,C) > p(A,C). The cells of A are indicated by *, the cells of B by E, and the cells of C by #. In this example p(B,C) = 3 while p(A,C) = 1, again using cruciform neighborhoods.









Figure 11-5



The next proposition is an observation of how the

choice of neighborhoods affects the value of the resulting metrics. Suppose that for every point x in X there are associated two neighborhoods N 1(x) and N2(x). Associated with these neighborhoods are two metrics p and p2 respectively.


Proposition 2.9. If N1(x) c N2(x) for every x E X, then P2(A,B) p1(A,B) for all nonempty subsets A and B of X. Proof. Let p 1(A,B) = j, and note that if j = n the result is immediate, so assume that j < n. Let E. (A) = ufN (a): a E A} for i = 1,2, and in general, E (A) =
k-i
E (E . (A)) for i = 1,2. Then E (A) c E (A) for all





17



nonnegative integers k and any nonempty set A since N 1(a) is a subset of N2(a) for all a E A. This implies that A c E3(B) c ED(B), and conversely that B c E3(A) c Ea(A), so that p2(A,B) 5 j = p 1(A,B).














CHAPTER III

TOPOLOGICAL CONNECTION


The definition of the metric p suggests a strong connection between it and the Hausdorff metric. In order to relate the two it is necessary to decide the neighborhoods of points in the discrete case and the underlying metric in the Hausdorff case. This chapter will establish a topological relationship between the unit square with the Minkowski metric and an appropriate choice of neighborhoods for the discrete version.


Let I denote the unit interval and consider the metric

space (I 2,d) where d((x1,x2),(yl' )) = max {1x. - y I:i = 1,2}. Let A denote the family of all nonempty closed subsets of I2. Then the Hausdorff metric pH defines a distance between elements of A.


For all positive integers n, let X be the set of all ordered pairs of binary sequences of length n; that is, Xn = {(a1...a nb1...bn): ab i = 0 or 1 for i = 1,2,...,n}. For simplicity of notation, denote elements of X
n
(a.. .an,b.. .bn) by (a.,b )n. Let An be the collection of all nonempty subsets of Xn for all positive integers n.


18






19


For points (a.,b.) of Xn , define the neighborhood to i i nn
be N((a.,b ) ) = f(u,v )n E X n: la ...an(2) - u .. un(2) s 1 and b 1...bnv n( ln(2) I l}, where a .. an(2)

denotes the integer equivalent to the base 2 number a1... an Let pn be the discrete Hausdorff metric defined on An x An with the above definition of neighborhoods. We now have a sequence of metric spaces (An'On) for n 1.


Observe that the binary sequences uniquely determine numbers L and M such that a1..an(2) = L, b ..bn(2) M, and 0 L,M ! 2 n - 1. Thus, there is a natural one-to-one correspondence between nonempty subsets of X and certain closed subsets of the unit square given by


D : A + A: n n
{(a.,b.) I - [L/2n (L+l)/2 n] x [M/2n (M+1)/2 n.
i i n

Define Dn of a general element of An by distributing over unions in the obvious manner. Consequently, there is a one-to-one correspondence between the elements of A and
2
certain closed subsets of I2. Applying n to neighborhoods
n
of points in X gives rise to the following correspondence:


Sn(N((a ,b )n =(M..)/ 2 : (L-l)/2n < x (L+2)/2n
(M-1)/2n < y (M+2)/2n1.


If (a.,b )n is an element of Xn , denote Ek (f(a,b )n 1) by E (a.,b ) for all nonnegative integers k. Then n i in





20


D (Ek (a,b.) ) = f(x,y) E I2 (L-k)/2n < x : (L+k+l)/2n n n ii n
(M-k)/2n < y (M+k+l)/2 n1.


Now let A be an element of A and define the n-th encoding of A, e n(A), by


e (A) = u{(a.,b )n e X A n D ((a.,b )n is nonemptyl.


Note that e (A) is an element of A . Denote n (e (A)) by (nenA.
n n

The n-th encoding of a set A can be interpreted as

placing a 2n x 2n grid over the unit square, and the cells of the grid which intersect A are the encoding. An illustration of this situation is given in Figure III-1.


One of the main results of this section is that for elements A and B of A, pH(A,B) = lim n (n (A),en(B)) n-o 2n

The following lemmas will be used to establish this result.


Lemma 3.1. pH(A,n e A) 1/2n


Proof. By the definition of e (A), A is a subset of D e A.
n n n Let x be an element of n e A. By the one-to-one correspondence and the definition of e (A), there is an element a of
n n
A such that d(a,x) : 1/2n. It follows that d(x,A) 1/2 and the result follows.



























e1(A) X


e3(A)


X3


Figure III-1


21


A


2 (A)


x2






22


Lemma 3.2. For elements A and B of A, if pH ne A,n e B)
nnn
k/2n, then p n (A),e (B)) k.


Proof. p H n e nA,n e nB) k/2n if and only if for every element a of nenA there is a b' in nenB so that d(a,b')

k/2n, and for every b in (nenB there is an a' in DnenA
n
so that d(a',b) k/2n. Let a = (ala 2) be an element of n e A. Let b = (1, 2) be an element of n e B such that d(a,b) k/2 n. Then by the one-to-one correspondence, there are points (a.,b )n in e n(A) and (q.,r ) in e n(B) so that the following is true:


a 1*. .an(2) L, b 1.. bn(2) = M,

q1.. qn(2) Q, r1. .. rn(2)
L/2n < a- (L+1)/2 , M/2n 2 a < (M+1)/2n

Q/2 n < 1 (Q+1)/2n R/2n 2n 2 (R+1)/2n


Then d(a,b) k/2n if and only if ja. - !. k/2n for i = 1,2. So (Q-k)/2n < % - k/2n < a 1S + k/2n <

(Q+k+l)/2n and (R-k)/2n < 2 - k/2n < a2 02 + k/2n < (R+k+l)/2 . This implies that a is an element of D (E (q ,r ). By the one-to-one correspondence, this n n -ii n
means that e (A) is a subset of E (e (B)). By a symmetric
n n n
argument, e (B) is a subset of Ek (e (A)). The result
n n n
follows from the definition of p . Lemma 3.3. For elements A and B of A, if p H( e A,D e B) H n n n n
> k/2n, then p (en (A) ,e (B) ) > k .






23


Proof. If PH( e n nA, e nB) > k/2 n, we can assume without loss of generality that there is a point a = (aa ,c2) in n e A so that for every point b = (61,32) in n e B, d(a,b)
n
> k/2 . Corresponding to b is the point (q.,r.)n in Xn where n ((qiri n) = [Q/2n, (Q+l)/2 n] x [R/2n, (R+1)/2 n. If d(a,b) > k/2 n, then Ia - > k/2n for some i = 1,2.
n
Assume without loss of generality that la - l > k/2 Then either a1 > 1 + k/2n or a1 < 1 - k/2 . But Q/2 n < :5 (Q+1)/2n implies that either a > (Q+k)/2n or

< (Q+l-k)/2 n Then a is not an element of 1(xy) E I2 (Q+l-k)/2n , x (Q+k)/2n, (R+l-k)/2n < y < (R+k)/2 cn (Ek (q.,r )n). This implies that e (A) is not a subset
k-1
of E (en (B)). So if P H n nA,n e nB) > k/2 , then pn (en (A),e n (B)) > k.

The main result of this section now follows from the lemmas.


Theorem 3.4. For elements A and B of A, pn (en (A),e (B))
PH (A,B) = lim n
n---o 2

Proof. Let pH(A,B) = r. If r = 0 then A = B and e (A) = en (B) for all n = 1,2,..., and the result is immediate. Assume that r is positive. Then because the dyadic rationals are dense in [0,1], there is a sequence {kn/2n kn is a positive integer,n > 1}, so that (k -l)/2n < r k /2n for
n
n = 1,2,..., and lim k /2 = r.
n-c n






24


By the triangle inequality and Lemma 3.1,

pH( n enA,DnenB) pH 0n enA,A) + PH(A,B) + pH(B,( n e nB) (k n+2)/2n. Also, pH n nA, nenB) pH:H(A,B) - pH(A,D nenA)
nn
- pH(BInenB) > (k n-3)/2 . By Lemma 3.2 and Lemma 3.3, this implies that k - 3 p (en (A),en (B)) 5 kn + 2. So
n n n n
(k n-3)/2n n (en (A),e n(B)))/2n < (k n+2)/2 . By taking the limit as n tends to infinity, the result follows.


Thus we see that the Hausdorff distance between two

nonempty closed subsets of I2 can be approximated by using the metric p on the encodings of the two sets. This suggests that a stronger connection exists between the Hausdorff space (A,pH) and the discrete metric p. In order to establish the topological connection, we must consider the direct limit of the spaces A . The following definitions are found in Dugundji [ 5 ].


Definition 3.5. A binary relation < on a set A is called a preorder if it is reflexive and transitive, that is,

(1) a < a for all a in A

(2) a < b and b < c implies that a < c for all a, b,

and c in A.

A set together with a definite preorder is called a preordered set. A preordered set A with the additional property that for all a and b in A there is a point c in A such that a < c and b < c is called a directed set.





25


Definition 3.6. Let D be a directed set, and let {X : a E D} be a family of spaces indexed by D. If for every pair of indices a and 6 with a < there exists a continuous map 6 : X + X such that whenever a < < y, then 6 = 6 o 6, then the family of spaces and maps
a,y , ,
(X ,6 I is called a direct spectrum over D. The maps 0 are called connecting maps, and the image of x E X under any connecting map is called a successor of xa.


Definition 3.7. Let fX ,6 I be a direct spectrum. Let R be the equivalence relation given by


xa E Xa' B E X , x aRx if and only if xa and x have

a common successor.


Then the quotient space EX /R is called the direct limit of
a a
the spectrum, and is denoted by X0.


We now define a direct limit of the spaces A . For all positive integers n, define a connecting map 6 : A - A
n,n+l n n+l
by 6 n,n+({(a.,b )n ) = uf(u,v )n+ ' X n+ n({(a.,b )n )

= Dn+l(u{(ui,v i)n+1}). For an arbitrary element of An' extend this map over unions of points in X by distributing

6 n,n+ over unions; i.e., Dn =n+ o 6 n,n+ for all elements of A . For integers j and k with j < k, define 6. : A + A
n t,k j k
to be the composition of the connecting maps:


6j,k ek-l,k 0 ek-2,k-1


* ''' ' Ojj+1'





26


If j k 1, then 0 = 6k,l o0 j,k, and because the spaces An are discrete, the maps 0j,k are continuous. Thus, fA n'n,m I forms a direct spectrum, and we can consider the direct limit A" of the spaces fA n}. The elements of Ao are equivalence classes of elements of ZA under the relation R hn
where xRy if and only if ej,k(x) = r,k (y) for some k, where x E A., y E Ar, and both j and r are less than or equal to k. Denote the elements of A by [x].


Lemma 3.8. [x] is an element of A if and only if [x] = [e n(A)] for some positive integer n and some A E A. Proof. If [x] is an element of A , then [x] is the equivalence class of some element a of A , [x] = [a 1. Then n n n

n (an) = U {[L /2n (L +1)/2n] x [M /2n,(M +1)/2n,
1 1 1

where 0 : L.,M. 2n - 1 for i = 1,2,...,k.


Let A = u k ((2L +1)/2n+l (2M +l)/2n+l) Then A is an element of A and e (A) = a n, which implies that [e n(A)] = [a n] = [x]. The opposite implication follows from the definition of A as the quotient space EA /R, and the fact
n n
that e (A) is an element of A
n n

Lemma 3.9. For all positive integers j and k, and elements A and B of A,


p. (6 (e.(A)), . (e.(B)))
p.(e.(A),e.(B)) = j+k j,j+k 3 _ ,3+k 3
j J j -





27


Proof. The proof follows by induction on k. Consider the case when k = 1. Let p.(e (A),e (B)) = m. If m = 0 then e . (A) = e i (B) and the result follows. Assume m is positive, J J
then e .(A) c E.(e .(B)), e .(B) c E(e .(A)), and we can assume
J J J J J J
without loss of generality that e .(A) is not a subset of
J
E. (e (B)). Then there exists an element (a.,b.) of e (A) such that (a.,b.) E E (q.,r.) for some (q.,r.) E e (B), but (a.,b.) is not an element of E. (u.,v.) for all elements (u.,v ) of e (B). Then


[L/2j,(L+1)/2j] x [M/2j,(M+1)/2j] =


((a ,b ) }) c: ( E (q ,r ) ) =


[ (Q-m) /2 ,(Q+m+)/2] x [ (R-m)/2j,(R+m+l)/2].


Then by the one-to-one correspondence and the equation n n+l 'e n,n+l' we have the following:

0.j+ (ej(A)) c E j(0 (e.(B))) and
j,1+l j j+l j~j+l j

0j (ej(B)) c E j +(0 (ej(A))).
j,j+l j j+l j'j+l j

This implies that p. (e j (j+1 (e j (A)), jj+ (e j (B))) 2m. If


({((a ,b i) }J) = (j+1(ej~ j+1 ( (a ,b j) }


j+l(E 2(j j( +l (f(u1,v.) .}))) =


[ (2U-2m+l) /2 j+1, 2U+2m) /2 j+ll x [(2V-2m+l) /2 j+1


(2V+2m)/2j+1], for some (u.,v ) c e (B),






28


then 2U - 2m + 2 2L 2U + 2m and 2V - 2m + 2 : 2M 2V + 2m. But then


U - m + 1 L U + m V - m + 1 M V + m
23 2i 2i 23 23 21


which contradicts the choice of (a.,b ) . Thus,


Oj+ (e ,e+1(ej(A)),.j~j+1(ej(B))) = 2m and case 1 is established.


Now assume the result is true when k = n, that is, p. (e. . (e.(A)),0. . (e.(B))) p.(e.(A),e.(B)) = j+n j+n j j,j+n .



By case 1,


Pj+n (j,j+n (ej (A)), j,j+n (j(B)))


j+n+ (8j,j+n+l j (A)),Oj,j+n+l (j(B)))
2


which implies that the result it true when k = n + 1. Thus, the result is true for all positive k.


We can now define a distance function p, from A x Ao into the nonnegative reals by



P,([e n(A)],[em(B)]) = n (n (A),em,n(em(B))) , where m n.






29


Claim. The function p, is well-defined.


Suppose that [en (A)] = [e(A')] and [e m(B)] = [ek(B')]. Then there is an integer q such that n, m, j, and k are all less than or equal to q, and e (e (A)) = 0. (e.(A')) and n,q n j,q j
emq (em(B)) = k,q (ek(B')). Assume that j k and m 5 n. By Lemma 3.9 we get that


p (en(A),0m nem(B))) p ([e (A)],[e (B)]) = n n mn m
n m2n


Pg (0 (e n(A)),mq (em (B)))

2 q


Pg (0. (e (A')),k,q (e k(B')))

2 q


Pk (j,k(e j(A')),ek(B'))

2 k - = m j.[e (A')],[e k(B')]).


Claim. The function p. is a metric on A x Am.


This follows directly from Lemma 3.9 and the fact that p is a metric for all positive integers n.


So we now have a metric space (Am,p.). Consider the Cauchy completion of A , denoted by C(A"). Extend p. to C(A) in the usual manner by defining


p,(lim[e (An )Ilim[en (Bn)]) lim p ([e (An ),[e (B n)M.
n n- n-o n-o n






30


The following lemma will be used to establish a homeomorphism between (A,p H ) and (C(A ),p0). Lemma 3.10. If m : n, then pn (en (A),em,n (em(B)))
pH n nAm mB =n


Proof. Because m e B = n m,n m (B))), problem to showing that


p (en (A),en (B)) PH n nA, n nB) = n n n


we can reduce the


Let


en (A) = u k=1(ai,b i)n


nnnA =.j[L /2n,(L +1)/2n] x [M /2 n, (M +1)/2n,} and

en (B) ut { (q,r ) }


n nB = t Q /2n,(Q +l)/2n, x [R /2n, (R +1)/2n Claim. There exists an integer z such that


pH (n enA, e B) = z/2n


Let pH n nA,n e nB) = r and suppose (z-l)/2n < r < z/2n for some integer z. Then for all j = 1,2,...,k and for some h


an element of {l,2,...,t},






31


D ({(a. ,b. ) n) = [L /2n,(L.+1)/2n1 x [M /2n,(M.+1)/2n,
J D
n n n Jn jn

C [Qh/2n - r,(Qh+1)/2n + r] x [Rh/2n - r,(Rh+1)/2' + r]. This implies that


(n ({(a .,b .)n 1
J J


[(Qh- z+l)/2 ,(Q h+z)/2 ] x [(Rh- z+l)/2 , (Rh+z)/2 n. So for every element x of nenA there is an element y of DnenB so that d(x,y) (z-l)/2 n, and a symmetric argument shows the converse. Thus,


PH n enA,n e nB) (z-l)/2 < r, which is a contradiction. So


PH (n enA,nenB) = z/2n for some integer z. By Lemma 3.2 and Lemma 3.3, this implies that


z - 1 pn (e n(A),e n(B)) z.


Suppose that pn (en (A),en (B)) = z - 1. Then by the one-to-one correspondence Dn' this implies that for all j = 1,2,...,k, there is an element h of fl,2,...,t} so that


[L /2 n(L +1)/2 n] x [M /2',(M +1)/2 n] c


[(Qh -z+l)/2 n' (Qh +z)/2 ] x [(Rh- z+l)/2 , (Rh+z)/2 n






32


Again we get that


PH (Dnen A,nenB) (z-l)/2n < z/2n


a contradiction. This implies that pn (e n(A),e n(B)) = z, and the conclusion follows.


We are now able to show the connection between (A,pH and the discrete spaces (An'On ' Theorem 3.11. (A,pH) is homeomorphic to (C(A ),p.). Proof. Define a map ': A -+ C(A ) by


Y(A) = lim [en (A)].
n+CO


Note that A = nn= n e nA. Suppose Y(A) = T(B). Then


lim [en (A)] = lim [e n(B)]
n_*0 n-*O


which implies that


0 = lim pOn([e (A)],[e n(B)]) = lim Pn(en(A),en (B))
n_*0 n_*0 n



PH (A,B)

by Theorem 3.4. But pH is a metric, so A = B and T is a one-to-one map. To see that T is onto, suppose that {[e n(A n)]: n 11 is a Cauchy suquence in C(Am). Then for every positive number c, there exists an integer N such that






33


p.(e n(An )],[e m(A m)]) < e if m and n are greater than N. But


p ([e (A )],[e(A)] nn m (A
00 n n m m n np0[ (A], m (A]m


pn (e n(A n),8m,n (em (Am)


PH RnnAn' me meAm) where m n.


Thus, {nnn e A n 11 is a Cauchy sequence in A and (A,pH) is complete [ 3 1, so there is an element A of A so that


lim eA = A.
nn n


But


lim nnA = A, n-co


so by the triangle inequality, for every positive c there is an integer N so that pH enAn' n nA) < c if n > N. Claim. lim [e (A )] = lim [e (A)].
n o nn


Let c be positive, then there is an N such that pH (n nAn'nenA) < 6 if n > N . Then


p0([en(A )],[en(A)]) = p.([e (An )], e n(A)]) =


p (en (An ' n(A))


= PH (n e A n'D e A) < c if n > N


Thus,






34


lim [e (Ar)] = lim [e (A)] = T(A),
n-* n n+-> n


proving that T is onto.


To show that T is a homeomorphism, it suffices to show that T is an isometry. But this follows from Theorem 3.4, as


p (T(A), T(B)) = lim p. ( [e (A)],[e (B)]) = n n


pn(en (A),e n(B))
lim PH (A,B).
nio 2


Thus, T is a homeomorphism from (A,p H) onto (C(Aw), p.).














CHAPTER IV

APPLICATION TO DIGITIZED GREY LEVEL IMAGES


This chapter examines the possible usefulness of the metric p in pattern recognition problems. In particular, graded patterns (digitized grey level images) are introduced and an algorithm for computing the discrete Hausdorff distance between two such graded patterns is given. Definition 4.1. Let Ni, N2, and N3 be positive integers. A graded pattern (or digitized grey level image) is an N x N2 matrix with entries from {0,1,2,...,N 3)


Such patterns are produced by high speed scanning devices, such as densitometers. These digitized images arise in areas such as land use studies, planetary observations, fingerprint analysis, X-ray diagnosis, and optical character recognition. An excellent example of the use of graded patterns is given in a study of the relationship between blood flow and brain function by Lassen et al. [10].


The discrete Hausdorff metric can be used to define a distance between graded patterns for fixed Ni, N2, and N3, thus giving a measure of similarity between patterns. This interset distance can then be used with a clustering algorithm for the purpose of pattern classification. The


35






36


following description of graded patterns as ordered triples of integers and the definition of neighborhoods of points is due to Bednarek and Ulam [ 2].


Fix integers Ni, N2, and N Let


X = {(i,j,k): 1 i N 1 1 j N2,l 5 k N }


Associate with every graded pattern A = [a .] a subset A of X in the following way:


A = {(i,j,h) e X: the ij-th entry of A is k e 0 and


1 h k}.


One can visualize A as stacking blocks on an N x N2 grid, where there are k blocks stacked on (i,j) if a.. = k, and no blocks on (i,j) if a.. = 0.
:LJ

For a point x = (i,j,k) E X, define the neighborhood of x, N(x) by


N(x) = {(p,q,r) E X: Ip - il + Iq - jj + r - k 5 1


and r > k}.


Thinking of this as blocks on an N1 x N2 grid, this forms a partial cruciform neighborhood which consists of all those blocks immediately to the left, right, front, back, and above the block x. This is shown schematically in Figure IV-1, where x is the center cell.






37


Figure IV-1






Given two graded patterns A and B, p is a metric on the associated sets A and B of X. An example of p applied to two 30 x 30 graded patterns with 15 grey levels using the above choice of neighborhoods is shown in the Appendix. The expansion of each pattern is displayed until the absorption of the other pattern is achieved.


Let x = (i,j,k) be an element of A, where A is a graded pattern and N(x) is defined as before. Proposition 4.2. En(fx}) = t(p,q,r) E X: Ip - il + Iq - ji + r - k n and r > kI.


Proof. The proof proceeds by induction on n. The case n = 1 is given by the definition of N(x). Assume the result






38


is true for case n and consider E n+ {(i,j,k)}) = E n+({xJ) = E(E ({x})) = u{N((p,q,r)): (p,q,r) E En(fx})} . If (p,q,r) is an element of E ({x}), then


Ip - il + Iq - j + r - k : n and r ' k


by the inductive hypothesis. So if (s,t,u) E N((p,q,r)), then Is - il + It - jI + u - k is - pl + Ip - il + It - qi + Iq - ji + u - r + r - k n + 1 and u r k. Thus


E n+ xl) c {(p,q,r) E X: Ip - il + Iq - jI + r - k


n + 1 and r k}.


Conversely, if (p,q,r) e X is such that Ip - il + Iq - ji + r - k n and r k, then (p,q,r) E En (fx}) c En+l ({xl). If fp - il + fq - j + r - k = n + 1 and r k, then there is an element (s,t,u) in X such that Ip - si + Iq - ti + r u =1, r ! u, and Is -il + It - jI + u - k = n, u k. To

see this, suppose k + 1 r k + n + 1 and let s = p, t = q, and u = r - 1. If r = k, then fp - il + Iq - ji = n + 1, and we must consider several cases. If p = i and q = j + n + 1, let s = i and t = j + n; it p = i and q = j - n - 1, let s = i and t = j - n. A similar argument establishes the case where q = j, so assume fp - il > 0 and Iq - ji > 0. If p i + 1 let s = i - 1 and t = q; if p i - 1 let s = p + 1 and t = q.






39


So (p,q,r) c N((s,t,u)) c E(E n({x})) = E n+ {x}), which implies that En ((i,j,k)}) = f(p,q,r) E X: Ip - ii + Iq - ji + r - k n and r kI for all nonnegative integers n.


Let A = [a El and B = [b .1! be two graded patterns.

Associated with A and B are subsets A and B of X. The discrete Hausdorff metric defines a distance between A and B:


p(A,B) = min fk: A c Ek(B) and B c Ek (A)).


We now provide an algorithm for computing p(A,B).


Proposition 4.3. Let


d = maximum minimum fli-pl + Iq-jf + fk-r if k > r
A (i,j,k)EA (p,q,r)EB 0 otherwise


d = maximum minimum fli-pl + lq-jl + {r-k if r > k
B (p,q,r)EB (i,j,k)EA 0 otherwise


then p(A,B) = max {dA'dB d


Proof. Let (i,j,k) be an element of A, where 1 k a...
J
For all elements (p,q,r) of B, where 1 r !b , compute m = minimum {ip-il + Jq-jI if k r, Ip-il + Iq-jl + k
(p,q,r)EB
m..
r if k > r}. Then (i,j,k) e E 13(B). Take the maximum over
d
all mi, dA = maximum fm 1, then A c E A B). A symmetric
(i,j,k)EA
dB
argument shows that B c E (A), which implies that p(A,B) < max {dA d B. Let m = max d A'd BI and suppose p(A,B) < m. Then A c Em- (B) and B c Em-1 (A), so for all elements






40


(i,j,k) of A there exists an element (p,q,r) of B such that (i,j,k) E E M-1 {(p,q,r)}). This implies that Ip-il + !q-jj + k - r ! m - 1 and k > r. By minimizing over all (p,q,r) in B and maximizing over all (i,j,k) in A, this implies that dA m - 1. A symmetric argument shows that dB m - 1, but this contradicts m being the maximum of dA and dB. Thus p(A,B) = m.


Note that the algorithm given in Proposition 4.3 agrees with the definition of the Hausdorff metric. In actual computer computations of the distance between graded patterns, the algorithm was not used. Rather, these distances were computed directly from the definition of p. Iterations of the operator E for the purpose of computing p are shown in the Appendix.


Other metrics on graded patterns that have been used in pattern recognition are the Hamming distance H given by


H(A,B) = . a.
1,J 1]


- b. .I,
J-J


and the metric M given by


M(A,B) = max ja.
i'j J


- b. .I,


or more generally,


M(A,B) = max {d(a. ,b. )} where d is a metric.
ij






41


It is our opinion that the metric p is a useful measure of similarity between graded patterns, and perhaps better than other metrics now in use as it seems to be less sensitive to small perturbations or changes. Such perturbations often result from noise or electronic interference in real data collecting and transmitting situations. Generally, graded patterns are subjected to an image enhancement procedure before being interpreted. We shall digress slightly to describe one such procedure- a procedure that seems particularly appropriate for digitized grey level images.














CHAPTER V

A SHARPENING TRANSFORMATION FOR GRADED PATTERNS


In this chapter we discuss a nonlinear transformation introduced by Kramer and Bruchner [9 1 that can be used for sharpening digitized grey level images. An alternative, and in our opinion simpler, proof of the Kramer-Bruchner Theorem is given.


As discussed previously, graded patterns often become distorted or "fuzzy" and a preprocessing sharpening may be performed in order to attempt to restore the pattern to its original state. Fourier and Laplacian transformations have been used for this purpose. A simple nonlinear transformation for sharpening digitized grey level images which depends on local operations was introduced by Kramer and Bruchner [91!. Basically the transformation replaces the value of an entry of a graded pattern A by the largest or smallest value in its neighborhood. The following definition given for the sharpening S is due to Kramer and Bruchner [9 ].


Rather than describe the transformation in terms of

graded patterns, it is best described in terms of real valued functions F on finite sets X. In the cases of interest to us though, X will be the cells of an N x N2 grid, that is,

X = {(i,j): 1 ! i N1,l j N21' 42





43


and F will be the function on X which assigns to the ij-th cell the value a.., F((i,j)) = a.., where A = [a..] is the graded pattern.


The definition of the sharpening transformation requires the notion of a neighborhood system for X. For every point x of X, associate with it a unique nonempty subset N(x) of X such that x E N(x), and we require that the neighborhoods satisfy a symmetry condition, that is, if x E N(y) then y E N(x) for all elements x and y of X. Notice that this definition of neighborhood is slightly more restrictive than that previously used. For every real valued function F on X, associate with it two other functions T and F, the local maximum and local minimum functions respectively; that is,


F(x) = max fF(y): y E N(x)j and


F(x) = min {F(y): y E N(x)}


for all elements x of X.


If X is a finite set with a neighborhood system and F

is a real valued function on X, then the sharpening transformation S is defined by


F(x) if F(x) - F(x) F(x) - F(x) S(F)(x) = (SF)(x) = {
F(x) otherwise.


Define S F = F, and for all positive integers n, S n+F = S(Sn F) is given by





44


n+l SnF(x) if SnF(x) - SnF(x) SnF(x) - SnF(x)
Sn+F(x) ={
SnF(x) otherwise.


Kramer and Bruchner prove the pointwise convergence of the sequence {SnF}. We provide a direct proof based on the cardinality of the range.


A point x c X is called a local maximum of F if F(x) = F(x). Dually, if F(x) = F(x) we say that x is a local minimum of F.


It is immediate that pointwise convergence of the

sequence {SnF} is equivalent to the assertion that there exists a positive integer N such that for each x E X, x is either a local minimum or local maximum of S NF. Theorem 5.1. If X is a finite set with a neighborhood system and F is a real valued function on X, then for every element x of X, there is an integer N such that n N implies that SnF(x) = SNF(x). Proof. If F is constant on X then T(x) = F(x) = F(x) for all x in X, and the result is immediate. If the cardinality of F(X), IF(X)I, equals two, then every point of X is a local maximum or local minimum of F, and again the result follows. The proof proceeds by induction on the cardinality of F(X). Assume that JF(X) ! 3, and that the result is true for all






45


functions F and finite sets X such that IF (X) I n. Let fF(X)j = n + 1.


Let u0 = max {F(x): x E X} and 10 = min {F(x): x E X}.

Define M(F) = {x E X: F(x) = u 0} and L(F) = {x E X: F(x) = 1 0 Note that if F(x) = u0 = F(x), then SkF (x) = sk-lF(x) = u0 for all positive integers k, and similarly, if F(x) = 10 F(x), then SkF (x) = Sk-lF (x) for all positive integers k. This implies that


M(F) c M(SF) c M(S F) c ... c M(S nF) c. c X, and


L(F) c L(SF) c L(S F) c c.. L(S nF) c ... c X.


Because X is finite there are integers N and N2 such that



M(SN F) = M(SN +kF) and L(S N2F) = L(S N2+kF)


for all nonnegative integers k. Let N = max {N,N21, and U = M(S NF), L = L(S NF).


Let x be an element of X - (U u L).


Claim. Either N(x) n U = $ or N(x) n L = c. Suppose that N(x) n U and N(x) n L are nonempty. Then there are points y1 and y2 such that y E N(x) and SNF(y1) = u0' and y2 E N(x) and SNF (y2) 10 But then






46


SN+l 1x NF (X) = u 0 or
NS F(x) = u0o
S F~x){sN(x) =10


which implies that x e U u L, a contradiction.


We now consider three cases.


Case 1. Suppose N(x) n U $. Then there is a y e N(x) such that SN F(y) = u0, and SN F(x) ; u0 so S NF(x) = S N-1F(x) < u0. Because S N+kF(x) = u0 for all k = 0,1,2 ... and x not in U, this means that S N+kF(x) = S N+k-F(x) for all nonnegative integers k. Then


SNF(x) = S FN-1F(x) S F(x) FN+1F(x) ... > 10*



So there is an integer N3 such that SN3 F(x) = S N3+kF(x) for all nonnegative integers k. Case 2. If N(x) n L is nonempty then there is a y E N(x) such that SN F(y) = SN+k F(y) = 10 for all k = 0,1,2,... . But x not in L implies that S N+kF(x) = S N+k-F(x) for all nonnegative k. So


SNF(x) : S N+1F(x) S N+2F(x) ... < u0'

N +k 4
Thus, there is an integer N such that S N4 F(x) = S NF(x) for all nonnegative integers k.






47


Case 3. Suppose that both N(x) n U and N(x) n L are empty. Let Y = X - (U u L), F* = Fly, and for all y E Y, let Ny(y) = N(y) n Y. Then IF*(Y)l < n, so by the inductive hypothesis there is an integer N5 such that



SN5F*(x) = SN5+kF*(x)


for all nonnegative integers k. But N(x) n U = = N(x) n L implies that N(x) n Y = N(x), so SF*(x) = SF(x). Moreover, SkF*(x) = SkF(x) since S F*(x) = SF(x) and Sk F(x) = S kF(x) for all integers k. Thus, S N5F(x) = SN5 F(x) = S N F+kF(x) = N +k
S F(x) for all nonnegative integers k, and the result is true for all x in X.


Remark. The Kramer-Bruchner claim of pointwise convergence of {SnF} to a function P in then established by letting P = SN*F, where N, is the maximum of the integers given by Theorem 5.1 for the individual elements of X.


We now provide an illustration of how the sharpening S can be applied to character recognition. Let X be an

8 x 8 matrix and let F be the function from X into f0,1,2,...,71 which represents the graded pattern A shown in Figure V-1 in which there are eight grey levels. We then distorted the pattern in a random fashion by using random numbers. The fuzzy pattern is represented by T in Figure V-1, where T: X -+ fO,l,...,7} is given by T(x) = F(x) + n(x) if





48


0 F(x) + n(x) 7, T(x) = F(x) otherwise. The noise n(x) introduced was determined by first generating a random number r from {O,1,...,9} and then using the following rule:


if r = 0 or 9 then n(x) = 0 if r = 1 or 5 then n(x) = 1

if r = 2 or 6 then n(x) = -l

if r = 3 or 7 then n(x) = 2

if r = 4 or 8 then n(x) = -2.


After two applications of the sharpening transformation S, the limit S2T was reached and found to very close to the original pattern, as shown in Figure V-l.






49


0 7 7 7 7 0 0 0 0 7 0 0 7 0 0 0 0 7 0 0 7 0 0 0 0 7 7 7 7 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 7 7 7 7 0 0


1 7 6 7 7 10 0 2 6 2 1 7 10 0 0 6 2 0 5 0 12 1 6 6 7 7 7 12 2 7 1 0 0 7 0 0 0 7 0 0 0 6 2 0 0 7 0 1 2 6 0 1 0 7 7 7 5 7 2 2


A


T


1 7 7 7 0 7 1 0 0 6 0 0 0 7 7 7 0 7 0 0 0 7 0 0 0 7 0 0 0 7 7 7


ST


7 0 0 0 7 0 0 0 7 0 2 2 7 7 0 2 0 7 0 0 0 7 0 0 0 7 0 2 7 7 0 2


0 7 7 7 0 7 0 0 0 7 0 0 0 7 7 7 0 7 0 0 0 7 0 0 0 7 0 0 0 7 7 7


7 0 0 0 7 0 0 0 7 0 2 2 7 7 0 2 0 7 0 0 0 7 0 0 0 7 0 2 7 7 0 2


s2T


Figure V-1














CHAPTER VI

SOME COMPUTATIONAL EXPERIMENTS


Both the metric p and the transformation S depend on the notion of neighborhoods of points of a set X. Due to this similarity, questions arise as to the relationship (if any) between the two. Several computer experiments were conducted in order to study this metric and its interaction with the sharpening of graded patterns.


Computation 1. The first set of computations involved twelve

8 x 8 matrices with random entries from {0,l,...,71. The Kramer-Bruchner sharpened limit was computed for each of the matrices A., i = 1,2,...,12; denote these limits LA(i). The average distance between the matrix A and its limit LA(i) was found to be 2.4167, and the average number of iterations to sharpen A was 4.3333. We then computed p(A.,A.) and
1 1 J p(LA(i)'LA(j)) for all i and j, i t j. The mean of p(AVA ) was 3.3333 with a standard deviation of .1667; the mean of p(LA(i),LA(j)) was 2.9090 with a standard deviation of .7986. Thus, sharpening seemed to decrease the distance between graded patterns. However, numerous examples were found where p(LA(i),LA(j)) > p(A.,A ), thus showing that no relationship of inequality could be stated between p(A,B) and p(LA,LB) for graded patterns A and B with sharpened limits LA and LB respectively.
50





51


Computation 2. As a comparison, the distances between A. and A., and LA(i) and LA(j) , i j, were computed using the Hamming distance; i.e., H(A.,A ) = l la -a . The k,1l ki ki
average distance between A. and A. for i e j was 87.8182
1 J
with a standard deviation of 11.5943; the mean for LA(i) and LA(j), i ; j, was 211.7121 with a standard deviation of 21.9333. Sharpening greatly increased the Hamming distance between graded patterns, whereas sharpening created little variance using the discrete Hausdorff distance. This could be expected though, as sharpening tends to increase the number of highest and lowest grey levels in the pattern. Because the range of values was so varied for the metrics H and p, they were normalized and are denoted by H and p respectively. The average of p(A.,A.) was .1667, while the
1 J
average of H(A.,A ) was .1960. The average of p(LA(i) ,L A(j) was .1455 and the average for H(LA(i)'LA(j)) was .4726.


In normalizing p it was noted that the maximum value of p is 20, and in general, if the size of the matrix is N 1x N2 with entries from f0,1,2,...,N3} then the maximum value of p is (N 1-l) + (N2- l) + (N3- l) N 1 + N2 + N3 - 3. This value is obtained, for instance, when A = [a..] where 1J
a1 = 1, a.. = 0 otherwise, and B = [b ], bNN = N3, b1 = D J Ntherwis
b. . = 0 otherwise.
1J






52


Computation 3. The next experiment generated one hundred 16 x 16 matrices, A. for i = 1,2,...,100, with random entries from {o,l,2,...,9}. We then computed the sharpened limit LA(i) for each pattern A., i = 1,2,...,100, and let s(i) be the number of iterations of the transformation S needed to sharpen A . Next we computed p(A ,LA(i)) for i = 1,2,...,100, and let p(i) be this value. The mean of p(i), i = 1,2,...,100, was 3.2900 with a standard deviation of .4777. The mean value of s(i) for i = 1,2,...,100 was found to be 7.7900 with a standard deviation of 2.2798. A histogram of the frequency of the sharpening index for the matrices A., i = 1,2,...,100, is shown in Figure VI-l.


Computation 4. It was conjectured that the discrete Hausdorff metric could be used as an estimator for the parameters s(i) and p(i) associated with A.. The estimation procedure was based on the nearest neighbor rule [ 41 which we now describe.


Let (X,d) be a metric space and suppose we are given n pairs (x, 1), (x212 ' .... n'en) where x is an element of X and . is a parameter associated with x which takes on values from fl,2,...,M}. Suppose we are given an x E X and wish to estimate its parameter 0. We say that the element x* of {x1,...,xn} is a nearest neighbor to x if d(x,x*) = min {d(x,x ): i = 1,2,...,n}. If x has only one nearest neighbor x* then we let 0 = 0* be the estimate. However, it






53


22

















12 11 10








5




2

1


4 5 6 7 8 9 10 11 1213 14 18

Frequency of values for sharpening index
for 100 16 x 16 x 10 graded patterns


Figure VI-1





54


is possible for a point x to have many nearest neighbors, x. , x. , ..., x. , in which case we let 6 = O. ,where the
1 2 k D
parameter 0 occurs with the highest frequency amongst
J
'1 12 ,.., }. In the case of ties, the choice is arbii 2 k
trary.


The nearest neighbor rule was applied to the matrices Ai, taking (A1,s(l)), (A2,s(2)), ..., (A50,s(50)) to be the sample set. We then computed the nearest neighbors to A51' A52, ..., A100 using three metrics: the discrete Hausdorff metric p, the Hamming distance H, and the maximum distance metric M given by M(A.,A.) = max f{a - a3 1}. Of the 50
1 J k,l k1 ki
estimates made, p correctly estimated only 9, as did M, and H correctly estimated 12. The procedure was repeated to estimate p(i) for i = 51,52,...,100 using (A.,p(j)), j =
J
1,2,...,50, for the sample set. This time the number of correct estimates for p(i) using p was 32, for M it was 29, and for H it was 28. The author's interpretation of the results was that p failed to be a good estimate for the parameters, particularly the sharpening index s(i), because the range of values of the metrics was too small. In particular, the values p(A.,A ) for i = 51,...,100, j = 1,2,...,50 all fell between 3 and 5, whereas the possible range of values for p was {O,l,2,...,38}. It was felt that the range of values was small because of the nature of the matrices; the






55


maximum entries were uniformly distributed throughout the matrix, so that E k(A.) = X for small k and for all i = 1,2,...,100.


Computation 5. In an attempt to make the data more meaningful as a pattern recognition problem, the next computation took three prototype 16 x 16 patterns with entries from {o,l,...,91, representing the letters A, B, and C, and created a class of A's, class of B's, and class of C's. This was done by introducing random noise into the three prototypes, creating one sharp version and nine fuzzy versions for each type. The parameter associated with each of the patterns was either 1, 2, or 3 depending on whether it was a member of the class of A's, B's, or C's respectively.


We then took the prototype A and created 46 fuzzy versions, F., i = 1,2,...,46, by adding random noise. The Kramer-Bruchner sharpened limits, LF(i), were computed for i = 1,2,...,46. The mean of the sharpening index s(i) was found to be 3.4348 with a standard deviation of 1.1086. For comparison with the frequency of sharpening given in Figure VI-l, a similar histogram is given in Figure VI-2 for F., i = 1,2,...,46. Note that the sharpening index mean dropped significantly from that given in Computation 3. This was most likely due to the fact that the entries of the patterns were no longer random, so the maximum entries were no longer uniformly distributed.






56


20
















11





8 5






1

2 3 4 5 6 7 s(i)

Frequency of values for sharpening index
for 46 16 x 16 x 10 graded patterns


Figure VI-2






57


The distances between F. and A., B., C., i = 1,...,46,
1 J J J
j = 1,.. .,10, were computed using both the discrete Hausdorff metric and the Hamming distance. Also, the distances between the sharpened versions, L F(i), i = 1,2,...,46, and A., B., C., j = 1,2,...,10, were computed. Based on these J J J
computations, we classified each F and LF(i) using the nearest neighbor rule. The Hamming distance correctly classified all F and LF(i), i = 1,2,...,46, and we note that the nearest neighbor of F and LF(i) was always A1. The mean of the nearest neighbor distance for F. using the Hamming distance was 154.7391 with a standard deviation of 12.5423; the mean nearest neighbor distance for the sharpened version was 120.4130 with a standard deviation of 14.3319. Thus, unlike Computation 2, sharpening tended to decrease distances.


When the discrete Hausdorff metric was used to classify the fuzzy A's, F , i = 1,...,46, based on nearest neighbors, 42 were correctly classified, or 91.304 percent. When the classification was made using the sharpened versions LF(i)' 44/46 or 95.652 percent were correctly classified. The average nearest neighbor distance for F. was 2.3478 with a standard deviation of .4815, and the average nearest neighbor distance for the sharpened versions decreased to 2.2826 with a standard deviation of .4552.





58


We observed that the only time the discrete Hausdorff metric misclassified an F or LF(i) was when there was one more nearest neighbor in the B class than in the A class. This was most likely because the distance between the prototypes A and B was small, p(A,B 1) = 3. We altered the pattern A1 so that p(A,B 1) = 4 and repeated the computations. This time both p and H correctly classified all F and LF(i)'


These computations show that the metric p might be

useful in character recognition problems, especially if the size of the matrices was sufficiently large enough to allow significant distances between pattern types.














CHAPTER VII

POSSIBLE APPLICATIONS AND PROBLEMS


The computations presented in Chapter VI suggest that the discrete Hausdorff metric could be useful in pattern recognition problems such as character recognition. Whether it is truely a better similarity measure on graded patterns than those in existence remains to be seen. It does have some advantages which seem to make it better, at least from a geometric standpoint. That is, the choice of neighborhoods for the discrete Hausdorff metric reflects a geometrical structure on the subsets of a set X. This is not the case for other similarity measures in use, for instance, the Hamming distance. The usefulness of a geometric structure is evident in such problems as photo interpretation and analysis. These problems arise, for instance, in land use studies where the photo is taken by satellite and transmitted to earth as a digitized grey level image. Such studies are conducted regularly by private and government agencies in an attempt to study the evolution of land use, and also to locate deposits of minerals and other natural resources (such as oil).


Another instance of the use of graded patterns comes

from satellite and telescopic photos of planetary observations.


59





60


For example, Strom and Strom [14] studied the evolution of disk galaxies with the aid of the interactive picture-processing system (IPPS) developed at the Kitt Peak National Observatory. Black and white photos are made with the use of a telescope in the ultraviolet to red regions of the spectrum. The various light spectrum photographs are weighted and composed to produce a digitized grey level image in which the various grey levels are color coded. The discrete Hausdorff metric provides a measure of similarity between such images.


In Chapter IV we alluded to the use of graded patterns in a study of the relationship between brain function and blood flow by Lassen et al. [10]. Specifically, they studied the changes in blood flow in areas of the human cerebral cortex in relation to specific sensory, motor, and mental activities performed by the subject. Their method of study was based on the idea that localized increased blood flow corresponds to an increase in local activity of the surrounding tissue. The study was done by injecting a small amount of a radioactive isotope, xenon 133, into the carotid artery in the neck, and measuring the arrival and subsequent washout of the radioactivity. The measurement was made by a gamma-ray camera consisting of 254 externally placed scintillation detectors, each detector measuring approximately one square centimeter of brain surface. The data were processed by computer and displayed on a color-television





61


screen, with different flow levels being assigned different colors or hues. These scientists were able to show a correlation between specific mental stimulation and actual activities being performed by the subject.


Because blood flow increases and decreases are localized, it is conjectured that the discrete Hausdorff metric could be of use in analyzing such data. Land use studies also share this attribute of localization which makes them particularly suited for a similarity measure, such as the discrete Hausdorff metric, which is dependent on a neighborhood system.


Other possible applications of the metric p arise in

taxonomy, in particular, as a paleontological dissimilarity measure as discussed by Bednarek and Smith in [1 ]. Many of the taxonomic distances in existence have very complex algorithms, and often they are not true metrics. The discrete Hausdorff metric is simply stated and generally easily computed.


Because of the generality of the metric p, it is conjectured that a number of metrics introduced earlier as evolutionary distances will be specific cases of p. Sellers [12-13] provides an algorithm for an evolutionary distance that was introduced by Ulam [15] and discussed further by Waterman, Smith, and Beyer [16]. This distance on finite





62


sequences is metric and is used to measure the degree of evolutionary divergence between homologous proteins or nucleic acid sequences. Basically, it compares two finite sequences A and B, not necessarily the same length, and finds the common subsequences. It can be interpreted as the smallest number of weighted changes necessary to bring the two sequences into coincidence.


Several questions concerning the discrete Hausdorff metric remain open. Due to the similar dependence on a neighborhood system, the question arises as to what relationship (if any) exists between the sharpening transformation S and the metric P. Specifically, what conditions will insure a relationship between p(A,B) and p(LA,LB), where A and B are graded patterns and LA and LB are their sharpened limits respectively? We have noted instances where p(A,B) > P(LA,LB) and p(A,B) < P(LALB), showing that some additional assumptions are necessary in order to predict a relationship. Also, what relationship exists between the sharpening index and the bounds N1, N2, and N3 on the graded pattern A?


Another area of investigation concerns metric semigroups. A metric semigroup is a triple (S,o,d) where (S,o) is a semigroup, (S,d) is a metric space, and a is continuous with respect to the topology induced by d. More generally, a topological semigroup is a topological space G that is also





63




a semigroup with operation o such that the mapping (x,y) + xoy is continuous. If (X,d) is a compact metric space, then (2X ,u,pH) is a metric semigroup, as the union operation is continuous with respect to the Hausdorff metric. Furthermore, in this space p(A u B,C u D) max {p(A,C),p(A,D), p(B,C),p(B,D)}. Is this the only metric semigroup having this property?














REFERENCES


1. A. R. Bednarek and T. F. Smith, A taxonomic distance
applicable to paleontology, to appear in Mathematical Biosciences.

2. A. R. Bednarek and S. M. Ulam, An integer valued metric
for patterns, Fundamentals of Computation Theory,
FCT 79, Akademic-Verlag, Berlin, 1979, 52-57.

3. K. Borsuk, On some metrizations of the hyperspace of
compact sets, Fundamenta Mathematicae, 41 (1955),
168-202.

4. T. M. Cover and P. E. Hart, Nearest neighbor pattern
classification, IEEE Trans. on Information Theory,
IT-13, 1 (1967), 21-27.

5. J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.

6. F. Hausdorff, Set Theory, Chelsea Pub. Co., New York,
1957.

7. N. Jardine and R. Sibson, Mathematical Taxonomy, John
Wiley and Sons, New York, 1971.

8. J. L. Kelley, General Topology, D. Van Nostrand,
Princeton, N. J., 1955.

9. H. P. Kramer and J. B. Bruchner, Iterations of a
nonlinear transformation for enhancement of
digital images, Pattern Recognition, 7 (1975),
53-58.

10. N. A. Lassen, D. H. Ingvar, E. Skinh~j, Brain function
and blood flow, Scientific American, 239, 4 (1978),
62-71.

11. E. Michael, Topologies on spaces of subsets, Trans.
Amer. Math. Soc., 71 (1951), 152-182.

12. P. H. Sellers, An algorithm for the distance between
two finite sequences, J. Combinatorial Theory (A),
16 (1974), 253-258.


64






65


13. P. H. Sellers, On the theory and computation of
evolutionary distances, SIAM J. Appl. Math.,
26 (1974), 787-793.

14. S. E. Strom and K. M. Strom, The evolution of disk
galaxies, Scientific American, 240, 4 (1979),
72-82.

15. S. M. Ulam, Some combinatorial problems studied
experimentally on computing machines, Applications
of Number Theory to Numerical Analysis, Academic
Press, New York, 1972.

16. M. S. Waterman, T. F. Smith, W. A. Beyer, Some
biological sequence metrics, Advan. Math., 20,
3 (1976), 367-387.















APPENDIX






L 9


0 0 0 0 0 0 0 0 0 C 0 0 C 0 0 0 c 0 0 0 0 0 0 0 3 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 C c 0 0 0 D 0 0 c 0 0 0 c 0 0 0 C 6 CIETEItTtIC 0 0




3ICI# t lItIOC 0
0 1? 1 Et It I tr 10 T
0 0 0 C 11 11 11 10 0 0 C. 0 G 0 Ti To 0 0 c 0 0 C 0 0 0 0 0 0 c 0C 0 0 0 0 C 0 c 0 0 c 0 0 c 0 0 c 0 0 0 0 0 0 0
0 c 0 0 0 0 0 0 0

0 0 0 c 0 0 0 0 0 0 00 c00 0300c


0 0 0 0 0 Di 0 0

0 0 0 c O0t't0 0 0


0 0 0 0 0 0 0 0 0 0 0 0 0 CV 0 c


0 0 c 0 0 c 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 C 0 0 0 0 0 0 c 0 C C


C c 0 0 C 0 0 c 0 0 C C 0 C C C C C C 0 0 0 0 0 C 0 0 C 0 C 0 0 0 G C 0 C c 0 c 0 0 c 0 0 0 0 0 c 0 0 0 0 C C 0 C 0
0 0 0 0 c 0 0 C c 0 0 0 c 0 0 c 0 C a c c C c 0 c P 9 G 0 c



C C 0 E 6Ef) 6 6 6 C C 0 5 CICTOIZIC16 C
C C C 5 21F 1C 6 C



0 0 0 C C 6 0:IC6 0 0 3 TTTT 016 0

10 1001 1 IZI0 1016 C
C t 'TtTE.TT TIT c6c C C t1tIIITCI6 C C 0 C0 CIT 016 C
0 C 0 C C I T1TC16 C
0 0 0 0 C 6 0C016 C
C 0 0 0 C C C


G0 C 0 c c 0 0 0 0 C 0 0 0 0 C 0 c 0 0

0 c 0 0 C C 0 C 0 C'

(V) I


0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 c 0
E 0 0 0 It It T 0

CTV0 0 0 0 It 10 0 0 0 0 0
0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 00 0 0 30


0 01't 0


C
C
0
C
C
C


C

c
0
0
C
C




C
C
C
0
C



0
C


0 0 0 0 C 0 c 0 0 C 0 C 0 0 c 0 0 0 C 0 c 0 0 c C 0 0 c c C 0 0 0 0 c 0 c 0 0 0 0 c c c r 0 0 0 c 0 c 0 0 0 c 0 0 0 0 0 c c 0 c 0 0 C 0 G c C0 c c c 0 c 0 0 O 0 c 0 0 c 0 0 0 0 C 0 c c C 6 2 0 0 0 c C 0 6 6 6 6 6 0 0 C c 0 0 0 F'21C1C'L C C C0 C0 111C16 0 C C
0 C E1?TTT106 C C C 0 t'EiT 11016 0 C C

0 '1ZT1It1016 0 C 0 0 0 C Z111016 0 C 0 0 C0 C 6 016 C 0 C C 0 C 0 C 5 6 0 C C
C 0 C C C 6 C 0 0 0 0 0 0 c c 0 0 0 c c 0 0 0 0 C 0 c 0 c D 0 C 0 C c 0 G 0 0 0 0 0 0 0 0 0 c 0 c 0 0 0 0 0 c 0 C 0 0 0 0 c 0 c C 0 c 0 0 c 0 0 0 C C 0 0 0 0 C c c 0 c c C c 0 c )
COCCO56Lvcj0300d






68


2(A)
1 0 0 0 0 0 3 0 0 0 0 0 0 0 3 0 0 a 0 0 0 0 0 3 0 0 0 0 3 J 0 0 0 a 0 0 0 0 1 0 J 0 0 0 0 3 0 0 0 0 3 0 0 a 0 0 0 0 0 3 0 3 a a 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 910 9 3 3 0 0 0 C 0 9101110 9 0 0 0 0 3 91011111112 0 0 0 0 C 9101112121313 0 0 0
0 910111213141414 0 0 0 91011121314141414 0 0 9101112131414141414 0 91011121314141414 141 C 9101112131414141414 0 91011121314141414 0 0 910121313141414 0 0 011121314141313 9 0 0 0 910121313121110 9 0 0 8 91012121010 9 0 0 3 0 8 9 s 9 9 9 0 0 0 0 0 0 8 8 8 0 0 0 0
3 0 0 3 C a 0 0 0 3 0 30 0 0 0 0 0 0 0 0 30 0 0 0 a 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 3 0 0 0 0 0 0 a 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Sa 0 0 0 0 0 0 0 0 0


E 3(A)
0 0 0 a 0 1) 0 0 0 0 0 0 C 0 0 0 c 0 1 0 0 0 20 1 0 0 0 0 3 0 0 0 3 0 0 C 0 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 9 0 0 0 0 a 0 0 c 3 0 910 9 1 0 0 0 0 3 3 0 9101113 9 0 0 0 0 0 0 91011121112 9 0 0 0 0 910111212121313 0 0 0 0 9 10 11121313141414 0 0 0 9101112131414141414 0 0 913111213141414141414 0 91011121314141414141414 910111213141414141414141
913 11121314141414141414 910111213141414141414 0 9131213141414141414 0 3
101213141414141414 9 0 0 9101213141413131110 9 0 8 910121313121110 9 0 0 0 8 91012121010 9 0 0 0 a 3 8 9 9 9 9 9 0 0 0 3 0 0 0 8 a 8 1 0 0 0 0 3 0 0 0 ) c0 3 0 0 0
0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 3 0 0 0 0 0 0 0
c 0 0 ' C 3 3 0 3 0 3
0 0 0 3 0 3 J 0 3 3 0 3


0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 a 0 0 0 0 0 0 a 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 01410 0 0 0 0 0 0 0 014141410 0 0 0 0 0 014141414141210 9 0 0 01414141414131210 9 014141414141414131213 0 0141414141414141310 0 0 014141414141310 9 0 0 0 01414131310 9 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a a 0 0 0 0 0 c 0 0 a 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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r 4(A)
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H 5(A)
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E 8(A)
c 0 0 0 9 0 0 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 0 910 9 0 0 030 3 0a 0 0 0 0 0 0 0 0 3 0 0 0 0 0 a 0
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E 9(A)
0 0 0 910 9 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 910113 9 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 C c c 0 91011121112 9 0 0 0 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 910111213121313 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 101112131413141414 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 11121314141414141414 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1213141414141414141414 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 131414141414141414141414 9 0 3 0 0 0 0 0 0 0 0 0 0 0 a 0 C 0
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E 4(B)
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E 5(8)
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9(s)
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78


E 10(B)
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79


E 12(8)
0 0 0 J 0 0 0 0 0 0 010111213121110 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 3 0 3 0101112131413121110 0 C 0 0 0 0 0 0 0 C C 0 0 0 3 C 0 0 0 01011121314141413121110 0 0 0 3 0 0 0 C 0 0 0 ) 0 J 0 3 0 01011121314141414141312111C 0 0 0 0 3 0 0 0 C 0 0 0 0 C 0 0101112131414141414141413121110 0 0 0 0 C C A
0 0 0 0 0 01311121214141414141414141413121110 C 0 0 0 0 0 0 C 0 0 0 010111213141414141414141414141413121113 0 C 0 0 0 C 3 0 0 01011121314141414141414141414141414131211100 0 0 0 0 0 0 01011121314141414141414141414141414141413121110 0 0 0 0 3 010111213141414141414141414141414141414141413121110 0 0 0 0101112131414141414141414141414141414141414141413121110 0 0 1)11121314141414141414141414141414141414141414141413121110 C 111213141414141414141414141414141414141414141414141413121110 121314141414141414141414141414141414141414141414141414131211 13141414141414141414141414141414141414141414141414141414312 121314141414141414141414141414141414141414141414141414131211 111213141414141414141414141414141414141414141414141413121110 1311121314141414141414141414141414141414141414141413121110 0 0101112121414141414141414141414141414141414141413121110 0 0 1 010111213141414141414141414141414141414141413121110 0 0 0 0 0 01011121314141414141414141414141414141413121110 0 0 0 0 0 0 0 3101112131414141414141414141414141413121110 0 0 0 0 0 C 0 0 0 010111213141414141414141414141413121110 0 C 0 0 C C 0 0 0 0 C 31311121314141414141414141413121110 0 0 0 0 0 0 0 0 0 0 3 0 3 0101112131414141414141413121110 C C 0 0 0 0 0 0 0 0 0 0 0 0 0 010111213141414141413121110 0 0 3 0 0 0 0 0 0 0 0 0 0 0 3 3 0 01011121314141413121110 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0101112131413121110 0 0 0 0 0 C 0 0 0 0 0 3 0 0 0 C 0 3 010111213121110 0 0 0 0 0 0 0 0 0 0 0 0C
0 0 0 3 0 J 0 0 0 0 0 01011121110 0 C 0 0 0 0 0 0 0 0 0 0 0

GPADED PATTERN A IS A SUNSET OF E 12(E)


DISTANCE BETWEEN A AND 8 [S 12














BIOGRAPHICAL SKETCH


Carolyn Jean Roche Johnson was born on March 2, 1954, in Boston, Massachusetts, to William Paul and Jean Leah Roche. In 1957, the Roche family moved to Florida and have remained since. Carolyn studied mathematics at the University of Florida, receiving her Bachelor of Arts degree with high honors March 1976, and her Master of Science degree in December 1977. Throughout her graduate studies she has taught mathematics at the University of Florida. After graduating she will be a Member of Technical Staff at Bell Laboratories in Holmdel, New Jersey.


Carolyn is married to Karl Bruce Johnson, a 1976 graduate of the University of Florida. She is a member of Phi Kappa Phi Honor Society, the American Mathematical Society, and the Mathematical Association of America.


80















I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.




A. R. Bednarek, Chairman
Professor of Mathematics





I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.




B. B. Baird
Assistant Professor of
Mathematics





I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.


M. P. Hale Associate Professor of Mathematics














I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.




S. Y. Su
Professor of Electrical
Engineering and Computer
and Information Sciences





I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.




S. M. Ulam
Graduate Research Professor of Mathematics





This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.


August 1980


Dean, Graduate School




Full Text

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AN INTEGER VALUED HAUSDORFF-LIKE METRIC BY CAROLYN ROCHE JOHNSON A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1980

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TO MY FAMILY AND ESPECIALLY TO KARL

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ACKNOWLEDGEMENTS The author would like to express her gratitude to all those who contributed, directly or indirectly, to the completion of this work. Above all, her sincere thanks go to the chairman of her supervisory committee. Dr. A. R. Bednarek, for his guidance throughout her graduate studies. This study was made possible by his many suggestions and intuitions. The author wishes to acknowledge the remainder of her supervisory committee: Dr. B. B. Baird, Dr. M. P. Hale, Dr. S. Y. Su, and Dr. S. M. Ulam for their contributions to her academic training. She also wishes to acknowledge Dr. J. E. Keesling and Dr. R. E. Osteen for their comments and suggestions . Finally, she wishes to thank her husband, Karl, whose encouragement and love have endured her throughout her graduate studies .

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TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS iii ABSTRACT V INTRODUCTION 1 CHAPTER I BASIC DEFINITIONS AND ELEMENTARY PROPERTIES OF THE HAUSDORFF METRIC 4 II A DISCRETE ANALOGUE 10 III TOPOLOGICAL CONNECTION 18 IV APPLICATION TO DIGITIZED GREY LEVEL IMAGES 35 V A SHARPENING TRANSFORMATION FOR GRADED PATTERNS 42 VI SOME COMPUTATIONAL EXPERIMENTS 50 VII POSSIBLE APPLICATIONS AND PROBLEMS 59 REFERENCES 64 APPENDIX 67 BIOGRAPHICAL SKETCH 80 iv

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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN INTEGER VALUED HAUSDORFF-LIKE METRIC By Carolyn Roche Johnson August 1980 Chairman: Dr. A. R. Bednarek Major Department: Mathematics Motivated by the need for a non-Euclidean metric between sets of objects and using the Hausdorff metric as a prototype, an integer valued discrete analogue of the Hausdorff metric is presented. Basic properties common to both metrics are examined, and a topological relationship is established. The Hausdorff space of nonempty subsets of the unit square is shown to be homeomorphic to the completion of the direct limit of a sequence of finite spaces having the discrete Hausdorff distance as a metric. The potential usefulness of the discrete Hausdorff metric in pattern recognition as applied to graded patterns, or digitized grey level images, is examined and an algorithm for computing the distance between two graded patterns is presented. Also, a nonlinear neighborhood dependent v

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sharpening transformation particularly suited to graded patterns is presented and an alternative proof of convergence of the sharpening procedure is provided. Several computations studying the interaction of the discrete Hausdorff metric with the sharpening transformation are reported. vi

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INTRODUCTION The evolution of the discipline of mathematical taxonomy [ 7 ] has increased the need for sensitive measures of similarity or distances between objects or classes of objects. The objects of interest may be biological sequences, digitized grey level images, individuals of a population, or elements of abstract point sets. Traditionally classification involved associating the objects to be classified with some n-dimensional vector in a Euclidean space, employing Euclidean distance as the measure of similarity and then clustering. The need for a non-Euclidean metric between sets of objects (clusters) was expressed in Jardine and Sibson [ 7 ] . The concept of distances between subsets of a metric space was first examined systematically by F. Hausdorff [6 ]. In particular, for nonempty closed subsets A and B of a compact metric space {X,d) the Hausdorff distance p„ between A and B is defined to be the maximum of the two values max min d{x,y) and max min d{x,y). Using the Hausdorff xeA yeB xeB yeA distance as a prototype, and motivated by the need for a nonEuclidean intertaxal distance, Bednarek and Smith [ 1 ] introduced an integer valued discrete analogue of the 1

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2 Hausdorff metric. The present work is devoted to a more detailed examination of this discrete Hausdorff metric and, in particular, to an examination of its potential for application in pattern recognition problems . In Chapters I and II we discuss the original Hausdorff metric, the discrete analogue and their basic properties. Chapter III contains one of the main results, namely, that the Hausdorff space of nonempty closed subsets of the unit square is the completion of the direct limit of a sequence of finite metric spaces having the discrete Hausdorff distance as a metric. This, in part, justifies the terminology and the use of the discrete metric for the study of digitized pictures. A particularization, first suggested by Bednarek and Ulam [ 2 ] , to the case of graded patterns (digitized grey level images) is examined more closely in Chapter IV. In particular, we provide an algorithm for the computation of the distance between two graded patterns and illustrate its application in the Appendix. In [ 9 ] , Kramer and Bruchner introduced a nonlinear neighborhood dependent sharpening transformation particularly applicable to the patterns considered in Chapter IV. In Chapter V we provide an alternate proof of the convergence of this sharpening procedure.

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3 We feel that the relationships between our metric and the Kramer-Bruchner sharpening transformation, both being highly neighborhood dependent and both being potentially applicable to the processing and recognition of digitized images, were worth examining. Toward this end some computations were carried out and their results are reported in Chapter VI. While not definitive, these studies suggest some possible applications and additional investigations. Some of these are delineated in Chapter VII. This V7ork was supported in part by NSF Grant No. MSC 75-21130.

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CHAPTER I BASIC DEFINITIONS AND ELEMENTARY PROPERTIES OF THE HAUSDORFF METRIC Let X be a set and let R^ denote the nonnegative reals. Definition 1.1 . A function d from the cartesian product XXX into r"*" satisfying the following conditions for all elements x, y, and z of X is called a metric: (1) d{x,y) = 0 implies that x = y (2) d(x,x) = 0 (3) d(x,y) = d(y,x) (4) d(x,z) < d(x,y) + d(y,z). The pair (X,d) is called a metric space . Let (X,d) be a compact metric space, and let 2 denote the family of all nonempty closed subsets of X. For elements A and B of 2^, defining Pjj(A,B) = max {max min d(x,y),max min d(x,y)} xeA yeB xeB yeA we obtain the well-known metric on 2 introduced by Hausdorf f [ 6 ] . A topologically equivalent definition of the Hausdorff metric is given by p„, , where H Pjj, (A,B) = max min d(x,y) + max min d(x,y). xeA yeB xeB yeA 4

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5 When it is necessary to identify the metric d underlying the Hausdorff metric we will write p„ instead of p„. n , ti d Let B be an element of 2 , and x an element of X, where (X,d) is a compact metric space. Definition 1.2 . The distance from x to B , d(x,B) is given by d(x,B) = inf {d(x,b): b e b} . For any positive number r, define U(B,r) = (x e X: d(x,B) < r} . If we let N^(b) denote the usual neighborhood about b of radius r, that is, N^(b) = {x e X: d(x,b) < r} , then we observe that U(B,r) = u{N^(b): b e b} . An equivalent and perhaps more intuitive definition of p„(A,B) is given by n p„(A,B) = inf (r: A c u(B,r) and B c u{A,r)}. n Note that the value of the infimum is not always realized. That is, if p„(A,B) = r, then it is possible that A is not a n subset of U(B,r) or B is not a subset of U(A,r), as these are open sets. See Example 1.7 for an illustration of this situation . Intuitively, the Hausdorff distance between sets A and B can be viewed as the smallest number e such that an "e-expansion" of the sets A and B will lead to mutual absorption, as shown in Figure I-l.

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6 Figure I-l Proposition 1.3 . If A and B are elements of 2 and A c b, then U(A,r) c u(B,r) . Proof . This follows from the fact that if A c b, then inf {d(x,b): b e B} < inf {d(x,a): a e a} . Proposition 1.4 . For A, B, C, and D elements of 2^, Pjj(A u B,C u D) < max { Pjj (A,C) , p^^ (A,D) , p^^ {B,C) ,Pjj (B,D) } . Proof. Let p^{A,C) = q, p^(A,D) = r, Pjj(B,C) = s, Pjj(B,D) = t, and let m = max {q,r,s,t}. Let e be a positive number By definition, A c U(C,q + e) c U(C,m + e) c u (C u D,m + e) and B c U{C,s + e) c U(C,m + e) c u(C u D,m + e), so that AuBcU(CuD,m+e). Analogously, CuDcU(AuB,m+e

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7 Then p„(A u B,C u D) < m + e for every positive number e, H which implies that Pjj(A u B,C u D) < m. The following example shows that strict inequality is possible . Example 1.5 . Let X = [0,1] and d(x,y) = |x yl for elements X and y of X. Let A = [0,1/4], B = [1/2,3/4], C = [1/4,1/2], and D = [3/4,1]. Then p^{A,C) = 1/4, p^(A,D) = 3/4, Pjj(B,C) = 1/4, Pjj(B,D) = 1/4, and p^{A u B,C u D) = 1/4 < max {pjj(A,C) ,Pjj(A,D) ,Pjj(B,C) ,Pjj(B,D) } = 3/4. Corollary 1.6 . For elements A and B of 2 , p^^ (A u B,A) ^ Pjj(A,B) . Example 1.7 . An example of strict inequality is given by letting X = [0,1] with the usual metric; i.e., d{x,y) = Ix y i . Let A = {1/2} u {1/2 + (1/2)^: k = 3,4, . . .} = {1/2,5/8,9/16,17/32,...} and B = {1} . Then clearly pjj(A,B) = 1/2, while pjj(A u B,A) = 3/8 because d(l,A) = inf {d(l,a): a e A} = inf {1/2,3/8,7/16,15/32,...} = 3/8. Corollary 1.8 . For elements A, B, and C of 2 , Pjj(A u B,C) < max {pjj(A,C) ,Pjj(B,C) } . Proposition 1.9 . For elements A, B, and C of 2^, Pjj(A u B,C) < Pjj(A,B) + Pjj(B,C). Proof. Let Pjj(A,B) = r, p^(B,C) = s, and E > 0. Then

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8 A c u(B,r + e) , B ^ u(C,s + e) c u(C,r + s + 2e) , and C c u(B,s + e) c u(A u B,r + s + 2e). This implies that for every a e A there exists a b e B such that d(a,b) < r + e, and for every b e B there exists a c e C such that d(b,c) < s + e. So d(a,c) < d(a,b) + d(b,c) < r + s + 2e, which implies that AuBcu(C,r+s+2e). Thus, p„(A u B,C) < r + s + 2e , but e was arbitrary, so that H p„(A u B,C) < r + s. H Definition 1.10 . If X and Y are topological spaces, then a continuous map f from X onto Y which is one-to-one and such that f is also continuous is called a homeomorphism . Definition 1.11 . Let (X^,d^) and (X2,d2) be metric spaces. An i some try is a map f from X^ onto X2 such that d^(x,y) = d2 (f (x) , f (y ) ) for all elements x and y of X^^. Note that every isometry is a homeomorphism. In particular, if {X,d) is a compact metric space, then the map f: x {x} is an isometry of X onto a subspace of 2 . An interesting property of the Hausdorff metric, though somewhat disappointing from a topological standpoint, is that the topology of the underlying space (X,d) does not determine the topology of (2" ,p„ ). That is, two metrics d^ and d_ "d 12 can generate the same topological spaces (X,d^) and (X,d2) while the spaces (2^,p ) and (2^,p„ ) can be different. ^1 ^2 For an example of this, see Kelley [ 8 ].

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9 Borsuk [ 3 ] also points out that p„ does not measure the difference in topological structure of sets A and B. Thus, Pjj(A,B) can be arbitrarily small although the topological structures of A and B are quite different. For a y broader discussion of the topological properties of 2 , the reader is referred to [11] .

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CHAPTER II A DISCRETE ANALOGUE In this chapter we introduce an integer valued metric which is analogous to the Hausdorff metric. Basic properties and similarities between it and the Hausdorff metric are examined. Several examples of the metric are included. This metric was first introduced in a paper by Bednarek and Smith [ 1 ] , and the description which follows is found in [ 1 ] . Let X be a set with cardinality Ix| equal to n. For most purposes, we only consider the case in which X is finite. We assume that with every x in X there is associated a unique nonempty subset N(x) of X called the neighborhood of X . The only restriction that we place on N(x) is that every point x is contained in its neighborhood, x e N(x). For subsets A of X, define E (A) to be the set E(A) = u{N(a): a e A}. Recursively, define E^ (A) = E (E (A) ) , k+1 k and in general, E (A) = E(E (A)) for any positive integer k. Define E^ (A) to be just A itself, E^ (A) = A, and note that A = E^ (A) c E (A) by definition. Moreover, E°(A) c E-^(A) c E^(A) c ... c (A) c e'^'^^ (A) c ... . We observe that the operator E is monotone , that is, if 10

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11 A c B then E(A) c E(B) . We also note that E is additive ^ that is, E(A u B) = E(A) u E(B). We now define an integral metric on the nonempty subsets of X. Let A and B be two nonempty subsets of X and define the distance between A and B, p {A,B) , by min {k: A c E^(B) and B c e'^ (A) } p(A,B) = { |x| otherwise and note that p (A,B) = |x| if and only if there is no k k positive integer k such that A c e (B) and B c e (A) . Theorem 2.1 . If X is finite, then p is a m.etric on the nonempty subsets of X. Proof. By definition p (A,B) is a nonnegative integer. If p(A,B) = 0 then A c E°(B) = B and B c E° (A) = A, which implies that A = B. Conversely, if A = B then A <= B = E^ (B) and B c A = E^ (A) , so that p(A,B) = 0. The symmetry of p, p(A,B) = p(B,A), follows directly from the definition. In order to prove the triangle inequality for p, let A, B, and C be nonempty subsets of X. We wish to show that p (A,B) < p(A,C) + p(C,B). If either P {A,C) = n or p (C,B) = n, where |x| = n, the result is true. Suppose that p (A,C) = j, P(C,B) = k, and that neither j nor k is equal to n. Then A c E^(C) and C ^ E^(B) so that A <= E^{E^(B)) = E^'''^(B). Also, B c e'^(C) and C c E^ (A) so that B c e'^ (E^ (A) ) = E^''"'^(A).

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12 By definition of p this implies that p(A,B) < j + k = p(A,C) + p(C,B), and triangularity is established. For the case in which X is infinite, we obtain an extended integer valued distance between sets A and B which fails to be a proper metric only in the sense that it can take on the value infinity. Corollary 2.2 . If X is infinite, p is an extended integer valued metric on the nonempty subsets of X. We shall refer to this metric as either p or the discrete Hausdorff metric throughout further discussions. We now give several particular examples of the metric p . Example 2.3 . A graph G is a nonempty finite set of points (or vertices ) , V, together with a set E of unordered pairs of distinct points of V, called edges. We say two points V, w are edged if (v,w) e E. We write G = (V,E) . Suppose G = (V,E) is a graph and let X = V. For points X in X, define the neighborhood of x, N(x), to be the set consisting of x and all those points edged with x; that is, N(x) = (x) u {y: (x,y) e El. Then p is a metric on graphs. In particular, consider the example given in Figure II-l. In Figure II-l, B c E^ (A) and A c E^ (B) = X, so p (A,B) = 4.

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13 Figure II-l Example 2.4 . Let and N2 be positive integers and let X be the cells of an x N2 grid. Formally, X= {(i,j): 1 < i
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14 5 4 3 2 1 2 3 4 4 3 2 1 * 1 2 3 3 2 1 * * 1 2 3 3 2 1 * * 1 2 3 4 3 2 1 * * 1 2 5 4 3 2 1 1 2 3 6 5 4 3 2 2 3 4 7 6 5 4 3 3 4 5 Figure II-3 Many properties of the Hausdorff metric also hold for the discrete Hausdorff metric. The following sequel is a partial list of these analogous properties. Because the proofs for the case in which X is infinite are the same as those for the finite case, we shall only consider sets X with cardinality Ix| = n where n is finite. Proposition 2.5 . Let A, B, C, and D be nonempty subsets of X. Then p (A u B,C u D) < max (p (A,C) , p (A,D) ,p (B,C) ,p (B,D) } . Proof . Let p(A,C) = h, p (A,D) = i, p(B,C) = j, and p(B,D) = k. If h, i, j, or k equals n, then the result follows as p (A u B,C u D) < n by definition. Suppose h, i, j, and k are all less than n. Let m = max {h,i,j,k}. Then A c E^(C) c e"^(C) c e"^(C u D) and B c (C) c e"^(C) c e"*(C u D) , so that A u B c e"^(C u D) . Similarly, C c e"'(A) c e"^(A u B) and D c e"^(A) c e"^ (A u B) , so C u D c e^ (A u B) . Thus p(A u B,C u D) < m.

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15 By letting C = D = A, we obtain the following corollary. Corollary 2.6 . Let A and B be nonempty subsets of X. Then p (A u B,A) < p (A,B) . Example 2.7 . A simple example showing that strict inequality can hold in Corollary 2.6 is shown in Figure II-4, where the cells of set A are indicated by a and the cells of B by b. Cruciform neighborhoods (see Example 2.4) are used in this example to compute p(A,B). In this example, p (A u B,A) = 1 < 4 = p (A,B) . Figure II-4 Proposition 2.8 . Let A, B, and C be nonempty subsets of X. Then p (A u B,C) < p (A,B) + p(B,C). Proof. Let p(A,B) = j and p(B,C) = k. If either j = n or k = n, then the result follows, so assume j and k are not equal to n. Then A c E^ (B) and B c e'^(C) so that A u B c E^(B) c E^(E^(C)) = E^'^^ (C) . Also, C c E^(B) and B c E^{A) so C c E^(E^(A)) = E^+^(A) c E^'^^ (A u B) . Thus p(A u B,C) < j + k.

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16 At this point it would seem that increasing the cardinality of sets decreases the resulting metric value. This is shown to be false in Figure II-5 where A c b but p(B,C) > p (A,C) . The cells of A are indicated by *, the cells of B by , and the cells of C by # . In this example p(B,C) = 3 while p (A,C) = 1, again using cruciform neighborhoods . # Figure II-5 The next proposition is an observation of how the choice of neighborhoods affects the value of the resulting metrics. Suppose that for every point x in X there are associated two neighborhoods (x) and N2 (x) . Associated with these neighborhoods are tv70 metrics p^ and respectively. Proposition 2.9 . If (x) c N2 (x) for every x e X, then P2(A,B) < p^{A,B) for all nonempty subsets A and B of X. Proof. Let p^(A,B) = j, and note that if j = n the result is immediate, so assume that j < n. Let (A) = u{N^(a): a e a} for i = 1,2, and in general, E^ (A) = E^(E :^(A)) for i = 1,2. Then E^ (A) c (A) for all

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17 nonnegative integers k and any nonempty set A since (a is a subset of N2 (a) for all a e A. This implies that A c E^(B) c E^(B), and conversely that B c (A) c E^ (A) so that p„(A,B) < j = p(A,B) .

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CHAPTER III TOPOLOGICAL CONNECTION The definition of the metric p suggests a strong connection between it and the Hausdorff metric. In order to relate the two it is necessary to decide the neighborhoods of points in the discrete case and the underlying metric in the Hausdorff case. This chapter will establish a topological relationship between the unit square with the Minkowski metric and an appropriate choice of neighborhoods for the discrete version. Let I denote the unit interval and consider the metric 2 space (I ,d) where d( (x^,X2) , (7-^,72) ) = max { |x^ y^ I : i = 1,2}. Let A denote the family of all nonempty closed subsets of 2 I . Then the Hausdorff metric p„ defines a distance between elements of A. For all positive integers n, let X be the set of all n ordered pairs of binary sequences of length n; that is, " Ha^. . .a^,b^. . .b^) : a^,b^ = 0 or 1 for i = l,2,...,n}. For simplicity of notation, denote elements of X n (a^. . .a^,b^. . .b^) by (a^,b^)^. Let A^ be the collection of all nonempty subsets of X^ for all positive integers n. 18

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19 For points (^^'^-j^^j^ °f ^n' ^^^^'^^ neighborhood to be N((a.,b.)^) = nu.,v.)^ e X^: |a^...a^(2) " ^1---%(2)I < 1 and |b^...b^(2) ^l---^n(2)l " ' ^^^""^ ^l---^n(2) denotes the integer equivalent to the base 2 number a^...a^ Let p be the discrete Hausdorff metric defined on A x A n n n with the above definition of neighborhoods. We now have a sequence of metric spaces (^^^/Pj^) n > 1. Observe that the binary sequences uniquely determine numbers L and M such that a,... a = L, b, . . .b = M, 1 n (2) '1 n (2) and 0 [L/2'^, (L+l)/2^] X [M/2'^, (M+1) /2''] . Define $^ of a general element of A^ by distributing over unions in the obvious manner. Consequently, there is a one-to-one correspondence between the elements of A and n 2 certain closed subsets of I . Applying $^ to neighborhoods of points in X^ gives rise to the following correspondence: $^(N((a^,b^)^)) = {{x,y) e I^: (L-l)/2'^ < x < (L+2)/2'^ (M-l)/2" < y < (M+2)/2"} . If (a. ,b. ) is an element of X , denote E'^{{(a.,b.) }) by j-j-n n iin ]^ E^(a^,b^)^ for all nonnegative integers k. Then

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20 ^n^^n^^i'^i^n^ = { (x,y) e : (L-k)/2" < x < (L+k+l)/2^, (M-k)/2^ < y < {M+k+l)/2^}. Now let A be an element of A and define the n-th encoding of A, e^(A) , by ej^(A) = u{(a^,b^)^ 6 X^: A n ^ ^^i '^i^ n^ nonempty). Note that e^ (A) is an element of A . Denote ^ (e (A) ) bv n n n n -'^^ $ e A. n n The n-th encoding of a set A can be interpreted as placing a 2^ x grid over the unit square, and the cells of the grid which intersect A are the encoding. An illustration of this situation is given in Figure III-l. One of the main results of this section is that for elements A and B of A, Pjj(A,B) = lim ^n^^n^^^ ^^n^^^ ^ . The following lemmas will be used to establish this result. Lemma 3.1 . Pn^^'^n^n^^ ~ 1/2^. Proof. By the definition of e^ (A) , A is a subset of $ e A. ^ n n Let X be an element of $^e^A. By the one-to-one correspondence and the definition of (A) , there is an element a of A such that d(a,x) < 1/2^. it follows that d(x,A) < 1/2^ and the result follows.

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21 Figure III-l

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22 Leirana 3.2. For elements A and B of A, if p„($ e A.f e B) — H n n n n < then p (e (A) ,e (B) ) < k. / ^n n n Proof . p„i^eAr^eB) < if and only if for every H n n n n element a of f e A there is a b' in $ e B so that d(a,b') n n n n < k/2", and for every b in $ e B there is an a' in $ e A n n n n so that d(a',b) < k/2^. Let a = {a^,a^) be an element of $„e A. Let b = (3^,3^) be an element of $ e B such that n n 12 n n d(a,b) ^ k/2". Then by the one-to-one correspondence, there are points (aifbi)^ ®n ^^i'^i^n ^n^^^ ^° ^^^^ the following is true: ^l-"^n(2) = ^' '^l'--^n(2) = ^' ^l"*^n{2) " ^l---^n(2) " ^' L/2'^ < < (L+l)/2^, M/2" ^ a2 ^ (M+l)/2'^, Q/2^ < < (Q+l)/2^, R/2" 2 ^2 (R+l)/2"Then d(a,b) < k/2^ if and only if B^l < k/2^ for i = 1,2. So (Q-k)/2" ^1 " ^/^^ < < 6^ + k/2'^ < (Q+k+l)/2^ and (R-k)/2'' < 6^ " k/2" < a2 < ^2 + k/2'' < (R+k+l)/2". This implies that a is an element of ^n ^^n ^^i'"'^i^n^ • one-to-one correspondence, this means that (A) is a subset of E|j(e^(B)). By a symmetric argument, e^ (B) is a subset of E^(e (A)). The result " n n follows from the definition of n . n Lemma 3.3 . For elements A and B of A, if p ($ e A,$ e B) — H n n ' n n > k/2^, then p^(e„(A),e (B) ) > k. n n n

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23 Proof. If p„i^e A,$ e B) > we can assiime without H n n n n loss of generality that there is a point a = (a^,a2) in $^e^A so that for every point b = {Q,,^^) in $ e B, d(a,b) n n 1' 2 n n ^ / ' > Corresponding to b is the point ('^i'^j^)^ where ^^<^ (^i^r^) ^) = [Q/2^ , (Q+l) /2^] x [R/2", (R+l)/2"] . If d(a,b) > then g^l > k/2^ for some i = 1,2. Assume without loss of generality that la^ > k/2'^. Then either > 3^ + k/2^ or < k/2^. But Q/2^ ^1 (Q+l)/2" implies that either > (Q+k)/2" or < (Q+l-k)/2^. Then a is not an element of { (x,y) e I^: {Q+l-k)/2'^ < X < {Q+k)/2^, (R+l-k)/2^ < y < (R+k)/2^} = k-1 ^n^^ n ^^i'^i^n^' "^^^^ implies that e^(A) is not a subset of E^^JJ^Ce^CB) ) . So if Pjj($^e^A,$^e^B) > k/2^, then Pj^(e^(A) ,e^(B) ) > k. The main result of this section now follows from the lemmas . Theorem 3.4 . For elements A and B of A, P^(e^{A),e^(B)) p„(A,B) = lim n _n n->-«> 2 Proof. Let Pjj(A,B) = r. If r = 0 then A = B and (A) = e^(B) for all n = 1,2, , and the result is immediate. Assume that r is positive. Then because the dyadic rationals are dense in [0,1], there is a sequence {k^/2^: k^ is a positive integer, n > 1} , so that (k^-l)/2^ < r < ky2" for n = 1,2,..., and lim k /2^ = r.

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24 By the triangle inequality and Lemma 3.1, p„($ e A,$ e B) < p„($ e A, A) + p„(A,B) + p„(B,$ e B) ^ n n n n H n n H n n (k +2)/2'^. Also, p„($ e A,$ e B) S p„(A,B) p„(A,$ e A) n H n n ' n n "^H ' '^H ' n n p„(B,$ e B) > (k -3)/2^. By Lemma 3.2 and Lemma 3.3, this ^H'nnn' ^ ' implies that k 3 ^ p (e (A) , e (B) ) ^ k +2. So f n '^n n 'n n (k^-3)/2^ < (p^(e^(A) ,e^(B) ) )/2" < (k^+2)/2^. By taking the limit as n tends to infinity, the result follows. Thus we see that the Hausdorff distance between two 2 nonempty closed subsets of I can be approximated by using the metric p on the encodings of the two sets. This suggests that a stronger connection exists between the Hausdorff space (A, p^) and the discrete metric p. In order to establish the topological connection, we must consider the direct limit of the spaces A^. The following definitions are found in Dugundji [ 5 ] . Definition 3.5 . A binary relation < on a set A is called a preorder if it is reflexive and transitive, that is, (1) a < a for all a in A (2) a < b and b < c implies that a < c for all a, b, and c in A. A set together with a definite preorder is called a preordered set . A preordered set A with the additional property that for all a and b in A there is a point c in A such that a < c and b < c is called a directed set.

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25 Definition 3.6 . Let D be a directed set, and let {X^: a 6 D) be a family of spaces indexed by D. If for every pair of indices a and B with a < B there exists a continuous map 6 o = X ^ Xp, such that whenever a < B < Y f then 9 = Qo ° 9 o/ then the family of spaces and maps o^/Y P/Y o. ,i5 {X ,9 q} is called a direct spectrum over D. The maps 9 „ are called connecting maps , and the image of x^ e X^ under any connecting map is called a successor of x^ . Definition 3.7. Let {X ,9 „} be a direct spectrum. Let R a a,B be the equivalence relation given by X e X , Xd e Xq , x RXq if and only if x and x^ have aa'BB aB a 3 a common successor. Then the quotient space EX^/R is called the direct limit of 00 the spectrum , and is denoted by X . We now define a direct limit of the spaces A^. For all positive integers n, define a connecting map 9 A A , , ^ n,n+l n n+1 ^,n+l(^(^i''^i)n^) = ^^(^i'^i^n+l ^ ^n+r ^ ^ ^ ^^i '^i^ n^ ^ = ^n+1 ^'^^ ^^i'^i^n+i "^^ arbitrary element of A^, extend this map over unions of points in X^ by distributing ®n,n+l unions; i.e., $^ = o 0^ for all elements of A . For integers j and k with j < k, define 9. , : A. ^A, D / k j k to be the composition of the connecting maps: 'j,k "k-l,k "k-2,k-l ° ••• °

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26 If j < k < 1, then 9 . ^ = 9, ^ o 9 . , , and because the ],i k,l DfJ^ spaces are discrete, the maps are continuous. Thus, {a ,9 } forms a direct spectrum, and we can consider the n n,m ^ 00 . . 00 direct limit A of the spaces i^^^ • The elements of A are equivalence classes of elements of EA under the relation R ^ n n where xRy if and only if 9. , (x) = 9 , (y) for some k, where 3 , K r ,K X e Aj, y e A^, and both j and r are less than or equal to 00 k. Denote the elements of A by [x] . Lemma 3.8 . [x] is an element of A if and only if [x] = [e (A) ] for some positive integer n and some A e A. n ' — Proof If [x] is an element of A , then [x] is the equivalence class of some element a of A , [x] = [a ] . Then n n' n ^^(a^) = uJ^^{[L./2'',{L.+l)/2^] x [M. /2'' , (M. +1) /2''] } , where 0 < L^,M^ < 2^ 1 for i = l,2,...,k. Let A = u^^^{ ( (2L^+l)/2^'^^, (2M^+l)/2^'*'^) }. Then A is an element of A and e (A) = a , which implies that [e (A) 1 = — n n n [a^] = [x] . The opposite implication follows from the definition of A as the quotient space ZA^/R, and the fact that e^ (A) is an element of A . n n Lemma 3.9 . For all positive integers j and k, and elements A and B of A, p . (e . (A) ,e . (B) ) =-l±iS — 1jJJ± — 2 J>3+k j 3 3 3 2^

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27 Proof. The proof follows by induction on k. Consider the case when k = 1. Let p ^ (e ^ (A) ,6^ (B) ) = m. If m = 0 then e^ (A) = e^ (B) and the result follows. Assume m is positive, then ej (A) c E^(ej(B)), e^ (B) = E^(ej(A)), and we can assume without loss of generality that e . (A) is not a subset of m 1 Ej (ej (B) ) . Then there exists an element (a^,b^)j of e^ (A) such that (a. ,b. ) e E^{q. ,r .) . for some (q. ,r . ) . £ e . (B) , ni~ 1 but (a.,b.) . is not an element of E. (u.,v.) . for all elements (u.,v.). of e . (B) . Then 1 1 D J [L/2^, {L+l)/2='] X (M+l)/2^] = $ . ({ (a. ,b. ) .}) c $ . (E"^(q. ,r. ) .) = [ (Q-m)/2^, (Q+m+l)/2^] x [ (R-m) /2^ , (R+m+1) /2^ ] . Then by the one-to-one correspondence and the equation = ^^+1 ° n+1' ^® have the following: e.^.^^(e.(A)) c E2^^(e.^.^^(e.(B))) and e.^.^^(e.(B)) c E2-^(6.^.^^(e.(A))). This implies that P • (6 . . (e. (A)),e. (e.(B))) < 2m. If J J I J"'"-'J J f J"''-'J $j(((a,,b.).}) = $j^i(ej^j^i({(a.,b,).})) c ^j,l(E2--l(e.^.^^(((u,,v,).}))) = [(2U-2m+l)/2^'^^, (2U+2m)/2^"^^] x [ (2V-2m+l) /2^"^^ , {2V+2m)/2^'^^] , for some (u^,v^). e e . (B) ,

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28 then 2U 2m + 2 < 2L < 2U + 2m and 2V 2m + 2 < 2M < 2V + 2m. But then U m + 1 < Ji_ < U + m V m + 1 < < V + oD -)]] 0^ which contradicts the choice of (a^,b^)j. Thus, Pj+l(ej,j+l(ej(A)),e.^.^3_(e.(B))) = 2m and case 1 is established. Now assume the result is true when k = n, that is, p.^ (e. (e.(A)),e. (e.(B))) p.(e.(A),e.(B)) = l2±IL_liJ±n_^3 J : : : 2^ By case 1, p (e . (e . (A) ) ,6 . . , (e . (B) ) ) = ''^+^^l'^,1^n.l'^^'^"-^,1^nM'"^'^'" 2 which implies that the result it true when k = n + 1. Thus, the result is true for all positive k. We can now define a distance function p from A°° x a°° 00 into the nonnegative reals by p (e (A) ,0 ^(e^(B))) P^([e^(A)],[e^(B)]) = ra^:_m ^ ^^^^^ ^ ^ ^_ 2

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29 Claim. The function p is well-defined. Suppose that [e^ (A) ] = [e. (A')] and [e^(B)] = [e, (B')]. n J in jc Then there is an integer q such that n, m, j, and k are all less than or equal to q, and 6 (e (A) ) = 8 . (e.(A')) and n,q n j ,q j 6 (e^(B)) = 9, _(e, (B')). Assume that j < k and m < n. m,q m k,q k -' By Lemma 3 . 9 we get that p„(e„(A),e_ „(e„(B))) p^([e„(A)], [e^(B)]) = n n m,n m <» n ' m _n p (9 (e (A)), 9 (e (B) ) ) "^q n,q n ' in,q m 2^ Pq^^,q^e.(A-)),9^^^(e^(B'))) 2^ p, (9 (e.(A')),e (B')) ^ ^-^ = P<„([e.(A')],[e,^(B')]). Claim . The function p^ is a metric on A°° x a°° . This follows directly from Lemma 3.9 and the fact that p n is a metric for all positive integers n. So we now have a metric space (A°°,p^). Consider the Cauchy completion of A , denoted by C(A°°). Extend o to 00 00 C{A ) m the usual manner by defining Poo(liin[e (A )],lim[e (B )]) = lim Poo ( fA ) ] , [e„ (B^) ] )

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30 The following lemma will be used to establish a homeo00 ^ morphism between (A,p„) and (C (A ),p ). — ri °° Lemma 3.10 . If m < n, then p^(e^(A) ^{e^(B))) /* -TV * n n m,n m p„ ($ e A, $ e B) = ' H n n m m ^n Proof. Because $ e B = $ (6 (e (B) ) ) , we can reduce the mm n m,n m problem to showing that p^ (e^ (A) ,e^ (B) ) p„($ e A,$ e B) = ^ ^ H n n n n _n Let e^(A) = u5^^{(a,,b,)^}, ^n^n^ " Uj^^{ [L./2'', (L.+l)/2''] x , (M^+l) /2^] } and e,(B) = u^^^{(q.,r.)^}, *n^n^ = ^j=l^fQj/2''.(Qj+l)/2''] X [Rj/2'', (Rj+l)/2'']}. Claim . There exists an integer z such that Let pH^^n^n^'^n^n^^ " ^ ^"'^ suppose (z-l)/2" < r < z/2^ for some integer z. Then for all j = l,2,...,k and for some h an element of {l, 2 , . . . , t} ,

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31 $j^({{a^ ,b. )^}) = [L. 72'', (L. +1)72''] X [M. 72", (M. +1)72''] c [0^/2^^ r, (0^^+1)72"^ + r] X [r^72" r, (Rj^+1)72" + r] . This implies that $ ({(a. ,b. ) }) c n 1 . 1 . n 3 3 [(Qj^-z+l)72'', (Qj^+z)72''] X [(Rj^-z+l)72'', (Rj^+z)72'^] . So for every element x of $ e A there is an element y of ^ n n ^ $^e^B so that d(x,y) < (z-l)72", and a symmetric argument shows the converse . Thus , pR^^n^n^'^n^n^^ (2-1)72"^ < r, which is a contradiction. So p {$ e A,$ e^B) = Z72" for some integer z. n n n n n By Lemma 3.2 and Lemma 3.3, this implies that z 1 < p (e^ (A) ,e^ (B) ) < z. n n n Suppose that p^ (e^ (A) , e^ (B) ) = z 1. Then by the one-to-one correspondence this implies that for all j = l,2,...,k, there is an element h of {1 , 2 , . . . , t} so that [L.72'', (L.+l)72''] X [M. 72'', (M. +1)72''] c [(0^-2+1)72'', (Qj^+z)72''] X [(R^-z+l)72'', (R^+z)72''] .

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32 Again we get that Pjj($^e^A,$^e^B) < (z-l)/2^ < 7,n^, a contradiction. This implies that (e^ (A) , e^ (B) ) = z, and the conclusion follows. We are now able to show the connection between (A,p ) — ri and the discrete spaces (A ,p ) . ^ n n 00 Theorem 3.11. (A,p„) is homeomorphic to (C (A ),p^). Proof . Define a map A -> C{A°°) by l'(A) = lim [e^(A) ] . n->-oo 00 Note that A = "n=l'^n^n^* Suppose Y (A) = ^ (B) . Then lim [e (A)] = lim [e (B) ] n^" n-><» which implies that 0 = lim p^ ( [e^ (A) ] , [e^ (B) ] ) = lim Pn^^n^^^ /e^(B)) ^ Pjj(A,B) by Theorem 3.4. But p„ is a metric, so A = B and • is a n one-to-one map. To see that • is onto, suppose that {[e^(A^)]: n > 1} is a Cauchy suquence in C(A°°). Then for every positive number e, there exists an integer N such that

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33 D ( fe (A ) ] , [e (A ) ] ) < c if m and n are greater than N. But 0° n n m iti ;<„([e^(A^)],[e^(A^)]) = P.([e^(A^)],[e^(Aj]) = p (e (A ) ^ ) ) _2 B 2 Sliil — SL-JI! = p ($ e A e A ) where m < n. ^n '^H n n n' m m m Thus, {$ e A : n > 1} is a Cauchy sequence in A and (A,p ) 'nnn — — H is complete [ 3 ] , so there is an element A of A so that lim $ e A = A. n n n But lim $ e A = A, ^ n n n-»-oo so by the triangle inequality, for every positive e there is an integer N so that p„($ e A ,$ e A) < e if n > N. ^ '^H nnn'nn Claim. lim [e (A ) ] = lim [e„ (A) ] ^ n n n Let e be positive, then there is an such that p„($ e A ,$ e A) < e if n > N . Then H nnn nn e p ([e (A )],[e (A)]) = p ( [e (A ) ] , [e (A)]) = p^ (e^ (A^) ,e„ (A) ) = p„($^e^A ,$ e^A) < e if n > N ^n Hnnnnn e Thus ,

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34 lim [e (A ) ] = lim [e (A)] = Y (A) , "-n n ^ n ' proving that f is onto. To show that 'i' is a homeomorphism, it suffices to show that • is an isometry. But this follows from Theorem 3.4, as p„ (1' (A) ,y (B) ) = lim p^([e„(A)], [e„(B)]) = p (e (A),e (B) ) lim ^ = p„(A,B) . n^2^ " Thus, 4' is a homeomorphism from (A,Py) onto (C(A°°),p^).

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CHAPTER IV APPLICATION TO DIGITIZED GREY LEVEL IMAGES This chapter examines the possible usefulness of the metric p in pattern recognition problems. In particular, graded patterns (digitized grey level images) are introduced and an algorithm for computing the discrete Hausdorff distance between two such graded patterns is given. Definition 4.1 . Let N^, N2 , and be positive integers. A graded pattern (or digitized grey level image ) is an X N2 matrix with entries from {0 ,1,2 , . . . ,N2} . Such patterns are produced by high speed scanning devices, such as densitometers. These digitized images arise in areas such as land use studies, planetary observations, fingerprint analysis. X-ray diagnosis, and optical character recognition. An excellent example of the use of graded patterns is given in a study of the relationship between blood flow and brain function by Lassen et al . [10]. The discrete Hausdorff metric can be used to define a distance between graded patterns for fixed , and N^, thus giving a measure of similarity between patterns. This interset distance can then be used with a clustering algorithm for the purpose of pattern classification. The 35

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36 following description of graded patterns as ordered triples of integers and the definition of neighborhoods of points is due to Bednarek and Ulam [ 2 ] . Fix integers N^, N2 / and N^. Let X = {(i,j,k): 1 < i < N^,l < j < N2 , 1 ^ k < N^) . Associate with every graded pattern A = f^ij^ ^ subset A of X in the following way: A = {(i,j,h) e X: the ij-th entry of A is k * 0 and 1 < h < k) . One can visualize A as stacking blocks on an x N2 grid, where there are k blocks stacked on (i,i) if a. . = k, and no blocks on (i,j) if a^^ =0. For a point x = (i,j,k) e X, define the neighborhood of X, N(x) by N(x) = {(p,q,r) 6 X: I p i | + I q j ! + r k < 1 and r > k} . Thinking of this as blocks on an x grid, this forms a partial cruciform neighborhood which consists of all those blocks immediately to the left, right, front, back, and above the block x. This is shown schematically in Figure IV-1, where x is the center cell.

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37 Figure IV-1 Given two graded patterns A and B, p is a metric on the associated sets A and B of X. An example of p applied to two 30 X 30 graded patterns with 15 grey levels using the above choice of neighborhoods is shown in the Appendix. The expansion of each pattern is displayed until the absorption of the other pattern is achieved. Let X = (i,j,k) be an element of A, where A is a graded pattern and N(x) is defined as before. Proposition 4.2 . E^({x}) = {(p,q,r) e X: |p ij + |q j| + r k < n and r > k} . Proof. The proof proceeds by induction on n. The case n = 1 is given by the definition of N(x). Assume the result

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38 is true for case n and consider E ({(i,j,k)}) = E ((xl) = E(E^{{x})) = u{N( (p,q,r) ) : (p,q,r) e e"{{x})}. If (p,q,r) is an element of E^({x}), then Ip-i| + Iq-jl +r-k k by the inductive hypothesis. So if (s,t,u) e N((p,q,r)), then |s-i| + |t-jl +u-k< |s-p| + |p-i| + |t-ql + Iq-jl +u-r+r-k r > k . Thus e"'^-'-({x}) c {(p,q,r) 6 X: Ip i| + jq j I + r k < n + 1 and r > k} . Conversely, if {p,q,r) e X is such that |p-il+|q-j|+ r k < n and r > k, then (p,q,r) e e"({x}) c e"'*"^({x}). If |p-i| + |q-j| +r-k=n+l and r > k, then there is an element (s,t,u) in X such that |p-s| + |q-t| +ru = 1 , r > u , and |s-i| + It-j| +u-k=n, u>k. To see this, suppose k+l 0 and Iq-jl > 0. If p > i + 1 let s = i 1 and t=q; ifp
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39 So (p,q,r) e N((s,t,u)) c E(E^({x})) = e'^"'"-'( {x} ) , which implies that E^({(i,j,k)}) ={(p,q,r) e X: |p i| + |q j| + r k < n and r > k) for all nonnegative integers n. Let A = [Sj^j] and B = [bj^jl be two graded patterns. Associated with A and B are subsets A and B of X. The discrete Hausdorff metric defines a distance between A and B: p(A,B) = min (k: A c E^ (B) and B c e^ (A) } . We now provide an algorithm for computing p (A,B) . Proposition 4.3 . Let d^ = maximum minimum {[i-pl + |q-i| + (l^"''^,?''^ ^} ^ (i,j,k)eA (p,q,r)eB 0 otherwise d„ = maximum minimum {|i-p| + Iq-il + {^~^ ^ .~ ^} ^ (p,q,r)eB {i,j,k)6A 0 otherwise then p(A,B) = max {d^,dg}. Proof. Let (i,j,k) be an element of A, where 1 < k < a. .. For all elements (p,q,r) of B, where 1 < r < b , compute m^. = minimum { I p-i I + |q-j| if k < r, |p-il + |q-j| + k (p,q,r)eB m. . r if k > r} . Then (i,j,k) e E (B) . Take the maximum over d all m^., d^ = maximum f m } , then A c e ^(B). A symmetric (i,j,k)eA ^B argument shows that B c e (A), which implies that p (A,B) < max {cl^/dg}. Let m = max {d^,dg} and suppose p (A,B) < m. Then A c e"^"^ (B) and B c e"'"^ (A) , so for all elements

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40 (i,j,k) of A there exists an element (p,q,r) of B such that (i,j,k) e f (p,q,r) }) . This implies that lp-i| + |q-j| +k-r r. By minimizing over all (p,q,r) in B and maximizing over all (i,j,k) in A, this implies that d^ < m 1. A symmetric argument shows that d^ < m 1, but this contradicts m being the maximvim of d^ and dg. Thus p (A,B) = m. Note that the algorithm given in Proposition 4.3 agrees with the definition of the Hausdorff metric. In actual computer computations of the distance between graded patterns, the algorithm was not used. Rather, these distances were computed directly from the definition of p . Iterations of the operator E for the purpose of computing p are shown in the Appendix. Other metrics on graded patterns that have been used in pattern recognition are the Hamming distance H given by H(A,B) = . la. . b. . I , and the metric M given by M(A,B) = max ja. . b. . | , i,j or more generally, M(A,B) = max {d(a^.,b^.)} where d is a metric.

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41 It is our opinion that the metric p is a useful measure of similarity between graded patterns, and perhaps better than other metrics now in use as it seems to be less sensitive to small perturbations or changes. Such perturbations often result from noise or electronic interference in real data collecting and transmitting situations. Generally, graded patterns are subjected to an image enhancement procedure before being interpreted. We shall digress slightly to describe one such procedure — a procedure that seems particularly appropriate for digitized grey level images.

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CHAPTER V A SHARPENING TRANSFORMATION FOR GRADED PATTERNS In this chapter we discuss a nonlinear transformation introduced by Kramer and Bruchner [ 9 ] that can be used for sharpening digitized grey level images. An alternative, and in our opinion simpler, proof of the Kramer-Bruchner Theorem is given. As discussed previously, graded patterns often become distorted or "fuzzy" and a preprocessing sharpening may be performed in order to attempt to restore the pattern to its original state. Fourier and Laplacian transformations have been used for this purpose. A simple nonlinear transformation for sharpening digitized grey level images which depends on local operations was introduced by Kramer and Bruchner [ 9 ] . Basically the transformation replaces the value of an entry of a graded pattern A by the largest or smallest value in its neighborhood. The following definition given for the sharpening S is due to Kramer and Bruchner [ 9 ] . Rather than describe the transformation in terms of graded patterns, it is best described in terms of real valued functions F on finite sets X. In the cases of interest to us though, X will be the cells of an N^ x n^ grid, that is, X = { (i, j) : 1 < i < N^,l < j < N2} , 42

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43 and F will be the function on X which assigns to the ij-th cell the value a^^, F((i,j)) = a^ ^ , where A = [a^j] is the graded pattern. The definition of the sharpening transformation requires the notion of a neighborhood system for X. For every point X of X, associate with it a unique nonempty subset N(x) of X such that X e N(x), and we require that the neighborhoods satisfy a symmetry condition, that is, if x e N(y) then y e N(x) for all elements x and y of X. Notice that this definition of neighborhood is slightly more restrictive than that previously used. For every real valued function F on X, associate with it two other functions F and F, the local maximum and local minimum functions respectively; that is, F(x) = max {F(y): y e N(x)j and F(x) = min {F(y) : y e N(x)} for all elements x of X. If X is a finite set with a neighborhood system and F is a real valued function on X, then the sharpening transformation S is defined by ^, , F(x) if F(x) F(x) < F(x) F(x) S(F) (x) = (SF) (x) = { F(x) otherwise. Define S°F = F, and for all positive integers n, S^''"-'-F = S (S^F) is given by

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44 s"f(x) if s"f(x) S^F(x) < S^F(x) S^F (x) S^'+^FCx) = { ^ S F (x) otherwise. Kramer and Bruchner prove the pointwise convergence of the sequence (S^f). We provide a direct proof based on the cardinality of the range. A point X € X is called a local maximum of F if F(x) = F(x). Dually, if F(x) = F(x) we say that x is a local minimijm of F . It is immediate that pointwise convergence of the sequence (s'^f) is equivalent to the assertion that there exists a positive integer N such that for each x e X, x is either a local minimum or local maximum of s'^F. Theorem 5.1 . If X is a finite set with a neighborhood system and F is a real valued function on X, then for every element x of X, there is an integer N such that n > N implies that s"f(x) = S^F(x). Proof. If F is constant on X then F(x) = F(x) = F(x) for all X in X, and the result is immediate. If the cardinality of F{X), |F(X)I, equals two, then every point of X is a local maximum or local minimum of F, and again the result follows. The proof proceeds by induction on the cardinality of F(X). Assume that |F(X)| > 3, and that the result is true for all

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45 functions F and finite sets X such that 1f{X)1 < n. Let |f(X) i = n + 1. Let Uq = max {F(x): x e x} and 1^ = min {F(x): x e X} . Define M{F) = {x e X: F (x) = u^} and L(F) = {x e X: F(x) = 1 Note that if F(x) = Uq = F (x) , then S^F{x) = S^"-'-F(x) = Uq for all positive integers k, and similarly, if F(x) = 1^ = F(x) , then s'^F(x) = S^'-'^F (x) for all positive integers k. This implies that M(F) c M(SF) c M(S^F) c ... c M(S^F) c ... c X, and L(F) c L(SF) c L(S^F) c ... c L{s"f) c ... c X. Because X is finite there are integers and N2 such that N, N +k N-+k M(S -^F) = M(S ^ F) and L(S ^F) = L(S ^ F) for all nonnegative integers k. Let N = max {N^,N2}, and U = M{S^F) , L = L(S^F) . Let X be an element of X (U u L) . Claim. Either N(x) n U = ({) or N(x) n L = (f) . Suppose that N(x) n U and N(x) n L are nonempty. Then there are points y^ and such that y^ e N(x) and S^F(y^) = Uq , and c N(x) and S^F(y2) = 1q. But then

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46 S^F(x) = or S^F(x) = 1q which implies that x e U u L, a contradiction, We now consider three cases. Case 1 . Suppose N(x) n JJ_ * <^ . Then there is a y e N(x) 1q, and s'^F(x) * Uq such that s'^F(y) = u^ , and s'^F(x) * u„ so S^F(x) = S^^_^(x) < Uq. Because S^^^F(x) = Uq for all k = 0,1,2,... and x not in U, this means that S^'^'^F(x) = S^^^ "*"F (x) for all nonnegative integers k. Then S^F(x) = S^"^F (x) > S^(x) > S^"^^F (x) > ... > 1q . NN-+k So there is an integer such that S F(x) = S F(x) for all nonnegative integers k. Case 2 . If N{x) n L is nonempty then there is a y e N(x) such that s'^F(y) = S^"*'^F(y) = 1^ for all k = 0,1,2,... . But x not in L implies that s'^''"'^F(x) = S^'''^"-'-F (x) for all nonnegative k. So sNf(x) . sN+1f(x) . s'^^hu) <_ ... < Uq. N.+k N. Thus, there is an integer such that S F{x) = S F(x) for all nonnegative integers k.

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47 Case 3 . Suppose that both N(x) n U and N(x) n L are empty. Let Y = X (U u L) , = Fly, and for all y e Y, let N^Cy) = N{y) n Y. Then [ F^ (Y) | < n, so by the inductive hypothesis there is an integer such that S ^F^(x) = S ^ F^(x) for all nonnegative integers k. But N(x) n U = <() = N(x) n L implies that N(x) n y = N{x), so SF^ (x) = SF{x). Moreover, s'^F^Cx) = s'^F(x) since S^F^ (x) = S^(x) and S^F* (x) = S^F(x) Nc N N +k for all integers k. Thus, S F(x) = S F^ (x) = S F^ (x) = Ng+k S F(x) for all nonnegative integers k, and the result is true for all x in X. Remark . The Kramer-Bruchner claim of pointwise convergence of fS^F} to a function P in then established by letting N P = S *F, where is the maximum of the integers given by Theorem 5.1 for the individual elements of X. We now provide an illustration of how the sharpening S can be applied to character recognition. Let X be an 8x8 matrix and let F be the function from X into (0,1,2, ... ,7} which represents the graded pattern A shown in Figure V-1 in which there are eight grey levels. We then distorted the pattern in a random fashion by using random numbers. The fuzzy pattern is represented by T in Figure V-1 where T: X -> f0,l,...,7} is given by T(x) = F(x) + n(x) if

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48 0 ^ F(x) + n(x) < 7, T(x) = F(x) otherwise. The noise n(x) introduced was determined by first generating a random number r from {0,1,..., 9} and then using the following rule: if r = 0 or 9 then n(x) = 0 if r = 1 or 5 then n(x) = 1 if r = 2 or 6 then n(x) = -1 if r = 3 or 7 then n(x) = 2 if r = 4 or 8 then n(x) = -2. After two applications of the sharpening transformation S, 2 the limit S T was reached and found to very close to the original pattern, as shown in Figure V-1.

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49 0 7 7 7 7 0 0 0 1 7 6 7 7 1 0 0 0 7 0 0 7 0 0 0 2 6 2 1 7 1 0 0 0 7 0 0 7 0 0 0 0 6 2 0 5 0 1 2 0 7 7 7 7 7 0 0 1 6 6 7 7 7 1 2 0 7 0 0 0 7 0 0 2 7 1 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 6 2 0 0 7 0 0 0 7 0 0 0 7 0 1 2 6 0 1 0 7 7 7 7 7 0 0 0 7 7 7 5 7 2 2 1 7 7 7 7 0 0 0 0 7 7 7 7 0 0 0 0 7 1 0 7 0 0 0 0 7 0 0 7 0 0 0 0 6 0 0 7 0 2 2 0 7 0 0 7 0 2 2 0 7 7 7 7 7 0 2 0 7 7 7 7 7 0 2 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 0 0 7 0 2 0 7 0 0 0 7 0 2 0 7 7 7 7 7 0 2 0 7 7 7 7 7 0 2 ST S^T Figure V-1

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CHAPTER VI SOME COMPUTATIONAL EXPERIMENTS Both the metric p and the transformation S depend on the notion of neighborhoods of points of a set X. Due to this similarity, questions arise as to the relationship (if any) between the two. Several computer experiments were conducted in order to study this metric and its interaction with the sharpening of graded patterns. Computation 1 . The first set of computations involved twelve 8x8 matrices with random entries from (0,1,..., 7}. The Kramer-Bruchner sharpened limit was computed for each of the matrices A., i = 1,2,..., 12; denote these limits L , ... The average distance between the matrix A. and its limit L, , . > 1 A(i) was found to be 2.4167, and the average number of iterations to sharpen A^ was 4.3333. We then computed p{A^,A^) and P j) ) for all i and j, i * j. The mean of p(A^,Aj) was 3.3333 with a standard deviation of .1667; the mean of P /L^( j) ) was 2.9090 with a standard deviation of .7986. Thus, sharpening seemed to decrease the distance between graded patterns. However, numerous examples were found where p (L^ ^ ^ ^ ) > p(A^,Aj), thus showing that no relationship of inequality could be stated between p{A,B) and ^^^A'^b'^ graded patterns A and B with sharpened limits and Lg respectively. 50 i

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51 Computation 2 . As a comparison, the distances between and A j , and L^^) ^Aij) ' ^ * ^' '^^^^ computed using the Hamming distance; i.e., H(A.,A.) = E \ait-, ~ a^i I • The ^ ^ k,l average distance between A^ and A^ for i j was 87.8182 with a standard deviation of 11.5943; the mean for , . . and A(i) L^^jj, i * j, was 211.7121 with a standard deviation of 21.9333. Sharpening greatly increased the Hamming distance between graded patterns, whereas sharpening created little variance using the discrete Hausdorff distance. This could be expected though, as sharpening tends to increase the number of highest and lowest grey levels in the pattern. Because the range of values was so varied for the metrics H and p, they were normalized and are denoted by H and p" respectively. The average of p"{A^,Aj) was .1667, while the average of H(A^,Aj) was .1960. The average of P^(L^ (i) '^^^ ( j ) ) was .1455 and the average for H(L ,..,L , .«) was .4726. In normalizing p it was noted that the maximum value of p is 20, and in general, if the size of the matrix is X N2 with entries from (O , 1, 2 , . . . ,N2} then the maximum value of p is (N^-1) + (N2-I) + (N3-I) = + N2 + 3. This value is obtained, for instance, when A = [a. .1 where ^11 " ^ij " ° otherwise, and B = [b^^], = N3, ^ij ~ ^ otherwise.

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52 Computation 3 . The next experiment generated one hundred 16 X 16 matrices, for i = 1,2,..., 100, with random entries from {0,1, 2, ...,9). We then computed the sharpened limit each pattern A^, i = 1,2,..., 100, and let s(i) be the number of iterations of the transformation S needed to sharpen A^ . Next we computed P (-^^ '^j^ ( ^ i = 1,2,..., 100, and let p(i) be this value. The mean of p(i), i = 1,2,..., 100, was 3.2900 with a standard deviation of .4777. The mean value of s(i) for i = 1,2,..., 100 was found to be 7.7900 with a standard deviation of 2.2798. A histogram of the frequency of the sharpening index for the matrices A^, i = 1,2,..., 100, is shown in Figure VI-1. Computation 4 . It was conjectured that the discrete Hausdorff metric could be used as an estimator for the parameters s(i) and p(i) associated with A^. The estimation procedure was based on the nearest neighbor rule [ 4 ] which we now describe . Let (X,d) be a metric space and suppose we are given n pairs (x, ,6,), (x„,e»), (x ,6 ) where x. is an element 1 1 ^ z n n 1 of X and 9^ is a parameter associated with x^ which takes on values from {1,2,...,M}. Suppose we are given an x e X and wish to estimate its parameter 9 . We say that the element x^ of {xj^,...,x^} is a nearest neighbor to x if d(x,x^) = min {d{x,x^): i = l,2,,..,n}. If x has only one nearest neighbor x^ then we let 9 = 6^ be the estimate. However, it

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53 22 12 11 10 -I 5 2 1 4 5 6 7 8 9 10 11 12 13 14 Frequency of values for sharpening index for 100 16 X 16 X 10 graded patterns 18 Figure VI -1

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54 is possible for a point x to have many nearest neighbors, which case we let 9=0. , where the ^1 ^2 parameter 9 . occurs with the highest frequency amongst (9. ,9. ,...,9. }. In the case of ties, the choice is arbi^1 ^2 ^k trary . The nearest neighbor rule was applied to the matrices A^, taking (A^,s(l)), (A2,s(2)), (A^q,s(50)) to be the sample set. We then computed the nearest neighbors to A^^, A^2' ^100 ^^^"^9 three metrics: the discrete Hausdorff metric p, the Hamming distance H, and the maximum distance metric M given by M(A. ,A.) = max (laj^ aj, |}. Of the 50 estimates made, p correctly estimated only 9, as did M, and H correctly estimated 12. The procedure was repeated to estimate p(i) for i = 51 , 52 , . . . , 100 using (Aj,p(j)), j = 1,2,..., 50, for the sample set. This time the number of correct estimates for p(i) using p was 32, for M it was 29, and for H it was 28. The author's interpretation of the results was that p failed to be a good estimate for the parameters, particularly the sharpening index s(i), because the range of values of the metrics was too small. In particular, the values p(A^,Aj) for i = 51,..., 100, j = 1,2,..., 50 all fell between 3 and 5, whereas the possible range of values for p was (0,1,2, ... ,38} . It was felt that the range of values was small because of the nature of the matrices; the

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55 maximum entries were uniformly distributed throughout the matrix, so that E (A^) = X for small k and for all i = 1,2, .. . ,100. Computation 5 . In an attempt to make the data more meaningful as a pattern recognition problem, the next computation took three prototype 16 x 16 patterns with entries from {0,1,..., 9}, representing the letters A, B, and C, and created a class of A's, class of B's, and class of C's. This was done by introducing random noise into the three prototypes, creating one sharp version and nine fuzzy versions for each type. The parameter associated with each of the patterns was either 1, 2, or 3 depending on whether it was a member of the class of A's, B's, or C's respectively. We then took the prototype A and created 46 fuzzy versions, F^, i = 1,2,..., 46, by adding random noise. The Kramer-Bruchner sharpened limits, L^^^^, were computed for i = 1,2,. ..,46. The mean of the sharpening index s(i) was found to be 3.4348 with a standard deviation of 1.1086. For comparison with the frequency of sharpening given in Figure VI-1, a similar histogram is given in Figure VI-2 for F^, i = 1,2,..., 46. Note that the sharpening index mean dropped significantly from that given in Computation 3. This was most likely due to the fact that the entries of the patterns were no longer random, so the maximum entries were no longer uniformly distributed.

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56 20 11 8 5 1 s(i) Frequency of values for sharpening index for 46 16 X 16 X IQ graded patterns Figure VI-2

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57 The distances between F. and A., B., C., i = 1,...,46, 1 3 3 3 j = 1,...,10, were computed using both the discrete Hausdorff metric and the Hamming distance. Also, the distances between the sharpened versions, L^^^, i = 1,2, ...,46, and Aj , By Cy j = 1,2,..., 10, were computed. Based on these computations, we classified each F. and L„ . . v using the 1 F V 1 j nearest neighbor rule. The Hamming distance correctly classified all F. and L„,.w i = 1,2,..., 46, and we note that the nearest neighbor of F . and L„ , . . was always A, . The mean of the nearest neighbor distance for F^ using the Hamming distance was 154.7391 with a standard deviation of 12.5423; the mean nearest neighbor distance for the sharpened version was 120.4130 with a standard deviation of 14.3319. Thus, unlike Computation 2, sharpening tended to decrease distances . When the discrete Hausdorff metric was used to classify the fuzzy A's, F^ , i = 1,...,46, based on nearest neighbors, 42 were correctly classified, or 91.304 percent. I^rhen the classification was made using the sharpened versions L F (i) 44/46 or 95.652 percent were correctly classified. The average nearest neighbor distance for F^ was 2.3478 with a standard deviation of .4815, and the average nearest neighbor distance for the sharpened versions decreased to 2.2826 with a standard deviation of .4552.

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58 We observed that the only time the discrete Hausdorff metric misclassif ied an F . or L_ , . v was when there was one 1 F (i) more nearest neighbor in the B class than in the A class. This was most likely because the distance between the prototypes A and B was small, p(Aj^,B^) = 3. We altered the pattern A^ so that p(A^,B^) = 4 and repeated the computations. This time both p and H correctly classified all F. and L„ , . . 1 F(i) These computations show that the metric p might be useful in character recognition problems, especially if the size of the matrices was sufficiently large enough to allow significant distances between pattern types.

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CHAPTER VII POSSIBLE APPLICATIONS AND PROBLEMS The computations presented in Chapter VI suggest that the discrete Hausdorff metric could be useful in pattern recognition problems such as character recognition. Whether it is truely a better similarity measure on graded patterns than those in existence remains to be seen. It does have some advantages which seem to make it better, at least from a geometric standpoint. That is, the choice of neighborhoods for the discrete Hausdorff metric reflects a geometrical structure on the subsets of a set X. This is not the case for other similarity measures in use, for instance, the Hamming distance. The usefulness of a geometric structure is evident in such problems as photo interpretation and analysis. These problems arise, for instance, in land use studies where the photo is taken by satellite and transmitted to earth as a digitized grey level image. Such studies are conducted regularly by private and government agencies in an attempt to study the evolution of land use, and also to locate deposits of minerals and other natural resources (such as oil) . Another instance of the use of graded patterns comes from satellite and telescopic photos of planetary observations. 59

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60 For example, Strom and Strom [14] studied the evolution of disk galaxies with the aid of the interactive picture-processing system (IPPS) developed at the Kitt Peak National Observatory. Black and white photos are made with the use of a telescope in the ultraviolet to red regions of the spectrum. The various light spectrum photographs are weighted and composed to produce a digitized grey level image in which the various grey levels are color coded. The discrete Hausdorff metric provides a measure of similarity between such images. In Chapter IV we alluded to the use of graded patterns in a study of the relationship between brain function and blood flow by Lassen et al. [10] . Specifically, they studied the changes in blood flow in areas of the human cerebral cortex in relation to specific sensory, motor, and mental activities performed by the subject. Their method of study was based on the idea that localized increased blood flow corresponds to an increase in local activity of the surrounding tissue. The study was done by injecting a small amount of a radioactive isotope, xenon 133, into the carotid artery in the neck, and measuring the arrival and subsequent washout of the radioactivity. The measurement was made by a gamma-ray camera consisting of 254 externally placed scintillation detectors, each detector measuring approximately one square centimeter of brain surface. The data were processed by computer and displayed on a color-television

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61 screen, with different flow levels being assigned different colors or hues. These scientists were able to show a correlation between specific mental stimulation and actual activities being performed by the subject. Because blood flow increases and decreases are localized, it is conjectured that the discrete Hausdorff metric could be of use in analyzing such data. Land use studies also share this attribute of localization which makes them particularly suited for a similarity measure, such as the discrete Hausdorff metric, which is dependent on a neighborhood system. Other possible applications of the metric p arise in taxonomy, in particular, as a paleontological dissimilarity measure as discussed by Bednarek and Smith in [ 1 ] . Many of the taxonomic distances in existence have very complex algorithms, and often they are not true metrics. The discrete Hausdorff metric is simply stated and generally easily computed. Because of the generality of the metric p , it is conjectured that a number of metrics introduced earlier as evolutionary distances will be specific cases of p . Sellers [12-13] provides an algorithm for an evolutionary distance that was introduced by Ulam [15] and discussed further by Waterman, Smith, and Beyer [16] . This distance on finite

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62 sequences is metric and is used to measure the degree of evolutionary divergence between homologous proteins or nucleic acid sequences. Basically, it compares two finite sequences A and B, not necessarily the same length, and finds the common subsequences. It can be interpreted as the smallest number of weighted changes necessary to bring the two sequences into coincidence. Several questions concerning the discrete Hausdorff metric remain open. Due to the similar dependence on a neighborhood system, the question arises as to what relationship (if any) exists between the sharpening transformation S and the metric p. Specifically, what conditions will insure a relationship between p(A,B) and p{L^,L_,), where A A B and B are graded patterns and L and L are their sharpened limits respectively? We have noted instances where p (A,B) > p{L^,Lg) and p(A,B) < p(L^,Lg), showing that some additional assumptions are necessary in order to predict a relationship. Also, what relationship exists between the sharpening index and the bounds , and on the graded pattern A? Another area of investigation concerns metric semigroups. A metric semigroup is a triple (S,o,d) where (5,°) is a semigroup, (S,d) is a metric space, and o is continuous with respect to the topology induced by d. More generally, a topological semigroup is a topological space G that is also

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63 a semigroup with operation " such that the mapping (x,y) -> xoy is continuous. If (X,d) is a compact metric space, then (2 ,u,p„) is a metric semigroup, as the union operation is continuous with respect to the Hausdorff metric. Furthermore, in this space p (A u B,C u D) < max {p {A,C) ,p (A,D) , p (B,C) ,p (B,D) } . Is this the only metric semigroup having this property?

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REFERENCES R. Bednarek and T. F. Smith, A taxonomic distance applicable to paleontology, to appear in Mathematical Biosciences. R. Bednarek and S. M. Ulam, An integer valued metric for patterns. Fundamentals of Computation Theory , FCT 79, Akademic-Verlag, Berlin, 1979, 52-57. Borsuk, On some metrizations of the hyperspace of compact sets, Fundamenta Mathematicae , 41 (1955), 168-202. M. Cover and P. E. Hart, Nearest neighbor pattern classification, IEEE Trans, on Information Theory, IT-13, 1 (1967), 21-27. Dugundji, Topology , Allyn and Bacon, Boston, 1966. Hausdorff, Set Theory , Chelsea Pub. Co., New York, 1957. Jardine and R. Sibson, Ma th ema t i ca 1 Taxo nomy , John Wiley and Sons, New York, 1971. L. Kelley, General Topology , D. Van Nostrand, Princeton, N. J., 1955. P. Kramer and J. B. Bruchner, Iterations of a nonlinear transformation for enhancement of digital images. Pattern Recognition, 7 (1975), 53-58. A. Lassen, D. H. Ingvar, E. Skinh(})j, Brain function and blood flow. Scientific American, 239, 4 (1978), 62-71. Michael, Topologies on spaces of subsets. Trans. Amer. Math. Soc . , 71 (1951), 152-182. H. Sellers, An algorithm for the distance between two finite sequences, J. Combinatorial Theory (A), 16 (1974) , 253-258. 64

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65 H. Sellers, On the theory and computation of evolutionary distances, SIAM J. Appl. Math., 26 (1974), 787-793. E. Strom and K. M. Strom, The evolution of disk galaxies. Scientific American, 240, 4 (1979), 72-82. M. Ulam, Some combinatorial problems studied experimentally on computing machines , Applications of N\amber Theory to Numerical Analysis , Academic Press, New York, 1972. S. Waterman, T. F. Smith, W. A. Beyer, Some biological sequence metrics, Advan. Math., 20, 3 (1976), 367-387.

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APPENDIX

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PATTERN A c 0 0 0 0 0 3 0 0 0 0 0 C 0 0 0 0 0 0 0 3 0 0 0 3 3 0 0 0 0 0 0 0 3 0 0 0 0 3 0 0 0 0 0 3 0 J 0 0 0 0 3 3 0 3 0 3 0 0 0 0 0 u 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 0 0 3 0 0 c 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 3 9 0 3 0 0 3 0 0 0 0 0 0 0 9 9 0 3 0 0 3 3 0 0 3 0 0 D 91 0 9 0 0 0 0 3 0 0 0 0 0 0 9101 11 2 0 0 0 0 0 0 0 J 0 9101 11213 0 0 tj 3 0 0 0 3 0 9 10111 13 14 0 c 3 0 3 D 3 n 9101 1 12131 41 4 c 0 3 3 -> J 3 6 91011 12131 4 0 0 0 0 0 0 0 3 910111213 0 0 0 3 0 3 J 0 9101112 0 0 0 0 3 0 '3 0 3 0 10 1212 0 0 0 0 0 3 0 0 0 0 0 9 9 9 9 9 0 0 0 3 0 0 3 0 0 3 e 0 0 0 0 0 0 0 0 3 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 c 0 0 3 0 0 0 0 0 3 c 0 0 0 0 0 0 0 0 0 c 0 0 0 0 3 0 0 0 0 0 0 0 w 0 0 0 c J 3 0 0 0 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 3 0 0 0 0 3 3 0 0 0 3 3 0 0 0 3 0 0 0 J 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c G 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 c 3 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 Q C 0 0 3 0 0 0 3 0 0 0 0 0 0 0 0 C 0 0 0 0 0 c 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 c 0 3 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 c c c 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 J J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Q 0 0 0 0 0 0 0 c c 3 0 0 0 c J 3 3 0 0 0 0 0 0 0 3 3 0 0 0 0 Q 0 0 0 0 0 0 0 0 c 3 0 0 0 0 r> J 3 0 0 C 0 0 3 0 0 c 3 0 0 c c J u 0 0 01 41 0 c 0 0 0 3 0 0 0 0 0 0 0 0 01413 0 0 0 0 0 0 0 0 0 0 0 0 01 4141313121 0 9 3 0 0 0 0 0 0 0 0 0141413131 0 9 3 0 0 0 0 0 U 0 0 0 0 Q 0 c 0 0 0 0 0 0 G u J 0 3 0 0 0 0 0 u 0 u 0 0 u 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 G c 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 c c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c c 0 0 0 0 C 0 0 w 0 0 3 3 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 ^ 0 0 c c E 1(A) c n C 3 0 3 0 0 0 3 0 3 3 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 0 0 3 3 0 0 0 0 0 0 0 3 0 3 0 0 (3 0 0 0 C 0 0 0 0 0 ^ u 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 3 c 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 3 0 0 0 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 3 c 0 0 0 0 0 c c 0 0 0 0 0 3 0 0 0 0 3 0 3 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 3 0 3 0 0 0 0 A 3 0 0 0 0 0 0 C 0 0 0 0 0 c 3 0 0 0 0 0 0 0 3 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 G 0 0 3 0 910 9 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 3 0 913 10 9 0 0 0 0 0 0 r> 0 0 0 0 0 C 0 0 0 c 0 0 0 c 0 c c 3 •J 9 10111112 0 0 0 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 3 0 9 10 1 1 121 31 3 0 c •J 3 0 0 0 0 c 0 0 0 0 0 c 0 0 3 0 0 0 0 3 9131112131414 0 0 3 0 0 0 0 0 0 0 0 0 c 3 0 3 0 0 0 c 3 913 111213141414 0 3 0 c 0 0 0 3 0 0 3 0 0 3 0 0 0 0 0 0 3 0 9 10 1 1121314141414 0 0 0 0 0 0 0 coco 0 0 0 0 0 0 0 0 0 0 9131 11213141414 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 9131112131414 c 0 3 0 0 0 0 0 0 01410 0 0 0 0 3 0 0 0 0 3 0 9 13 1 21213 13 0 0 0 0 0 0 0 0 0 014141410 c 0 0 0 0 0 c 0 c 1012131312 9 0 0 0 0 0 0 0 01 4141413121 0 9 0 0 0 c c 3 3 9 13 12 1213 13 c 0 0 3 0 0 0 0 01414141414131210 9 0 0 0 0 3 3 3 9 9 9 9 9 0 3 0 3 0 0 0 0 0 01414141414131 C S 0 c c c 0 3 0 3 6 3 0 0 0 0 0 3 0 3 0 0 0 0 0141413131 0 9 3 0 0 0 0 0 0 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 3 0 J 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 3 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 3 3 0 3 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 c c 3 0 3 3 3 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 c 0 0 0 3 C 0 0 0 rj 0 0 0 0 3 0 0 0 0 G 0 0 0 0 0 0 3 0 0 0 0 67

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68 E 2(A) 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v 0 0 0 0 0 0 0 0 Q 0 0 3 c c 0 0 0 0 0 0 0 0 0 0 0 0 V 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Q 0 0 0 0 0 0 0 0 0 c 0 0 0 0 c 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 •J J n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 Q 0 0 c 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 c 0 0 0 91 0 <; 0 0 0 0 0 3 n Q n Q 0 0 0 0 0 0 0 0 0 9 101110 9 0 0 0 0 0 0 0 w Q w n 0 0 0 0 Q 0 0 0 0 0 0 0 91011111112 0 0 0 0 0 n n \J A w \j n \J 0 c 0 0 0 Q w n w w Q c 910 111 2121 313 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 5 9 10 11 1213141414 0 0 0 0 Q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 91011 I 2131 4141414 0 3 n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 91011 12131414141414 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9101112 13 141414141414 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Q c 91011 12131414141414 0 0 3 0 0 0 01410 0 0 0 0 0 0 0 0 0 0 910 111 21314141414 0 0 0 0 0 0 0 01 414141 0 0 0 0 0 0 0 0 0 0 910121313141414 0 0 0 0 3 0 014141414141210 9 0 0 0 0 310121314141313 9 0 0 3 0 0 3 0 0141 414141413121 0 9 0 0 0 0 0 9 10 1213131211 10 9 0 0 0 0 0 014141 41414141413121 0 g 0 0 0 0 8 91012121010 9 0 0 0 c 0 0 01 41 414141414141 310 9 0 0 0 0 0 0 9 9 9 9 9 0 0 0 0 0 0 0 0 0 01 41 414141 4131 0 9 0 0 3 0 0 0 0 8 8 8 3 0 0 0 0 0 0 0 0 0 0 0 0141413131 C 9 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 a 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 H 3(A) 00000 0 000 0 0 0 0 cooocoooo 0 3 3 0 300000000 0 0 0 0 C00000 3 00 0 0 0 0 oooocoooo 0 0 0 0 000090000 0 0 0 0 000 91 09000 0 0 0 0 3 0 9 10 1110 9 0 0 0 0 0 0 0 9101 1 1 21 11 2 9 0 0 0 0 0 910111212121313 0 0 0 0 0 9 10 1 1 121 313 141414 0 0 0 0 913 1112131414141414 0 0 0 9 10 11 1213141414141414 0 0 9101 I 121314141414141414 0 9101112131414141414141414 913 11 12 1314141414141414 0 91011 1213141414141414 0 0 91312 13 141414141414 0 0 0 10 121314 1414141414 9 0 0 0 91012131 41 41313 1110 9 0 0 a 9 10 12 1313121110 9 0 0 0 0 8 91012121010 9 0 0 0 0 003999990 0 0 0 0 0 0 0888300 0 0 3 0 OOOOCOOOO 0 0 3 0 003300 3 03 0 0 3 0 000000000 0 0 0 0 COOOCOOOO 3 0 0 0 000300000 3 0 0 0 COOOCOOOO 0 0 0 0 ooooooooooooooooc 00000000000000000 ooooocooooooooooo ooooocooooooooooo ooooocooooooooooo ooooooooooooooooc 00000000030000000 ooooocooooooooooo 00000000000300000 ooooocooooooooooo ooooocooooooooooo 0000000000 0 000000 ooooocooooooooooo OOOOOOOOOOOCOOOGO 0 0 0 0 0 01410 000030000 0 0 0 0 014141410 00000000 3 0 0 014141414141213 9 3 0 0 0 0 0 0 0141414141414141210 9 0 0 0 0 0 0 014141414141414131210 9 0 C 0 0 0141414141414141414131210 9 0 0 0 0 01414141414141414141310 9 0 0 0 0 0 0141414141414141310 9 0 C C 0 0 0 0 014141414141310 9 0 0 0 0 0 0 0 0 0 01414131310 9 0 0 0 C 0 OOOOOOOOOOOOOOOOC OOOOOCOOOOOOOOOOO 0 3 000000 OCOOOOOOO 00000300000000 0 00 OOOOOODCOOOOOOOOO OOOOOCOOOOOOOOOOO

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70 6(A) 3 0 0 0 0 3 0 3 0 0 0 0 0 0 0 0 0 ? 0 0 0 0 09109 0 0 0 9101 11 0 9 0 C 9 10 11 12111 2 9 910 11 1213121313 0 0 0 0 0 0 0 101112131413141414 9 1 1 12131414141414141 4 0 0 0 0 0 3 0 0 0 0 3 0 0 0 12 13 1414141414141414141414 14 141414141 41 41414141210 9 1213141414141414141414141414141 41 4141414141414141210 12131414141414141414141414141414141 41 41 4 14 14 14141 41 21 0 0 0 0 0 0 0 0 0 0 0 1213141414141414 1414141414 0 1213141414141414 141414141414 121314 14 1414141<»14141414141414 0 0 0 0 0 0 0 0 0 1213141414141414141414 0 1213 1414141 41 4141 4141414 0 J 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 w 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 01410 0 01 4141410 3 0 0 c 0 0 0 0 0 0 0 0 0 0141 4141414121 0 0 0 0 0 0 0 0 0 c 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1314141414141414141414141414141414141414141414141414121 0 141414141414141414141414 914141414141 41 414141 41 4141413121 13 141414 14141414 14141411 141414141414141414141414141414131 3 0 0 0 0 0 c 0 0 c 0 0 0 0 0 0 0 c 9 1213141414141 4 1414141 110 13 12131414141414141110 9 9131213 141413 13 1110 9 a 9 10121 3131 211 1 0 9 3 91012121310 9 8 9 9 9 9 9 0 3 0 0 0 0 8 3 3 3 0 0 0 0 0 0 0 914 141414141 41414141414141414141 0 014141414141414141414141414131 0 0 014141414141414141414141310 3 01414141414141414141310 9 21 0 31 0 0 9 0 0 3 0 0 0 0 0 014141414141414131 0 0 01 414141414131 C 9 0 01414131310 9 C 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) 7( A 0 0 3 3 0 0 3 0 91 0 3 910111 0 9 0 0 3 0 3 9 10 11 121 11 91011 1213121 3 3 3 0 0 0 101112131413141414 9 11121314141414 14 1414 0 0 0 0 9 13 3 0 3 0 0 9 3 3 3 3 3 3 3 0 1213141414 14 1414 141414 9 131414141414141414141414 1314141414141414141414141 1214141414141414141414141 1314141414141414141414141 1 314141414141414141414141 1314141414141414141414141 1314141414141414141414141 1 31 4141414141414141414141 1414141414141414141414141 1414141414141414141414141 1414141414141414141414141 1314141
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78 ; 10 ( 8 » C 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J 0 0 0 0 0 0 c 0 0 0 0 0 10 CI 31 1 0 0 0 0 0 OOOOOlOlllOOOCOOO 0 0 0 01011121110 0 C 0 0 0 0 0 0 10 1 1 121 3121 1 1 C C 0 0 0 0 010 111213 1413121110 0 0 0 01011121314141413121110 C 0 0 10 1 1 12 1314141414141 31 21 1 1 C 0 101112131414141414141 41 31 21110 1112121414141414141414141312111 1213141414141414141414141413121 110 0 0 0 0 0 101112 13 14 14 14141414 14 14 14141414 1413121110 0 0 0 0 0 010 111213 14 1414 141414 14 14 14 1414141414 1413 121110 0 0 C 010 1112131414 141414141414 14 14 1414141414141413121110 0 010111213141414141414 14 14141414141414 14 1414141 41 3121110 10111213141414141414141414141414141414141414141414131 21 11 11121314141414141414141414 14 141414141 41 4141414141414131 21 1011121314141414141414141414141414141 41 414141414141312111 0101 1121 3 14141414141414141 41414141414141414141 41 3121110 0 10 1112131414141414141414141414141414141414131211 10 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 c 0 0 0 0101 1121314141414141414141414141414141413121 110 0 01011121314141414 1414 1414141414141413121110 0 0 010 11121314 14 14 1414141414 1414 1413121110 C 0 0 0 0 0 0 0 0 0 0 D1011121314141414141414141413121110 0 0 0 0 0 3 0 3 0 0 10 0 0 11 1213141 414 1414141 413121 11 10 11 1213141414141413121110 0 0 01011121314141413121110 0 0 0 0 0 10 111213 14 13121110 0 0 0 0 3 0 010111213121110 0 0 0 3 3 0 0 0 01011121110 0 0 0 0 00 3000 0 10 11100 0 000 0 0 0 0 0 3 0 010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 c 0 0 c 0 0 0 0 c 0 0 c 0 0 0 0 c 0 0 c 0 0 0 00030300000 01011121110 OCOOOOOOOOO 0003030000 0 10 111213121110 OOCOOOCOOO 3 0 0 0 C 3 0 0 0 0101112131413121110 0 0 0 0 0 0 0 0 0 00000300 0 1011121314141413121110 00000000 3 0 0 0 0 3 0 0 10111213141414141413121110 0 0 0 0 0 0 0 0 0 0 0 0 3 3 10 111213141414 14 1414 1413121110 0 0 0 0 0 0 0 0 0 3 0 010 11121314141414141414141413121110 C C 0 0 C 0 0 0 3 010111213141414141414141414141413121110 0 0 0 0 0 0 0 3 ion 1213 14 1 4 14141 414 14 14 m 1 41 4141413121 1 1 0 0 0 0 C 0 01311121314141414141414141414141414141413121110 0 C 0 3 101112131414141414141414 14 141414141414141413121110 0 0101112131414141414141414141414141414141414141413121110 101112131414141414141414141414141414141414141414141312111 11121314141414141414141414141414141 41 41 414141414141413121 12 13 14141414141414141414141414141414141414141414141414131 111213141414141414141414141414141414141414141414141413121 131112131414141414141414141414141414141414141414141312111 3101112131414141414141414141414141414141414141413121110 0 0 10 111213 14 1414 14 141414 14 14 14 14 14141414141413121110 0 0 0 0 13 111213 14 14 14 141414 1414 14 14 14141414 1413121110 0 COO 0101112131414141414141414141414141413121110 0 0 0 0 0 010 1112 1314 14141414 14 14 14 14141413 121110 3 0 0 0 0 3 13 111213141414 14 1414 14141413121110 0 0 0 0 0 C 3 310111213141414 14 14 14 1413121110 0 0 3 0 0 0 0 0 010111213141414141413121110 0 C 0 0 0 0 0 0 0 01011121314141413121110 C 3 3 3 0 0 0 3 0 0 3 0101112131413121110 0 0 0 3 0 0 3 0 3 0 0 3 0 0131112 1312111C 0 0 0 0 03000330303 31011121110 3 0 0 0 0 0000 0000000001011100 3 0000 110 21 1 110 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c c 0 0 0 0 0 0 c 0 0

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79 E 12(B> 0000000000 010111213121110 OOOOOOOOOOOO 0 0 0 0 0 0 0 0 0 01011121314131211100 C 0 0 0 00 0 0 C C OOOOCOOO 01011121314141413121110 OOOOOOOCOO 0 0 0 0 0 0 0 0 10 1 1 1213141414 14 1 4 1 31 21 I 1 C OOOOOOOCC 0 0 0 0 C 0 0101112131414 14 14 1414 1413121110 OOCOOCCO 0 0 0 0 0 0 10 1112121414141414 14 14141413121110 C 0 0 0 0 0 0 C 0 0 0 010111213141414141414141414141413121110 0 C 0 0 0 C 0 0 0 010111213141414141414 14 14141414141413121110 0 0 0 0 0 0 0 0 10 111213 14 14 14 1414 14 14 141414 14141^14 1413121110 0 0 0 C 0 010111213141414141414141414141414141414141413121110 0 0 0 0 101112131414141414141414141414141414141414141413121110 0 0 10 11 12 13141414 14 14141414141414 14 1414141414141414141312111 0 C 11 12 1314141 41 414141414141414141414141414141414141414131 21 110 12131414141414141414141414 14 1414141 41 41 4 14 14 14141 41 4141 31 211 131414141 41 41414141414141414 14 1414141 41 4141414141 4141 41 41 312 12 12 14141414141414141414141414141414141414141414141414131211 1112131414141414141414141414 1414141414 14 1414141 41 41 41 31 21 110 1011121 3 14141414 14141414 1414141414141 41414141414141 3 121 110 0 0 1011121214 14 14 1414 141414141414141414141414141413121110 0 0 0 0 10 11121314 14 14 14 1414 14 14 1414 1414141414141413121110 0 0 0 0 0 01011121314141414141414141414141414141413121110 0 0 0 C 0 0 0 01O1H2131414141414141414141414141413121 1 I 0 0 0 0 0 0 C 0 0 0 010111213141414141414141414141413121110 0 C 0 0 C C 0 0 0 0 C 01011121314141414 14 1414141413121110 0 0 0 0 C 0 0 0 0 0 0 0 0 0 10 1112 131414141414 14 1413121110 CCOOOOOO 0 0 0 0 0 0 0 010111213141414141413121110 000000000 0 0 0 0 0 0 0 0 01011121314141413121110 0 0 0 0 0 0 0 C 0 0 00000 0 000 0 10 111213 14 13121110 OOOOOCOOOOO 0 0 0 0 C 0 0 0 0 0 010111213121110 0 0 0 0 0 0 0 0 0 0 0 0 00000000000 01011121110 ocooooooooooo GRADED PATTERN A IS A SUBSET OF E 12(E) DISTANCE BETWEEN A AND B IS 12

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BIOGRAPHICAL SKETCH Carolyn Jean Roche Johnson was born on March 2, 1954, in Boston, Massachusetts, to William Paul and Jean Leah Roche. In 1957, the Roche family moved to Florida and have remained since. Carolyn studied mathematics at the University of Florida, receiving her Bachelor of Arts degree with high honors March 1976, and her Master of Science degree in December 1977. Throughout her graduate studies she has taught mathematics at the University of Florida. After graduating she will be a Member of Technical Staff at Bell Laboratories in Holmdel, New Jersey. Carolyn is married to Karl Bruce Johnson, a 1976 graduate of the University of Florida. She is a member of Phi Kappa Phi Honor Society, the American Mathematical Society, and the Mathematical Association of America. 80

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. A. R. Bednarek, Chairman Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. B. B. Baird Assistant Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. M. P. Hale Associate Professor of Mathematics

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. S. Y. Su Professor of Electrical Engineering and Computer and Information Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. S. M. Ulam Graduate Research Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1980 Dean, Graduate School


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