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Partial Differential equations-based image processing in the space of bounded variation using selective smoothing functionals for noise removal

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Partial Differential equations-based image processing in the space of bounded variation using selective smoothing functionals for noise removal
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PARTIAL DIFFERENTIAL EQUATIONS-BASED IMAGE PROCESSING IN THE SPACE OF BOUNDED VARIATION USING SELECTIVE SMOOTHING
FUNCTIONALS FOR NOISE REMOVAL












By

THOMAS WUNDERLI


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


UNIVERSITY OF FLORIDA


2003




























Copyright 2003 by


Thomas Wunderli















This is dedicated to Mom, Dad, and Cristina.














ACKNOWLEDGEMENTS


I would like to thank my advisor, Yunmei Chen, for introducing me to this subject and for her constant guidance and support. I would also like to thank my committee members for their input and advice. A special thank you goes to the Department of Mathematics office staff for their assistance throughout my graduate career. And finally, thanks go to my wife Cristina and my parents for their undying moral support.














TABLE OF CONTENTS




ACKNOWLEDGEMENTS ............................ iv

A BSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

CHAPTER

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

1.1 Purpose of Investigation ....................... 1
1.2 A Brief Overview .......................... 1
1.3 Basic Function Spaces and Notation ................ 2

2 SELECTIVE SMOOTHING USING A DAPTIVE TOTAL VARIATION 4

2.1 Introduction ................................ 4
2.2 Preliminaries and Definitions .................... 9
2.3 Minimization Problem ........................ 15
2.4 Flow Related to the Minimization Problem ............ 15
2.5 Stability and Asymptotic Behavior ................. 23
2.6 Numerical Results .......................... 27
2.7 Updated Selective Smoothing .................... 33
2.8 Other Selective Smoothing Models ..................... 38

3 SELECTIVE SMOOTHING AND PARTIAL REGULARITY ..... ..40

3.1 Introduction ... ............................. 40
3.2 Proof of Theorem 3.1.1 and Theorem 3.1.2 ............ 42
3.3 Partial Regularity for the p-Laplacian ............... 53

4 A QUESTION FOR FURTHER STUDY ..................... 65


REFERENCES ........ ................................... 68

BIOGRAPHICAL SKETCH ................................... 70















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




PARTIAL DIFFERENTIAL EQUATIONS-BASED IMAGE PROCESSING IN THE SPACE OF BOUNDED VARIATION USING SELECTIVE SMOOTHING FUNCTIONALS FOR NOISE REMOVAL By

Thomas Wunderli

August 2003

Chairman: Dr. Yunmei Chen
Major Department: Mathematics

In this thesis we study two different models for PDE-based image processing. Both model the removal of noise, also referred to as smoothing, from digital images while retaining essential features, such as edges, and both take the restored image, represented as a function defined on a rectangle Q C R , to be the solution to a minimization problem over BV space.

The first model uses an adaptive total variation (ATV) functional defined on BV space. We first define the ATV functional for functions that are not necessarily in any Sobolev space. This space is the a-BV space, where a is a chosen function to locally control the amount of smoothing. Then we derive important approximation and compactness theorems concerning functions in a-BV. Having defined our functional and proven existence and uniqueness of a solution, we then study the associated time evolution problem. Here we define a weak solution u(x, t) to this problem









and prove its existence, uniqueness, stability, and asymptotic behavior as t --+ oo. We prove that u(x, t) weakly converges in L2(Q) to the solution u, of the original stationary problem. In addition, we demonstrate some numerical results of the time evolution ATV model as well as prove the existence of a solution for an updated ATV functional. Also discussed is an updated version, where the parameter function a depends on the solution u and not on initial noisy image.

The second model uses a functional which smoothes the image where its gradient norm is below a certain threshold E, that is where IVul < c, using either the Laplacian or a regularized p-Laplacian for 1 < p < 2, and retains edges where its gradient norm is above the threshold (jVu > c). We in fact prove that the solution u is smooth where IVul













CHAPTER 1
INTRODUCTION

1.1 Purpose of Investigation

As mentioned, this dissertation is the study of two methods of digital image processing from a partial differential equation (PDE) approach. However, instead of working in the more traditional context of Sobolev spaces, we work in the space of functions of bounded variation, or BV space. As will be explained in the first chapter, BV space is a more natural space for images, represented by functions defined on some rectangle Q c R', to belong. Both problems studied here relate to image denoising, that is, removing noise from corrupted images while retaining essential features of the image. And both models involved remove noise while retaining edges in the image by utilizing a built in "selective smoothing" feature.

1.2 A Brief Overview

In the second chapter, we introduce the total variation and adaptive total variation models for image denoising and some of the relevant results. We then define a-BV space as a generalization of BV space, given an appropriate function a. Then we define a minimization problem over BV using the a-BV semi-norm, the solution to which is the restored image. We then investigate an associated time evolution problem and prove existence, uniqueness, and stability of the solution as well as show that its asymptotic time limit is the restored image. A few numerical results are presented for this model as well as a brief investigation of another adaptive total variation model, this time using an "updated" version of the selection parameter function a.









For the third chapter we study another selective smoothing model, this one based on a model for plasticity. As in the first problem we take the restored image to be the solution to a minimization problem. The main focus of this chapter is proving partial regularity of solutions. We first consider the simpler case involving the 2Laplacian, then focus on the model using a regularized p-Laplacian. Some results from the 2-Laplacian case are also used for the p-Laplacian case.

1.3 Basic Function Spaces and Notation

In order to discuss problems in PDE-based image processing, we must first include the necessary Banach spaces and notation related to the study of PDE's. These include Sobolev spaces, H6lder spaces, and of particular importance to image processing, the space of functions of bounded variation, or BV space. It is assumed that the reader is familiar with the embedding and compactness theorems as relates to the above spaces, as well their use in PDE theory. For convenience we include a brief summary of these spaces. We start with weak derivatives and Sobolev spaces. Let Q C R' be open.

Definition 1.3.1 If u, v c L'oC(Q) and a (a,...,,), then v is defined to be the ath-weak derivative of u, written v = D'u if


j uDa�dx = (-1)'i j vbdx

for all test functions � G CO.

From this, the Sobolev WkP(Q) spaces for integer k > 0 and real p > 1 are defined by

Definition 1.3.2

Wk'p(Q) - {u : Q - RI Vlal : k, D'u exists in the weak sense and D'u E LP(Q)}








The Sobolev norm is
1
JU!IWkP.,(Q) z IDaulPdx for 1 < p < cc


and

IIUIlWk,-(r) = ID'uIL-() for p = oo.
Ial We note for the special case of p = 2 we write Hk(q) = Wk,2(Q).
Next we have the H6lder spaces. Let k be any nonnegative integer, 0 < 7 < 1, and a= (c, ..., ,~) as before. Define

Iul() max ul, ul = IDju(�), and


IuI(-) Suplu(x) - u(y)l xul yE) su

Then we say u E Ck'y (Q) if u E Ck (Q) and
k
5Hu[j+ E ID'uj(- with norm
k
IIUIc k,(Q) - Iula + E IDaul ).
j=0 IaI=k
The most important Banach space we will use is the space of bounded variation, or BV space. This is essentially an extension of W,'. BV functions, however, now may include jumps and their gradients are interpreted as measures. Definition 1.3.3 A function u E L'(Q) is in BV(Q) if

[U]BV(Q) = sup {j 1u div((p)dx : p E Cd(Q, Rn), kPIL o() - 1}













CHAPTER 2
SELECTIVE SMOOTHING USING A DAPTIVE TOTAL VARIATION

2.1 Introduction

This chapter is an investigation of the mathematical theory for adaptive total variation (ATV) regularization, a powerful technique in edge preserving and noise removal, which has been effectively applied to image restoration. Let u0, defined on a rectangle Q C R2, be an observed image which is the result of the true image Uoriginal with added noise 7j, i.e., U0 " Uoriginal � 77.

Our goal is to try to approximate Uoriginal as best as possible from the observed image U0.

In recent years much work in image restoration has been done using total variation (TV), see for example Rudin, Osher, and Fatemi [22], and the results have been promising. The restored image is then taken to be the solution to min aTV(u) + -lu - 2 for the unconstrained problem, and min TV(u),
U

subject to

UoL2(Q)

for the constrained problem. Here a > 0, a > 0 are chosen positive parameters, and TV(u) is defined by

TV(u) =sup { udiv( o)dx: C(Q, R), IL-(Q) < 1} (2.1)






5

for any function u E L1(Q). We note that (2.1) is precisely the BV(Q) seminorm of u, as stated in the introduction, if the above is finite. This definition for the TV functional does not require differentiability or even continuity of u. In fact one of the main advantages of using TV functional for image restoration is that jump discontinuities, that is possible edges, are allowed. The example below demonstrates this. In fact the only solution to the problem below is one with edges.

Example 2.1.1 Let (2 = (0, 2) and define the functional F on L2(Q) by

F(u) = TV(u) + j(u- Uo)2dx with no~) /0 if 0< x

Later we will prove that there is a unique solution u to this problem with TV(u) < oo. We claim that the solution to the problem min F(u)
uEL2(1Q)

is not in W1,1(Q).
In fact, let f C (Q).
Case 1: There exists an x E (0,1) such that f(x) < 0 but f(x) < 2 for all x E [1,2)
a. If f(x) < 0 for all x E (0, 1) then let

SSUPXE(,1)f(x) y C (0,1), supXE[1,2) f(x) y E [1, 2) if supXE(o,1) f(x) supXE[1,2) f(x). Then we can see that F(f) > F(u). If

sup f(x) sup f(x) xE(0,1) xE[1,2)


set u(y) = 0 and apply Case 4.









b. If f(x) > 0 for some x E (0, 1), then f(x*) = 0 for some x* E 0. Then let

0 y E (0, 1), u(y) =
SsupxE,2) f(x) y c [1,2) if SUPXE[1,2) f(x) 0. Again we have F(f) > F(u). If SUPXE[1,2) f(x) = 0 set u = 0 and apply Case 4 to f = u.
Case 2 : There exist x E (0, 1) such that f(x) < 0 and x E [1, 2) such that f(x) > 2.
Then take u = uo, so F(f) > F(u).
Case 3: f(x) >_ 0 for all x E (0,1) and there exists an x E [1, 2) such that f(x) > 2.
a. If f(x) = 0 for some x E (0, 1), let u = uo. Then we clearly have F(f) > F(u).
b. If f(x) > 0 for all x E (0,1) define

uny) = / i f(x ) y G (0, 1),
u (y) {
L 2 yE [1,2) if infxE(o,1) $ 2. Again we see that F(f) > F(u). If infxG(o,1) 2, let u 2 and apply Case 4 to f = u.
Case 4: f(x) > 0 on (0,1) and f(x) < 2 on [1,2).
So let f be continuous on (0,2) satisfying the above. Let a = minf(x), (0,1)
b = min(2- f(x)), [1,2)

and
= a y E (0, 1), 2 - b y E [1,2). if a : 2 - b. Then clearly F(f) > F(u). If a = 2 - b then


F(f) > F(u) = a2 + b2 = 4- 4b + 2b2 > 2 = F(uo).








Thus we conclude that if f is any continuous function on (0,2) then there exists a function u with a jump discontinuity such that F(f) > F(u). Thus by uniqueness, no f E C(Q) can be a minimizer of F.
Now if f E W1"1(Q), then by the Sobolev embedding theorem f is a.e. equal to an absolutely continuous function f and hence F(f) = F(f). Thus by the above argument f and hence f can not be a minimizer for F.

As an improvement on the TV functional, Strong and Chan [23, 9] introduced the weighted TV functional

TV, a (x)IVu(x) dx

for spatially adaptive (selective) image restoration. The function a is chosen so that a is larger away from possible edges and smaller near a likely edge. Hence we allow for greater smoothing away from edges and less smoothing at the edges. Certain choices of a(x) were given by Strong and Chan [23, 9], and their numerical results were very promising. However, many theoretical questions such as the existence and uniqueness for the minimizer of the weighted TV norm with some penalized term, and for the related evolution problems when a is a function on Qi (not only piecewise constant) remain. The question whether the solution of the evolution equation converges to the minimizer or not as t --+ oc also remains open. The goal of this chapter is to investigate these problems. Here we would like to point out in the case of constant a, results were obtained by Chambolle and Lions [8] for the minimization problem

minimize j VuI

with /Au=fuo and fn Auuo12=cr2 where o2 is known and A is a continuous linear operator on LP(Q), and by Acar and Vogel [3] for the problem

minimize 1 JAu - u012 + aJ,3(u)








where A is a linear operator on LP(Q) and


J(u) f X'Vu2 + 0

is defined on BV space. Results were also obtained, for instance, by Vese [29] for the functional

F. = j(Ku - uo)2dx + a fj (JVu) and its corresponding flow. Here a > 0 is constant and W : R -+ R+ is a convex, even function nondecreasing in R+ with linear growth, and K : LP(Q) -+ L2(Q) is a linear, continuous, injective operator. However, the results for the flow are only in the dimensions one and two, due to the methods employed there, that is, using general results on maximal monotone operators and evolution operators in Hilbert spaces.
In addition, Hardt and Zhou [18] consider the flow related to


min fa (IVuD),
UEBV(m), u J( V=gul)

that is
au = div.Wp(Vu), (pp = V)


with Dirichlet boundary data u = g on o9 for a bounded domain Q, u = uO on Qx {0}, and any convex linear growth functional W. In their analysis, they approximated the above solution by the flow associated with W'(p) = 71, * W(p) where 77, is the usual mollifier on R'.
In this work we shall extend the results of Chambolle and Lions, Acar and Vogel, as well as those of Hardt and Zhou, to the adaptive TV scheme. In particular, we shall develop mathematical theories for the problem of


minF(u) = TV, + JJu- (2.2)
2 UIL2(Q-)(2)








over BV(Q), and its corresponding evolution equation,

au
- = divr,(a(x) op(Vu)) - 0(u - uo) on Q x RT, (2.3)
at
a- = 0 onOQxRT, (2.4)

u(O) = u0 on Q, (2.5) where Q is an open, bounded Lipschitz domain, RT = [0, T], 0 > 0 is a parameter, (p(p) = IpI on Rn, pp(p) - Vv(p) = p/jpj on Rn, and a(x) is a parameter (smooth) function used for edge detection and to control the speed of smoothing. In image processing problems a is often chosen as
1
(x) 1 + k VG �u2 (2.6) k being a parameter, and G, being the Gaussian filter with parameter a. The definition of a weak solution to (2.3)- (2.5) is similar to that in Zhou [30] or Hardt and Zhou [18]. However, our method of proving existence differs from Hardt and Zhou in two ways, the first being the approximation equation. Second, and most importantly, is the way most of the necessary estimates of the approximate solution are obtained. Finally, our use of an approximate PDE to the corresponding flow is in contrast to Vese. Also, our result holds for all n.

2.2 Preliminaries and Definitions

We start with the definition of f1 ajVfI for functions in f E

Definition 2.2.1 Let Q be a bounded open subset of Rn. Let f be a real valued function on Q such that f E L'(Q). Also let a(x) > 0 be a continuous real valued function on Q. Then we define the a-total variation of f or a-TV of f to be


f alVf= sup {ffdiv(O)dx' 0(x) wr Qiave - f(,n)
where � is a vector-valued function �(1..�)







Definition 2.2.2 We define f E L1(Q) to be in a-BV if

sup { f div(o)dx: q(x), <_ a(x) Vx E Q} < 00.
�ECo (Q,rtn) U
Definition 2.2.3 Iff E a-BV we define the a-BV seminorm by
jaVfI= sup { f div(O)dx: 1(x) < a(x) Vx E Q} and the a-BV norm to be

lf Ila-BV faIVfI + If IL'(Q)In the sequel we will write the above norm as Ilf I. Remark 2.2.4 It is easy to show that if f E W'1(Q) then L/ajV I = fI ajV dx.

Remark 2.2.5 Note that if f E BV(Q) and functions a and 0 both satisfy the conditions of Definition 2.2.1 where a(x) < 3(x) for every x E Q and f E /-BV, then we have f E a-BV and
(aIVfI 1 jIlVf1.

This follows directly from the Definition 2.2.3 since 10(x) < a(x) implies 10(x)I <



The theorem below establishes lower semicontinuity for the a-BV seminorm. Theorem 2.2.6 If {fj} C L1(Q) and f c L'(Q) is such that fj -+ f in L'(Q), then jaVfl 1 lim inf j aIVfjI. Proof: Let 0 E Cl'(Q, R) be a vector valued function such that IO(x)l _< a(x) Vx E Q, then
l j0jdiv(o)dx = im ffjdiv(O)dx








Now take the supremum over � to get fal Vfi < liminf J a[VfjI. l 3-+00 Lf

Next is an important approximation result for functions in BV.

Theorem 2.2.7 Let f E BV where a(x) > 0 is continuous on n. Then there is a sequence {fj} of functions from C'(Q) such that im njIf - f dx = 0
j--+ 00 I

and

lim falVfjldx= f&alVf1. Proof: We essentially apply the argument of Giusti [13] with an important modification. Given c > 0 we construct the covering {Ai} of Q where Ai = - i+1 - n" i-1, A 1

with

Qk= {xCQ:dist(x,OQ) > k =0,1,2, and where m is large enough such that jaVf < F. (2.7) Next we construct the sequence {f } so that c* (f Oi)
i1

where 77i is the usual mollifier on R' and {oi} is a partition of unity subordinate to {Ai}. We then choose the ci's such that the following four conditions hold simultaneously for each ci:


1. ci < f








2. fj ?17c * (f Ot) - fqi Jdx
3. f in, * (fV~i) - fVo, dx < E2-' 4. support q,, * (f 0) C Qi+2 -i-2. Summing over all i we get If - fjdx < 177, * (f� ) - foildx < E, thus
f, --+ f in L'(Q),

and by Theorem 2.2.6, f alVf I < liminf j alVfEldx. (2.8) Now let g E C (Q,Rn) be such that 1g(x)I < a(x) Vx E Q. Then

fdiv(g)dx f (n, * (f �$))div(g)dx = f�idiv(rE, * g)dx, so


j fLdiv(g)dx f fdiv(�1,q,1 * g)dx + f fdiv(0qr/ * g)dx (2.9)
n fo i=2

/ g(q, * (fVq�) - f VO))dx. Denote the three terms on the right side of (2.9) by 1, 11, and III respectively. Note III< EjjajjL by our choice of the E's.
By uniform continuity of a, there exists an increasing function w such that w(r) -+ 0 as r -+ 0 and I a(z) - a (z')I < w(r) for all z, z' such that Iz - z'j < r. Consequently a(z) < w (fi)�+a(x) for all Ix-z 1 :i. Now write g = ag' where g'= 0








if a = 0 and [g'j < 1 Then for i

I O (77(i,* g)(W I =


> 1 and any x C Q, Ii(x)(77i * ag')(x) J Oi(x)?rh(x - z)a(z)lg'(z)ldz fnj i(x)77i(x - z)(a(x) +w(,i))Ig'(z)Idz �5,(x) a(x) f 77(x - z)Ig'(z)Idz +iW(x)W('Ei) fj 7i(x - z)Ig'(z)Idz a(x) + w(c).


So we get, for i = 1,

I = fdiv(017,Ei * g)dx < aVfI +W(C) IVfI and also

II~ = S fdiv(05j, * g)dx < 3 f ajVf I + 3w() Vf < & + 3w( ) 11 IVf,

with the last inequality following from (2.7). Therefore

f fdiv(g)< 14f aVfI + w(El) f2 VfI + + 3w(E) fn Vf+I +E� eL

for every g E CoJ(Q, Rn) with Ig(x)I < a(x) Vx E Q. Hence taking the supremum over g and then taking the limsup as E -+ 0 we get lim sup / a VfE[dx < f aIVf1. (2.10)
f-+ 0 fo
From (2.8) and (2.10) one finally has lim alVflIdx =faVf1. 0
E--* 0








Remark 2.2.8 If f E BV(Q) nl L2(Q), a E C(Q), and 9Q Lipschitz, then there exists a sequence of functions {f} c C(n) such that

f,, -+ f in L2( Q) and I llVfldx- / aIVf1. (2.11) And if f E L'(Q), we also have

IlfM IL-(Q) '5 C(Q)llfllL-(S2). (2.12) In fact, in the proof of Theorem 2.2.7 we choose the c's to satisfy 1-4 and in addition S177i * (fqOi) - f Oi2dx fj -- f in L2(Q) and jn alVfjidx -+ f alVf1. (2.13) Since C'(K) is dense in W"1(Q) n L2(Q) there exists for each fj a sequence {fj,k} E C' (KI) such that as k -+ 0

Ilfjk - fjIL2(Q) -+ 0 and Ilfj,k - fjjjw1,1(Q) -+ 0. (2.14) Then (2.11) follows from (2.13), (2.14) and a standard diagonal argument applied to {fj,k} to obtain the desired sequence {fn}. By the construction of {fj} and {fj,k} if in addition f E L'(Q) we obtain (2.12).

Theorem 2.2.7 now allows us to prove a compactness theorem.

Theorem 2.2.9 Let {ff} be a bounded sequence in a-BV where a C C(Q) and in addition a(x) > J > 0 Vx E Q. Also assume that Q C R' is such that oQ is Lipschitz. Then there is a subsequence of {fj}, also denoted by {fj}, and an f E LP(Q) such that fj -> f strongly in LP(Q) where 1 < p < 'l, and weakly in L- (Q).

Proof: Since 0 < J < a(x) Vx E Q and by remark 2.2.5 we have Sf lVfjl < f aIVflj < C.








Therefore

jVfjI < C.

Thus fj is bounded in BV norm and the theorem follows from the compactness result in Giusti [13]. LI

2.3 Minimization Problem

We now consider the minimization problem

B(n/2(a) - avul (u _ Uo)2dx. (2.15) Here u0 is the initial noisy image, / is a positive parameter, and Q is a bounded open subset of R' with Lipschitz boundary. In the sequel we will always assume that a is a smooth function satisfying the conditions of Theorem 2.2.9. In practice we can take a(x) in (2.6) for some o, > 0 and k > 0. Assuming u0 E LI(Q), we see

1
la1 + CUO011H2( )

The constant 6 in Theorem 2.2.9 can therefore be chosen as

6 =I
1+ Clu0 12 (Q

To conclude this section we verify that (2.15) does have a unique minimizer. Lemma 2.3.1 The functional in problem (2.15) has a unique solution in BV(Q) n L2 (Q).

Proof: Clearly, the functional is convex, coercive in BV(Q) n L2(Q) and by Theorem 2.2.6 is lower semicontinuous. So by standard results, (2.15) has a solution in BV(Q)n L2(Q). The uniqueness follows from strict convexity of the functional in (2.15). LI

2.4 Flow Related to the Minimization Problem

To motivate the definition of a weak solution to (2.3)-(2.5) where p(p) = p, Vp(p) = and RT = [0, T] we follow Zhou [30]. Assume the solution u is sufficiently








smooth to justify the following calculations and that p is as mentioned above. For arbitrary v E L 2([0, T]; H'(Q)), we multiply the equation in (2.3)-(2.5) by v - u to get, after integrating by parts and using the formula W(p) - V(q) > pp(q) . (p - q) (due to the convexity of W),


j it(v - u)dx + jaw(Vv)dx > j aw(Vu)dx - j 03(u - uo)(v - u)dx. (2.16) Then integrate with respect to t to get

j j (v - u)dxdt + jS j c(Vv)dxdt > j j f p(Vu)dxdt (2.17)

- j' j. O(u - uo)(v - u)dxdt.


On the other hand if (2.16) holds, by selecting v = u+ A for � E C'(Q) we get that u is a solution to (2.3) in the sense of distributions.

We are thus led to the following definition of a weak solution to (2.3)-(2.5): Definition 2.4.1 A function u E L2([0, T]; BV(Q)) is called a weak solution of (2.3)(2.5) if it - Otu E L2(Q x RT), u(O) = uo, and u satisfies (2.17) for every v E L2([O,T];BV(Q)), a.e. s E [0, T].

Before we continue we list some properties of the smooth approximating function of p,

W (p) = p2 + c2,

defined on Rn, which will be used in the subsequent discussion:

1. W'(p) is convex in p,

2. Wp(p) .p > 0 Vp,

3. p' W uniformly with respect to p as c -- 0.

In fact 0 < 'J(p) - o(P)I < f Vp E R'. Here we use the notation V '. The proofs of the above properties are all straightforward.








Consider the following approximation problem of (2.3)-(2.5):
Ou
- =cAu + div(aVp(Vu)) - 0(u-u) on Q x RT, (2.18)
Ou
- 0 on OQ x RT, (2.19)
an
u(0) = u0 on Q, (2.20) where
u0 Cc�() with uo - uO in L2(Q), (2.21)


Iu OIlL- < C(Q)lluolL, (2.22) Ia(Vu) _ C(Q)fua(Vuo). (2.23) The existence of u5 is from Remark (2.2.8) if uo E BV(Q) n LI(Q). The idea is to prove an existence result for the above quasilinear uniformly parabolic PDE, obtain bounds for the solution independent of c and 6, and pass the limit as E -- 0 and 6 - 0. Indeed, the lemma below provides an existence and uniqueness result for (2.18)-(2.20).

Lemma 2.4.2 The approximation problem (2.18)-(2.20) admits a unique weak solution uE'6 where u',6 E L'([O,T];H1(Q)), it ' E L2([O,T];L2(Q)) and

jj( 6)2 dxt+ j u I VE(t)12 dx + jaso(Vu'6(t))dx


SVu + (Vu)dx + for a. e t E [0, T].

Proof: By using the Galerkin method, the fact that pp is a monotone operator ([6] and [20]) we have a weak solution to (2.18)-(2.20) such that it/'J E L2([0, T]; L2(Q)), ur'6 E L- ([0,T];H1(Q)), and
j Eic6)2dx( + j, (jv (t) 12 + aOE (VU,,6 (t)) + 0 (ucd - 64)2)d
L fu + 2V2
f-E- lVU012 + Ce (Wg) dx.








Since W(p) < V(p) < W(p) + E for any p we arrive at

f j + Vu 6(t)12dx + j

VjVu11 + aw(Vuj)dx + E. L


We also have the following L'(Q) bound for the solution to (2.18)-(2.20) obtained above:

Theorem 2.4.3 Suppose uo c L"O(Q)nBV(Q) and uE,6 is a solution of initial boundary value problem (2.18)-(2.20). Then we have H~U'6ILoo(SIRT) C(Q)IIUOHIL-(Q)jo Proof: Let M = IJ1ju6L(Q). For any A > 0, multiply (2.18) by e-At(e -Atuf,- m)+, where
w e-Atu 4, - M if e-Atu ,' - M > 0 (e-At U C6 - M
S0 otherwise, and integrate over Q to get

j Ce't(e -Atuc,6 - M)+dx + Ef Vuf'J e-At . e-AtVu,"dx

+ j c (Vu',-)e t e-\tVu',Jdx + 2 j(', - U o U)e -( , M)+dx 0. Then since the last three integrals are non-negative we see


-- eA(e uf6 - M)+dx < 0. Let

1(t) - fj I(e-A tut, - M)+12dx.
I 2t =


Then


- -1 xte u - M)+12dx -(e- - M)+(-Ae-Atu4'6 + e-at - -)dx
2 J dt' 1t

eAt OU'6 (eA- - + Ae-Atu E'(e-AtuE'6 - M)+dx < 0.
at L+








Therefore I(t) > 0 is decreasing in t with 1(0) = 0. Hence J I(e-Atu'j- M)+12dx 0

Vt, and then u ,6(t) < Me)'t L - a.e on Q, VA > 0 and Vt > 0. Letting A -* 0 we obtain
_ ' t < M -- IIOI0lL-( )Similarly, u','(t) > -M = -Ilu0 l1L (Q) by multiplying (2.18) by e-At(-M-e-AtuE,5)+ and using M = HluojlL-(Q). Thus IuE'jlLo(QxRT) < IIlu01L o(D) _ C(Q)IuolIL-(Q). l
Before we state the existence theorem we need the following lemma:

Lemma 2.4.4 Assume the weak solutions {uE',} of (2.18)-(2.20) have uniformly bounded L'(Q x RT) and L2(Q x RT) norms in E for u',6 and it'E respectively. Then there is a subsequence of {u',6}, still denoted by {u,6}, such that as c -* 0

1. it'6 --j h weakly in L2(Q x RT) for some h

2. u'6- u6 weakly in L2 (Q x RT) for some u' where h = itj and u6(0) = uo.

For the proof see Zhou [30] or Temam [25]. Theorem 2.4.5 Suppose uo C L'(Q) n BV(Q), then there exists a unique u C L-([O, T]; BV(Q) n L- (Q)), it c L2(Q x RT), and u(O) uo such that u satisfies (2.17) for a.e. s E [0, T] and every v E L2([O, T]; BV( )). Proof: Let u',6 be the solution to (2.18)-(2.20). By Lemma 2.4.2 and Theorem 2.4.3 it satisfies


E'6 'o(ILRT) -u<~ C(Q)HIuoHLoo(f2(224


(2.24)








and


LRT itEJ12 dxdt + j (Vu',')dx (2.25)

1 VU6120dx + cj &(Vu')dx 12 0 < - /Vu912dx + C fo(Vuo)dx + E.
- 2

By (2.25)


(litC'51 + o(Vu'6))dxdt < !IU"'6IBV(QxRT) CIIUolH1(n)nLoo(),

with C = C(a, Q, T). So for fixed 6 > 0, there exists a subsequence of {u"'6} such that as c -+ 0,

u E-* u6 strongly in LI(Q x RT) and a.e in Q x RT and
i -' itj weakly in L2(QXRT). (2.26) Notice that by letting E -+ 0 in (2.24) with fixed 6 we have IIU6IIL�o(QxRT) < C(Q)luoHlLo(n), (2.27) since from (2.26) uEj -4 u5 a.e. in Q x RT. By (2.26) we can also extract a subsequence, still denoted by {uEJ}, such that

u--+ u6 strongly in L'(Q) for a.e. t E [0, T]. (2.28) Also notice that

u6'6 -+ u6 in L2( Q x RT) as c -+ 0 (2.29) as well since

j I u' - u112 dxdt < C(Q) luolILo(Q) j Z uRT - u'l dxdt by (2.24) and (2.27).








As uf,6 is also a weak solution to (2.18)-(2.20) we have as in the motivation of Definition 2.4.1
f j it'(v - u',6) dxdt + f jV12 dx + j j (v~xd

> fj j lVu,'b12dxdt + f j a, o(Vu E6)dxdt

~Ij(u',6 - u6 v- u'6) dxdt

> f f ao(Vu'6)dxdt - f f (u',6 - u5)(v - u,')dxdt

for all v E L2([0, T]; HI(Q)). Now let c --* 0 in the above inequality to arrive at
j j it6 (v - u6) dxdt + jja~o(vv)

> liminf (J f c,((Vu',6)dxdt- JSJ(6 - Uo)(v- u',')dxdt)

j j" f a(Vzt6)dt - Of" j j(u' - u)v- u')dxdt.

for all v E L2([O, T]; H1 (Q)) and hence also for all v E L2([O, T]; BV(Q)) by Proposition 2.2.8. Here we used (2.26), (2.28), (2.29), Theorem 2.2.6, the fact that V(p) > o(p) for all p, and uniform convergence of V' to V. This shows that u6 is a weak solution of (2.3)-(2.5) with initial data u0.
Additionally from (2.28) and Theorem 2.2.6 it follows that

jp(Vu6) < liminfjV(Vu',6)dx for a.e. t E [0, T]. (2.30) Thus letting c -- 0 in (2.25),

L �a iL,2dxdt + �Jc(Vu) < Cjf (VuO). So u6 E LO([0, T]; BV(Q) n LOO(Q)) and d E L2(Q X [0, T]).
Again we extract a subsequence in 6 to get as 6 --+ 0

u6 -+ u in LI(Q x RT), hence in L2(Q x RT) from (2.27),
u -+ u in L'(Q) for a.e. t E [0, t], and (2.31)

/t6 it in L2(Q x RT).







Finally pass to the limit as 5 -+ 0 in the inequality
j j i? (v - uj) dxdt + jja O(Vv) > av(Vu')dt

/3j (u6 - Uo)(V - u')dxdt

to get

jjit(v - u)dxdt + j j c(Vv) jjaV(Vu)dt

f.~ j(U - UO v- u) dxdt for all v E L2([0, T]; BV(Q)). Therefore we have the existence of a weak solution u to (2.3)-(2.5). Using (2.27), (2.30), and (2.31) we see as before that u E L-([0, T]; BV( t) n L-(Q) ).
To prove uniqueness, consider two weak solutions ul, u2 to (2.3)-(2.5) with u1(0) = u2(0) u0. We have the two inequalities j ji 'd(U2 -ui)dxdt + j c(VU2)dt > j j c(Vuj)dt f j j(U - UO) (U2 - ul)dxdt and
j j 7i2(u1 - u2)dxdt + jo j c(Vul)dt > j j av(Vu2)dt f j ,(U2 UO) (U1 -U2) dxdt.

Adding the above inequalities and combining we get
jo j(i2 - zil)(ul - U2)dxdt > j j O(ul - u2)2dxdt.

And hence

j (ul - U2 )2 dxdt < - j' i3(ul - U2 )2 dxdt < 0 giving
Hu1 (., s) - u2(', s) IL2(p) = 0 for L-a.e. s.


Therefore ul= u2. L








2.5 Stability and Asymptotic Behavior

Lemma 2.5.1 If u, and u2 are two weak solutions of (2.3) with initial data ujo and u20, then for �-a. e. t E [0, T], ]]1 - U21IL-(Q) HIU10 - U201IL-([). Proof: Let M = IIuio - U20lLO (Q). For 1-a.e. t we have

j 7il(v - ul)dx + ja~p(vv) > j aV(Vul) - j 03(u, - ujo)(v - ul)dx (2.32) and

jn 7i2(v - U2)dx + jn ap(vv) > jn a O(VU2) - j /(U - U20) (V - U2) dx. (2.33) Define
v =u, - (u'l - U2, - M)+ and
2 u +(u - M)+ where ul and u' are the approximation functions from Theorem 2.2.7. Inserting v and wE into (2.32) and (2.33) respectively we obtain

fo til(v' - ul)dx + jn Qa(VvE) j c(Vu1) - j0~(u, - ujo)(vE - ld and

j 7i2(W' - U2) x + fj a,(Vw,) > jf (Vu2) - j (U2 - U20)(wE - U2)dx. Now add the above two inequalities to get

jzil(vE - ul)dx + 12 li2(wE - u2)dx > jn aW(Vu1) + j a(VU2)

- j &w(vE) - jn ca(Vw')

- j 13(u, - ujo)(vE - ui)dx

-f (U2 - U2o)(wE- u2)dx.








Observing that

j1 ao(vw') + f2 c~(Vv6) j cp(Vu6) + j ac(Vu') with v' -4 v and w' -+ w in L2(Q) where V = Ul - (ul - U2 - M)+ (2.34) and
W = U2 + (Ul -U2- M)+ (2.35) we see after letting E --* 0 that

zil(v - u)dx +joZi2 (W -U2) dx> - 1(uj - uo)(v - u)dx

- j ,(U2 - u2o)(W - u2)dx. But the right hand side of the above inequality satisfies

j(Ul - u2 + U20 - U1o)(ul - U2 - M)+dx > 0. Thus
jzil(v- ui)dx + Ii2(w- u2)dx > 0. Hence, using equalities (2.34) and (2.35) for v and w in the above and combining, j (il - it2)(ul - - M)+dx < 0 which implies
d i(Ul - u2 -M)+12dx < 0 and therefore

j j(Ul - U2 - M)+12dx < j (UIo - U20 - M)+I2dx = 0 by the choice of M. Similarly ul - u2 > -M. I
To conclude, we investigate the asymptotic behavior of the weak solution to (2.3) by showing that the solution converges weakly in L2 (Q) and strongly in L' (Q) to a minimizer of (2.15).








Remark 2.5.2 It is straightforward to show that if inequality (2.17) is satisfied for all v E L2([O,T];BV(Q)), then (2.16) holds for all v E L2([O,T];BV(Q)) a.e. t. Then by using Young's inequality, inequality (2.16) implies

j t(v - u)dx + j (Vv) + 5 j(V _ uo)2 dx > acp(Vu) + 5 f(U _ UO)2dx for all v C BV(Q), a.e. t.

Finally we prove the following theorem concerning the asymptotic convergence of our solution u to (2.3) as t -4 oc:

Theorem 2.5.3 The solution u to (2.3) weakly converges in L2(Q) to a minimizer Uo of (2.15).

First, let F be defined on BV(Q) n L2(Q ) by F(u) j cVuj + f (U - Uo)2dx.

Second, we recall the definition of the subdifferential of a proper convex functional G : H -* (-oo, oo] where H is a Hilbert space with inner product < -,. >: the subdifferential of G at u, written OG[u], is defined as OG[u] ={w E H I G(v) > G(u)+ < w,v - u > Vv E H}. We also let D(OG) C H be the set of all u with OG[u] =A 0. Noting that the above F is defined on a subspace of L2 (Q), let the operator OF(u) be the subdifferential of F at u so that F(v) > F(u) + fn w(v - u)dx Vw c OF(u), Vv E BV(Q) nL2(Q). By Brezis [6], OF is a maximal monotone operator and by the above remark our solution u to (2.3) satisfies for a.e. t du
u OF(u).


Using the above, we can prove Theorem 2.5.3 by the following lemma ([7]): Lemma 2.5.4 Let p : H -4 (-oc, +oo] be a proper lower semicontinuous convex function which assumes a minimum in H. Then for any xo E D(Oa), there exists a








unique function x : [0, oo) -+ H which is absolutely continuous on [6, oc) for all 6 > 0 and which satisfies

x(t) E D(Dco) for all t > 0,

J(t) E -O9o(x(t)) a.e.,

X(0) = x0,

and w-limt-, x(t) exists and is a minimum point of o.

To prove Theorem 2.5.3 we take H = Lc(p), = P where, F(u) if u E BV(Q)
F(u) =
0c u C L2(Q) \ BV(Q)

x = u, and x0 = u0. As in the proof of Theorem (2.2) ([18]), the function u [0, oo) --+ L2(Q ) is absolutely continuous for all nonnegative t. So by Lemma 2.5.4 we have u(t) -- um, weakly in L2(Q) as t --+ o and u, a minimizer of F in L2(Q ). By uniqueness, u, E L2(Q ) n BV(Q). Since u(t) is uniformly bounded in BV(Q), we may also conclude that any sequence {u(tn)} has a convergent subsequence still denoted by {u(tn)} converging to um, strongly in Ll(Qi). Hence u(t) -4 u, strongly in L (Q). The theorem is now proved.
Note in fact that the minimizer u = u, from problem (2.15) is actually in L'(Q). To see this, note that the inequality in Lemma 2.4.2 and the proof of Theorem 2.4.5 imply that IIu(t)IIBV(Q) _ C(Q,6)11uoIIBV(Q) and IIU(t)IILOO(Q) , C(Q)II u0Loo(IL ) for a.e t c [0, T]. Hence by compactness, we can extract a subsequence {u(t )} such that u(t ) -+ u in L1(Q2) and u(t,) - u a.e. on Q. Thus u G L (Q) with
_lUIL-(Q) C(Q)IIuoHL-Q).









2.6 Numerical Results

For the numerical experiments we approximate (2.3) with
Ou
- div(c ,(Vu)) - (u - u') in Q x RT,

--- 0 on Qx �RT, On
u(O) = u on Q x {0}. We write the above as

(9 V u Vu 4)
t d + Vu[2 + V .u2 - /3(u - u0).

Using forward time differences and the Neumann boundary condition we compute u~t+1, n = 1, 2,..., N, N = number of iterations, by U ,t = Un + tg(Un)


where u9. = Uo(i, yj), and uVu Vu 9(u) =oa div ( Vf+ I Vu P)+Va-2 T+V UI - O(u - uO The term



is discretized using the following scheme developed by Osher and Sethian [21] to permit the development of discontinuities at object boundaries. Here we let !A2ij = ui+l,j - Uij, Ayui,j = ui,j+j - uij, Ui+l,j - Ui-lj
2

AX ui,j = ui,j - ui-l,j, AY Ui,j = Ui,j - ti,j-1, = Ui,j1 - Uij_1 AY~i' ----2

























Figure 2.1: Original 256 x256 image Then we use

(Va " Vu)i,j = max(A/;ai,j, O)Ajui,j + min(Aai,j, O)A+ui,j max(AYa?,j, O)A-u,j + min(Ayai,j, O)A+uij as in Osher and Sethian [21], while the term V/c + IVu2 is computed using the central differences Axui,j and Ayujj.
For the term
div ( Vu
E 7C+ I VuF,
we use the scheme from Rudin, Osher, and Fatemi [22] with central differences, that is

div (- Vu2) ++
Vre + -IU P V +(AX+Ui;)2 + AUj2 +A
V/ f + (A Y+u ,j + (AXU ,)2


These figures demonstrate the implementation of the above numerical scheme using 256 by 256 gray level images, with pixel values 0 to 255. Figure 2.1 shows






























Figure 2.2: Noisy image with SNR=1:1


Figure 2.3: Restored image using ATV with k=0.001, 7=0.5, E=0.01, 0=0.001, 300 iterations





























Figure 2.4: Restored image using TV with 0 = 0.001, 300 iterations


50 100 150 200

Figure 2.5: Noisy image at y=200



































140-


50 100 150 20 250 30




Figure 2.6: Dashed line is restored image from Figure 1 using ATV at y=200, dotted line is plot of original image at y=200


0 s0 100 150 200 250 300




Figure 2.7: Dashed line is restored image using TV, 0=0.001, 300 iterations at y=200, dotted line is plot of original image at y=200


II


160


140-i II


120 I





80






























Figure 2.8: Original 256 x 256 image


Figure 2.9: Noisy image, SNR=1:1.5


























Figure 2.10: Restored image using ATV with k=0.005, u=0.7, c=0.01, 0=0.002, 300 iterations

the original image; figure 2.2 shows the original image with added noise; figure 2.3 the restored image using the above ATV scheme; and figure 2.4 the restored image using the above scheme with a - 1 (restoration using TV). The next three figures examine a cross section of the images from figures 2.1 through 2.4. Figure 2.5 is a cross section of the noisy image which includes the bottom tip of the thin ellipse. Figures 2.6 and 2.7 demonstrate the effectiveness of the ATV scheme in preserving the tip of the ellipse. By plotting the cross sections of the restored images and the original image, we see that the ATV scheme is better able to preserve the tip of the ellipse. Figures 2.8 through 2.10 also demonstrate the effectiveness of the method to preserve thin lines while reducing noise. For better display, we threshold the restored images to be between the gray levels 0 and 255.

2.7 Updated Selective Smoothing

In this section we will investigate problem (2.15) with our choice of a being

1
a (x) =1 + k 1V(Gi � u)]2" (2.36)








So in the time dependent case, the selective smoothing functional now uses a continuously updated version of the image u. Fix a and denote G, by G. Assuming sufficient smoothness we can formally derive the Euler-Lagrange equation for (2.15) with a as above to be


Z ( 2kG,,(y - x)wj(y)jVujdy) + div (a&pp(Vu)) - 03(u - uo) =0 with
G1u =aGj.*u, - = 0 (1 + klV(G * u)12)2 n and where G is the partial derivative of G with respect to its ith argument. The first integral on the left hand side was obtained by using the definition of convolution and then reversing the order of integration. In order to extend the solution to the space BV(Q) n L2( Q) we multiply the above by v - u, use convexity of o, and integrate to get

(I Z j 2k(y - x) wi(y) IVu) dx) (v - u) + j ~ (v

> j2 co(vu) + jn (u - ua)(v - u)dx. Notice that in the first integral on the left hand side the coefficient function of IVul is not necessarily nonnegative as in Definition 2.2.1. However, the definition of f2 &jVuj for arbitrary & E C(Q) can be extended to be


fjivui =j&f+Ivul - j& Ivul

where &+(x) = max(&(x), 0) and &-(x) - min(&(x), 0). One can easily verify that if u E W1'(Q) then

fn &Vudx= f &+IVuldx - f &-IVujdx. Note that if a is not assumed to a nonnegative function then we no longer have lower semicontinuity of j &IVuj. From this definition combined with Remark 2.2.5 we have the simple result below.









Theorem 2.7.1 If & C(Q) n L'(Q) and u E BV(Q), then jn &IVuj < oo and



Instead of studying the complicated flow corresponding to the above EulerLagrange equation, we may instead consider a simplified version of this flow as follows:
Ou
-t = div.(oe~op(Vu)) - 0(u - uo) on Q x RT
au
- 0 on 9Q x RT
On

u = uoon Qx{0}

with a as in (2.36). Note that a(x, t) now depends on u(x, t). Although the definition of a weak solution of the above PDE is the same as Definition 2.4.1, the dependence of a on t greatly complicates an existence argument. So here we will only prove the existence of a minimizer of (2.15) with the above choice of a. Uniqueness is complicated by the observation that F is no longer a convex function. We thus only prove an existence result for the stationary problem. Theorem 2.7.2 There exists a solution belonging to BV(Q)fnL2(Q) to (2.15) where a is given by
1
Oa 1 + kIV(G * u)12 Proof: Let

F(u) = f a(x)V(u), + n(U- Uo)2 dx and let {u'} be a minimizing sequence for F. Then IjunIL2(Q) < M, M depending only on Q. Now for any u E L2(Q) we have
1
1(x)1 + JVG *U2 where

VG * U12 < (G, (x-y)dy) (u2 dy) + (jG2(X-y)dy) (jU2dy) < CIUI12-,








with C depending only on Q, a.

Denote
1
1 + IVG * Un12
by a,,(x). Then since [IUn[IL2(Q) < M by the above we have a, > 6 > 0 for some 6 depending only on Q, a. Thus

F(u,) > j 6Vu, I + j(Un - Uo)2dx > 6j I JVUnj.


Therefore j IVu, I < M and by compactness there exists a subsequence which we still denote by {un} and a u in BV(Q) n LP(Q), for any 1 < p < 2, such that u" -+ u in LP(Q). Let

a(X)1
1 + IVG *u12
Since u, -+ u in L1 (Q) we have

jGxi (X - Y) Un(y) dy Gx, jG(x - y) u(y) dy

for every x E Q. Hence an(x) -+ a(x) and la,(x)l _5 1 for every x E Q. So a. -* a in Lq(Q) for any 1 < p < oc by Lebesgue's Dominated Convergence Theorem. Now computing

2 ( Gx (x - y)un(y)dy) (jGxx (x - Y)un(y)dy) Oxl (1 + {VG* ul2)

2 ( G2(x - Y)Un(y)dy) (jGx2X(x - y)un(y)dy) (1 + {VG* U{2)2


gives
ax-- 9aw pointwise on Q, and similarily for Ox

Since
a, Mf
axi








for i = 1, 2 and M' depending only on Q, G we again have by Lebesgue's Dominated Convergence Theorem, a,, -+ a in Wl'q(Q) for any 1 < q < oo.

By definition,


fjaVun. = sup {nudiv(0)dx " �(x).

sup {j u,(Vao,(x)' -) + a(x)div(o))dx: li(x)l < 1 V(x) E }
1 1
We choose p, q with 1 < p < 2 such that - + - 1. Then p q

u, - u in LP(Q) and a, -+ a in W1'q. Thus for any 0 E CO(Q)n with j0i(x)l < 1 V(x) E Q we have

f u(Va. + adiv(�))dx = liminf un(Va," - + andiv(�))dx = liminfj undiv(a,�)dx < lim inf j I Vu, .

Hence taking the supremum over all 0 of the left hand side of the above we have ajVuj < liminf a, IVu"I. Finally using this result,

infF(u ) = liminf ( anlVunl + j(un - UO)2dx)

> liminfj a IVu ,I liminfj(un,-Uo)2dx > faVuj + L (U- Uo)2dx. The last inequality follows by convexity of the second integral and by the fact that we can also choose {un} to converge weakly to u in L2(Q). Thus F has a minimizer in L2( ) n BV(2). I









2.8 Other Selective Smoothing Models

We mention here some other proposed selective smoothing models used for image restoration.

One such model was proposed by Chambolle and Lions [8] which uses a combination of TV and isotropic diffusion. Here edge preservation is achieved using total variation where the image gradient is above a certain threshold and smoothed isotropically where the gradient is below that threshold. The model is the minimization problem


min f IVuldx + (1vu - +) + (U I)2dx.
BV(n)nL2(Q) 2f JuI 2 2J"

with given corrupted image I E BV(Q) nLL2(Q). This model will be discussed further in chapter 3.

A model proposed by Chen, Levine, and Stanich [10] is min j (x, Du) + A 1)2 BV(f )nL2 (Q) L-2

where
�(x, z) = / P(X)Iz( if Izl /3 0 jzI ()- if IzI >/

where 3 > 0 and 1 < a < p(x) < 2. For example we can use

1
p(x) + 1 �
1 + kIVG, * I(x)l"

This model uses intermediate values of the exponent of of IVul for anisotropic diffusion, that is 1 < p < 2 as well as TV-based diffusion (p = 1) along edges and isotropic diffusion in homogeneous regions (p = 2). The anisotropic diffusion is used for where the difference between noise and edges is unclear.

Another model proposed is by Blomgren, Chan, Mulet, and Wong [4] which avoids the difficulty of choosing the threshold c or /3 as used in the above models.






39

The problem is min j IVuIP(Ivul)dx where limlvul0p = 2 and limlwl_,p = 1. This model also uses a combination of edge preserving TV-based diffusion as well as anisotropic and isotropic smoothing.














CHAPTER 3
SELECTIVE SMOOTHING AND PARTIAL REGULARITY

3.1 Introduction

In the first two sections of this chapter, we focus on the problem

min , f (Du) + 'f(U- _I)2dx (3.1) uESV(fl)nL2(p 2 "2

where p is the following C' convex function defined on R' 1Ip12 if jpI < 1

I- 2 if Ipl > 1,

Q C R' is a bounded domain with Lipschitz boundary, and I E LcO(Q) n BV(Q) is given. Such a functional for image restoration was considered in Chambolle and Lions [8]. Here the restored image is taken to be the minimizer of a combination of the total variation and the squared L2 (Q) norm of the gradient. That is we minimize


f IV2dx + (Vul - ) + f (u -I)2dx.
2c- 1uj6 22

Thus we expect to have isotropic diffusion where the image is uniform (1VuI < E), and edge preserving via TV-based diffusion where edges are more pronounced (1Vu I > f). Without loss of generality we take c 1 as in (3.1).

In order to define V for u E BV(Q) we note that since the gradient of u is a measure Du, we can decompose Du into its absolutely continuous and singular parts with respect to Lebesgue measure, that is Du = Vu dx + D'u.

See Evans and Gariepy ([12]) for a complete discussion. Then we define ([15]) J(u) =_ f (Du) =_ f (Vu)dx + f IDSul









with

f ID'ul - f d[Du[ = ID'uI(Q). It is important to note ([30] or [15]) that the functional J can also be defined by

J(u)= sup ){-f (1 �2+udiv())dx:q(x)l OECo'(9,R-) f
Using this definition, we see that the functional J is lower semicontinuous with respect to convergence in L' (Q). The proof is similar to that of the proof of lower semicontinuity of the BV seminorm. Since we have established lower semicontinuity of J, is now straightforward to show that there is a unique solution to (3.1) by standard methods.

We now state the two main regularity results.

Theorem 3.1.1 If u is the solution to (3.1), then for any given 0 < z < 1 there exist positive constants Eo and io depending only on n and p such that if

1 f Du-l11

holds for some B,(a) CC Q and for some I E R', with rC (1 + IIIIL-(O)) < ro and Il < 1 - 2y, for some constant C depending only on n and Q then,

ID'ul(Br/2(a)) = 0 and IVu[ < 1 - on Br/2(a) and u solves

-Au = I - u on Br/2(a). Hence u E C' (B/2 (a)) for any c < 1. Theorem 3.1.2 Let u be as in Theorem (3.1.1). If �n({IVuI < 1}) > 0, then there exists a nonempty open region E on which u is C1',, IVul < 1 and u solves


-Au=I-u on E.








In addition we have IVu I > 1 a.e. on Q \ E.


It is actually straightforward to show that Theorem 3.1.2 is a direct consequence of Theorem 3.1.1 using standard results from measure theory and analysis. Thus from Theorem 3.1.2, we do indeed have smoothing where IVu] < 1.

Here we should point out that regularity results were obtained in Anzellotti and Giaquinta [1] for minimizers in BV(Q) of functionals of the form


j (F(x, Du) + G(x, u))

where F(x,p) is a convex function in p with cipl < F(x,p) !_ c2(1 + JPJ) for all p E R' and G(x, z) satisfies certain continuity conditions in both x and z. In our case, G(x, z) = 1/2(z - I(x))2 with only the stated assumption on I.

The proof of the above theorems follows the ideas of Tonegawa [26], where the above theorems were proved for the minimizer to the plasticity functional


min {j p(Du) - j fudx}
uEBV(Q )nL2 (Q)

for p as above and f E L0.

3.2 Proof of Theorem 3.1.1 and Theorem 3.1.2

First we will show that the solution u to (3.1) is in L'(Q). To prove this we could consider the time evolution problem corresponding to (3.1), as in the case of the previous selective smoothing problem with parameter function a, prove an L' bound for the time dependent solution u(x, t), and then consider the time asymptotic limit u. Thus we would conclude as in that case that u E LI(Q). The next lemma however provides a proof of this without having to consider the time evolution of (3.1).

Lemma 3.2.1 If u is the solution to (3.1), then u E L-(Q). In fact, we have







Proof: Let o be defined on R' by


E(P)= {


(pj2 + ) if jpI j 1 )I - if IPl > 1,


and consider the minimization problem min {f (Vu) + 1 I(u -I)2dx} uEWII( )flL2(12) ~ 2

By standard methods, there is a unique solution u, to the this problem. We follow a standard truncation argument where we fix E and t > 0 and let v = min(u,, t). Noting that v E W''(2) n L2( Q) with


V:u, ifu, t,


we have


fE(VuE) +1 + j(U( I)2 dx < j(Vv) + 1f(V I)2dx,


(3.2)


and thus after subtracting


E,(Vu,) dx + j


(uC - I)2dx < fuft- I)2dx.


Hence

But - I)2dx < /lu> (t _ h)2dx. But setting t =IIIlL-o(Q) we see that if ess sup u, > t then


(t - I)2dx < f >j


(U, - I)2dx


which contradicts the above, hence ess sup u, < IIIIL-(Q). Applying a similar argument to v = max(u, -t) for t = IIIIL-(Q) we get ess inf u, > -HIIIL-(p) and thus IIUEIIL-(Q) < IIIILo(Q). Furthermore, letting v = 0 in (3.2) we see that u, is bounded in W1,1(Q) n L2( Q) C BV()) n L2( Q) independent of c. Thus there is a


flu >tj








i! C BV(Q) n L2(Q) and a subsequence of {u }, still denoted by {u(}, such that u, --+ fi strongly in LI(Q), u, i i weakly in L2(Q), and u, --+ ii a.e in Q. Letting f -- 0 in (3.2), noting that 7o - uniformly, lower semicontinuity of the functional f o(Vv) defined on BV(Q), and weak lower semicontinuity of the second term on the left hand side, we get
f 1f
j O(Vf) + f ](ii _ I)2 dx < (Vv + f I(V _ I)2 dx n 2 fu2 "1

for all v E WI'1(7)fnL2((). We now note ([15]) that for any v E BV(Q)fnL2(Q) there exists a sequence v, in C'(n) such that j o(Vv,)dx --* j (Vv) and v,, - v in L'(Q), and since v E L2(Q) from the construction of v, ([15]) we can also take v, -+ v in L2( Q). Therefore we see that the above holds for all v E BV(Q) n L2( Q) as well. Hence ii solves (3.1). By uniqueness, l = u. By the uniform LI bound for u, and the convergence of u, to u a.e. in Q we have u E L'(Q) with IlujIL(Q) _< IIIJIL (Q). l

We note here that in the above proof, we could have chosen instead, for instance, a regularization of (3.1) using an appropriate smoothing I, of I with




and I, -+ I in L2(Q), instead of o,. We also mention that in passing to the regularized problem, we avoid having to consider the singular measures in the truncation argument.

Throughout the rest of this section, we fix IL > 0 and let c1, c2,..� represent constants that depend only on n, p, u, Q, o and possibly I. Unnumbered constants will be clearly labeled on what they depend.

From Tonegawa [26] we have

Lemma 3.2.2 Let u E BV(Br(a)) and h E C'(Br(a)) with

sup IVhj < 1- p,
B,(a)








then

Br(a) p(Du) - B(a) p(Vh)dx > p fB,(a) p DSuj + !B(a) V(u - h) " Vhdx


+ j u. Du Vh + 2 , IVuldx + j
B~a B(){W ll} ,(a)n{IWujlsl}
We fix B2,(a) Cc Q. Let v be a Lipschitz function defined on B2r(a) and assume there exists an 1 E Rn with IlI < 1 - 2/t, such that SUPB2,(a) IVV - < /326 for 6 > 0 and 0 < fl < 1 to be chosen later. Also let ;u be defined by TU(x) = v(x) - Ix Let r7, be the usual mollifier on Rn and denote g ro * -7 and v3 = * v. We also have the estimates ([24])

sup IVvO - 11 = sup [VQ, < o326, (3.3)
B,(a) B,(a)
sup V13 sup VII= - _< r/ sup Iwzl _< r031,29,
Br(a) B,(a) B,(a)
r sup IX - yI6IVv1(x) - vi,3(y)I
B,(a)
< clr6 sup IVy - l[ sup Ix' - y'l-6Irh((r)-lx') - ?7l((rO)-')[
Br (a) x/ Oy,
< c2326 - 6 = C06.

Now for any f E [1,r] there exists a unique solution ([14]) w E HI(Bf(a)) n C1 6(B(a)) with 6 E (0,1) for the problem

-Aw = I -w on Bf(a) (3.4) w = vo on OBf(a).

Lemma 3.2.3 The solution w to (3.4) satisfies

PWIILB (.)) <- be theIL(nBd (a)) + Cn, Q) IIIsLccta Proof: Let I, be the standard smoothing of I such that


jj4EjjL-(f2) -< IIIJIL-(Q),








a.e. in Bf(a), and 1, -> I in L2(Bf(a)). There exists a smooth solution ([14]) w, to


-Aw, = I,-w, onBF(a)


(3.5)


w. = vo on OBf(a)

with the estimate HW, L- (B(a)) - I V/3L- (OBf(a)) + C(n, )I I L-(0). Let w be the solution to (3.4). Since we can also bound w, in H'(Bi(a)) independent of c, there exists a subsequence of {wE}, still denoted by {w }, such that w, - w a.e. in Bf(a). Let f --+ 0 in the above estimate for w, and the lemma is proved. II
We now have, using the above lemma, the following estimates ([1, 14]):


Note that from


IWIIL- (B-(a)) I IViIIL-(aBi(a)) + C(n, Q)IIlL-o(Q), sup rT6x - yl-6IVw(x) - Vw(y)j + sup IVw - 11 (3.6)
Bj;(a) Bf (a) < c3(06 + rII - WIIL-(Bf(a))),

sup IVw(x) - Vw(y) x,yEBf/2(a) IX - y11

< c4 ( l> fSf(a)) Ivl d7'-1 + r1/2111 _ WIIL(B(a))) (3.3) we also have I I-WIIL-(Bf(a)) < IIVIIL(9BF(a))+C(n, Q)lIlHL-(Q).


Lemma 3.2.4 Suppose there is a v E C�'l(B2,(a)) and 1 E Rn with Ill < 1 - 2pa,

sup IVv - 1 < 2,
B2r(a)

and SUPB2(a) Ivt < Cu where Cu is a constant depending only on I1uH1Lo(Q). Let v3, f, and w be as in the previous discussion. Then there exists constants c5 and c6 such that if < c5 and r(Cu + C(n,Q)III I Lo()) < c6 then we have


I~ a (u IBf(a)


so(Vw)dx > JBf(a


(u - vo) 09- d- 1
On


(u- w)(I- w)dx + pi f


ID~ut +-2 /Bf:(a)n{IVu>1}


IVuldx


B;(a)


j;(a)









21IfB(a)fvu<11 IV(u- w)12dx n 1 f (W -- 1)2d-(u - v,) "y dW +
IB,;(a) (u 2 Ow 2 B(a IDsui + 2 IB(a)n(l{v>i IVukdx � 2 /Bf(a)nf{IVu,_1}

Proof: From (3.6) and by assumption on 1 we see that


(u - I)2dx a)
IV(U- _W)12dx


sup IVwl sup [Vw - 11 + Ill < C3(9 + rIIi - wlILoO(B?(a))) + 1 - 2pi
BF (.) Bp(a)
< c('3 + r(ivflLo(aBf(a)) + C(n, Q)iI(lL-(Q)) + 1- 2y

< C(6 + r(Cu + C(n, Q)IIIIIL(-))) + 1 - 2A.

Choose c5 and c6 such that /3 < c5 and r(Cu + C(n, Q) IIIIL(Q)) < c6 imply c3(0'6 + r(Cu + C(n, Q)IIIILO(Q)) < A. Thus sup IVw < 1 - it.
B, (a)

The conditions of Lemma (3.2.2) now hold for v = w. Substituting in w for v in the inequality in Lemma (3.2.2), integrating by parts, and using Young's inequality for (u - w)(I - w) = -(u - I)(w - I) + (I - w)2 the Lemma is proved. Ii

We now introduce the following definition, notation, and results ([15] and [16]).

Definition 3.2.5 A function u E BV(Q) is a local solution in Q if

j cp(Du) + I f (u - I)2dx < j p(D(u + ()) + 1 j((u + J) - I)2dx

for any ( E BV(Q). Thus the solution u to (3.1) is a local solution. The following first variational formula is from Hardt and Kinderlehrer [15]: if u is a local solution then
j u- V~dx + j a. IDu= - (u - I)(dx (3.7)


a








where ( is any function in BV(Q) with D'( << ID'ul, is the Radon-Nikodym derivative of D'( with respect to JDsuI, and a E L1(Q) is the stress tensor defined by
{p (VU) in Qa
0'(u)
D'u/ I D'u Iin Q .
Here D'u/jDsuj denotes the Radon-Nikodym derivative of D'u with respect to JD'u and Q =a U Q, is the decomposition of Q with respect to the mutually singular measures L' and ID'uj. Clearly Ia(u)l < 1. Note that a(u) depends only on the local solution u. In the sequel we will write a instead of or(u) and write the left hand side of (3.7) as
f9 r. D(.

Also note if

Jn - j(u - )d holds for arbitrary E BV(Q) for some u where a is defined as above, then u solves (3.1). In fact, for arbitrary v E BV(Q) we take ( = v - u, noting that by convexity of o we have W(Vv) - a(Vu) >_ V(v - u)- p(Vu) on Q, and that on , we have f [DS~fD8~Dsu IDCQ, IIn U1Q IDlul' Next, define the functional

Jr.a(v) j p(Dv) - j a . v n-1 + _ I ) v
JrBv = (a) fB,(a)o'*f -a2B,()

We have a second variational formula ([16]) involving J,a. Theorem 3.2.6 The function u E BV(Q) is a local solution if and only if

Jr,a(U) < Jr,a(v)


for every v E BV(Q) and r < dist(a, 0aQ).








We use the above theorem, Lemma 3.2.4, and estimates (3.3) to obtain the following inequality for the solution u to (3.1): Lemma 3.2.7 Let v, 1 be as in Lemma 3.2.4 with r(C, + C(n, Q)I L-(Q)) < c6, W as in (3.4), and u a solution to (3.1). Then

j;f(-) ID~ul + jmiqo)nIVui} IVuldx + /BF(a)n{,W,

< c7 f lu - vI dl-t'1 + csr,31+26.
JoB (a)

Proof: Clearly a solution u to (3.1) is also a local solution. By the second variational formula and Lemma (3.2.4) we have
fo x- a u vz dT-'-l ox- a (u-w) d'-Hna x _ a " lx a n-1
aBf (a) Bf(a)
> / s(Du) + - J(U _ I)2 dx - J p(Vw)dx -32J (W _I)2 di
SBjz(a) (Bf(a) Bf(a) f (a)
>I (u -v,) "wd-H n- +pf ID'ul + e f lVuldx
JB a nB (a) 2B(a)fl{ivuI 1}
2 B f(on{u<)1 IV(u - v)l dx.

The lemma is thus proved by then using the estimate for IVwl obtained in Lemma (3.2.4), the estimate for Iv - v3I from (3.3), and by noting that loI < 1. 0

The proof of the following lemma is based on Hardt and Kinderlehrer [16], and Tonegawa [26] with some necessary modifications. Lemma 3.2.8 . Suppose u is a local solution in Q of our minimization problem, B2,(a) CC Q, r(Cu + C(n, Q)ll-ll n(0)) < c6, and v E C�'1(B2r (a)) with sup VvI < 1 -,
B2,r(a)

suPB2 (a) IvI < C, and

,C({u : v} n Bp(a)) IBpj for all r < p< 2r.
2r








Then there exists positive constants c9 and clo such that if n({u = v} n B2r(a)) < cqrn then

IU- VIILOO(Br(a)) < c1O (Ln({u 5 V} fl B2r(a))).

Proof: First we note that the function o satisfies Ip - A < (p) < IP for all p E R , some A > 0. By convexity of p we have W(p) _ op(p).p + o(0) for all p E Rn. Hence we have
IDul = IVuldx + ID'uj < V(Vu)dx + IDsuj + Adx

< pp(Vu) " Vudx + IDSul + (A + W(O))dx = o . Du + Adx.

Let 0 : R -+ R be a bounded, increasing, piecewise differentiable function with O'(t) < 1 for almost all t. Let 0 < p < h and
1 in Bp(a)
(x) = (h - p)-l(h -Ix - al) in Bh(a)\Bp(a)

0 in Q\Bh(a). Now apply the first variational formula to 7 7 0(u - v) to get

7a.- D[O(u - v)] = (h - p)-i a -- a(u-v)dx
Bh() Bh(a)\Bp(a)

Bh (a) 0(u - v)(u- 1dx. In order to obtain a lower bound for 77o,. D[O(u - v)] we use the above properties of W. We have D[O(u - v)] = 0'(u - v)D(u - v) and hence by noting the bound of jVv

fp(a lD[O(u - v)]I < f(o 0'(u - v)IDuI + f(o 0'(u - V) !Bp(a) '(u - v)W(Du) + fB (A + 1)0'(u - v) IBp(a) 0'(u - v)u. Du + IB()(A + 1)0'(u - v)









I 1(u v) �j - O V '(u -v)oa.-Dv
B,, f~p+ f (A + 1)0'(u - v) B p(a)

<1h ro. D[O(u- v)] +f C\O'(u- v) for some constant A depending only on A. Therefore


fBh(a) qlD[O(u - v)]j < (h - p)-I fj O(u - v)dx + CAlsupp 71(u - v)I
- IILo(Q) IBh(a) 0(u - v)ldx. For 0 < k < s we choose 0 as

0 for t < k

0(t)= t- k for k < t < s s-k for t > s.

Now let A(k, h) R=h fn {u - v > k}. Clearly support [710(u - v)] C A(k, h). Thus


fBp(a) - v)] I ((h - 1 J Bh(a) 1(u - v)ldx + CAA(k, h)I By assumption, IA(O, p)I < 11Bp(a)I for r < p < 2r. Thus we see that nff{o(u -v) = 01 f Bp(a)} > 1 JBp(a)I - 2 Then apply the isoperimetric inequality for s > k > 0 to get


(s-k)A(s,p) (f<_ O(u -v) dx cil /p(a) ID[O(u - v)]I

< C12((h - p)-1 + IIIJlL-(Q))B 10(u - v)ldx + c13A(k, h)1. So since h < 2r we have

(s - k)A(s, p) I < c14(h - p)-' ( 10(u - v)dx + C14 1A(k, h)1.









And since

Bh(a) 1O(u - v)ldx < (s - k)[A(k, h)1, we arrive at

JA(s, p) I -- < C14((h -p)-1 + (s - k)-)IA(k, h)l

for every r < p < h < 2r and s > k > 0. We now apply Lemma 2.1 ([16]) to obtain the upper bound.

The lower bound for u - v is obtained by using a similar argument for 0 < k < s <00o,
0 for t> -k O(t) -t-k for -s
Now define the energy function

'1(r,l,x)- 1 IB,(x)nvu>1} IVux


+ If I VU- l2dx + f IDsul} The next theorem provides a decay estimate for (D. Theorem 3.2.9 If u solves (3.1) with Br(a) CC Q, 11 E Rn with 111 < 1 - t, then there exist positive constants w, c, K, c15, c16, and c17 such that 4)(4r, 11, a) c

and



implies

1 (wr, 12, a) < I4(4r, 1, a) + c,5r
2
where


111 - 121 _ c16((4r, 11, a) 2 + c17r.








The above theorem (a proof for a similar result will be given in the next section) and subsequently Theorems 3.1.1 and 3.1.2 now follow. See for instance Tonegawa [26] or Hardt and Tonegawa [17].

3.3 Partial Regularity for the p-Laplacian

Instead of restricting our model to the Laplacian as in the last section, we may also include the p-Laplacian. So we can consider the problem min po(Du) + I(u I)2dx (3.8) uEBV(n1)flL2(Q) - 2

where W0 is the following C' convex function defined on RI I1I if xl< 1

(PO( { W q
1X0x) q1if 1xl > 1,

where 1 < p < 2, 1/p + 1/q = 1, I E L'(Q) n BV(Q), and Q C R n is a bounded domain with Lipschitz boundary. Again as in Anzellotti and Giaquinta [1] or Hardt and Kinderlehrer [15] we may define the above functional on BV as j o(Du) j- L o(Vu)dx + j IDsul. For applications to image restoration this functional is a combination of anisotropic diffusion (1 < p < 2) and TV diffusion.

Due to the singular nature of p0 we instead consider a regularized version of the above minimization problem, namely


min (p, { (Du) + (u- I)2dx (3.9)
uCBV(f )nL2(q ) fn2

for the following C' convex function W, c > 0, defined on R' by { (Ix12 + ) 2 if 1xI < 1 (1 + C)p/2-1 x - -(1 + )p/2-1 (p _ 1- E) if I > 1,
P








where Q C R', I, and p, are as before. As above, j (p(Du) j f v(Vu)dx + j ID'ul We note that for E > 0, , E C1 and is C2 on the interior of the unit ball B1(0).
First we show that f9 W,(Du) is lower semicontinuous in L'(Q) for any e > 0. Lemma 3.3.1 For any E > 0, the functional fj cp(Du) is lower semicontinuous in L1(Q).

Proof: Let V {q$ E C1(Q,Rn) : q0(x)] I< 1 Vx E Q}. Without loss of generality we can adjust yp if necessary so that W,(O) = 0 and V,(x) = IxI - K where lxi > 1 for some constant K. From Eckland and Temam [11] we have for each x E Rn,

(x)y=lsup}xy-p*(y)'yEaR, ]Y
W0(x) sup{x y - iyP/q) : y E Rn, I < 1}.

The linear growth property of W actually gives finite values for W[(y) only when ]y] < 1 as well as the fact that

W:(y)=sup{x.y-E(x) :xER, Ix] <1}.

We note that we actually have for E > 0, W(y) =x* (y) � y - W,(x*(y)) where x*(y) is a continuous function of y. From Brezis [5] we have for any g E L'(Q)n J W,(g)dx sup (g- 0 ()) dx (3.10)
2 (P~qEV tf








Now define the following functional on BV(Q): J(u) = sup {f udivo + p* (0) dx}


= sup { Vu - *(�) dx + j ' Dsu} where the last equality follows from integration by parts. From the above discussion, we easily see that for every 0 E V

J Vu " 0- v*(O) dx + j qD'u < j o,(Du) giving J(u) < j oE(Du).
For the reverse inequality we follow, for example, Chen et.al. [10], noting the continuity of o*. Fix e > 0. For any u E BV(Q) there exists an open set 0, such that support(D'u) C 0, and 1O1 < E. We can also find a Ol E C (Q,Rn) with loll < 1 and
J D'u.l > jD'udx - E (3.11) from the definition of the TV norm. By (3.10) there exist a �2 E CJ(Q,,Rn) with 1�21< 1 such that

Vu - *(�2)dx > f/ p(Vu)dx - c. (3.12) Now define
�1 on O

02 on Q - 0,
Let % be the standard mollifier on R' and let , = �. Note that 0, has compact support for sufficiently small a. Then
J(u) > Vu " � , ,)dx +


Letting a -* 0 in the above inequality we then have J f) Vu - p*()dx+j .Du
( _)> f + D'u
Jn fn








>jVu.2 WE(02)dx + 01 Dsu -(E) > j W(Vu)dx + j ID'%u - L(c) - 2c where
A(00 = jo IVuldx + IIEIIL(B,(O))IOelClearly /t(E) -4 0 as c -4 0. The reverse inequality is now proved. Lower semicontinuity then easily follows as in Theorem 2.2.6. D
We also have an approximation lemma as in Theorem 2.2.7.

Lemma 3.3.2 Let u E BV(Q) n L2(Q). Then for any c > 0, there exists a sequence of functions {un} C BV(Q) n L2(Q ) n COO (Q) such that un -* u in L2(Q) and

j W(Dun) dx -+ j Wo(Du). Proof: Fix E > 0 and for simplicity write WE as V. Consider the function (x) o(x) + IxI/n. Then ([1]) there exists a un E BV(Q) n C (Q) such that

1. I un - UlIL2(Q) < 1/n

2. j /l + 1(Dun)2dxj- l+ (Du)2 < 1/n

3. o n(DU)dxj n (Du) < 1/n.

The proposition there is actually stated for functions in BV, but an adjustment of the proof on which this proposition is based ([2]) as in Remark 2.2.8 gives us estimate
1 for u E BV(Q) n L2(Q). Then

jv(Du) - j'Wo(Dun)dx < j ,,(Du) - jf n(Dun)dx +1/n (j lDunldx + j1 Dul)









< l/n + 1/n (f iDuidx + fj Du])� From 2, j IDunldx is bounded. Letting n -* 0 proves the theorem. 0

These lemmas then imply

Theorem 3.3.3 For any E > 0, there exists a unique solution u c LI(Q) to problem

3.9. In fact we have IluIL-(Q) < IIIIIL-(r). Proof: The proof is standard using lower semicontinuity and convexity. The L' bound for u follows as in Lemma 3.2.1 using both of the above lemmas. D

Now we state the regularity theorem.

Theorem 3.3.4 If u solves (3.9) for c > 0 and Cn({IVuI < 1}) > 0, then there exists a nonempty open region f on which u is C1', IVul < 1, and u solves

-div(pp(Vu)) = I - u on Q.

In addition we have IVul > 1 a.e. on Q\ . For simplicity we consider the case where c = 1 and we let o = oj. To prove the above theorems, we use a method similar to that in Hardt and Tonegawa [17] for proving partial regularity for weak solutions u E L2([0, oc], BV(Q)) to problems of the form
au
-- = divo(Vu)
at
on Q C R' or R2 where W is a convex linear growth function satisfying local ellipticity and continuity assumptions.

The essential part of this result is Theorem 3.3.7, but first we need some preliminary lemmas. We have ([17]) Lemma 3.3.5 Let u E BV(E) with open region E CC Q with smooth boundary. Then there exist constants C1, C2 < 1/2 such that if P E B1 (0) and h E C1(E)








with SUPE lVh - PI < clo, then for any vector pi c Bcl,(P),

E p(Du) - fE (Vh)dx - fE Wu(pI) - Vh D(u - h)

+ [(pli(p)pi- v(pi))" D(u-h)+supw(IVh-Pl)2j
JE E fE

> C2 (fEnjiuV IDu - + E D(u - h)12) I where w : R -- R is a non-decreasing, nonnegative function with Recall that


IEn{DuEB, (j)}


(3.13)


lVh - PI12dx limtow(t) = 0.


JD(u - h)]2 + IEnlDuOBc(p)} IDu-PI


means


IEn{BVu (B, )} IV(u - h)12dx + j VVB,(P)} [Vu - PI + DE D'u1.

Throughout the rest of this discussion u will be the solution to problem (3.9). Lemma 3.3.6 Let E CC Q be an open elliptical region. Now suppose h c C'(E) satisfies suPE IVh - ] < Cl, and


-- pi) 7i = I - h on E

h = vo on aE


(3.14)


for some Pi E B ,,(P) and smooth vo. Then we have


jf{Du B,(p)} [Du - pl + If{DuEB (()} JD(u - h)12
< C3 (f l Eu - Vol d-1 +supw(lVh _pu)2E Vh-PiI2dx).

Proof: Such a solution h exists by elementary linear theory. A simple modification of the proposition in Hardt and Kinderlehrer [16] yields the variational formula


JE(U) < JE(V)








for the minimizer u and for all v E BV(Q) where JE(v) is defined for all v E BV(Q) n L2 (q) by

JE(v) L v(Dv) - a- ftvd-I1 - + (v -I)2dx,

with stress tensor a defined as in the previous section. Using this, integrating by parts, and using Young's inequality for (u - h)(I - h) = -(u - I)(h - I) + (I - h)2 the lemma is proved. Li
The energy functional for this problem is given by

C(rp,a) - JBr. (fB,(a)n{Du B,(5)} Du - PI + fBr(a)n{DuEB (T)} IDu _p- 2 Our next goal is to prove

Theorem 3.3.7 There exist constants E, and E2 depending on W and P such that if 1D(ro,p1,xo) < Ej for some ro, xo Q Q and some P, E B,,(p), then there exists p E B,(p) such that limr-D P(r,p, xo) 0. Furthermore, p = Du(xo). From now on we drop the "a" in B, (a) and in 1 unless noted otherwise. In order to prove Theorem 3.3.7 we will obtain estimates for JE Iu - vol dT7in-1, IVh - pi1 and j IVh - P112dx for a suitable Lipschitz function v. These will then be used to prove a decay estimate for 1p on a smaller ball and a different p, whose difference can be estimated. This decay estimate will be used to prove the theorem.
Now choose a Lipschitz function v defined on B,/2(a) ([26]). From the proof of Proposition 3 there we have

Lemma 3.3.8 If B,/2 cc Q and for v defined as above, then we have sup IVV- Pl D (r, pl)26
B,/2
n+l 1 1
with 6 such that (1 - 46) = 1 + -, that is 6 - In addition we have n 2n 8(n + 1)








From the above lemma we get an L' estimate for u - v. Lemma 3.3.9 If Ip - PI < cia, then there exists an E > 0 such that T(r,pl) < implies
sup IVV -Pl -< 4(r, pl)2J
B,/2
1
for v defined as above with 6 = and with the following estimate for r suffi8(n + 1)
ciently small:

1 IIU - VIj.Coo(B,4) - C5 (112n({A v} nl B,/2) 1/nl
1 /n
c5�(I({u 0 v} n Br/2)
r

Proof: This follows from the estimate of In (Br/2 n {u 0 v}) in Lemma 3.3.8, the bound IVvj < IP1 + clo + ,D(r, pl)26, and from Lemma 3.2.8 where we use the above bound for IVvI in that proof instead of the bound IVvI < 1 used there. LI

We can now estimate IVh - pil on E for any f < r/2 for the solution h to (3.14). Let vo be the smoothing of v as described in the previous section with all the accompanying estimates. Then from the linear theory ([14]) as in the last section, taking = D(r,pl), we have IlhlIL(E) < c6 (see Lemma 3.2.3) and

sup IVh - pil < c7(4(r,pj)' + rnih - IJIL-(E)) 5 c8(41(r, p) 6 + r).
E

Let T : Q - Rn be an appropriate transformation such that if h h o T-', f= u o T-1, vp = vo o T-1, _ = I o T-', E is an ellipse centered at a, T(E) = B' is a ball of radius f centered at T(a) and

- Ah=_i-honB' (3.15) h =#v on OB'.

We note that the Jacobian of T, which depends only on the eigenvalues of the matrix [jj (pl)], is bounded from above and away from 0. Furthermore these bounds can








be made independent of a and Pi due to uniform ellipticity. We have ([14])

sup IVh(x) - Vh(y)I < C9 o I dnn-1 + f11h- iiLl Ix _ y11/2- f n+1/2 IB'
where b C B' is concentric with B' with radius f/2.
Choose E C Br/4 centered at a with diam E = r/8 and such that both folu - vj dTW n-I < l-�f lu - vldx' JE r C10fI4

and

f IU - uB/4 - P1(x -a) Id -Hn-l <_ C10- Iu-uB,/4-pl(x-a)ldx (3.16)
8oE - "fBr/4
hold, where UBr/4 denotes the average of u over B,/4. Lemma 3.3.10 Let v be as in Lemma 3.3.8 with its smoothing vo. Then


fE o - ul d'-1 < clirn('T(r,pI)1+21 + (r, P)1+1/(2n)).

Proof: By the properties of vp we have suPE Ivo - vI < r1(r,pl)1+6, and from Lemma 3.3.9 with the above integral boundary for estimate for Iu - vi


E u - vi d rn-1 < Cl rnP(Tp )1+1/(2n). Hence the lemma is proved. [] Lemma 3.3.11 If h solves (3.14) and [p, -pi < clo- then

fE jVh <_ P2dx C20(r n1i1h - IIIL- (E) + r'T (r, pi)).

Proof: Multiply (3.14) by h - vf, integrate by parts, use the fact that (3.14) is a linear equation with constant coefficients, and then use Young's inequality to arrive at
j Vh - p112dx < c12 f(h - I)(h - v,)dx + c13 f IVv - p112dx.








By the uniform bounds of Vh and VvO we see that Ih - V I L-(E) : c14(diam E) < c15r.

For other part of the estimate we use ([24]) j VO _ PiI12dx < C16 If 1,V _ Pj2dX.
fBr/4 B r/2 Finally, to estimate j Vv - Pi 2dx, the construction of v gives
Jr/2
f[r2 JV?) -- p,12 dx < C17 f~r/2n} IDu - pl 2 + c18I(r, pI) < cI9I(r, pI).
iJBr/2 -j d IBr/2f{DuEB.( )}
Combining the above estimates proves the lemma. LI
We now arrive at our decay estimate.

Theorem 3.3.12 There exist positive constants E, C20, and K depending only on n, Q, and u such that if D(r, pl) < c and r < C20, then there exists P2 E Rn such that 1(KTr,p2) < I(r,pj) � c31r and JP2 - p1 1 5C35(D(r, p1)1/2 + c34r. Proof: Using Lemma 3.3.6 and the estimates obtained in Lemmas 3.3.10 and 3.3.11 we have

LDu Bc, - I +ElDUEB,(p)} ID(u - h) 2
< c21 (r nD(r, pl)+1/(2n) + w(c8(D(r,pl)' + r))2(rn+'1h - IIILo(E) + r J(r,pu))

Letting P2 = Vh(a), we now estimate sup IVh(x) - Vh(a)I over a ball Br C SC E. Since h - ZBr/4 - P1 ' (x - a) also satisfies PDE (3.15) we see
IVh(x) - Vh(y)l ___ - if _J 3.7
sup- I C22 f - iiB,/4 -B Pl" (x - a)I dH-n-1 (3.17)
i3 I - Y11/2 - ?+1/2 JOB' r4 ( +C22 1/ A - 4IL .

Recall that the ball 3 is concentric with B' and with radius F/2 and center T(a). Changing back to the original variables, using (3.16), and finally Poincare's inequality, the right side of the above can be estimated as
C23 f ivO - UB,/4 - Pi (x - a)[ d n-1 + c221/2[[A - IHLOC
- +l/2--- JE








C24 fvo - uj dW -u + / U - UBi4 - p, (x - a)ldx
- rn+3/ B r/4
+C24r1/21h- IHLO(E)
C25 f Ivo - ul dh n-1 + C25--- IDu -pidx+C25r/211h -IL E)
Jrn+2 'E JBr/4
Let k = T-1(/3) and restrict K as necessary so that Br C k C E. Hence we easily see, after changing variables in the left hand side of (3.17), using Lemma 3.3.10,

suplVh(x)-IVh(y)l v C26(61 [ v - ul dM4- + r- IDu - plldx
Bnr rn J E rn JBr4 + K 1/2 rjh - IIILOC(E))

< c27 1/2 (,(r, pi)1+26 + P(r, )1+1/(2n) + I-f IDu - pil + rllh - IIIL-(E)).
r r/4
We thus obtain

supIVh(x) - Vh(y)l < C28K 1/2('D(r, pl)1+26 + P,)1+1/(2n)
B,
+ (D(r, P1)112 + rlh - IlIL-(E)).

This is our desired estimate for lVh(x) - Vh(y)l.
Now use the inequality IVu - Vh12 > 1Vu - P212 - jVh - P212 we arrive at


Ikrf{DuOB,7(T)} IDu - fil + J'krflDuEB,() IDu - P2 12
< c29(r n((r, p)1+1/(2n) + w(c8('D(r, p) + r))2 r+'llh - IIL(E))

+C29(w(c8((r, pl) + r))2 r' r,pi) + fj IVh - p212dx).
For the rest of the proof we denote (I(r,pl) by (. Using the estimate for IVh(x) - Vh(y)l, recalling that P2 = Vh(a), and dividing the above inequality by K nrn we have

FD(nr,p2) _ c30-d 1+1/(2n) + c3K-o . (csV5 + csr)2(D + C30K

+ C30(w(c8-V + c8r)2K-n + rr)rllh - IlIL(E).








Restrict K again so that K < -. Then restrict 4) and r so that C3on11/(2n) �
--4C30
C30K- W(C84i6 + c8r)2 < 1/4. This proves the decay estimate for 1. Finally we derive the estimate for JP2 - p i. From the linear theory ([14]) as applied to h we have

Pl IvJE - Pi (x - a) - UB IdlH -1 JP2 - P IlI = I V h(a) - p il C32 11 _f E


+C33f 11h - IHIL

1 f v, - uldl n-1 + C32 f u - pi (x -- a) -- EEJd-n-1 +c34rIjh - IJIL-.

Then using, the boundary estimates, Poincare's inequality, and H16der's inequality we get
IP2 - P1 I c35D(r, pI)I/2 + c34rjh - IIL-. 0

By using this decay estimate iteratively, we then have Theorem 3.3.7, as in Tonegawa [26]. We actually have Theorem 3.3.7 holding for all x E Br/2(Xo) if T(ro,p1,xo) is sufficiently small by noting that 1(r/2,pl,x) < 2 I(r,pl,x0) for all x E Br/2(Xo). Theorem 3.3.4 then follows ([17], [26], or [1]).














CHAPTER 4
A QUESTION FOR FURTHER STUDY In the previous chapter, we mentioned the partial regularity problem for the (unregularized) p-Laplacian. For convenience we restate the problem here. We examine the solution to

min fpo(Du) + f (U- I)2dx (4.1) uEBV(Q)NL2() - 2 J

with 0 being the C1 convex function defined on R' by I IxlP if IXI < 1

1 if IxI > 1,

where 1 < p < 2, and 1/p + 1/q - 1. We wish to obtain a partial regularity result similar to Theorems 3.1.2 and 3.3.4. We note that the method used for the regularized p-Laplacian would not give regularity on the set {VuI < 1} for the minimizer u, but only on {0 < IVuI < 1}, as 0 is singular where Vu = 0. An alternative approach would be to apply the method used for the Laplacian in the first section of chapter

3. A lemma similar to Lemma 3.2.2 can be obtained. In fact we have Theorem 4.0.13 Let u E BV(Br(a)) and h E Cl(Br(a)) with SUPB(a) Vhp-1 < 1-tI. Then


I o (DTu) o(Vh)dx > I~rC)D'uI + J V(u - h) V Pd
fB (a) 4,(a) fB,(a) B,(a) hj + f Du Vh
JB,(a) jVhJ2

+f(A) fBr(a)nvu>i1} IVuldx

+C IBr(a)fljVul<1} IV (u - h)I2dx.








for some increasing function f defined on [0, 1] such that f(p) > 0 for p $ 0 and a constant c depending only on p.

Proof: By definition, where IVul > 1 we have Vh
=0(Vu) - p0(Vh) - V(u - h) � IVh2_p


p p hVhl2up S p Vhl2-p > jvul - + - IVhIP - IVuJIVhIP-1.
P P

Nowletg(x)a- + xp - axP-' for a =Vul and x E [0, 1]. Then since
p p
a>1,

g(x) = a(1-xp-1) - a(p- - I xp) p p
1 p-li
> a(p - p -P- xP-1).
p p

Letting j(x) = a(- + P- Ixp - xP-l) we easily see that � is strictly decreasing on
p p
[0,1] with j(O) = 1/p and j(1) = 0. Now let f(p) = �(1 - M). Where IVul < 1 there holds
Vh
p,(Vu) - o,(Vh) - V(u - h) > iVhi2 + )2-p > cIV(u - h)12

where W, is defined as in (3.9) and c depends only on p. This follows from the properties of D20,, the restriction IVul < 1, and the assumption of Vh. Letting f -4 cc we get

WO(Vu) - (p0(Vh) - V(u - h) - > cIV(u - h)12.
1'7hl2-p

For the singular measure we see
IDsul > D'u IVh -p + ID ul(1 - IVhIP-1)
(r() Vh B J (a)
f D%' pVhI2_ + Br D19 B,(.) jrJ B.(a)






67

Combining the above results proves the theorem. []

However, in considering the auxiliary PDE Vw
-div(- VW- I-w on B(a) (4.2) w vo on 9Bf(a).

sufficient estimates such as (3.6) are difficult to obtain. There are regularity results ([28, 27, 19]) for the p Laplacian with 1 < p < 2, but it is not clear if the bounds provided are sufficient for obtaining the decay estimate.













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BIOGRAPHICAL SKETCH


Thomas Wunderli was born in Birkenfeld, (formerly West) Germany, in 1968, and spent most of his childhood in Colorado Springs, Colorado, where his parents still live. He received his bachelor's degree in mathematics in 1991 from Occidental College in Los Angeles, California. He was a Peace Corps volunteer from 1993 to 1994 as a math teacher in The Gambia, a small country in western Africa. In 1995 he entered the University of Florida, and received his Master of Science degree in mathematics in 1997. He then continued his studies as a Ph.D. student under Dr. Yunmei Chen.








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.


nmei Chen , Chairman
rofessor 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.


Murali Rao
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.


William Hager
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.


P' aulEhlc
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.


Mark Yan
Professor o Stics

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 School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.

August 2003
Dean, Graduate School




Full Text

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PARTIAL DIFFERENTIAL EQUATIONS-BASED IMAGE PROCESSING IN THE SPACE OF BOUNDED VARIATION USING SELECTIVE SMOOTHING EUNCTIONALS EOR NOISE REMOVAL By THOMAS WUNDERLI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OE THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OE THE REQUIREMENTS EOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OE ELORIDA 2003

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Copyright 2003 by Thomas Wunderli

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This is dedicated to Mom, Dad, and Cristina.

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ACKNOWLEDGEMENTS I would like to thank my advisor, Yunmei Chen, for introducing me to this subject and for her constant guidance and support. I would also like to thank my committee members for their input and advice. A special thank you goes to the Department of Mathematics office staff for their assistance throughout my graduate career. And finally, thanks go to my wife Cristina and my parents for their undying moral support. IV

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TABLE OF CONTENTS ACKNOWLEDGEMENTS iv ABSTRACT vi CHAPTER 1 INTRODUCTION 1 1.1 Purpose of Investigation 1 1.2 A Brief Overview 1 1.3 Basic Function Spaces and Notation 2 2 SELECTIVE SMOOTHING USING A DAPTIVE TOTAL VARIATION 4 2.1 Introduction 4 2.2 Preliminaries and Definitions 9 2.3 Minimization Problem 15 2.4 Flow Related to the Minimization Problem 15 2.5 Stability and Asymptotic Behavior 23 2.6 Numerical Results 27 2.7 Updated Selective Smoothing 33 2.8 Other Selective Smoothing Models 38 3 SELEGTIVE SMOOTHING AND PARTIAL REGULARITY 40 3.1 Introduction 40 3.2 Proof of Theorem 3.1.1 and Theorem 3.1.2 42 3.3 Partial Regularity for the p-Laplacian 53 4 A QUESTION FOR FURTHER STUDY 65 REFERENCES 68 BIOGRAPHICAL SKETCH 70

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PARTIAL DIFFERENTIAL EQUATIONS-BASED IMAGE PROGESSING IN THE STAGE OF BOUNDED VARIATION USING SELEGTIVE SMOOTHING FUNGTIONALS FOR NOISE REMOVAL By Thomas Wunderli August 2003 Chairman: Dr. Yunmei Chen Major Department: Mathematics In this thesis we study two different models for PDE-based image processing. Both model the removal of noise, also referred to as smoothing, from digital images while retaining essential features, such as edges, and both take the restored image, represented as a function defined on a rectangle D C R", to be the solution to a minimization problem over BV space. The first model uses an adaptive total variation (ATV) functional defined on BV space. We first define the ATV functional for functions that are not necessarily in any Sobolev space. This space is the a-BV space, where cv is a chosen function to locally control the amount of smoothing. Then we derive important approximation and compactness theorems concerning functions in a-BV. Having defined our functional and proven existence and uniqueness of a solution, we then study the associated time evolution problem. Here we define a weak solution u{x, t) to this problem VI

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and prove its existence, uniqueness, stability, and asymptotic behavior as t — > oo. We prove that u{x,t) weakly converges in to the solution u^o of the original stationary problem. In addition, we demonstrate some numerical results of the time evolution ATV model as well as prove the existence of a solution for an updated ATV functional. Also discussed is an updated version, where the parameter function a depends on the solution u and not on initial noisy image. The second model uses a functional which smoothes the image where its gradient norm is below a certain threshold e, that is where |Vu| < e, using either the Laplacian or a regularized p-Laplacian for 1 < p < 2, and retains edges where its gradient norm is above the threshold {\Vu\ > e). We in fact prove that the solution u is smooth where |Vw| < e. vii

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CHAPTER 1 INTRODUCTION 1.1 Purpose of Investigation As mentioned, this dissertation is the study of two methods of digital image processing from a partial differential equation (PDE) approach. However, instead of working in the more traditional context of Sobolev spaces, we work in the space of functions of bounded variation, or BV space. As will be explained in the first chapter, BV space is a more natural space for images, represented by functions defined on some rectangle D C R", to belong. Both problems studied here relate to image denoising, that is, removing noise from corrupted images while retaining essential features of the image. And both models involved remove noise while retaining edges in the image by utilizing a built in ’’selective smoothing” feature. 1.2 A Brief Overview In the second chapter, we introduce the total variation and adaptive total variation models for image denoising and some of the relevant results. We then define a-BV space as a generalization oi BV space, given an appropriate function a. Then we define a minimization problem over BV using the a-BV semi-norm, the solution to which is the restored image. We then investigate an associated time evolution problem and prove existence, uniqueness, and stability of the solution as well as show that its asymptotic time limit is the restored image. A few numerical results are presented for this model as well as a brief investigation of another adaptive total variation model, this time using an ’’updated” version of the selection parameter function a. 1

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2 For the third chapter we study another selective smoothing model, this one based on a model for plasticity. As in the first problem we take the restored image to be the solution to a minimization problem. The main focus of this chapter is proving partial regularity of solutions. We first consider the simpler case involving the 2Laplacian, then focus on the model using a regularized p-Laplacian. Some results from the 2-Laplacian case are also used for the p-Laplacian case. In order to discuss problems in PDE-based image processing, we must first include the necessary Banach spaces and notation related to the study of PDF’s. These include Sobolev spaces. Holder spaces, and of particular importance to image processing, the space of functions of bounded variation, or BV space. It is assumed to the above spaces, as well their use in PDF theory. For convenience we include a brief summary of these spaces. We start with weak derivatives and Sobolev spaces. Let 0 C R" be open. Definition 1.3.1 If u,v E and a = (ai, a„), then v is defined to be the a^^-weak derivative of u, written v = if for all test functions (f E C^. From this, the Sobolev kF*’^(D) spaces for integer k > 0 and real p > 1 are defined by Definition 1.3.2 = {u : Q ^ R| V|a| < k, D^u exists in the weak sense and D°‘u E L^{H)} 1.3 Basic Function Spaces and Notation that the reader is familiar with the embedding and compactness theorems as relates

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3 The Sobolev norm is l|^||iy'=’P(n) and for 1 < p < oo ||u||H^*,oc(n) = |D“u|£,oc(n) forp = oo. |a| < 00 Mnv(Q) |y ^ div{(f)dx : p G Co(D,R"), \p\ic with norm |iu||Li(n) + Mijv(n)We also write the seminorm [u]BV(n) as / |Vu|. This is the notation we will Jq. More will be said about the gradient measure in chapter 3. use.

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CHAPTER 2 SELECTIVE SMOOTHING USING A DAPTIVE TOTAL VARIATION 2.1 Introduction This chapter is an investigation of the mathematical theory for adaptive total variation (ATV) regularization, a powerful technique in edge preserving and noise removal, which has been effectively applied to image restoration. Let uq, defined on a rectangle 12 C R^, be an observed image which is the result of the true image Uoriginai with added noise 77 , i.e.. Our goal is to try to approximate Uoriginai as best as possible from the observed image Uq. In recent years much work in image restoration has been done using total variation (TV), see for example Rudin, Osher, and Eatemi [22], and the results have been promising. The restored image is then taken to be the solution to for the constrained problem. Here a > 0, a > 0 are chosen positive parameters, and for the unconstrained problem, and min TV{u), U subject to 11“ “o||i2(n) = TV {u) is defined by TV {u) = sup U ( 2 . 1 ) 4

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5 for any function u G L^{Q). We note that (2.1) is precisely the BV{Q.) seminorm of u, as stated in the introduction, if the above is finite. This definition for the TV functional does not require differentiability or even continuity of u. In fact one of the main advantages of using TV functional for image restoration is that jump discontinuities, that is possible edges, are allowed. The example below demonstrates this. In fact the only solution to the problem below is one with edges. Example 2.1.1 Let Q, — (0,2) and define the functional F on LP‘{Ll) by F{u) = TV{u)-\I {u — Uo)‘^dx with Jq fO ^/0 F{u). If sup f{x) i=sup fix) xe(o,i) xe[i,2) set u{y) = 0 and apply Case 4 .

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= z<^qz + qf-f = ^q + ^v^ {n)j < (/)^ usy; q-’z, = vfj -(n)^ < {f)j fijMsp udyx q-Z^'ofi (s‘i] q-z ] \ = ‘(t‘ 0) 3 o J puv ‘{{^)f-z) uira ‘(^)/ (i‘o) uim q V P7 -daoqv a?/^ Suififsi^vs (g‘o) uo snonuiiuoo sq / p/ og '(S‘rJ uo z> {X)f puv (r‘o) UO 0^ (x); f ssbq n = / 0? f 9 SVJ fijddv puv 2 = n 79/ ‘z = // '{n)j < (/)t7»?/7 99s 9 ffi -g ^ /? (g‘T] 3 ^ 2 1 I = {fi)n ‘(l‘ 0 ) 3 /i (x)/ (^‘o) 9 ^jui j 9 UJ/ 9 P (t ‘0) 3 X IP jof 0 < (x); f I -q •(™)^ < {f)j f^ixvvp dm udijx Pt '(l ‘o) 3 x diuos jo/ q = (x)/ fj ts z < {x)I 7077 yons (g‘x] 3 X uv spixd 9^977 puv (x ‘o) 3 x gv jo/ 0 < (x)/ : £ asBQ < U)d os 'On = n gyv; uoy^ 'Z < {x)I 7»?/7 yons {z'\] 3 x puv q > [x)/ py; yons (x ‘o) 9 x pixd dJdyx Z 9sbq •n = / o'l f dsvQ dfddv puv 0 = n 79s 0 = (x)/ // '{n)j < {/)j davy dm uivBy -g ^ (x)/ /i (s‘l]3/i (x)/ 'I } = ‘(t‘0) 3/1 0 J 797 udyjp -Q 9 *x dmos jo/ q = (*x)/ udy^ '(x ‘o) 9 x diuos jo/ q < {x)/ /j -q 9

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7 Thus we conclude that if f is any continuous function on (0,2) then there exists a function u with a jump discontinuity such that F{f) > F{u). Thus by uniqueness, no f ^ C{fl) can be a minimizer of F. argument f and hence f can not be a minimizer for F. As an improvement on the TV functional, Strong and Chan [23, 9] introduced the weighted TV functional for spatially adaptive (selective) image restoration. The function a is chosen so that a is larger away from possible edges and smaller near a likely edge. Hence we allow for greater smoothing away from edges and less smoothing at the edges. Certain choices of a{x) were given by Strong and Chan [23, 9], and their numerical results were very promising. However, many theoretical questions such as the existence and uniqueness for the minimizer of the weighted TV norm with some penalized term, and for the related evolution problems when a is a function on VL (not only piecewise constant) remain. The question whether the solution of the evolution equation converges to the minimizer or not as t — > oo also remains open. The goal of this chapter is to investigate these problems. Here we would like to point out in the case of constant a, results were obtained by Chambolle and Lions [8] for the minimization problem Now if f E PT^’^(f2), then by the Sobolev embedding theorem f is a.e. equal to an absolutely continuous function f and hence F{f) = F{f). Thus by the above minimize with where cr^ is known and A is a continuous linear operator on IF{Tl), and by Acar and Vogel [3] for the problem

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8 where A is a linear operator on L^(fi) and Jn(u) = [ ^\Vu\‘ + D Jn is defined on BV space. Results were also obtained, for instance, by Vese [29] for the functional and its corresponding flow. Here a > 0 is constant and (/? : R — ) R+ is a convex, even function nondecreasing in R+ with linear growth, and K ; is a linear, continuous, injective operator. However, the results for the flow are only in the dimensions one and two, due to the methods employed there, that is, using general results on maximal monotone operators and evolution operators in Hilbert spaces. In addition, Hardt and Zhou [18] consider the flow related to with Dirichlet boundary data u = g on dVt for a bounded domain fl,u = Uq on flx{0}, and any convex linear growth functional cp. In their analysis, they approximated the above solution by the flow associated with = rje * v{p) where is the usual mollifier on R”. In this work we shall extend the results of Chambolle and Lions, Acar and Vogel, as well as those of Hardt and Zhou, to the adaptive TV scheme. In particular, we shall develop mathematical theories for the problem of that is = divxP>p{Vu), {(fp = V(/?) minf («) = TV^ + f 11“ “ollii(n) ( 2 . 2 )

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9 over BV{Vl), and its corresponding evolution equation, dvi — = diw:,{a{x)ipp{Vu)) P{u-uq) du — = 0 on dO, X Rt’, on u{0) = uq on 0 , on n X Up, (2.3) (2.4) (2.5) where 0 is an open, bounded Lipschitz domain, = [0, T], ,5 > 0 is a parameter, (p{p) = \p\ on R”, (pp{p) = Vp{p) = p/|p| on R", and a{x) is a parameter (smooth) function used for edge detection and to control the speed of smoothing. In image processing problems a is often chosen as a{x) 1 1 + k\WG(j * Uq\^ ( 2 . 6 ) k being a parameter, and being the Gaussian filter with parameter a. The definition of a weak solution to (2.3)(2.5) is similar to that in Zhou [30] or Hardt and Zhou [18]. However, our method of proving existence differs from Hardt and Zhou in two ways, the first being the approximation equation. Second, and most importantly, is the way most of the necessary estimates of the approximate solution are obtained. Finally, our use of an approximate PDF to the corresponding flow is in contrast to Vese. Also, our result holds for all n. 2.2 Preliminaries and Definitions We start with the definition of / a\Vf \ for functions in / G L^(fi). Jn Definition 2.2.1 Let LI be a bounded open subset of R". Let f be a real valued function on such that f G L^{L1). Also let a{x) > 0 be a continuous real valued function on O. Then we define the a-total variation of f or a-TV of f to be / a\Vf\= sup < / f div{(/))dx : \(f{x)\ < a{x) \/x e Ll\ Jn 0 eci(n,R") Un J where 4> is a vector-valued function
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10 Definition 2.2.2 We define f G to be in a-BV if sup W / div{(j))dx : \4>{x)\ < a{x) Vx G Q > < oo. 0eci(f2,R") 1 7a J Definition 2.2.3 If f & a-BV we define the a-BV seminorm by / a|V/| = sup iff div{(t))dx : \4>{x)\ < a{x) Vx G 7a 1 7a and the a-BV norm to be a-BV = f o|V/| 7a + z-i(a)In the sequel we will write the above norm as Remark 2.2.4 It is easy to show that if f E [ a\Vf\ = [ a\Vf\dx. 7a 7a Remark 2.2.5 Note that if f E BV{Q) and functions a and fi both satisfy the conditions of Definition 2.2.1 where a{x) < fi{x) for every x E fl and f E fi-BV, then we have f E a-BV and [ a|V/|< f l3\Vf\. 7a 7a This follows directly from the Definition 2.2.3 since |^/>(x)| < a{x) implies |
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11 Now take the supremum over (j) to get / ci|V/| < liminf [ a\Vfj\. Jn Jn Next is an important approximation result for functions in BV. Theorem 2.2.7 Let f € BV where a{x) > 0 is continuous on fl. Then there is a sequence {fj} of functions from C°°{Ll) such that lim [ \ fj f\dx = Jn and lim f o;|V/,|dx = f q:|V/|. j^co Proof: We essentially apply the argument of Giusti [13] with an important modification. Given e > 0 we construct the covering {Ai} of where Ai A\ = O2, with Ojt = G : dist{x, dQ) > = 0, 1, 2, and where m is large enough such that Next we construct the sequence {/J so that (2.7) i=l where rj^. is the usual mollifier on R" and {f>i} is a partition of unity subordinate to {Ai}. We then choose the efs such that the following four conditions hold simultaneously for each ef 1. Ci < e

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12 ix < e2' 2[ IVci * f^i\da JQ 3[ \Ve.*{fV(^>^)-fV(|>,\dx / in L^(Q), and by Theorem 2.2.6, / q;| V/ l < lim inf [ o;|V/e|dx. Jn ^-*0 Jn Now let g G Co(Q,R") be such that |^(x)| < a{x) Vx G Then n OO n 00 n / /ediv(i;)da; = V / (^e^ * (/0i))div(i/)dx = V] / /idiv(r/,. * an an an (2.8) so / /ediv(£/)dx = / /div(0i?7,, * + Y] / /div((;^i77,; * g)dx (2.9) an an an OO ^ X] / f^(l>i))dx. Z=1 Denote the three terms on the right side of (2.9) by I, II, and III respectively. Note III< ellallioo by our choice of the e^s. By uniform continuity of a, there exists an increasing function ca such that uj{r) — > 0 as r 0 and |o:(2) — Oi{z')\ < cu(r) for all z, z' such that \z — z'\ < r. Consequently a{z) < u>{ei) + a{x) for all |a; — < e,. Now write g = ag' where g' = Q

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13 if q; = 0 and \g'\ < 1 Then for i > 1 and any x e \(i>i{x){r]i* g){x)\ = \(j)i{x){qi * ag'){x)\ < / (j)i{x)gi{x z)a{z)\g' {z)\dz Ju < / 4>i{x)gi{x z){a{x) uj{ti))\g' {z)\dz Jn = (j)i{x)a{x) / gi{x z)\g' {z)\dz Jn +(j)i{x)uj{ei) / gi{x z)\g' {z)\dz Jn < a{x)+uj{e). So we get, for z = 1, 1= f * g)dx < f a\Vf\+u{ei) / |V/| Jn Jn Jn and also 11 = ^ / fdiv{(j)ig,. * g)dx < 3 / a\Vf\ + 3oj{e) [ \Vf\ i =2 ^ Jn—no Jn < 3e + 3u{e) [ |V/|, Jn with the last inequality following from (2.7). Therefore [ fcdW{g)< [ a|V/|+a;(ei) [ \V f\ + 3e + 3oo{e) [ |V/| + Jn Jn Jn q\a\Loc for every g € with | 5 '(x)| < a{x) Vx G fl. Hence taking the supremum over g and then taking the limsup as e ^ 0 we get limsup / a|V/(:|da; < f a|V/|. 0 Jn Jn From (2.8) and (2.10) one finally has J|m f a\Vf,\dx = j a|V/|. (2.10)

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Remark 2.2.8 If f E BV{n) n a G C(fi), and dQ Lipschitz, then there exists a sequence of functions {/«} C such that ( 2 . 11 ) And if f E L°°{Q), we also have ll/n||L°°(n) < C'(^)||/|U°°(n)( 2 . 12 ) In fact, in the proof of Theorem 2.2.7 we choose the e’s to satisfy l-f and in addition \ve, * if (pi) f(pi\^dx < e2 \ Then we can take fj E C°°(fl) n VK^’^(Jl) n L^(fi), Since C°°(Sl) is dense in n L^(f2) there exists for each fj a sequence {fj,k} € C°°{Sl) such that as k ^ 0 Then (2.11) follows from (2.12>), (2.1A) and a standard diagonal argument applied to {fj,k} to obtain the desired sequence {fn}By the construction of {fj} and [fj^k] if in addition f E we obtain (2.12). Theorem 2.2.7 now allows us to prove a compactness theorem. Theorem 2.2.9 Let {fj} be a bounded sequence in a-BV where a E C(Q) and in addition a{x) > 5 > 0 ^x E Q. Also assume that C R” is such that 09. is Lipschitz. Then there is a subsequence of {fj}, also denoted by {fj}, and an f E IT{9) such that fj f strongly in Lp{ 9) where 1 < p < and weakly in Jn such that (2.13) \\fj,k ~ fj\\L^{n) -> 0 and \\fj^k ~ /j||wci(n) -> 0. (2.14) L—^{9). Proof: Since 0 < 5 < a[x) \/x E 9 and by remark 2.2.5 we have

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15 Therefore f |V*| < C. Jn Thus fj is bounded in BV norm and the theorem follows from the compactness result in Giusti [13]. 2.3 Minimization Problem We now consider the minimization problem min [ alVuj f (u — Uo)^dx. (2-15) BV{n)nLHa) Jn 2 Here uq is the initial noisy image, /3 is a positive parameter, and is a bounded open subset of R" with Lipschitz boundary. In the sequel we will always assume that a is a smooth function satisfying the conditions of Theorem 2.2.9. In practice we can take a{x) in (2.6) for some cr > 0 and A; > 0. Assuming uq G we see 1 “ 1 + C'||uo|licx>(n) The constant 5 in Theorem 2.2.9 can therefore be chosen as 1 1 + C'll“o|lioo(n) To conclude this section we verify that (2.15) does have a unique minimizer. Lemma 2.3.1 The functional in problem (2.\b) has a unique solution in BV{Q.) n L2(Q). Proof: Clearly, the functional is convex, coercive in BV (fl) nL^(fi) and by Theorem 2.2.6 is lower semicontinuous. So by standard results, (2.15) has a solution in BV{Q,)C\ The uniqueness follows from strict convexity of the functional in (2.15). 2.4 Flow Related to the Minimization Problem To motivate the definition of a weak solution to (2.3)-(2.5) where (p{p) = p, P
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16 smooth to justify the following calculations and that (p is as mentioned above. For arbitrary v E L^([0, T]; we multiply the equation in (2.3)-(2.5) by u — u to get, after integrating by parts and using the formula p(p) — p(g) > pp(g) (p — g) (due to the convexity of p), / u(v — u)dx + / ap(Vv)dx > / ap{Vu)dx— / P{u — uq){v — u)dx. (2.16) Jci Jo. Jn Jn Then integrate with respect to t to get n u{v — u)dxdt + / ap{V ! Jo Jn v)dxdt > n apiyu)dxdt (2-17) ! — I I /3{u — Uq){v — u)dxdt. Jo Jn On the other hand if (2.16) holds, by selecting v = u + X4> for (f) E C^(Q) we get that u is a solution to (2.3) in the sense of distributions. We are thus led to the following definition of a weak solution to (2.3)-(2.5): Definition 2.4.1 A function u E L^([0,T]; BV(fi)) is called a weak solution of (2.3)(2.5) if ii = dfU E L‘^{Cl x Rr), u(0) = uq, and u satisfies (2.17) for every v E L\[0,T]-BV{Q)), a.e. sG [0,T]. Before we continue we list some properties of the smooth approximating function of p, p^{p) = defined on R”, which will be used in the subsequent discussion: 1. p^^{p) is convex in p, 2. p],{p) p>0 Vp, 3. p^ ^ p uniformly with respect to p as e — ) 0. In fact 0 < |
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17 Consider the following approximation problem of (2.3)-(2.5): du m eAu + div(o;(pp(Vu)) — /3(u — Uq) on x R1—1 oo du dn 0 on dTl X Rp, (2.19) u(0) = Uq on n. (2.20) where Uq G C°°(0) with Uq uq in (2.21) ||'f^olU°° — C'(fI)||uo||L°°) (2.22) 1 a(f{V ul) < C {fl) f a(f{'Vuo). Jn Jn (2.23) The existence of Uq is from Remark (2.2.8) if uq G BV{Tt) n L°°(fl). The idea is to prove an existence result for the above quasilinear uniformly parabolic PDE, obtain bounds for the solution independent of e and 6, and pass the limit as e — > 0 and 5^0. Indeed, the lemma below provides an existence and uniqueness result for (2.18)-(2.20). Lemma 2.4.2 The approximation problem (2.\%)-(2.2Q) admits a unique weak solution u^'^ where G L°°([0, T]; i7^(Q)), G L^([0, T]; and f f {u^’^)‘^dxdt f ^\Vu^’^{t)\‘^dx-\[ ap{Vu^'^{t))dx Jo Ju Jn 2 Jn < / + o;(p(Vuo)d2; + e Jn 2 for a.e t E [0, T], Proof: By using the Galerkin method, the fact that pp is a monotone operator ([6] and [20]) we have a weak solution to (2.18)-(2.20) such that G L^([0, Tj; L^(fl)), ^e,s ^ L°°([o,T];/7i(fi)), and ^ J (u'’‘)‘‘dxdt + j + aip'(Vu'^’\t)) + ^(u‘* -ulf^dx I Jn -|VuqP + o(p^(VMo)dx.

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18 Since (p{p) < ip^{p) < (p{p) + e for any p we arrive at f f {u^’^)'^dxdt+ f + f ap{Vu^'^{t))dx Jo Jn Jn 2 Jq — / Jq 2 We also have the following L°°(n) bound for the solution to (2.18)-(2.20) obtained above: Theorem 2.4.3 Suppose Uq 6 L°°{Q.)f]BV{fl) andu^’^ is a solution of initial boundary value problem (2.lS)-(2.20). Then we have ||^^’‘^||L°°(nxRT) ^ C'(^)||^o||L°°(n)Proof: Let M = ||uolU°°(n)For any A > 0, multiply (2.18) by — M)+, where g At^£,(5 __ g At^£,5 _ > Q 0 otherwise. and integrate over fl to get du^’^ r du^>^ r J —e-^\e-^*u^’^-M)+dx + eJ Vu^’^ e~^^ e~^^Vu^’^dx + f app{Vu^'^)e~^* e~^^Vu^’^dx + P f {u^’^ uo)e~^\e~^^u"’^ M)pdx = 0. Jq Jq Then since the last three integrals are non-negative we see du^’^ Let Then f ^ — e — M)+dx < 0. Jq nt) = ljHe-V-MM^dx. e,6 2 Jo dt Fin Q dt )dx I JQ -At £.5 du dt e ;r— (e M)pdx f -Xe M)+dx < Q. Jq

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19 Therefore I{t) > 0 is decreasing in t with /(O) = 0. Hence [ |(e-^V’-^-M)+|"dx = 0 Jn Vt, and then C — a.e on Q, VA > 0 and Vt > 0. Letting A ^ 0 we obtain u^’\t) < M = ||4l|L~(n)Similarly, > —M = — ||uo||£)-(2.2D) have uniformly bounded L°°{Q x Rj^) and L^{Q x R^) norms in e for and respectively. Then there is a subsequence of still denoted by such that as e — )• 0 1. ^ h weakly in L^(f 2 x R 7 -) for some h 2. weakly in L‘^{Q, x Rj’) for some where h = ii^ and u'^(O) = Ug. For the proof see Zhou [30] or Temam [25]. Theorem 2.4.5 Suppose uq £ L°°{Q) n BV{Q.), then there exists a unique u € L^{[0,T]] BV{fl) n L°°(f2)), ii G x Rt), and u{0) = uq such that u satisfies (2.17) for a.e. s G [0,T] and every v G L^([0,T]; RH(fl)). Proof: Let be the solution to (2.18)-(2.20). By Lemma 2.4.2 and Theorem 2.4.3 it satisfies ||'t^^’‘^||L°°(nxR 7 ’) ^ ||^olU°°(n) < C'(f2)||Mo||L°°(n) (2.24)

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20 and I \ii^’^\^dxdt -f / a^piy u^'^)dx J UxRt’ Ju < 1 [ \Vul\‘^dx + C 'n < 1 [ \v4\'^dx + c 'u (2.25) / a(p{Vul)d. Ju X + e X + e. By (2.25) / \u nxRx + ^{Vu^'^))dxdt < ||u^’'||By(nxR^) < C|h-' with C — C(q;, Q,T). So for fixed 5 > 0, there exists a subsequence of such that as e ^ 0, > vr strongly in L^(fi x R^) and a.e in f] x R-b and u ^ if weakly in L^[Q, x R^). (2.26) Notice that by letting e ^ 0 in (2.24) with fixed 5 we have ||'w‘^||L°°(nxRT) ^ C'(^)lko|U°°(n)) (2.27) since from (2.26) a.e. in fi x R^. By (2.26) we can also extract a subsequence, still denoted by such that u ^.<5 ^ u" strongly in L^{Q) for a.e. t E [0, T]. Also notice that as well since — > u'^ in L^(Q X R^) as e — >• 0 (2.28) (2.29) / \u^'^ — HxRt’ dxdt < C(n)||uo||L°°(u) f \u^’^ — u^\ dxdt t/s7 X Rt by (2.24) and (2.27).

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21 As is also a weak solution to (2.18)-(2.20) we have as in the motivation of Definition 2.4.1 n u'^’^{v — u^’^)dxdt + ^ f f \Vv\^dx+ f f aip'^{Vv)dxdt ^ Jo Jn Jo Jn > / /* \Vu^’^\‘^dxdt + f f aip^{Vu^'^)dxdt ^ Jo Jn Jo Jn — f f {u^’^ — Uq){v — u^’^)dxdt Jo Jn > j I aip^ (Vu^’^)dxdt — ^ f [ ~ ul){v — u^'^)dxdt Jo Jn Jo Jn for all V € L^([0,T]; Now let e -> 0 in the above inequality to arrive at n ii^{v — u^)dxdt + f f a(p{Vv) ! Jo Jn >liminf^y J a(p^(Vu^’^)dxdt ~ (3 J J — Uq){v — u^’^)dxdt^ > f f aip{Vu^)dt — P f [ {u^ — Uq){v — u^)dxdt. Jo Jn Jo Jn for all V G L^([0, T]; and hence also for all v G L^([0, T]; 5V(D)) by Proposition 2.2.8. Here we used (2.26), (2.28), (2.29), Theorem 2.2.6, the fact that (p{p) for all p, and uniform convergence of to p. This shows that is a weak solution of (2.3)-(2.5) with initial data Un. Additionally from (2.28) and Theorem 2.2.6 it follows that / p{Vu^) < liminf f p{Vu^’^)dx for a.e. t G [0,T]. J Q, t/ (2.30) Thus letting e — )• 0 in (2.25), lii^pdxdt + 0 X Rj* f 0 u u u u in L^(D X Rr), hence in LP{VL x R 7 -) from (2.27), — > w in L^{Q) for a.e. t G [0,f], and ^ « in L^(0 X Rr). (2.31)

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22 Finally pass to the limit as 0 in the inequality n ii\v u^)dxdt + f f a(p{Vv) > f f aip{'Vu^)dt ! Jo Jn Jo Jn n {u^ — uo)(v — u^)dxdt ; to get n ii{v-u)dxdt+ f f a(p{Vv) > f f a(p{'Vu)dt ! Jo Jn Jo Jn ~P I / {u — Uq){v — u)dxdt Jo Jn for all V e L2([0,r]; W(f2)). Therefore we have the existence of a weak solution u to (2.3)-(2.5). Using (2.27), (2.30), and (2.31) we see as before that u E L°°([0,T];BV(n)nL°°(Q)). To prove uniqueness, consider two weak solutions Ui, U 2 to (2.3)-(2.5) with Ui(0) = «2(0) = Uq. We have the two inequalities n u\{u2 Ui)dxdt + / / aip{Vu2)dt > a(f{Vui)dt ! Jo Jn Jo Jn — / P{ui — Uq){u 2 — Ui)dxdt Jo Jn and n U2{ui U2)dxdt + f f a(p{Wui)dt > f f aip{Vu2)dt ' Jo Jn Jo Jn / /3{u 2 ~ Uo){ui U2)dxdt. Jo Jn Adding the above inequalities and combining we get n {u2 — ui){ui — U2)dxdt > / / /3{ui — U2)^dxdt. 1 Jo Jn And hence giving [ ^ / (^1 U2fdxdt < [ [ !J{ui U2f‘dxdt < 0 Jo dt Jq Jq ||ui(-, s) U 2 {-, s)||l 2 ( 0 ) = 0 for £-a.e. s. Therefore Ui = U 2 D

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2.5 Stability and Asymptotic Behavior Lemma 2.5.1 If Ui and U 2 are two weak solutions of (2.3) with initial data uio and U 20 , then for C-a.e. tE [0,T], ll'^l ~ U2\\L<^(n) < ll^^io — 'i^2o||L«>(n)Proof: Let M = ||uio — U 2 o||L°°(n)For £-a.e. t we have / Ui{v — Ui)dx ~\/ aip{Vv)> / aip{Vui)— / ^{u\ — uiq){v — Ui)dx (2.32) Jri Jn Jn Jn and / U 2 {v — U 2 )dx + / acp{Vv)> / aip{'Vu 2 ) — / I3{u — U 2 o){v — U 2 )dx. (2.33) Jn Jn Jn Jn Define = u{ — {u\ — ^2 — M)_|_ and = U2 + {u\ — U2 — M)+ where u\ and are the approximation functions from Theorem 2.2.7. Inserting and w^ into (2.32) and (2.33) respectively we obtain / ui{v^ — ui)dx+ / aifCVv^) > / aip(Vui) — / ^{ui — uiq){v^ — ui)dx Jn Jn Jn Jn and / U2{w^ — U2)dx + / ai^[Vw^)> / a^p{Vu2)— / ^{u2 — U2 q){w^ — U2)dx. Jn Jn Jn Jn Now add the above two inequalities to get / ui{v^ — ui)dx+ / U2{w^ — U2)dx > / a^p{Vui)+ / a(p(Vu2) Jn Jn Jn Jn

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24 Observing that / aip{S/w^) + / aip{Vv^) = / a(f{Vu\) + / o;(^(V'Ui) >/n Jo. Jn Jn with ^ V and ^ to in L?{Q) where V = U\ — {Ui — U2 — M)^ and (2.34) (2.35) W = U2 + {Ui U2~ M) + we see after letting e — > 0 that I Ui{v — Ui)dx + / U 2 {w — U 2 )dx > — I P{ui — Uio){v — Ui)dx Jn Jn Jn / P{U2 ~ U2q){w U2)dx. Jn But the right hand side of the above inequality satisfies p / (ui tt 2 + U 20 Uio){ui -U 2 M)+dx > 0. Jn Thus / ui{v — U\)dx + / U2{w — U2)dx > 0 . Jn Jn Hence, using equalities (2.34) and (2.35) for v and to in the above and combining, / (ui — U 2 ){ui — U 2 — M)+dx < 0 Jn which implies / |(tii — U2 — M)+pdx < 0 Jn A dt and therefore / |(tti — ti2 — M)+|^dx < / \{uio — U 20 — M)^\^dx = 0 Jn Jn by the choice of M. Similarly U\ — U 2 > —M. To conclude, we investigate the asymptotic behavior of the weak solution to (2.3) by showing that the solution converges weakly in L^(fi) and strongly in L^(0) to a minimizer of (2.15).

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25 Remark 2.5.2 It is straightforward to show that if inequality (2.17) is satisfied for all V G L‘^{[0,T]\ BV{il)), then (2.1Q) holds for all v G L‘^{[0,T]; a.e. t. Then by using Young’s inequality, inequality (2.1Q) implies )dx [ f / (^ Jn ^ Jn uo)‘^dx > for all V G BV{fl), a.e. t. Finally we prove the following theorem concerning the asymptotic convergence of our solution u to (2.3) as t — ) oo: Theorem 2.5.3 The solution u to (2.3) weakly converges in L?‘{VL) to a minimizer Uoo of (2.13). First, let F be defined on BV{Q.) n L^(n) by F{u) = f q;|Vu| f{u — Uo)^dx. Jn 2 Jq Second, we recall the definition of the subdifferential of a proper convex functional G : H (— 00 , 00 ] where H is a Hilbert space with inner product < •, • >: the subdifferential of G at u, written dG[u], is defined as dG[u] = [w G H\G{v) > G{u)+ < w,v — u > Vu G H}. We also let D{dG) C H he the set of all u with dG[u] 0. Noting that the above F is defined on a subspace of F^(0), let the operator dF{u) be the subdifferential of F at u so that F{v) > F{u) + Jqw(v — u)dx Yw G dF{u), \/v G BV{Q) riL^(f2). By Brezis [6], dF is a maximal monotone operator and by the above remark our solution u to (2.3) satisfies for a.e. t Using the above, we can prove Theorem 2.5.3 by the following lemma ([7]): Lemma 2.5.4 Let (/?://—) (— 00 , + 00 ] be a proper lower semicontinuous convex function which assumes a minimum in H . Then for any Xq G D{dp), there exists a

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26 unique function x : [0, oo) — > H which is absolutely continuous on [5, oo) for all 5 > 0 and which satisfies x{t) G D{d(p) for all t > 0, x{t) G —dcp{x{t)) a.e., x(0) = Xq, and iw-limt^oo exists and is a minimum point of ip. To prove Theorem 2.5.3 we take H = p = F where, r F(u) \iueBV{Q) F{u) = { ( oo ue L‘^{n)\BV{n) X — u, and Xq = uq. As in the proof of Theorem (2.2) ([18]), the function u : [0,oo) -> L^(fl) is absolutely continuous for all nonnegative t. So by Lemma 2.5.4 we have u{t) — ^ Uqo weakly in L^(fl) as t — > oo and Uqo a minimizer of F in Z/^(f]). By uniqueness, Uoo € 1/^(0) fl BV{Q). Since u{t) is uniformly bounded in BV{^), we may also conclude that any sequence {u(t„)} has a convergent subsequence still denoted by {u(t„)} converging to u^o strongly in L^(0). Hence u{t) — >• Uoo strongly in L^(fl). The theorem is now proved. Note in fact that the minimizer u = Uqo from problem (2.15) is actually in To see this, note that the inequality in Lemma 2.4.2 and the proof of Theorem 2.4.5 imply that ||u(t)||Bv(n) < ^'(Q, (5)||uo||By{n) and ||w(t)||L°°(n) < C'(f^)||^to||L°°{n) for a.e t G [0,Tj. Hence by compactness, we can extract a subsequence {-{/(tn)} such that u{tn) — > u in L^{fl) and u(t„) — ) u a.e. on fl. Thus u G L°°(fi) with ll'a||L°°(n) < C'(fl)||?^o|lL°°(n)-

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27 2.6 Numerical Results For the numerical experiments we approximate (2.3) with du = div(o;(^p(Vu)) — P{u — Ug) in x R^, = 0 on do, X Rt-, u(0) = Uq on Q X {0}. dt du dn We write the above as du dt = adiv Vu \A + |v + Va Vti u\ Ve+|V p{u ul). Using forward time differences and the Neumann boundary condition we compute n = 1,2, . . . , N , N = number of iterations, by = ul + Ai6W,) where uF = uo{xi,yj), and Q{u) = a div Vu The term y/e + I Vu| Va • Va • Vu Vu i/e-b |Vup p{u-ul). V^e+TV u\ is discretized using the following scheme developed by Osher and Sethian [21] to permit the development of discontinuities at object boundaries. Here we let Aj. Uij Ax Uij Ay UiJ AyUiJ 2 ^ '^hj ^ij + 1 ~ 2

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28 Figure 2.1: Original 256 x256 image Then we use (Va • = max(Aj;Q;ij, 0) Uij + min(Aa;Q;i,j, 0)A+Ujj + max(Aj,Q;ij, 0)A~Uij + min(Aj/0;jj, 0)A+Ujj as in Osher and Sethian [21], while the term + [Vup is computed using the central differences AxUij and AyUij. For the term \^e+\Vu\\ we use the scheme from Rudin, Osher, and Fatemi [22] with central differences, that IS div Vu y/e-h |Vu|^ +a: These figures demonstrate the implementation of the above numerical scheme using 256 by 256 gray level images, with pixel values 0 to 255. Figure 2.1 shows

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29 hMiMmsiMms Pm%W• Figure 2.2: Noisy image with SNR=1:1 Figure 2.3: Restored image using ATV with A:=0.001, cr=0.5, e=0.01, /3=0.001, 300 iterations

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30 Figure 2.4: Restored image using TV with ^ — 0.001, 300 iterations Figure 2.5: Noisy image at j/=200

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31 Figure 2.6: Dashed line is restored image from Figure 1 using ATV at y=200, dotted line is plot of original image at y=200 Figure 2.7: Dashed line is restored image using TV, /3=0.001, 300 iterations at y=200, dotted line is plot of original image at y=200

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32 Figure 2.8: Original 256 x 256 image Figure 2.9: Noisy image, SNR=1:1.5

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33 Figure 2.10: Restored image using ATV with A;=0.005, (7=0.7, e=0.01, ^=0.002, 300 iterations the original image; figure 2.2 shows the original image with added noise; figure 2.3 the restored image using the above ATV scheme; and figure 2.4 the restored image using the above scheme with a = 1 (restoration using TV). The next three figures examine a cross section of the images from figures 2.1 through 2.4. Figure 2.5 is a cross section of the noisy image which includes the bottom tip of the thin ellipse. Figures 2.6 and 2.7 demonstrate the effectiveness of the ATV scheme in preserving the tip of the ellipse. By plotting the cross sections of the restored images and the original image, we see that the ATV scheme is better able to preserve the tip of the ellipse. Figures 2.8 through 2.10 also demonstrate the effectiveness of the method to preserve thin lines while reducing noise. For better display, we threshold the restored images to be between the gray levels 0 and 255. 2.7 Updated Selective Smoothing In this section we will investigate problem (2.15) with our choice of a being 1 a{x) = l + k\V{G^*u)\^' (2.36)

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34 So in the time dependent case, the selective smoothing functional now uses a continuously updated version of the image u. Fix a and denote by G. Assuming sufficient smoothness we can formally derive the Euler-Lagrange equation for (2.15) with a as above to be ‘2^kGi,{y x)wi{y)\V u\dy^ + div (q;(^p(Vu)) /3(u uq) = 0 with Wi Gi * u 1% 2r^ n a Gi^*U, ^ = 0 an {l + k\V{G*u)\^Y and where Gl^ is the partial derivative of G with respect to its zth argument. The first integral on the left hand side was obtained by using the definition of convolution and then reversing the order of integration. In order to extend the solution to the space BV{Q.) n we multiply the above hy v — u, use convexity of cp, and integrate to get “^kGi^iy x)wi{y)\Vu\^ dx^ {v u) + j ap{Vv) > / a(/?(Vu) -h / P{u — Uo){v — u)dx. Jn Jn Notice that in the first integral on the left hand side the coefficient function of | Vu| is not necessarily nonnegative as in Definition 2.2.1. However, the definition of / Of|Vu| Jo. for arbitrary a G C(D) can be extended to be / dlVul = / d'^'lVul — a |Vu| Jn Jn Jn where dc^{x) — max(d(a:), 0) and a (x) = — min(o;(x), 0). One can easily verify that if u G then / a\Vu\dx= / oi^\Vu\dx— / a~\Vu\dx. Jn Jn Jn Note that if a is not assumed to a nonnegative function then we no longer have lower semicontinuity of [ d|V Jn u\. From this definition combined with Remark 2.2.5 we have the simple result below.

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35 Theorem 2.7.1 If a ^ C(fi) n L°°(r2) and u G BV{Q), then / q;|Vm| < oo and Jn [ «|V Jn u\ < ||q;| L°°{n) f |V«|. Jn Instead of studying the complicated flow corresponding to the above EulerLagrange equation, we may instead consider a simplifled version of this flow as follows: du dt du dn u = diVi(o;(/Jp(Vu)) — P{u — Uq) on x Rp = 0 on 50 X Rp = uo on 0 X {0} with a as in (2.36). Note that a{x, t) now depends on u{x, t). Although the deflnition of a weak solution of the above PDE is the same as Deflnition 2.4.1, the dependence of Of on t greatly complicates an existence argument. So here we will only prove the existence of a minimizer of (2.15) with the above choice of a. Uniqueness is complicated by the observation that F is no longer a convex function. We thus only prove an existence result for the stationary problem. Theorem 2.7.2 There exists a solution belonging to BV (Tl) C\ (il) to (2.1b) where a is given by 1 l + k\V{G*u)\^' Proof: Let F{u) = I Q!(a:)| V(u)| + f {u — uq)^ dx Jn Jn and let {«„} be a minimizing sequence for F. Then ||u„||p2(n) < M, M depending only on D. Now for any u G L^(Q) we have where a{x) 1 1 + |VG*u|2 < <

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36 with C depending only on 0, a. Denote 1 1 + |V(j * Un\^ by an{x). Then since ||iin||z,2(n) ^ M hy the above we have o;„ > <5 > 0 for some 5 depending only on 0, a. Thus F{Un)> / (5|Vu„l+ / {Un~ Uofdx > 5 / iVu„ 7n JQ. Jn Therefore / |Vu„| < M and by compactness there exists a subsequence which we Jn still denote by {u„} and a u in BV{Q.) fl i/(D), for any 1 < p < 2, such that Un u in U’{Q). Let a{x) Since — > u in LJ{Q) we have 1 1 + / G^.{x y)un{y)dy ^ y)u{y)dy Jn Jn for every x E Q. Hence an{x) o;(x) and |o;„(a:)| < 1 for every x E G,. So o;„ — >• o; in I/^(D) for any 1 < p < oo by Lebesgue’s Dominated Convergence Theorem. Now computing dan dxi 'iUo,,(x~v)uMdy )(/. Gx,xA^ y)un{y)dy {l + \VG*Un\^)^ ^ Un {x y)un{y)dy )(/. Gx 2X\ {x y)un{y)dy 2\2 (1 + |VG*u„|^) gives dan dxi da dxi pointwise on Q, and similarily for dan dx2 ' dan dxi < M' Since

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37 for i = 1, 2 and M' depending only on G we again have by Lebesgue’s Dominated Convergence Theorem, > a in for any 1 < g < oo. By definition, / anlVunl = sup < U„div(0)dx : |0i(x)| < an{x) V(a;) Jn <^>ec^(n)" Un e 0 = sup < / Un{Van{x) • (j)) + an{x)div{(f)))dx : \(f)i{x)\ < 1 V(a;) € D (j>ec^{n)^ I in We choose p, q with 1 < p < 2 such that + = 1. Then P Q Un ^ u in L^{Q) and ^ ct in Thus for any (j) G with |0j(a;)| < 1 V(x) G D we have / u{V a 4> + adiv {(/))) dx = liminf / ttn(Vo!„ • 0 + o;„div(0))dx in in = liminf / u„div(o;„)dx n->oo < liminf / 0 !„|Vu„|. rn.oo Hence taking the supremum over all (f) of the left hand side of the above we have / q;|Vu| < in Finally using this result, infF(u„) = liminf ( / o;„|Vu„| + {un-Uo)^dx Vin in y > liminf / a„| Vu„| + lim inf / {un — Uofdx n^oo n-)-oo > / q;|Vu| + (u — Uo)^dx. in in liminf / o;„|Vu^ n-Fcx) In The last inequality follows by convexity of the second integral and by the fact that we can also choose {«„} to converge weakly to u in L^(D). Thus F has a minimizer in L2(Q) nBC(D).

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38 2.8 Other Selective Smoothing Models We mention here some other proposed selective smoothing models used for image restoration. One such model was proposed by Chambolle and Lions [8] which uses a combination of TV and isotropic diffusion. Here edge preservation is achieved using total variation where the image gradient is above a certain threshold and smoothed isotropically where the gradient is below that threshold. The model is the minimization problem min ^ f iVuPdx-f f (iVnl — |) -b ^ f (u — lYdx. BV{n)nL^{u) 2e J\yu\<( J\vu\>e 2 ^ Jn with given corrupted image I € BV (fl) nL^(n). This model will be discussed further in chapter 3. A model proposed by Chen, Levine, and Stanich [10] is min f (bix.Du) + —{u — iV BV(n)nL2(Q) 2 where (j){x,z) = < where ,5 > 0 and 1 < a < p{x) < 2. For example we can use ^ ^ l + k\VG^*I{x)\' This model uses intermediate values of the exponent of of |Vn| for anisotropic diffusion, that is 1 < p < 2 as well as TV-based diffusion (p = 1) along edges and isotropic diffusion in homogeneous regions (p = 2). The anisotropic diffusion is used for where the difference between noise and edges is unclear. Another model proposed is by Blomgren, Chan, Mulet, and Wong [4] which avoids the difficulty of choosing the threshold e or ^5 as used in the above models.

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39 The problem is min [ Jn where lim|v«|-).oP = 2 and lim|vu|->.ooP = 1This model also uses a combination of edge preserving TV-based diffusion as well as anisotropic and isotropic smoothing.

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CHAPTER 3 SELECTIVE SMOOTHING AND PARTIAL REGULARITY 3.1 Introduction In the first two sections of this chapter, we focus on the problem (3.1) where is the following convex function defined on R" 0 C R" is a bounded domain with Lipschitz boundary, and I G L°°(D) Cl BV{fl) is given. Such a functional for image restoration was considered in Chambolle and Lions [8]. Here the restored image is taken to be the minimizer of a combination of the total variation and the squared L^(Q) norm of the gradient. That is we minimize edge preserving via TV-based diffusion where edges are more pronounced (|Vu| > e). Without loss of generality we take e = 1 as in (3.1). In order to define for u G BV (D) we note that since the gradient of u is a measure Du, we can decompose Du into its absolutely continuous and singular parts with respect to Lebesgue measure, that is See Evans and Gariepy ([12]) for a complete discussion. Then we define ([15]) Thus we expect to have isotropic diffusion where the image is uniform (|Vu| < e), and Du = Vu dx -tD^u. 40

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41 with [ \D^u\ = [ d\D^u\ = |Il"u|(Q). Ja Jn It is important to note ([30] or [15]) that the functional J can also be defined by J{u)= sup \~ [ ( dx : \(f){x)\ < 1 yx E Q. ec^(n,R") I JO \2 / Using this definition, we see that the functional J is lower semicontinuous with respect to convergence in The proof is similar to that of the proof of lower semicontinuity of the BV seminorm. Since we have established lower semicontinuity of J, is now straightforward to show that there is a unique solution to (3.1) by standard methods. We now state the two main regularity results. Theorem 3.1.1 If u is the solution to (3.1), then for any given 0 < /r < 1 there exist positive constants cq o,nd kq depending only on n and /j, such that if ^ [ \Du-l\< eo ^r\ J Brio.) \Br\ JBr{a) holds for some Br{a) CC and for some I G R”, with rC (l -I|]/]|L(n)) < kq and |/| < 1 2/r, for some constant C depending only on n and Q then, D^u\{Br/ 2 {a)) = 0 and |Vu| < 1 — p, on Br/ 2 {o) and u solves —Au = I — u on Br/ 2 {a). Hence u € C^’°‘{Br /2 (a)) for any a < 1. Theorem 3.1.2 Let u be as in Theorem (3.1.1/ 7/£"({|Vu| < 1}) > 0, then there exists a nonempty open region E on which u is |Vr/| < 1 and u solves —Au = I — u on E.

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42 In addition we have |Vu| > 1 a.e. onH\E. It is actually straightforward to show that Theorem 3.1.2 is a direct consequence of Theorem 3.1.1 using standard results from measure theory and analysis. Thus from Theorem 3.1.2, we do indeed have smoothing where |Vu| < 1. Here we should point out that regularity results were obtained in Anzellotti and Giaquinta [1] for minimizers in BV (fl) of functionals of the form f {F{x, Du) -f G{x, u)) Jn where F{x,p) is a convex function in p with Ci|p| < F{x,p) < C 2 (l + |p|) for all p e R" and G{x, z) satisfies certain continuity conditions in both x and z. In our case, G{x,z) = \j2[z — I{x))‘^ with only the stated assumption on I. The proof of the above theorems follows the ideas of Tonegawa [26], where the above theorems were proved for the minimizer to the plasticity functional min < / (fiDu) — / fudx > u€BV{n)nL^Q) j for p as above and / E L°° . 3.2 Proof of Theorem 3.1.1 and Theorem 3.1.2 First we will show that the solution u to (3.1) is in L°°{D). To prove this we could consider the time evolution problem corresponding to (3.1), as in the case of the previous selective smoothing problem with parameter function a, prove an L°° bound for the time dependent solution u(x, t), and then consider the time asymptotic limit u. Thus we would conclude as in that case that u E L°°(Q). The next lemma however provides a proof of this without having to consider the time evolution of (3.1). Lemma 3.2.1 If u is the solution to then u E L°°{D). In fact, we have ||«||L°°(n) < ll^||L°°(n)-

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43 Proof: Let be defined on R” by ^e{p) IpI if bl > 1, and consider the minimization problem min I [ (Pf(Vu) + [ (u — I)^dx\. uew^^{u)nLHQ) [J^ 2 Ja j By standard methods, there is a unique solution to the this problem. We follow a standard truncation argument where we fix e and f > 0 and let v = Noting that v G fl with { Vue if Ue < ^ 0 if > t, we have [ ^ f {u^-lfdx< [ (pe(Vu) + i [ {v-lfdx, Jn ^ Jn Jn ^ Jq and thus after subtracting / PeC^u^)dx+ / {u^ — I)‘^dx< / {t — I^dx. J {Ue>t} J {Uc>t} J {Ue>t} (3.2) Hence f (Ue — I)^dx < [ {t — ly J {uc>t) J {ue>i\ dx. Hue>t} J {Ue>t) But setting t = ||/||L°°(n) we see that if ess sup u^> t then ^dx which contradicts the above, hence ess sup < ||/||L°°(n)Applying a similar argument to u = max(Uf, — t) for t = ||/||L°°{n) we get ess inf > — ||/||Loo(n) and thus |b6||L°°(n) < ||L|U°°{n)Furthermore, letting u = 0 in (3.2) we see that is bounded in n L'^{Q) C BV{VL) fl L?(Vt) independent of e. Thus there is a

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44 u G BV{Q) n and a subsequence of {ue}, still denoted by {ue}, such that ^ u strongly in L^(Q), ^ u weakly in and ^ u a.e in Q. Letting e — > 0 in (3.2), noting that (/?£->• / v?(Vu) and ^ u in Jn Jn L^(Q), and since v G L‘^{Q) from the construction of ([15]) we can also take > n in Z/^(fl). Therefore we see that the above holds for all v G BV (Q) fl L^(fi) as well. Hence u solves (3.1). By uniqueness, u = u. By the uniform L°° bound for and the convergence of Uj to u a.e. in Q we have u G L°°{Q) with ||n||Loo(n) < ][/[|L°°(n)D We note here that in the above proof, we could have chosen instead, for instance, a regularization of (3.1) using an appropriate smoothing of I with ||-fe||L«>(n) < PI|L«>(n) and If I in T^(fl), instead of We also mention that in passing to the regularized problem, we avoid having to consider the singular measures in the truncation argument. Throughout the rest of this section, we fix ^ > 0 and let ci, C 2 , . . . represent constants that depend only on n, /x, u, f2, (/? and possibly I. Unnumbered constants will be clearly labeled on what they depend. From Tonegawa [26] we have Lemma 3.2.2 Let u G BV{Br{a)) and h G C^{Br{a)) with sup |V/i| < 1 — /X, Br{a)

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45 then / ip{Du) — / (p{Vh)dx >11 \D^u\ + V{u — h) Vhdx J Br{a) J Br{a) J Br{a) J Br{a) + [ D^u-Vh + ^ [ \Vu\dx + l [ \\/{u-h)\^ J Br{a) 2 yBr(a)n{|Vu|>l} 2 ys^(a)n{| Vu| 0 and 0 < /3 < 1 to be chosen later. Also let v be defined by v{x) — v[x) — I • x. Let Tje be the usual mollifier on R" and denote = rjrp*v and = r]r^*v. We also have the estimates ([24]) sup \Vvp — ^1 = sup iVn^l < , Br(a) Br{a) sup \vp — v\ = sup \vp — v\(n) < ||^||L“°(n), (3.4) le^I

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46 a.e. in Bf{a), and 4 — > / in L'^{Bf{a)). There exists a smooth solution ([14]) to —Aw^ = — We on Bf{a) (3.5) We = vp on dBf{a) with the estimate ||n;e||/,oo(B,-(a)) < ||^’^|U°°(aiJj={a)) + C'(?^, f^)||^IU~(o)Let w be the solution to (3.4). Since we can also bound We in H^{Bf{a)) independent of e, there exists a subsequence of {we}, still denoted by {we}, such that We ^ w a.e. in Bf{a). Let e — > 0 in the above estimate for We and the lemma is proved. We now have, using the above lemma, the following estimates ([1, 14]): ||'^||L°°(B;(a)) < \\vp\\L°°{dBf(a)) + C'(?^) ^^) ||Ll|L°°(n) , sup r^\x — y\~^\Vw{x) — Vu;(y)| + sup |Vru — /| (3.6) Bf[a) Bf{a) < + r\\I wWi^i^Bfia))), |Vtu(3:) — Vw{y)\ x,yeBf/2{a) \x-y\^ Note that from (3.3) we also have \\I -w\\Loo(^Bf{a)) < ll^^||L°°(aSj=(a))+C'(?^, L!)||/||Loo(n). Lemma 3.2.4 Suppose there is a v E C^’^{B 2 r{a)) and I G R” with |/) < 1 — 2/u, sup I Vu — /| < B2t{o) and sup^ 2 ^(„) |n| < C„ where Cu is a constant depending only on ||u||£,oo(q). Let vg,f, and w he as in the previous discussion. Then there exists constants C 5 and Ce such that if P < C 5 and r{Cu + C{n, fl)||/||i f {u — d'W' ^ JBfia) JBf{a) J dBf{a) f {u — w){I — w)dx -\p, f \D%\ + ^ f \Wu\dx J Bf{a) JBf{a) 2 J Bf{a)n{\Vu>l}

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47 I [ \V{u-w)\‘^dx > [ {u-vp)^dH^ ^ f {wIfdx [ {u-I) JdBf{a) C'n I JBf{a) ^ J Bf {a) dx r(“) +n [ \D^u\ + f^ [ \Vu\dx+^ f JBf(a) ^ ./Br(a)n{|Vu|>l} ^ J Bi |V(u — w)\'^dx 'Bf{a) ^ JBr(a)n{|Vu|>l} ^ JSf(a)n{|Vu|(n))) + 1 — 2^. Choose C 5 and ce such that < C 5 and r{Cu + C{n, i7)||/||x,oo(f^)) < Ce imply C 3 (/ 5 ‘* + /(Cu + C(n, f2)||/||L°°(n))) < fJ>Thus sup |Vu;| < 1 — /i. 5f(a) The conditions of Lemma (3.2.2) now hold for v = w. Substituting in w for v in the inequality in Lemma (3.2.2), integrating by parts, and using Young’s inequality for {u — w){I — w) — —{u — I){w — /) + (/ — the Lemma is proved. We now introduce the following definition, notation, and results ([15] and [16]). Definition 3.2.5 A function u G BV{Q) is a local solution in Q if [ if{Du) + l[ {u-I)^dx< [ (p{D{u + C)) + ^ [ {{u + Q-lfdx Jn ^ Jn Jci ^ Jn for any f G BV{Vt). Thus the solution u to (3.1) is a local solution. The following first variational formula is from Hardt and Kinderlehrer [15]: if u is a local solution then j a -V(^dx + I a-^|D*u| = — j {u — I)Qdx fc/ 57 */ 57 «/ 57 (3.7)

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48 where C is any function in BV{Cl) with << \D^u\, ^ is the Radon-Nikodym derivative of with respect to \D^u\, and a G L^{fl) is the stress tensor defined by { (fp{Vu) in Qq D^u/\D^u\ in fisHere D^u/\D^u\ denotes the Radon-Nikodym derivative of D^u with respect to \D^u\ and = Qa U r^s is the decomposition of Q with respect to the mutually singular measures £” and \D^u\. Clearly lo'(n)| < 1. Note that cx{u) depends only on the local solution u. In the sequel we will write a instead of cr{u) and write the left hand side of (3.7) as a DC,. / Jn Also note if / Jo. a DC, = — j {u — I)C,dx Jn holds for arbitrary ^ £ BV (17) for some u where a is defined as above, then u solves (3.1). In fact, for arbitrary v G BV{Q) we take ( = v u, noting that by convexity of (f we have (p{Vv) — t^(Vu) >V{v — u) ipp{Vu) on f7a, and that on Dg we have " , , J / \D^v\Jn, Ji \D^u\ > / rts ^ • D‘‘lv — u) — i>. \D-n\ Next, define the functional = [ (p{Dv) f J Br(a) J dBr(a o :v dn (a) a\ “ -ft ^ J Br (a) dx. r(a) J dBr{a) H ^1 ^ J Br (a) We have a second variational formula ([16]) involving Jr,aTheorem 3.2.6 The function u G BV{C1) is a local solution if and only if for every v G BV{fl) and r < dist{a,dQ).

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49 We use the above theorem, Lemma 3.2.4, and estimates (3.3) to obtain the following inequality for the solution u to (3.1): Lemma 3.2.7 Let v, I be as in Lemma 3.2.4 with r(C„ + C(n, f2)l|/||ioo(n)) < cq, w as in (3A), and u a solution to (2>,\). Then f \D^u\+ f \Vu\dx+ j \V{u w)\‘^dx < ct f \u — v\ d'H”'~^ + c^r"' . JdBf{a) Proof: Clearly a solution u to (3.1) is also a local solution. By the second variational formula and Lemma (3.2.4) we have [ 0^ 7 “ — —Au — VR)dTr~^= [ cr • -r ^Au — w) dW^~^ JdBf{a) P ~ JdBf[a) P “ > f ip{Du) + 1: [ {u — lYdx — I cp{Vw)dx — ^ / {w — I)^dx J Bf{a) 2 J Bf{a) J Bf(a) 2 J Bf{a) >f {u V 0 )^ d'H'^-^ + [ |L>*u| + ^/ \Vu\dx JdBf{a) on JBf(a) 2 Vu|>l} +l [ \V{u-v)\^dx. The lemma is thus proved by then using the estimate for \Vw\ obtained in Lemma (3.2.4), the estimate for \v — v^\ from (3.3), and by noting that |cr| < 1. The proof of the following lemma is based on Hardt and Kinderlehrer [16], and Tonegawa [26] with some necessary modifications. Lemma 3.2.8 . Suppose u is a local solution in Q, of our minimization problem, B 2 r{a) CC Ll, r{Cu + C{n,Ll)\\I\\L°°{n)) < and v € C^’^{B 2 r{a)) with sup |Vu| < 1 — pt, B2r{a) suPB 2 .(a) \v\
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50 Then there exists positive constants Cg and Cio such that if ^v}r\ B2r{a)) < cgr"' then \u — v\ L°°(Br(a)) < Cio (>C"({u ^v}n 52r(a)))" Proof: First we note that the function p satisfies |p| — A < ip{p) < \p\ for all p G R", some A > 0. By convexity of p we have cp{p) < pp{p) p +
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51 j 9'{u — v)a D{u — v) + j 9'{u — v)a-Dv J Bp{a) J Bp(a) + f (A + 1)9' {u v) JBp{a) < f rja D[9{u — u)] + f C\9'{u — v) J Bh(a) J Bh{a) ' Bh{a) JBh(a) for some constant C\ depending only on A. Therefore f t]\D[9{u v)]\ < (h-p) ^ f t;)|cix + C'A|suppr 7 ^( JbM j Bh{a)\Bp(a) U — V) ' Bh{a) + \\u For 0 < A: < s we choose 9 as \9{u — v)\dx. Bh(a) 0 for t < A: 9{t) = '^ t — k lor k < t < s s — k for t > s. Now let ^(A:, h) = Bh C\ {u — v > k}. Clearly support [r]9{u — v)] C ^(A:, h). Thus [ |T»[0(n-t;)]|<((h-p)-i + 2||/|Uoo(n)) [ \9{u v)\dx + C,\A(k,h)\ J Bp{a) J Bh{a) By assumption, |t4(0,p)| < \\Bp{a)\ for r < p < 2r. Thus we see that C^{{9{u-v) = fl}nBp{a)} ^ 1 \Bp[a)\ 2 Then apply the isoperimetric inequality for s > A: > 0 to get n— 1 {s k)\A{s,p)\'^ < I [ \9{u-v)\^dx] < Cn [ \D[9{u-v)]\ \JBp(a) J JBp{a) < Ci 2 ((/i + ||/|U-(n)) [ \9{u v)\dx + ci 3 \A{k,h)\. JBh(a) So since h < 2r we have (s — A;)|^(s, p)| " < Cu(h — p) ^ I \9{u — v)\dx + Cu\A{k,h)\ JBh(a)

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52 And since f \6{u — v)\dx < {s — k)\A{k, h)\, we arrive at |A(s, < cu{{h p)-^ + {sk)-^)\A{k, h)\ for every r < p < h < 2r and s > A: > 0. We now apply Lemma 2.1 ([16]) to obtain the upper bound. The lower bound for n — n is obtained by using a similar argument for 0 < k < s < oo, 0 for t > —k m = -t — k for — s < t < — A: s — k for A < — s, and A{k, h) = Bh ^ {u — v < —A:}. The lemma then follows. Now define the energy function 1 iVnjd. + \Br\ t^Br(i)n{|Vu|>l} [ \Vu-l\^dx+ [ J Br(x)nUVu\(4r, Ai, a) < e and implies where r < K ^{cor, I 2 , a) < -(4r, h,a) + ci^r \h -h\< Ci6‘J>(4r, Ai, a) 2 + ciyr.

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53 The above theorem (a proof for a similar result will be given in the next section) and subsequently Theorems 3.1.1 and 3.1.2 now follow. See for instance Tonegawa [26] or Hardt and Tonegawa [17]. 3.3 Partial Regularity for the p-Laplacian Instead of restricting our model to the Laplacian as in the last section, we may also include the p-Laplacian. So we can consider the problem min I f (fo{Du )-\ — [ (u — I)^dx\ «esv{n)nL2(n) 2 j where ipo is the following convex function defined on R" (po(^) where 1 < p < 2, 1/p -h 1/q = 1, / e L°°{Q) fl and f2 C R" is a bounded domain with Lipschitz boundary. Again as in Anzellotti and Giaquinta [1] or Hardt and Kinderlehrer [15] we may define the above functional on BV as (3.8) ) p if |2;| 1 , , 1 if jxj 1 q / (po{Du) = / po{Vu)dx+ / \D‘ J ^ t/ V rz u\. For applications to image restoration this functional is a combination of anisotropic diffusion (1 < p < 2) and TV diffusion. Due to the singular nature of cpo we instead consider a regularized version of the above minimization problem, namely [ 0, defined on R" by -(jrrp -Iif jx] < 1 (p,{x) = { ^ I (1 + e)P/^~La:| (1-l-e)^/^ ^(p— 1 — e) if jx] > 1, P (3.9)

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54 where $7 C R", I, and p, are as before. As above, We note that for e > 0, (/?£ G and is on the interior of the unit ball Ri(0). First we show that / p^{Du) is lower semicontinuous in L^{Q) for any e > 0. Jn Lemma 3.3.1 For any e > 0, the functional / Pe{Du) is lower semicontinuous in Jn L^{n). Proof: Let V = {(p E C'o(fl,R") : \(p{x)\ < 1 Vx G f7}. Without loss of generality we can adjust if necessary so that 1 for some constant K. From Eckland and Temam [11] we have for each x G R", 0,
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55 Now define the following functional on BV{Q): J(m) = sup < — / u div 4> + dx> 0€V [ JQ J = sup / Vu 4> — ip*{(j)) dx / 0. For any u G BV{Cl) there exists an open set such that support(D^u) C and \Of\ < e. We can also find a 0i G Co(fl,R") with |(^i| < 1 and [ D^u (j)i > f \D^u\dx — Jn Jn ( 3 . 11 ) from the definition of the TV norm. By (3.10) there exist a (/>2 G C'd(fl,R") with \(f> 2 \ < 1 such that / Vu • 02 > (p^{Vu)d: Jn Jn X — e. (3.12) Now define 01 on 0 = 02 on fl — Oe, Let Tja be the standard mollifier on R" and let 0^ = T]a* / Vu 0a • 0 in the above inequality we then have J{u) > I Vu (f) — ip*{^)dx + f


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56 > f \/u (f )2 ip*{(j) 2 )dx + f (j)i D% fi{e) Jn Jn > / ip^(Vu)dx+ / \D^u\ — fj,{e) — 2e Jq Jn where /^(e) = [ \^u\dx+\\(t>*\\L°o[Br(0)}\Oe\. JOe Clearly /r(e) -> 0 as e -> 0. The reverse inequality is now proved. Lower semicontinuity then easily follows as in Theorem 2.2.6. We also have an approximation lemma as in Theorem 2.2.7. Lemma 3.3.2 Let u G BV{fl) n L‘^{Q). Then for any e> 0, there exists a sequence of functions {un} C BV{fl) fi L^(f2) n C°°{Ll) such that Uji ^ u in and / (fe{Dun)dx^ / ipe{Du). Jn Jn Proof: Fix e > 0 and for simplicity write as (/?. Consider the function
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57 0, there exists a unique solution u G L°°{Q.) to problem 3.9. In fact we have ||u||/,(n) < ||/||L«>(n)Proof: The proof is standard using lower semicontinuity and convexity. The L°° bound for u follows as in Lemma 3.2.1 using both of the above lemmas. Now we state the regularity theorem. Theorem 3.3.4 If u solves (S.9) for e > 0 and £”({|Vu| < 1}) > 0, then there exists a nonempty open region on which u is |Vrt| < 1, and u solves — div{(pp{Vu)) = I — u on Cl. In addition we have |Vu| > 1 a.e. on Cl\Cl. For simplicity we consider the case where e = 1 and we let (p = (pi. To prove the above theorems, we use a method similar to that in Hardt and Tonegawa [17] for proving partial regularity for weak solutions u G L‘^{[0,oo], BV{Cl)) to problems of the form du dt = div^p(Vu) on C or where is a convex linear growth function satisfying local ellipticity and continuity assumptions. The essential part of this result is Theorem 3.3.7, but first we need some preliminary lemmas. We have ([17]) Lemma 3.3.5 Let u G BV{E) with open region E CC LI with smooth boundary. Then there exist constants Ci,C 2 < 1/2 such that if p E Ri(0) and h G C^{E)

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58 with sup^; \Vh p\ < Cia, then for any vector pi € -Bcia(p); f p{Du) f (p{Vh)dxf (pii{pi) -VhD{uh) (3.13) Je Je Je [ {pii{pi) Pi Pi{pi)) D{uh) + sup uj{\Vh Pi\)^ \Vh-pifdx J E ^ d E + > C 2 ( / \Du I Er\{Du^B„(p)} P\+ [ J El \D{u-h)\'^ , ' En{DueB„{p)} where ci; : R — > R is a non-decreasing, nonnegative function with limt^o^(0 = 0Recall that L En{DueB„(p)} \D{u h)p -I[ \Du-p\ J En{Du^B^{p)} means f ~ h)\‘^dx -If JEniVuGB^m) J E\ I Vu p1+ / JE ' En{VuGBa{p)} J En{Vu^B^{p)} Throughout the rest of this discussion u will be the solution to problem (3.9). Lemma 3.3.6 Let E CC Ll be an open elliptical region. Now suppose h E C^{E) satisfies sup^; | Vh — p| < Cicr and d'^h PhijiPi) dxidxj I — h on E (3.14) h = vp on dE for some pi E Rci
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59 for the minimizer u and for all v G BV (f2) where Je{v) is defined for all v G BV (f2) fl L^{^) by Je{v) = [ [ a hv + ]r f {v ~ I^dx, Je JdE ^ J E with stress tensor a defined as in the previous section. Using this, integrating by parts, and using Young’s inequality for {u — h){I — h) = —{u — I){h — /) + (/ — the lemma is proved. The energy functional for this problem is given by $(r,p, a) = -^ ( / \Du-p\+ \Du-pi \^r\ \J Br{a)n{DuiB„{p)} J Br{a)n{Du€B^{p)} Our next goal is to prove Theorem 3.3.7 There exist constants Ci and 62 depending on p and p such that «/ $(ro,Pi, xo) < Cl for some Tq, xq E Q and some p\ G B(:^{p), then there exists p G B^{p) such that limr_>oo = 0. Furthermore, p = Du{xq). From now on we drop the ”a” in Br{a) and in $ unless noted otherwise. In order to prove Theorem 3.3.7 we will obtain estimates for / \u — vp \ dTF~^, \Vh — pi\, and JdE / I V/i — pipdx for a suitable Lipschitz function v. These will then be used to prove Je a decay estimate for $ on a smaller ball and a different p, whose difference can be estimated. This decay estimate will be used to prove the theorem. Now choose a Lipschitz function v defined on Brj 2 [o) ([26]). From the proof of Proposition 3 there we have Lemma 3.3.8 If 5^/2 CC Q, and for v defined as above, then we have sup I Vu — Pi| < Bt/2 with S such that (1 = 1 + that is 5 = 777 -^— 77 . In addition we have n 2 n 8(n + l) C^{Br/2 F{u^ t;}) < C4r”$(r,pi)^"‘‘‘^.

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60 From the above lemma we get an L°° estimate for u — u. Lemma 3.3.9 If \pi — p| < Cicr, then there exists an e > 0 such that $(r,pi) < e implies sup I Vn — Pi I < $(r, pi) 25 B r/2 for V defined as above with 5 = ciently small: 1 8(n + 1) and with the following estimate for r suffi/ 1 \ 1/n ^5 ( f Br/ 2 ] C5^(£"({u^r;}n5,/2)'/". Proof: This follows from the estimate of £”(5^/2 Fl 7^ n}) in Lemma 3.3.8, the bound |Vn| < |p| + cicr + (r, pi)^^, and from Lemma 3.2.8 where we use the above bound for |Vw| in that proof instead of the bound |V?;| < 1 used there. We can now estimate |V/i — pi| on E for any f < r j2 for the solution h to (3.14). Let be the smoothing of v as described in the previous section with all the accompanying estimates. Then from the linear theory ([14]) as in the last section, taking /? = $(r,pi), we have ||/i||l°°(£:) < Ce (see Lemma 3.2.3) and sup|V/i-pi| < C7(4>(r,pi)‘^ + r||/i/|| loc(£;)) < C8($(r,pi)'^ + r). E Let T : fl ^ R” be an appropriate transformation such that \ih = ho u = uo T“\ vg = v /3 o T~^, i = I o E is an ellipse centered at a, T{E) = B' is a ball of radius f centered at T{a) and — Ah = I — h on B' h = vg on dB' . (3.15) We note that the Jacobian of T, which depends only on the eigenvalues of the matrix [Pii/;(Pi)j) is bounded from above and away from 0. Furthermore these bounds can

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61 be made independent of a and pi due to uniform ellipticity. We have ([14]) \Vh{x)-Vh{y)\ ^ C 9 sup B < \x yjl/2 ^n+1/2 [ \v0\dn^-^ + f\\h-i\\L. JdB' where B C B' is concentric with B' with radius f/2. Choose E C Brf 4 centered at a with diamE = r/8 and such that both [ \u — v\ d'H" ^ < — [ \u — v\dx, JdE f Jb^/4^ and [ \uub.^ Pi {x a)\d'W ^ < — [ \u ub.^ Pi {x a)\dx (3.16) JdE f JB^/i hold, where Ub^^ denotes the average of u over Sr/ 4 Lemma 3.3.10 Let v be as in Lemma 3.3.8 with its smoothing vg. Then [ \v^ u\ dU^-^ < Cur”((r,pi)). J E Proof: Multiply (3.14) by h — vp, integrate by parts, use the fact that (3.14) is a linear equation with constant coefficients, and then use Young’s inequality to arrive at / |V/i — pipdx < Ci 2 / {h — I){h — vp)dx + Ci 3 / \Vvp — pi\^dx. Je Je Je

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62 By the uniform bounds of V/i and we see that \\h-vp\\L °°{E) < cu{diam E) < C\^r. For other part of the estimate we use ([24]) / I Vu /3 — < Ci6 / \Vv—pi\^dx. Finally, to estimate / |Vu —pi^dx, the construction of v gives J B^/2 / \Vv -pi\^dx u^B^{p)} Combining the above estimates proves the lemma. We now arrive at our decay estimate. Theorem 3.3.12 There exist positive constants e, C20, and k depending only on n, n, and u such that if ^{r,pi) < e and r < C20, then there exists p2 € R" such that ^{Kr,p2) < ^^{r,pi) + C3ir and \p2 pi\ < C35$(r,pi)^/^ + C34r. Proof: Using Lemma 3.3.6 and the estimates obtained in Lemmas 3.3.10 and 3.3.11 we have f \Du-p\+ f \D{u-h)\^ J En{Du^B^{p)} J En{DueB„{p)} < C 21 +o;(c8($(r,pi)'' + r))2(r"+^||/i-/||L-(E) +r'^$(r,pi)) . Letting p2 = Vh{a), we now estimate sup |Vh(2;) — Vh(a)| over a ball B^r C E C E. Since h — ub^^^ — p\ • {x — a) also satisfies PDF (3.15) we see \Vh{x) -Vh{y)\ ^ C22 sup B -y\m < f \v~y JdB' JdB' +C22r^/^||h /||l-. UB^/^-pi-{x-a)\dH"‘ (3.17) Recall that the ball B is concentric with B' and with radius r/2 and center T{a). Changing back to the original variables, using (3.16), and finally Poincare’s inequality, the right side of the above can be estimated as C23 < r.ri+1/2 f JdE ub,,, -Pi{x -a)\ dTE ^ + C22r^^^||h I\\l

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63 ^ 1^^ ^1 pi • (^ +C24r-^/^||/i I\\l°-{e) ^:;:^[ [ IDu-pildx + c^^r^^^hi\\lc.^e)Let E = T~^{B) and restrict k as necessary so that 5«;r C E C E. Hence we easily see, after changing variables in the left hand side of (3.17), using Lemma 3.3.10, ^1/2 f ^1/2 r sup |Vh(a:) — V/i(y)| < C 2 e ( — — / \v ^ — u\ dl-E~^ -\ — / \Du — pi\dx JdE + K^^^r\\hI\\loo(^e)) f' Jb,/^ We thus obtain sup \Vh{x) Vh{y)\ < C 2 ^K^I'^{^{r,pif^^^ + Bkt + ^{r,piYl‘^ -^r\\h~ I\\l^^e))This is our desired estimate for |Vh(x) — Vh{y)\. Now use the inequality | Vu — Vhp > ^|Vu — P 2 p — |V/i — P 2 p we arrive at [ \Du-p\+ f |Du-p 2 p J BKrn{Du^B„{p)} J B^r<^{DueB„(p)} < C29(r"$(r,pi)^+^/(^'“) +ta(c8((r,pi)'^ + r))V"‘^^||/i /||l«>(e)) +C29(o;(c8($(r,pi)‘^ + r))^r"$(r,pi) + f \Vh p2\‘^dx). J B^r For the rest of the proof we denote $(r, pi) by $. Using the estimate for \Vh{x) — Vh(p)|, recalling that p 2 = Vh{a), and dividing the above inequality by we have $(ACr,P2) < C30K "$^+V(2n) "o;(c8'h'^ + C8r)^
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64 Restrict k again so that k < . Then restrict <4> and r so that + 4c3o C3o«:“"'a;(c8<4''* + c^rY < 1/4. This proves the decay estimate for $. Finally we derive the estimate for \p 2 — Pi|From the linear theory ([14]) as applied to h we have \P2 -Pil = |V/i(a) -pij < 0327^7 / \vp -pi • (x a) -UB'\dW~^ \^\ JdE +C33f||h — < C32t47 [ \v^ u\drr~'^ + ^32]^ I \u-pi{x-a) UE\d'H'^~'^ \^\ JdE rI JdE +C34r||/l — Then using, the boundary estimates, Poincare’s inequality, and Holder’s inequality we get |P2 -Pil < C35$(r,pi)^/^ + C34r||/l7|| LOO. By using this decay estimate iteratively, we then have Theorem 3.3.7, as in Tonegawa [26]. We actually have Theorem 3.3.7 holding for all x G 5^/2 ( 2 ^ 0 ) if $(ro,pi,xo) is sufficiently small by noting that d>(r/2,pi,x) < 2"(r,pi,xo) for all X e Br/ 2 {xo). Theorem 3.3.4 then follows ([17], [26], or [1]).

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CHAPTER 4 A QUESTION FOR FURTHER STUDY In the previous chapter, we mentioned the partial regularity problem for the (unregularized) p-Laplacian. For convenience we restate the problem here. We examine the solution to min I [ ipo{Du) + [ (u — I)^dx\ ueBV{n)nL^n) [J^ 2 ' j with p j JBr{a I J Br(a \D^u\ + [ V(u h) J Br{a) V/i |V/l|2-P dx Vh +fih) [ J Br(a / J Br(a {Vuldx (a)n{|Vu|>i} I V(u — h)\‘^dx. {a)n{|V«|
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66 for some increasing function f defined on [0, 1] such that /(//)> 0 for ^ 0 and a constant c depending only on p. Proof: By definition, where |Vii| > 1 we have iPq{Vu) — ipo{Vh) — V{u — h) Vh |Vh|2-p |Vu|-^ — ^--|Vh|PV(M-h) p p |Vh|2-p iv^i ^ + ivhr p p |Vh|2-p > |Vm| + ?~~\Vh\P |Vu||Vhr\ j) — 2) 1 Now let g{x) = a 1 — — ax^~^ for a = |Vn| and x G [0, 1]. Then since P P a > 1, g{x) — a{l — x^ ^) — a{— — ^x^) > a{+ ^^xP -xP-^). P 1 J) — 1 Letting g{x) = a{H x^ — x^^^) we easily see that g is strictly decreasing on [0, 1] with ^(0) = 1/p and ^(1) = 0. Now let /(/r) = ^(1 — /r). Where |Vu| < 1 there holds iPi{Vu) — <^c(Vh) — V(u — h) Vh > c|V(m — h)|' (y/|Vh|2 + e)2-P where is defined as in (3.9) and c depends only on p. This follows from the properties of the restriction |Vu| < 1, and the assumption of Vh. Letting e — > oo we get yjo(Vn) cpoi'^h) V(u h) V/i |Vh|2-P > c|V(n — /i)p For the singular measure we see / |D-«| > / D‘u--^+f |D*«|(l-|Vftr') J Br{a) J Br[a) J Br{a) > f J Br [d (a) |V/l|2-P p f \D^u\. J Br{a)

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Combining the above results proves the theorem. However, in considering the auxiliary PDE 67 -div( Vw |Vu;| 2 -p' W I — w on Bf{a) V 0 on dBf{a). (4.2) sufficient estimates such as (3.6) are difficult to obtain. There are regularity results ([28, 27, 19]) for the p Laplacian with 1 < p < 2, but it is not clear if the bounds provided are sufficient for obtaining the decay estimate.

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REFERENCES [1] G. Anzellotti, M. Giaquinta, Convex functionals and partial regularity, Arch. Rat. Mech. Anal. 102 (1988), pp. 243-272. [2] G. Anzellotti, M. Giaquinta, Funzioni BV e tracce, Rend. Sem. Mat. Univ. Padova 60 (1978), pp. 1-21. [3] R. Acar, C.R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems 10 (1994), pp. 1217-1229. [4] P. Blomgren, T. Chan, P. Mulet, C.K. Wong Total Variaton Image Resoration: Numerical Methods and Extensions Proceedings of the 1997 IEEE International Conference on Image Processing. Vol. 3, pp. 384-387. [5] H. Brezis, Integrates convexes dans les espaces de Sobolev, Israel J. Math. 13 (1972), pp. 9-23. [6] H. Brezis, Operateurs maximaux monotone. North Holland, Amsterdam (1993). [7] R.E. Bruck, Asymptotic convergence of nonlinear contraction semi-groups in Hilbert space, J. Fund. Anal. 18 (1975), pp. 15-26. [8] A. Chambolle, P.L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik 76 (1997), pp. 167-188. [9] T. Chan, D. Strong, Relation of regularization parameter and scale in total variation based image denoising (1996), preprint. [10] Y. Chen, S. Levine, J. Stanich, Functionals with p{x)growth in Image Restoration (2003), preprint. [11] I. Ekeland, R. Temam, Convex analysis and variational problems. North Holland, Amsterdam (1976). [12] L. Evans, R. Gariepy, Measure theory and fine properties of functions, GRC Press, Boca Raton (1992). [13] E. Giusti, Minimal surfaces and functions of bounded variation, Monogr. Math. 80, Birkhauser, Basel-Boston-Stuttgart (1984). [14] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, 2nd ed., SpringerVerlag (1983). [15] R. Hardt, D. Kinderlehrer, Elastic plastic deformation, Appl. Math. Optim. 10 (1983), pp. 203-246. 68

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69 [16] R. Hardt, D. Kinderlehrer, Variational problems with linear growth, PDEs and Cal. Var., Vol. 2, Birkhauser (1989), pp. 633-659. [17] R. Hardt, Y. Tonegawa, Partial regularity for evolution problems with discontinuity (199?), preprint. [18] R. Hardt, X. Zhou, An evolution problem for linear growth functionals, Commun. in Partial Differential Equations 19(11&12) (1994), pp. 1879-1907. [19] F. Hua Lin, Yi Li, Boundary C^’°‘ regularity for variational inequalities, Comm, on Pure and Applied Math. 44 (1991), pp. 715-732. [20] J.L. Lions, Quelques methodes de resolution des problems aux limites non lineaires, Dunod-Gauthier-Villars, Paris (1969). [21] S. Osher, J. Sethian, Fronts propigating with curvature dependent speed, algorithms based on the HamiltonJacobi formulation, J. Comp. Physics 79 (1988), pp. 12-49. [22] L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60, North-Holland (1992), pp. 259-268. [23] D. Strong, Adaptive total variation minimizing Image Restoration (Ph.D. Thesis, University of California at Los Angeles, August 1997). [24] R. Schoen, L. Simon, A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals, Indiana Math. J. 31(3) (1982), pp. 415-434. [25] R. Temam, Navier-Stokes equations, theory and numerical analysis, NorthHolland-Elsevier, Amsterdam (1977). [26] Y. Tonegawa, A regularity result for plasticity (1994), preprint. [27] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Eqs. 51 (1984), pp. 126-150. [28] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), pp. 219-240. [29] L. Vese, A study in the BV space of a denoisingdeblurring variational problem, Appl. Math. Optim. 44 (2001), pp. 131-161. [30] X. Zhou, An evolution problem for plastic antiplanar shear, Appl. Math. Optim. 25 (1992), pp. 263-285.

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BIOGRAPHICAL SKETCH Thomas Wunderli was born in Birkenfeld, (formerly West) Germany, in 1968, and spent most of his childhood in Colorado Springs, Colorado, where his parents still live. He received his bachelorÂ’s degree in mathematics in 1991 from Occidental College in Los Angeles, California. He was a Peace Corps volunteer from 1993 to 1994 as a math teacher in The Gambia, a small country in western Africa. In 1995 he entered the University of Florida, and received his Master of Science degree in mathematics in 1997. He then continued his studies as a Ph.D. student under Dr. Yunmei Chen. 70

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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. nmei Chen , Chairman rofessor 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. Murali Rao 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. William Hager 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. Paul Ehrlich 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 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 School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 2003 Dean, Graduate School