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Title: Cumulative residual entropy, a new measure of information & its application to image alignment
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Title: Cumulative residual entropy, a new measure of information & its application to image alignment
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Cumulative Residual Entropy, A New Measure of Information & its Application
to Image Alignment *

B. C. Vemuri F. E. Wang', M. Rao2 and Y Chen2


1Dept. ofCISE,
University of Florida,


Abstract

In this paper we use the cumulative distribution of a ran-
dom variable to define the information content in it and
use it to develop a novel measure of information that paral-
lels Shannon entropy, which we dub cumulative residual en-
tropy (CRE). The salientfeatures of CRE are, (1) it is more
general than the Shannon Entropy in that its definition is
valid in the continuous and discrete domains, (2) it possess
more general mathematical properties than the Shannon en-
tropy and (3) it can be easily computedfrom sample data
and these computations asymptotically converge to the true
values. Based on CRE, we define the cross-CRE (CCRE)
between two random variables, and apply it to solve the
uni- & multi-modal image alignment problem for parame-
terized (rigid, aifine and projective) transformations. The
key 'i,. rili of the CCRE over using the now popular mu-
tual information method (based on Shannon 's entropy) are
that the former has ',,-,r ... Iid\' larger noise immunity and
a much larger convergence range over the field ofparame-
terized transformations. We demonstrate these i, ,.,1rih, I ,
experiments on synthesized and real image data.

1. Introduction

The concept of Entropy is central to the field of Informa-
tion Theory and was originally introduced by Shannon in
his seminal paper [16], inthe context of communicationthe-
ory. Since then, this concept and variants thereof have been
extensively utilized in numerous applications of science and
engineering. To date, one of the most widely benefiting ap-
plication has been for data compression and transmission.
Shannon's definition of entropy originated from the discrete
domain and its continuous counterpart called the dittrrential
entropy is not a direct consequence of the definition in the
discrete case. It is well known that the Shannon definition of
Entropy in the discrete case does not converge to the con-
tinuous definition [7]. Moreover, the definition in the dis-
*This research was in part funded by the NIH grant NS42075.
Manuscript submitted to ICCV'03. Also, a technical report dept. of CISE
TR03-005.


2Dept. ofH /itih'en ti \
Gainesville, Fl. 32611


create case, which states that the entropy H(X) in a random
variable, X, is H(X) = E, p(x)log(p(x) is based on
the density of the random variable p(X), which in general
may or may not exist [7]. Several alternative measures have
been defined in literature [13, 1, 8, 9] to overcome some
of these drawbacks. In this regard, all of the methods either
simply replace the summation with an integral or use the di-
rected divergence from the uniform distribution. The use of
directed divergence i.e., comparing the uncertainty in a ran-
dom variable to that in one which maximizes the Shannon
entropy namely, the uniformly distributed random variable
seems to overcome the difficulty of leaping from the entropy
definition in the discrete random variable case to that of the
continuous case. For more details, we refer the reader to [9].
However, this approach is not a direct solution to the prob-
lem i.e., uses a comparative/relative measure. In this paper,
we present a new measure of information in a random vari-
able that will overcome the aforementioned drawbacks of
the Shannon entropy and has very general properties as a
consequence. This new measure is a fundamental depar-
ture from all the existing measures of entropy in that it is
based on the probability distribution of a random variable
rather than its density function. We will also present some
interesting properties of this measure and then state some
theorems which are proved elsewhere [5]. Following this,
we will define a new matching criterion based on our in-
formation theoretic measure for application to the image
alignment problem and compare it to methods that use the
Shannon entropy in defining a match measure.


1.1 Previous Work on Image Alignment

In the context of the image alignment problem, informa-
tion theoretic measures for comparing image pairs differing
by an unknown coordinate transformation has been popular
since the seminal works of Viola & Wells [20] and Col-
lignon et.al., [6]. There are numerous methods in literature
for solving the image alignment problem. Broadly speak-
ing, these can be categorized as feature-based and direct
methods. The former typically compute some distinguish-










ing features and define a cost function whose optimization
over the space of a known class of coordinate transforms
leads to an optimal coordinate transformation. The latter set
of methods involve defining a matching criterion directly on
the intensity image pairs. We will briefly review the direct
methods and refer the reader to a recent survey [12] for oth-
ers.
Sum of squared differences (SSD) has been a popular
technique for image alignment [2, 18, 19, 10]. Variants of
the original formulation have been able to cope with the de-
viations from the image brightness constancy assumption
[10]. Other matching criteria use of statistical information
in the image e.g., correlation ratio [14] and maximum likeli-
hood criteria based on data sets that are pre-registered [11].
Image alignment is achieved by optimizing these criteria
over a set of parameterized coordinate transformations. The
statistical techniques can cope with image pairs that are not
necessarily from the same imaging modality.
Another direct approach is based on the concept of max-
imizing mutual information (MI) defined using the Shan-
non entropy reported in Viola and Wells [20], Collignon
et al., [6] and Studholme et al., [17]. MI between the source
and the target images that are to be aligned is maximized
using a stochastic analog of the gradient descent method in
[20] and other optimization methods such as the Powells
method in [6] and a multiresolution scheme in [17]. Re-
ported registration experiments in these works are quite im-
pressive for the case of rigid motion. In [17], Studholme
et.al., presented a normalized MI scheme for matching
multi-modal image pairs misaligned by a rigid motion. Nor-
malized MI was shown to be able to cope with image pairs
not having the same field of view (FOV), an important and
practical problem. Most of the effort in the recent past has
been spent on coping with non-rigid deformations between
the source and target multi-modal data sets [15, 4].


2 Cumulative Residual Entropy: A
new measure of information

In this section we define our new information theoretic mea-
sure and derive some properties/theorems. We do not delve
into the proofs but refer the reader to a more comprehensive
mathematical -unpublished technical -report [5].
The key idea in our definition is to use the cumulative
distribution in place of the density function in Shannon's
definition of entropy. The distribution function is more reg-
ular because it is defined in an integral form unlike the den-
sity function, which is computed as the derivative of the
distribution. Moreover, in practice what is of interest and/or
measurable is the distribution function. For example, if the
random variable describes the life span of a light bulb, then
the event of interest is not whether the life span equals t,


but whether it exceeds t. Our definition also preserves the
well established principle that the logarithm of the proba-
bility of an event should represent the information content
in the event. We dub this measure as cumulative residual
entropy henceforth abbreviated CRE.
Definition: Let X be a random vector in RZN, we define
the CRE of X, by :

8(X) =- f P(X > A) logP(IXI > A)dA (1)

Where X (X1,X2, ...,XN), A = (A1, ...AN) and |X >
Means Xi > A, and R = (X ; X; > 0). CRE
is easily computed for various distributions (in some cases
numerically). For example, in the uniform distribution case,


1
p(x)= o

The CRE computes to,


8(X)


0 < x < a
O< o.W


S>)loP(X )d
SP(JX > x)logP( X > x)dx
Jo


(1 x)log(1
a


x
-)dx
a


In the case of the exponential distribution with mean 1/A
and density function: p(x) = Ae- A, the CRE computes to:


8(x)


Se- Aloge-x dt
oAt
Xte-)Adt


A k"
For the case of the Gaussian distribution, the expression for
CRE will involve the error function erf
Proposition 1 8(X) < oo if for all i and some p >
N, L| P] < oo; where E is the expectation operator.
Proposition 2 IfXi are independent, then

F8=(X = Xj 1)) S (Xi)

Proposition 3 (Weak Convergence). Let the random vec-
tors Xk converge in distribution to the random vector X;
by this we mean
lim E[p(Xk)] = E[p(X)] (5)
k-+oo
for all bounded continuous function ) on "ZN, if all the Xk
are bounded in LP for some p > N, then
lim 8(Xk) = 8(X) (6)
k-4oo









Definition: Given random vectors X and Y E JZN, we de-
fine the conditional CRE (X Y) by :

(X\Y) =- P( X > x|Y)logP(|X| > xlY)dx
(7)
Proposition 4 For any X and Y

E[(X Y)] < (X) (8)

Equality holds iff X is independent of Y. This is analo-
gous to the Shannon entropy case. Essentially, it states that
c. i i, r. i i. reduces CRE.
Definition: The continuous version of the Shannon entropy
called the differential entropy [7] 'H(X) of a random vari-
able X with density f is defined as

'H(X) = -EI- L. f] = f f(x) log f (x)dx

The following proposition describes the relationship be-
tween CRE and the differential entropy and we prove that
the CRE is exponentially larger than the differential entropy.
This in turn will have an influence on relationship between
quantities derived from S (X) and 7/(X) such as cross-CRE
(CCRE) and mutual information (MI) respectively. CCRE
and MI will be used in estimating the image alignment prob-
lem subsequently.

Proposition 5 Let X > 0 have density f, then,

(X) > C. exp(7-(X)), (9)

C = exp( log(x log x)dx)

Proof: Let G(x) = P[X > x] = '7 f(u)du using the
Log-Sum inequality [7] we have,


Finally a change of variable gives:

Sf (x) log (G(x) log G(x) dx= log (XI log x dx

Using the above and exponentiating both sides of (10), we
get (9) 0
Definition: The mutual information I(X, Y) of two con-
tinuous random variables X and Y using Shannon entropy
is defined as :


(11)


This measure for the discrete random variable case is now
widely employed in assessing the misalignment between a
pair of uni- or a pair of multi-modality image data sets.
We now define a quantity called cross-CRE (CCRE)
given by


(12)


Note that I(X, Y) is symmetric but C(X, Y) need not be.
We define the symmetrized version of C as,


C(X, Y) = (X)

2+ (Y)


E[S(Y/X)])

E[(X/Y)])


(13)


From Proposition 4, we know that C is non-negative. In our
experiments, we found that the non-symmetric CCRE given
by C was sufficient to yield the desired results. We empiri-
cally show the superior performance of CCRE over MI and
normalized-MI under low signal to noise ratio (SNR) con-
ditions and also depict its larger capture range with regards
to the convergence to the optimal parameterized transfor-
mation.


S f(x) log f(x) d
f(x) log G(x) ilog G(x)idx,
G (x)I log G(x) da


1
= log )
(X)
The left hand side in (10) equals

-H/(X)- f(x) log(G(x) log G(x) )ddx

so that,


2.1 Estimating Empirical CRE
In order to compute CRE of an image, we use the histogram
of an image to estimate the P(X > A) where X corre-
sponds to the image intensity which is considered as a ran-
dom variable. Note that as a consequence of proposition
3, empirical CRE computation based on the samples will
converge in the limit to the true value. This is not the case
for the Shanon entropy computed using histograms to esti-
mate the probability densityfunctions, as is usually done in
current literature. In the case of CRE, we have,


$(X)


/L(X) + I f(x) log(G(x) log G(x) )dx < log S(X)


/P(X > A) logP(X > A)dA

- P(X > A) logP(X > A)
A


(14)


I(X, Y) = -H(X) E[-H(XiY)]


C(X, Y) = (X) E[(YiX)]




























-20 0 20 40 -40 -20 0 20
Rotation Angle(degree) Rotation Angle(degree)


30

25

2O
20

15

10 -

40 -40 -20 0 20
Rotation Angle(degree)


Figure 1: Comparison of the magnitude of C and I over a range of rotations, for a pair of images shown in Figure (2). (a) Traditional MI
which is computed by H(f)+H(r)-H(f,r); (b) Normalized MI which is computed by (H(f)+H(r))/H(f,r); (c) CCRE.


Hence, using a histogram to compute the CRE is well de-
fined and justified theoretically.
Note that estimating (X/Y) is done using the joint his-
togram and then marginalizing it with respect to the condi-
tioned variable.




3 The Alignment Problem


The alignment problem is defined as: Given a pair of images
f(x,y) and r(x',y'), where (x',y')' = T (x,y)' where
T is the matrix corresponding to the unknown parameter-
ized transformation to be determined, define a match met-
ric M (f (x, y), r(x', y')) and maximize/minimize M over
all T. In our case, the matching criterion M is defined
by CCRE. The class of transformations that we consider
are, rigid motions, affine motions and projective transfor-
mations.
To show the marked contrast in the range of values taken
by C and I, we compare the ranges for a given pair of reg-
istered images over a range of rigid motions applied to one
of the two given pair of registered images.
Note the significant difference in the range of values of
C and I shown in Figure 1. As evident from the experiments
described later, this characteristic of CCRE will prove to be
very useful in demonstrating a large range of convergence
and noise immunity for a given optimization procedure over
the traditional MI defined using the Shannon Entropy. This
we believe is a significant strength of our approach to image
alignment using CCRE.


Figure 2: Aligned a) TI weighted MR and b) T2 weighted MR
images used in the computation of CCRE, MI and NMI over the
range of rotations.


4 Experiment Results

In this section we demonstrate alignment by maximization
of CCRE for a variety of transformations. The performance
of the CCRE was evaluated for each set. The first exper-
iment (with 30 image pairs) was done for synthetic mo-
tions, where we compare the estimated alignment with the
ground-truth alignments. The second experiment (two pairs
of data sets) is done on the real data image pair. In all of
the following experiments, bi-linear interpolation was used
when needed for non-integral indexing into the image.


4.1 Synthetic Motion Experiments

In this section, we demonstrate the robustness property
of CCRE and hence justifying the use of CCRE over MI
and NMI (normalized-MI) in the alignment problem. This












mc aII 11i1-- I I


Y Y

Figure 3: Registration example for rigid motion using our algorithm. Leftmost:The source image, Rightmost: Target image, obtained by
applying a synthetic rigid motion to the source image. The sizes of both images are: 240x320. Middle: Overlay of the target edge and the
transformed source image by applying the estimated rigid motion using CCRE.

noiseo-2 true motion CCRE traditional MI normalized MI
10 10 5.0 5.0 9.998 5.016 4.996 9.993 4.999 5.007 10.002 5.256 2-.
15 9.998 5.077 i 11i" 0 6.003 -3.000 10.132 5.046 5.998
19 9.998 5.006 5.001 FAIL 0 -15.890 19.222
30 9.998 5.256 2.. FAIL
59 10.027 5.124 4.995
60 0 -3.003 0
61 FAIL

Table 1: Comparison of the registration results between CCRE and other MI algorithms for a fi xed synthetic motion. Note that the image
intensity range before adding noise is 0-255.


is demonstrated via experiments depicting superior perfor-
mance in matching under noisy inputs and larger capture
range in the estimation of the motion parameters.


4.1.1 Rigid Motion

In order to compare the robustness property of CCRE ver-
sus traditional MI and NMI, we designed a series of exper-
iments as follows: with a 2D aerial image as the source, the
target image is obtained by applying a known rigid trans-
formation to the source image. The source and target image
pair along with the result of estimated transformation using
CCRE applied to the source with an overlay of the target
edge map are shown in Figure 3. The registration is quite
accurate as evident visually. Quantitative assessment of ac-
curacy of the registration is presented subsequently.
Next, we applied CCRE together with other MI algo-
rithms to estimate motion parameters, with 30 randomly
generated rigid transformations. These are normally dis-
tributed around the values of (0, 5pixel, 5pixel), with
standard deviations of ( 8, 3pixel 3pixel) for rotation and
translation in x and y respectively. Table 4.1.1 shows the
statistics of errors resulting from the 3 different methods.


In each cell, the leftmost value is the rotation angle (in de-
grees), while the right two values show the translations in
x and y directions. Out of the 30 trials, the traditional MI
failed 3 times while CCRE and Normalized MI both failed
only once ("failed" here means that the optimization algo-
rithm sequential quadratic programming (SQP) primar-
ily diverged). If we only count the cases which gave reason-
able results, as shown in the first (for CCRE), second (for
traditional MI) and third (for normalized MI) rows, CCRE
and the traditional MI have comparable performances, all
being very accurate. Thus, in terms of accuracy, CCRE and
NMI are comparable and are both better than MI.
mean standard deviation
1 0.0570 0.456 0.286 0.0220 0.236 0.079
2 0.1650 0.645 0.478 0.0670 0.271 0.204
3 0.122 0.397 0.466 0.0400 0.093 0.077

Table 3: Comparison of estimation errors for rigid motion between
CCRE, MI and normalized MI.

In the second experiment, we compare the robustness of
the three methods (CCRE, MI and normalized MI) in the
presence of noise. Still selecting the aerial image from the
previous expt. as our source image, we generate the target


I sa. iI ., 1 1 I I


I ML II n 1 j+-- : I










noiseao2 true motion CCRE traditional MI normalized MI
13 5 6 6 4.997 6.002 5.997 5.008 ,.'.,7 6.004 5.003 6.007 6.022
5 7 7 4.995 7.004 7.012 0.087 6.988 7.018 5.384 7.995 6.541
10 10 10 10.015 i-)'.: 9.972 FAIL 0 -18.748 -21.041
20 10 10 20.002 '.) '.) 9.990 FAIL FAIL
30 13 13 30.002 12.990 12.998
32 13 13 31.950 14.037 12.974
35 14 14 19.840 1.119 -9.942

Table 2: Comparison of the convergence range of the rigid registration between CCRE and other MI schemes for fi xed noise variance.


Figure 4: An affi ne motion estimation example of our algorithm. Leftmost: The source image, which is a T1 weighted MR image. Right-
most: Target image, obtained by apply a synthetic affi ne motion to the T2 weighted MR image. The sizes of both images are:256x256.
Middle: Overlay of the target edge map on the transformed (using affi ne motion computed by CCRE) source.


image by applying a fixed synthetic motion. We conduct
this experiment by varying the amount of Gaussian noise
added and then for each instance of the added noise, we reg-
ister the two images using the three techniques. We expect
all schemes are going to fail at some level of noise. By com-
paring the noise magnitude of the failure point, we can show
the degree to which these methods are tolerant. We choose
the fixed motion to be 100 rotation, and 5 pixel translation
in both x and y direction. The numerical schemes we used
to implement these registrations are all based on sequential
quadratic programming (SQP) technique. Table 1 show the
registration results for the three schemes. From the table,
we observe that the traditional MI fails when the variance of
the noise is increased to 15. It is slightly better for normal-
ized MI, which fails at 19, while CCRE is tolerant until 60,
a significant dittlrence when compared to the traditional
MI and the normalized MI methods. This experiment con-
clusively depicts that CCRE has more noise immunity than
both traditional MI and the normalized MI.
Next, we fix the variance of noise and vary the magnitude
of the synthetic motion until all of them fail. With this ex-
periment, we can compare the convergence range for each
registration scheme. From Table 2, we find that the con-
vergence range of traditional MI and normalized MI is es-


timated at (50, 6, 6) and (90, 10, 10) respectively, while our
CCRE-based algorithm has a much larger capture range at
(320, 13, 13). It is evident from this experiment that the cap-
ture range for reaching the optimum is significantly larger
for CCRE when compared with MI and NMI in the pres-
ence of noise. Note that in all the cases, the same numerical
optimization scheme SQP was used.


4.1.2 Affine Motion

The affine motion experiment was designed as follows: in
every experiment, we applied a known affine transformation
to the target image shown in Figure 2. One example of the
pair of source and transformed target image are displayed
in Figure 4.
For the purpose of comparison, we separate the affine
motion into three parts, rotation, translation and scaling.
Three sets of 10 randomized transformations have been
used. They are normally distributed around the values
of (50, 1.0, 5pixel), (70, 1.0, 7pixel) and (10, 1.0, 9pixel)
respectively, with standard deviations of 50, 0.2 and 2pixel
for rotation, scale and translation respectively. For a quan-
titative assessment of the accuracy of the registration, we
computed the mean and standard deviation of the errors


'I


' --~L~LI~


~-1131 ~CI I











for the six parameters of the affine motion. It should be
noted that in all the three sets of experiments, our CCRE
method has yielded superior performance over the other
two methods. Out of the 30 trials, the traditional MI failed
6 times, the normalized MI 3, while CCRE failed only 2
times. ("failed" here means that the results diverged).

mean standard deviation
S 0.0020 0.0068 0.0732 0.0000 0.0005 0.0233
0.0098 0.0029 0.0395 0.0011 0.0001 0.0017
2 0.0460 0.0163 0.3945 0.0155 0.0005 0.2200
0.0231 0.0432 0.4743 0.0007 0.0130 0.2537
0.0078 0.0076 0.1260 0.0001 0.0001 0.0132
0.0089 0.0069 0.1443 0.0001 0.0001 0.0149

Table 4: Comparison of estimation errors between CCRE, and
other MI-based methods in estimating the affi ne motion.


The second test on affine motion is similar to the one
for the rigid motion (table refconverg), we registered the
source and target images while varying the synthetic affine
motion until the methods fail to find the motion. Each mo-
tion parameter is evaluated independently, Table 5 summa-
rizes the results of applying our CCRE algorithm as well as
the other MI schemes. The values shown are the maximum
capture range (from zero) for each parameter in each algo-
rithm. As evident, our algorithm has a significantly larger
convergence range.

algorithm Rotation Translation Scaling
CCRE 390 30 3.2
Traditional MI 180 15 2.2
Normalized MI 210 14 2.6

Table 5: Convergence range of different algorithms for affi ne mo-
tion. Here we divide the affi ne motion into 3 parts. Each part is
evaluated independently.


The last test for the affine motion is to vary the amount
of Gaussian noise while fixing the synthetic affine motion.
Table 6 depicts the noise variance which causes each algo-
rithmto fail. Again, observe superior performance of CCRE
over the other MI-based methods.


algorithm noise variance(-2)
CCRE 19
Traditional MI 6
Normalized MI 5
Table 6: Comparison of the registration results between CCRE
and other MI-based methods for the fi xed affi ne motion,
(1.4772, -0.2605, 5.0000, 0.2605, 1.4772, 5.0000) and varying
noise levels


4.2 Real Data Experiments


In this section, we demonstrate the algorithm performance
for a pair aerial images taken over time. The transformation
between the two images is assumed to be a projective trans-
formation. Our data is approximated by a planar surface in
motion viewed through a pinhole camera. This motion can
be described as 2D projective transformation.


) aox aiy + a2
u(x,y) =-----
a6x + a7y + 1

v(x,y) = +a4+a5
a6x + a7y + 1


This projective transformation requires us to estimate
eight parameters for each image pair. For brevity, only one
registration result is shown in Figure 5. Here, the source
and target images are shown in the top row, and the lower
left image is the overlay of the transformed source with the
source edge map (showing the change in the source due
to the applied transformation), while the lower right image
shows the overlay with the target edge map showing the
registration. As evident, the registration is visually quite
accurate.


Figure 5: Registration results for the projective transformation.
Upper left, the source image; Upper right, the target image; Lower
left, the transformed source overlayed with the source edge map;
Lower right, the transformed source overlayed with the target edge
map.


(15)











5 Summary


In this paper, we presented a novel measure of informa-
tion that we dub cumulative residual entropy (CRE). This
measure has several advantages over the traditional Shanon
entropy whose definition is based on probability density
functions which are hard to estimate accurately. In con-
trast, CRE can be easily computed from the sample data
and these computations asymptotically converge to the true
value. Unlike Shanon entropy, the same CRE definition is
valid for both discrete and continuous domains.
We defined the cross-CRE denoted by CCRE and applied
it to estimate the parameterized misalignments between im-
age pairs and tested it on synthetic as well as real data sets
from mono (video) and multi-modality (MR T1 and T2
weighted ) imaging sources. Comparisons were made be-
tween CCRE and traditional MI and normalized MI both of
which were defined using the Shanon entropy. Experiments
depicted significantly better performance of CCRE over the
other MI-based methods currently used in literature.



Acknowledgements

Authors would like to thank Dr. Wen Masters of ONR for
providing the Aerial images.



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