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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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Wunderli, Thomas
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vii, 70 leaves : ; 29 cm.

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Data smoothing ( jstor )
Image processing ( jstor )
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Mathematical independent variables ( jstor )
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Thesis (Ph.D.)--University of Florida, 2003.
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Vita.
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by Thomas Wunderli.

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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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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 Â— > 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 Â’Â’selective smoothingÂ” 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 Â’Â’updatedÂ” 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 PDFÂ’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, aÂ„), 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*Â’^(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 DÂ°Â‘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'=Â’P(n) and for 1 < p < oo ||u||H^*,oc(n) = |DÂ“u|Â£,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 11Â“ Â“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, Â— (0,2) and define the functional F on LPÂ‘{Ll) by F{u) = TV{u)-\I {u Â— Uo)Â‘^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-Â’z, = vfj -(n)^ < {f)j fijMsp udyx q-Z^'ofi (sÂ‘i] q-z ] \ = Â‘(tÂ‘ 0) 3 o J puv Â‘{{^)f-z) uira Â‘(^)/ (iÂ‘o) uim q V P7 -daoqv a?/^ Suififsi^vs (gÂ‘o) uo snonuiiuoo sq / p/ og '(SÂ‘rJ uo z> {X)f puv (rÂ‘o) UO 0^ (x); f ssbq n = / 0? f 9 SVJ fijddv puv 2 = n 79/ Â‘z = // '{n)j < (/)t7Â»?/7 99s 9 ffi -g ^ /? (gÂ‘T] 3 ^ 2 1 I = {fi)n Â‘(lÂ‘ 0 ) 3 /i (x)/ (^Â‘o) 9 ^jui j 9 UJ/ 9 P (t Â‘0) 3 X IP jof 0 < (x); f I -q Â•(Â™)^ < {f)j f^ixvvp dm udijx Pt '(l Â‘o) 3 x diuos jo/ q = (x)/ fj ts z < {x)I 7077 yons (gÂ‘x] 3 X uv spixd 9^977 puv (x Â‘o) 3 x gv jo/ 0 < (x)/ : Â£ asBQ < U)d os 'On = n gyv; uoy^ 'Z < {x)I 7Â»?/7 yons {z'\] 3 x puv q > [x)/ py; yons (x Â‘o) 9 x pixd dJdyx Z 9sbq Â•n = / o'l f dsvQ dfddv puv 0 = n 79s 0 = (x)/ // '{n)j < {/)j davy dm uivBy -g ^ (x)/ /i (sÂ‘l]3/i (x)/ 'I } = Â‘(tÂ‘0) 3/1 0 J 797 udyjp -Q 9 *x dmos jo/ q = (*x)/ udy^ '(x Â‘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 Â— > 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^Â’^(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\Â‘ + 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 Â— ) 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 RÂ”. 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 (Â«) = TV^ + f 11Â“ Â“ollii(n) ( 2 . 2 )

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9 over BV{Vl), and its corresponding evolution equation, dvi Â— = diw:,{a{x)ipp{Vu)) P{u-uq) du Â— = 0 on dO, X RtÂ’, 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 RÂ”, (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 PAGE 17 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 | PAGE 18 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 Â€ BV where a{x) > 0 is continuous on fl. Then there is a sequence {fj} of functions from CÂ°Â°{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 PAGE 19 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(Â£/)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) Â— > 0 as r 0 and |o:(2) Â— Oi{z')\ < cu(r) for all z, z' such that \z Â— z'\ < r. Consequently a{z) < u>{ei) + a{x) for all |a; Â— < e,. Now write g = ag' where g' = Q PAGE 20 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 ^ JnÂ—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 Â€ 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) PAGE 21 Remark 2.2.8 If f E BV{n) n a G C(fi), and dQ Lipschitz, then there exists a sequence of functions {/Â«} C such that ( 2 . 11 ) And if f E LÂ°Â°{Q), we also have ll/n||LÂ°Â°(n) < C'(^)||/|UÂ°Â°(n)( 2 . 12 ) In fact, in the proof of Theorem 2.2.7 we choose the eÂ’s to satisfy l-f and in addition \ve, * if (pi) f(pi\^dx < e2 \ Then we can take fj E CÂ°Â°(fl) n VK^Â’^(Jl) n L^(fi), Since CÂ°Â°(Sl) is dense in n L^(f2) there exists for each fj a sequence {fj,k} Â€ CÂ°Â°{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 RÂ” 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) LÂ—^{9). Proof: Since 0 < 5 < a[x) \/x E 9 and by remark 2.2.5 we have PAGE 22 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 Â— 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 Â“ 1 + C'||uo|licx>(n) The constant 5 in Theorem 2.2.9 can therefore be chosen as 1 1 + C'llÂ“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 PAGE 23 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 Â— u to get, after integrating by parts and using the formula p(p) Â— p(g) > pp(g) (p Â— g) (due to the convexity of p), / u(v Â— u)dx + / ap(Vv)dx > / ap{Vu)dxÂ— / P{u Â— uq){v Â— u)dx. (2.16) Jci Jo. Jn Jn Then integrate with respect to t to get n u{v Â— u)dxdt + / ap{V ! Jo Jn v)dxdt > n apiyu)dxdt (2-17) ! Â— I I /3{u Â— Uq){v Â— 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 LÂ‘^{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 RÂ”, 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 Â— ) 0. In fact 0 < | PAGE 24 17 Consider the following approximation problem of (2.3)-(2.5): du m eAu + div(o;(pp(Vu)) Â— /3(u Â— Uq) on x R1Â—1 oo du dn 0 on dTl X Rp, (2.19) u(0) = Uq on n. (2.20) where Uq G CÂ°Â°(0) with Uq uq in (2.21) ||'f^olUÂ°Â° Â— C'(fI)||uo||LÂ°Â°) (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 LÂ°Â°(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 Â— > 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 LÂ°Â°([0, T]; i7^(Q)), G L^([0, T]; and f f {u^Â’^)Â‘^dxdt f ^\Vu^Â’^{t)\Â‘^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 ^ LÂ°Â°([o,T];/7i(fi)), and ^ J (u'Â’Â‘)Â‘Â‘dxdt + j + aip'(Vu'^Â’\t)) + ^(uÂ‘* -ulf^dx I Jn -|VuqP + o(p^(VMo)dx. PAGE 25 18 Since (p{p) < ip^{p) < (p{p) + e for any p we arrive at f f {u^Â’^)'^dxdt+ f + f ap{Vu^'^{t))dx Jo Jn Jn 2 Jq Â— / Jq 2 We also have the following LÂ°Â°(n) bound for the solution to (2.18)-(2.20) obtained above: Theorem 2.4.3 Suppose Uq 6 LÂ°Â°{Q.)f]BV{fl) andu^Â’^ is a solution of initial boundary value problem (2.lS)-(2.20). Then we have ||^^Â’Â‘^||LÂ°Â°(nxRT) ^ C'(^)||^o||LÂ°Â°(n)Proof: Let M = ||uolUÂ°Â°(n)For any A > 0, multiply (2.18) by Â— M)+, where g At^Â£,(5 __ g At^Â£,5 _ > Q 0 otherwise. and integrate over fl to get du^Â’^ r du^>^ r J Â—e-^\e-^*u^Â’^-M)+dx + eJ Vu^Â’^ e~^^ e~^^Vu^Â’^dx + f app{Vu^'^)e~^* e~^^Vu^Â’^dx + P f {u^Â’^ uo)e~^\e~^^u"Â’^ M)pdx = 0. Jq Jq Then since the last three integrals are non-negative we see du^Â’^ Let Then f ^ Â— e Â— M)+dx < 0. Jq nt) = ljHe-V-MM^dx. e,6 2 Jo dt Fin Q dt )dx I JQ -At Â£.5 du dt e ;rÂ— (e M)pdx f -Xe M)+dx < Q. Jq PAGE 26 19 Therefore I{t) > 0 is decreasing in t with /(O) = 0. Hence [ |(e-^VÂ’-^-M)+|"dx = 0 Jn Vt, and then C Â— a.e on Q, VA > 0 and Vt > 0. Letting A ^ 0 we obtain u^Â’\t) < M = ||4l|L~(n)Similarly, > Â—M = Â— ||uo||Â£)-(2.2D) have uniformly bounded LÂ°Â°{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 Â— )Â• 0 1. ^ h weakly in L^(f 2 x R 7 -) for some h 2. weakly in LÂ‘^{Q, x RjÂ’) for some where h = ii^ and u'^(O) = Ug. For the proof see Zhou [30] or Temam [25]. Theorem 2.4.5 Suppose uq Â£ LÂ°Â°{Q) n BV{Q.), then there exists a unique u Â€ L^{[0,T]] BV{fl) n LÂ°Â°(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^^Â’Â‘^||LÂ°Â°(nxR 7 Â’) ^ ||^olUÂ°Â°(n) < C'(f2)||Mo||LÂ°Â°(n) (2.24) PAGE 27 20 and I \ii^Â’^\^dxdt -f / a^piy u^'^)dx J UxRtÂ’ Ju < 1 [ \Vul\Â‘^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^Â’'||By(nxR^) < C|h-' with C Â— 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 ||'wÂ‘^||LÂ°Â°(nxRT) ^ C'(^)lko|UÂ°Â°(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 Â— > u'^ in L^(Q X R^) as e Â— >Â• 0 (2.28) (2.29) / \u^'^ Â— HxRtÂ’ dxdt < C(n)||uo||LÂ°Â°(u) f \u^Â’^ Â— u^\ dxdt t/s7 X Rt by (2.24) and (2.27). PAGE 28 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'^Â’^{v Â— u^Â’^)dxdt + ^ f f \Vv\^dx+ f f aip'^{Vv)dxdt ^ Jo Jn Jo Jn > / /* \Vu^Â’^\Â‘^dxdt + f f aip^{Vu^'^)dxdt ^ Jo Jn Jo Jn Â— f f {u^Â’^ Â— Uq){v Â— u^Â’^)dxdt Jo Jn > j I aip^ (Vu^Â’^)dxdt Â— ^ f [ ~ ul){v Â— u^'^)dxdt Jo Jn Jo Jn for all V Â€ L^([0,T]; Now let e -> 0 in the above inequality to arrive at n ii^{v Â— u^)dxdt + f f a(p{Vv) ! Jo Jn >liminf^y J a(p^(Vu^Â’^)dxdt ~ (3 J J Â— Uq){v Â— u^Â’^)dxdt^ > f f aip{Vu^)dt Â— P f [ {u^ Â— Uq){v Â— 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^Â’^)dx for a.e. t G [0,T]. J Q, t/ (2.30) Thus letting e Â— )Â• 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), Â— > w in L^{Q) for a.e. t G [0,f], and ^ Â« in L^(0 X Rr). (2.31) PAGE 29 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^ Â— uo)(v Â— 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 Â— Uq){v Â— 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 LÂ°Â°([0,T];BV(n)nLÂ°Â°(Q)). To prove uniqueness, consider two weak solutions Ui, U 2 to (2.3)-(2.5) with Ui(0) = Â«2(0) = Uq. We have the two inequalities n u\{u2 Ui)dxdt + / / aip{Vu2)dt > a(f{Vui)dt ! Jo Jn Jo Jn Â— / P{ui Â— Uq){u 2 Â— 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 Â— ui){ui Â— U2)dxdt > / / /3{ui Â— U2)^dxdt. 1 Jo Jn And hence giving [ ^ / (^1 U2fdxdt < [ [ !J{ui U2fÂ‘dxdt < 0 Jo dt Jq Jq ||ui(-, s) U 2 {-, s)||l 2 ( 0 ) = 0 for Â£-a.e. s. Therefore Ui = U 2 D PAGE 30 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 Â— 'i^2o||LÂ«>(n)Proof: Let M = ||uio Â— U 2 o||LÂ°Â°(n)For Â£-a.e. t we have / Ui{v Â— Ui)dx ~\/ aip{Vv)> / aip{Vui)Â— / ^{u\ Â— uiq){v Â— Ui)dx (2.32) Jri Jn Jn Jn and / U 2 {v Â— U 2 )dx + / acp{Vv)> / aip{'Vu 2 ) Â— / I3{u Â— U 2 o){v Â— U 2 )dx. (2.33) Jn Jn Jn Jn Define = u{ Â— {u\ Â— ^2 Â— M)_|_ and = U2 + {u\ Â— U2 Â— 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^ Â— ui)dx+ / aifCVv^) > / aip(Vui) Â— / ^{ui Â— uiq){v^ Â— ui)dx Jn Jn Jn Jn and / U2{w^ Â— U2)dx + / ai^[Vw^)> / a^p{Vu2)Â— / ^{u2 Â— U2 q){w^ Â— U2)dx. Jn Jn Jn Jn Now add the above two inequalities to get / ui{v^ Â— ui)dx+ / U2{w^ Â— U2)dx > / a^p{Vui)+ / a(p(Vu2) Jn Jn Jn Jn PAGE 31 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\ Â— {Ui Â— U2 Â— M)^ and (2.34) (2.35) W = U2 + {Ui U2~ M) + we see after letting e Â— > 0 that I Ui{v Â— Ui)dx + / U 2 {w Â— U 2 )dx > Â— I P{ui Â— Uio){v Â— 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 Â— U\)dx + / U2{w Â— U2)dx > 0 . Jn Jn Hence, using equalities (2.34) and (2.35) for v and to in the above and combining, / (ui Â— U 2 ){ui Â— U 2 Â— M)+dx < 0 Jn which implies / |(tii Â— U2 Â— M)+pdx < 0 Jn A dt and therefore / |(tti Â— ti2 Â— M)+|^dx < / \{uio Â— U 20 Â— M)^\^dx = 0 Jn Jn by the choice of M. Similarly U\ Â— U 2 > Â—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 LÂ‘^{[0,T]\ BV{il)), then (2.1Q) holds for all v G LÂ‘^{[0,T]; a.e. t. Then by using YoungÂ’s inequality, inequality (2.1Q) implies )dx [ f / (^ Jn ^ Jn uo)Â‘^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 Â— ) oo: Theorem 2.5.3 The solution u to (2.3) weakly converges in L?Â‘{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 Â— Uo)^dx. Jn 2 Jq Second, we recall the definition of the subdifferential of a proper convex functional G : H (Â— 00 , 00 ] where H is a Hilbert space with inner product < Â•, Â• >: the subdifferential of G at u, written dG[u], is defined as dG[u] = [w G H\G{v) > G{u)+ < w,v Â— 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 Â— 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 (/?://Â—) (Â— 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) Â— > 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 Â—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 LÂ‘^{n)\BV{n) X Â— 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) Â— ^ Uqo weakly in L^(fl) as t Â— > oo and Uqo a minimizer of F in Z/^(f]). By uniqueness, Uoo Â€ 1/^(0) fl BV{Q). Since u{t) is uniformly bounded in BV{^), we may also conclude that any sequence {u(tÂ„)} has a convergent subsequence still denoted by {u(tÂ„)} converging to u^o strongly in L^(0). Hence u{t) Â— >Â• 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)||LÂ°Â°(n) < C'(f^)||^to||LÂ°Â°{n) for a.e t G [0,Tj. Hence by compactness, we can extract a subsequence {-{/(tn)} such that u{tn) Â— > u in L^{fl) and u(tÂ„) Â— ) u a.e. on fl. Thus u G LÂ°Â°(fi) with ll'a||LÂ°Â°(n) < C'(fl)||?^o|lLÂ°Â°(n)-

PAGE 34

27 2.6 Numerical Results For the numerical experiments we approximate (2.3) with du = div(o;(^p(Vu)) Â— P{u Â— 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 Â• Va Â• 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 Â• = 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%Wâ€¢ 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 ^ Â— 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 Â‘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 Â— u, use convexity of cp, and integrate to get Â“^kGi^iy x)wi{y)\Vu\^ dx^ {v u) + j ap{Vv) > / a(/?(Vu) -h / P{u Â— Uo){v Â— 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 Â— a |Vu| Jn Jn Jn where dc^{x) Â— max(d(a:), 0) and a (x) = Â— min(o;(x), 0). One can easily verify that if u G then / a\Vu\dx= / oi^\Vu\dxÂ— / 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 LÂ°Â°(r2) and u G BV{Q), then / q;|Vm| < oo and Jn [ Â«|V Jn u\ < ||q;| LÂ°Â°{n) f |VÂ«|. 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)) Â— P{u Â— 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 Â— uq)^ dx Jn Jn and let {Â«Â„} be a minimizing sequence for F. Then ||uÂ„||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;Â„ > <5 > 0 for some 5 depending only on 0, a. Thus F{Un)> / (5|VuÂ„l+ / {Un~ Uofdx > 5 / iVuÂ„ 7n JQ. Jn Therefore / |VuÂ„| < M and by compactness there exists a subsequence which we Jn still denote by {uÂ„} and a u in BV{Q.) fl i/(D), for any 1 < p < 2, such that Un u in UÂ’{Q). Let a{x) Since Â— > 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;Â„(a:)| < 1 for every x E G,. So o;Â„ Â— >Â• o; in I/^(D) for any 1 < p < oo by LebesgueÂ’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*uÂ„|^) 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 LebesgueÂ’s Dominated Convergence Theorem, > a in for any 1 < g < oo. By definition, / anlVunl = sup < UÂ„div(0)dx : |0i(x)| < an{x) V(a;) Jn <^>ec^(n)" Un e 0 = sup < / Un{Van{x) Â• (j)) + an{x)div{(f)))dx : $$f)i{x)\ < 1 V(a;) Â€ 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!Â„ Â• 0 + o;Â„div(0))dx in in = liminf / uÂ„div(o;Â„)dx n->oo < liminf / 0 !Â„|VuÂ„|. 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(uÂ„) = liminf ( / o;Â„|VuÂ„| + {un-Uo)^dx Vin in y > liminf / aÂ„| VuÂ„| + lim inf / {un Â— Uofdx n^oo n-)-oo > / q;|Vu| + (u Â— Uo)^dx. in in liminf / o;Â„|Vu^ n-Fcx) In The last inequality follows by convexity of the second integral and by the fact that we can also choose {Â«Â„} to converge weakly to u in L^(D). Thus F has a minimizer in L2(Q) nBC(D). PAGE 45 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 Â— |) -b ^ f (u Â— lYdx. BV{n)nL^{u) 2e J\yu\<( J\vu\>e 2 ^ Jn with given corrupted image I Â€ 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) + Â—{u Â— 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. PAGE 46 39 The problem is min [ Jn where lim|vÂ«|-).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. PAGE 47 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 LÂ°Â°(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 PAGE 48 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 RÂ”, 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 Â— p, on Br/ 2 {o) and u solves Â—Au = I Â— u on Br/ 2 {a). Hence u Â€ C^Â’Â°Â‘{Br /2 (a)) for any a < 1. Theorem 3.1.2 Let u be as in Theorem (3.1.1/ 7/Â£"({|Vu| < 1}) > 0, then there exists a nonempty open region E on which u is |Vr/| < 1 and u solves Â—Au = I Â— u on E. PAGE 49 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 Â— I{x))Â‘^ 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) Â— / fudx > uÂ€BV{n)nL^Q) j for p as above and / E LÂ°Â° . 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 LÂ°Â°{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 LÂ°Â° 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 LÂ°Â°(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 LÂ°Â°{D). In fact, we have ||Â«||LÂ°Â°(n) < ll^||LÂ°Â°(n)- PAGE 50 43 Proof: Let be defined on RÂ” by ^e{p) IpI if bl > 1, and consider the minimization problem min I [ (Pf(Vu) + [ (u Â— 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^ Â— I)Â‘^dx< / {t Â— I^dx. J {Ue>t} J {Uc>t} J {Ue>t} (3.2) Hence f (Ue Â— I)^dx < [ {t Â— ly J {uc>t) J {ue>i\ dx. Hue>t} J {Ue>t) But setting t = ||/||LÂ°Â°(n) we see that if ess sup u^> t then ^dx which contradicts the above, hence ess sup < ||/||LÂ°Â°(n)Applying a similar argument to u = max(Uf, Â— t) for t = ||/||LÂ°Â°{n) we get ess inf > Â— ||/||Loo(n) and thus |b6||LÂ°Â°(n) < ||L|UÂ°Â°{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 PAGE 51 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 Â— > 0 in (3.2), noting that (/?Â£->Â• / v?(Vu) and ^ u in Jn Jn L^(Q), and since v G LÂ‘^{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 LÂ°Â° bound for and the convergence of Uj to u a.e. in Q we have u G LÂ°Â°{Q) with ||n||Loo(n) < ][/[|LÂ°Â°(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||LÂ«>(n) < PI|LÂ«>(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 Â— /X, Br{a) PAGE 52 45 then / ip{Du) Â— / (p{Vh)dx >11 \D^u\ + V{u Â— 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) Â— v[x) Â— I Â• 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 Â— ^1 = sup iVn^l < , Br(a) Br{a) sup \vp Â— v\ = sup \vp Â— v\(n) < ||^||LÂ“Â°(n), (3.4) le^I PAGE 53 46 a.e. in Bf{a), and 4 Â— > / in L'^{Bf{a)). There exists a smooth solution ([14]) to Â—Aw^ = Â— We on Bf{a) (3.5) We = vp on dBf{a) with the estimate ||n;e||/,oo(B,-(a)) < ||^Â’^|UÂ°Â°(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 Â— > 0 in the above estimate for We and the lemma is proved. We now have, using the above lemma, the following estimates ([1, 14]): ||'^||LÂ°Â°(B;(a)) < \\vp\\LÂ°Â°{dBf(a)) + C'(?^) ^^) ||Ll|LÂ°Â°(n) , sup r^\x Â— y\~^\Vw{x) Â— Vu;(y)| + sup |Vru Â— /| (3.6) Bf[a) Bf{a) < + r\\I wWi^i^Bfia))), |Vtu(3:) Â— Vw{y)\ x,yeBf/2{a) \x-y\^ Note that from (3.3) we also have \\I -w\\Loo(^Bf{a)) < ll^^||LÂ°Â°(aSj=(a))+C'(?^, L!)||/||Loo(n). Lemma 3.2.4 Suppose there is a v E C^Â’^{B 2 r{a)) and I G RÂ” with |/) < 1 Â— 2/u, sup I Vu Â— /| < B2t{o) and sup^ 2 ^(Â„) |n| < CÂ„ where Cu is a constant depending only on ||u||Â£,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 Â— d'W' ^ JBfia) JBf{a) J dBf{a) f {u Â— w){I Â— w)dx -\p, f \D%\ + ^ f \Wu\dx J Bf{a) JBf{a) 2 J Bf{a)n{\Vu>l} PAGE 54 47 I [ \V{u-w)\Â‘^dx > [ {u-vp)^dH^ ^ f {wIfdx [ {u-I) JdBf{a) C'n I JBf{a) ^ J Bf {a) dx r(Â“) +n [ \D^u\ + f^ [ \Vu\dx+^ f JBf(a) ^ ./Br(a)n{|Vu|>l} ^ J Bi |V(u Â— w)\'^dx 'Bf{a) ^ JBr(a)n{|Vu|>l} ^ JSf(a)n{|Vu|(n))) + 1 Â— 2^. Choose C 5 and ce such that < C 5 and r{Cu + C{n, i7)||/||x,oo(f^)) < Ce imply C 3 (/ 5 Â‘* + /(Cu + C(n, f2)||/||LÂ°Â°(n))) < fJ>Thus sup |Vu;| < 1 Â— /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 YoungÂ’s inequality for {u Â— w){I Â— w) Â— Â—{u Â— I){w Â— /) + (/ Â— 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| = Â— j {u Â— I)Qdx fc/ 57 */ 57 Â«/ 57 (3.7) PAGE 55 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 Â£Â” 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, = Â— j {u Â— I)C,dx Jn holds for arbitrary ^ Â£ 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) Â— t^(Vu) >V{v Â— u) ipp{Vu) on f7a, and that on Dg we have " , , J / \D^v\Jn, Ji \D^u\ > / rts ^ Â• DÂ‘Â‘lv Â— u) Â— i>. \D-n\ Next, define the functional = [ (p{Dv) f J Br(a) J dBr(a o :v dn (a) a\ Â“ -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). PAGE 56 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(CÂ„ + 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)\Â‘^dx < ct f \u Â— v\ d'HÂ”'~^ + 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 Â“ Â— Â—Au Â— VR)dTr~^= [ cr Â• -r ^Au Â— w) dW^~^ JdBf{a) P ~ JdBf[a) P Â“ > f ip{Du) + 1: [ {u Â— lYdx Â— I cp{Vw)dx Â— ^ / {w Â— 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 Â— 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\\LÂ°Â°{n)) < and v Â€ C^Â’^{B 2 r{a)) with sup |Vu| < 1 Â— 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 Â— v\ LÂ°Â°(Br(a)) < Cio (>C"({u ^v}n 52r(a)))" Proof: First we note that the function p satisfies |p| Â— 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 Â— v)a D{u Â— v) + j 9'{u Â— v)a-Dv J Bp{a) J Bp(a) + f (A + 1)9' {u v) JBp{a) < f rja D[9{u Â— u)] + f C\9'{u Â— 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 Â— V) ' Bh{a) + \\u For 0 < A: < s we choose 9 as \9{u Â— v)\dx. Bh(a) 0 for t < A: 9{t) = '^ t Â— k lor k < t < s s Â— k for t > s. Now let ^(A:, h) = Bh C\ {u Â— v > k}. Clearly support [r]9{u Â— v)] C ^(A:, h). Thus [ |TÂ»[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 nÂ— 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 Â— A;)|^(s, p)| " < Cu(h Â— p) ^ I \9{u Â— v)\dx + Cu\A{k,h)\ JBh(a)

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52 And since f \6{u Â— v)\dx < {s Â— 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 Â— n is obtained by using a similar argument for 0 < k < s < oo, 0 for t > Â—k m = -t Â— k for Â— s < t < Â— A: s Â— k for A < Â— s, and A{k, h) = Bh ^ {u Â— v < Â—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\< Ci6Â‘J>(4r, Ai, a) 2 + ciyr. PAGE 60 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 )-\ Â— [ (u Â— I)^dx\ Â«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 LÂ°Â°{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+ / \DÂ‘ 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)^/^ ^(pÂ— 1 Â— e) if jx] > 1, P (3.9) PAGE 61 54 where 7 C R", I, and p, are as before. As above, We note that for e > 0, (/?Â£ 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, PAGE 62 55 Now define the following functional on BV{Q): J(m) = sup < Â— / u div 4> + dx> 0Â€V [ JQ J = sup / Vu 4> Â— 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 Â— 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 Â• 02 > (p^{Vu)d: Jn Jn X Â— e. (3.12) Now define 01 on 0 = 02 on fl Â— Oe, Let Tja be the standard mollifier on R" and let 0^ = T]a* / Vu 0a Â• 0 in the above inequality we then have J{u) > I Vu (f) Â— ip*{^)dx + f PAGE 63 56 > f \/u (f )2 ip*{(j) 2 )dx + f (j)i D% fi{e) Jn Jn > / ip^(Vu)dx+ / \D^u\ Â— fj,{e) Â— 2e Jq Jn where /^(e) = [ \^u\dx+\\(t>*\\LÂ°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 LÂ‘^{Q). Then for any e> 0, there exists a sequence of functions {un} C BV{fl) fi L^(f2) n CÂ°Â°{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 PAGE 64 57 0, there exists a unique solution u G LÂ°Â°{Q.) to problem 3.9. In fact we have ||u||/,(n) < ||/||LÂ«>(n)Proof: The proof is standard using lower semicontinuity and convexity. The LÂ°Â° 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 Â£Â”({|Vu| < 1}) > 0, then there exists a nonempty open region on which u is |Vrt| < 1, and u solves Â— div{(pp{Vu)) = I Â— 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 LÂ‘^{[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) PAGE 65 58 with sup^; \Vh p\ < Cia, then for any vector pi Â€ -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^BÂ„(p)} P\+ [ J El \D{u-h)\'^ , ' En{DueBÂ„{p)} where ci; : R Â— > R is a non-decreasing, nonnegative function with limt^o^(0 = 0Recall that L En{DueBÂ„(p)} \D{u h)p -I[ \Du-p\ J En{Du^B^{p)} means f ~ h)\Â‘^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 Â— p| < Cicr and d'^h PhijiPi) dxidxj I Â— 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 YoungÂ’s inequality for {u Â— h){I Â— h) = Â—{u Â— I){h Â— /) + (/ Â— the lemma is proved. The energy functional for this problem is given by $(r,p, a) = -^ ( / \Du-p\+ \Du-pi \^r\ \J Br{a)n{DuiBÂ„{p)} J Br{a)n{DuÂ€B^{p)} Our next goal is to prove Theorem 3.3.7 There exist constants Ci and 62 depending on p and p such that Â«/$(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 Â”aÂ” in Br{a) and in $unless noted otherwise. In order to prove Theorem 3.3.7 we will obtain estimates for / \u Â— vp \ dTF~^, \Vh Â— pi\, and JdE / I V/i Â— pipdx for a suitable Lipschitz function v. These will then be used to prove Je a decay estimate for$ 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 Â— Pi| < Bt/2 with S such that (1 = 1 + that is 5 = 777 -^Â— 77 . In addition we have n 2 n 8(n + l) C^{Br/2 F{u^ t;}) < C4rÂ”$(r,pi)^"Â‘Â‘Â‘^. PAGE 67 60 From the above lemma we get an LÂ°Â° estimate for u Â— u. Lemma 3.3.9 If \pi Â— p| < Cicr, then there exists an e > 0 such that$(r,pi) < e implies sup I Vn Â— Pi I < $(r, pi) 25 B r/2 for V defined as above with 5 = ciently small: 1 8(n + 1) and with the following estimate for r suffi/ 1 \ 1/n ^5 ( f Br/ 2 ] C5^(Â£"({u^r;}n5,/2)'/". Proof: This follows from the estimate of Â£Â”(5^/2 Fl 7^ n}) in Lemma 3.3.8, the bound |Vn| < |p| + cicr + (r, pi)^^, and from Lemma 3.2.8 where we use the above bound for |Vw| in that proof instead of the bound |V?;| < 1 used there. We can now estimate |V/i Â— pi| on E for any f < r j2 for the solution h to (3.14). Let be the smoothing of v as described in the previous section with all the accompanying estimates. Then from the linear theory ([14]) as in the last section, taking /? =$(r,pi), we have ||/i||lÂ°Â°(Â£:) < Ce (see Lemma 3.2.3) and sup|V/i-pi| < C7(4>(r,pi)Â‘^ + r||/i/|| loc(Â£;)) < C8($(r,pi)'^ + r). E Let T : fl ^ RÂ” be an appropriate transformation such that \ih = ho u = uo TÂ“\ vg = v /3 o T~^, i = I o E is an ellipse centered at a, T{E) = B' is a ball of radius f centered at T{a) and Â— Ah = I Â— h on B' h = vg on dB' . (3.15) We note that the Jacobian of T, which depends only on the eigenvalues of the matrix [Pii/;(Pi)j) is bounded from above and away from 0. Furthermore these bounds can PAGE 68 61 be made independent of a and pi due to uniform ellipticity. We have ([14]) \Vh{x)-Vh{y)\ ^ C 9 sup B < \x yjl/2 ^n+1/2 [ \v0\dn^-^ + f\\h-i\\L. JdB' where B C B' is concentric with B' with radius f/2. Choose E C Brf 4 centered at a with diamE = r/8 and such that both [ \u Â— v\ d'H" ^ < Â— [ \u Â— v\dx, JdE f Jb^/4^ and [ \uub.^ Pi {x a)\d'W ^ < Â— [ \u ub.^ Pi {x a)\dx (3.16) JdE f JB^/i hold, where Ub^^ denotes the average of u over Sr/ 4 Lemma 3.3.10 Let v be as in Lemma 3.3.8 with its smoothing vg. Then [ \v^ u\ dU^-^ < CurÂ”((r,pi)). J E Proof: Multiply (3.14) by h Â— vp, integrate by parts, use the fact that (3.14) is a linear equation with constant coefficients, and then use YoungÂ’s inequality to arrive at / |V/i Â— pipdx < Ci 2 / {h Â— I){h Â— vp)dx + Ci 3 / \Vvp Â— pi\^dx. Je Je Je PAGE 69 62 By the uniform bounds of V/i and we see that \\h-vp\\L Â°Â°{E) < cu{diam E) < C\^r. For other part of the estimate we use ([24]) / I Vu /3 Â— < Ci6 / \VvÂ—pi\^dx. Finally, to estimate / |Vu Â—pi^dx, the construction of v gives J B^/2 / \Vv -pi\^dx u^B^{p)} Combining the above estimates proves the lemma. We now arrive at our decay estimate. Theorem 3.3.12 There exist positive constants e, C20, and k depending only on n, n, and u such that if ^{r,pi) < e and r < C20, then there exists p2 Â€ R" such that ^{Kr,p2) < ^^{r,pi) + C3ir and \p2 pi\ < C35$(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{DueBÂ„{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;) Â— Vh(a)| over a ball B^r C E C E. Since h Â— ub^^^ Â— p\ Â• {x Â— 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"Â‘ (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 PoincareÂ’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 Â• (^ +C24r-^/^||/i I\\lÂ°-{e) ^:;:^[ [ IDu-pildx + c^^r^^^hi\\lc.^e)Let E = T~^{B) and restrict k as necessary so that 5Â«;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:) Â— V/i(y)| < C 2 e ( Â— Â— / \v ^ Â— u\ dl-E~^ -\ Â— / \Du Â— 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,piYlÂ‘^ -^r\\h~ I\\l^^e))This is our desired estimate for |Vh(x) Â— Vh{y)\. Now use the inequality | Vu Â— Vhp > ^|Vu Â— P 2 p Â— |V/i Â— P 2 p we arrive at [ \Du-p\+ f |Du-p 2 p J BKrn{Du^BÂ„{p)} J B^r<^{DueBÂ„(p)} < C29(r"$(r,pi)^+^/(^'Â“) +ta(c8((r,pi)'^ + r))V"Â‘^^||/i /||lÂ«>(e)) +C29(o;(c8($(r,pi)Â‘^ + r))^r"$(r,pi) + f \Vh p2\Â‘^dx). J B^r For the rest of the proof we denote$(r, pi) by $. Using the estimate for \Vh{x) Â— Vh(p)|, recalling that p 2 = Vh{a), and dividing the above inequality by we have$(ACr,P2) < C30K "$^+V(2n) "o;(c8'h'^ + C8r)^ PAGE 71 64 Restrict k again so that k < . Then restrict <4> and r so that + 4c3o C3oÂ«:Â“"'a;(c8<4''* + c^rY < 1/4. This proves the decay estimate for$. Finally we derive the estimate for \p 2 Â— Pi|From the linear theory ([14]) as applied to h we have \P2 -Pil = |V/i(a) -pij < 0327^7 / \vp -pi Â• (x a) -UB'\dW~^ \^\ JdE +C33f||h Â— < C32t47 [ \v^ u\drr~'^ + ^32]^ I \u-pi{x-a) UE\d'H'^~'^ \^\ JdE rI JdE +C34r||/l Â— Then using, the boundary estimates, PoincareÂ’s inequality, and HolderÂ’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 Â— 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 Â— h)\Â‘^dx. {a)n{|VÂ«|
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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) Â— ipo{Vh) Â— V{u Â— h) Vh |Vh|2-p |Vu|-^ Â— ^--|Vh|PV(M-h) p p |Vh|2-p iv^i ^ + ivhr p p |Vh|2-p > |Vm| + ?~~\Vh\P |Vu||Vhr\ j) Â— 2) 1 Now let g{x) = a 1 Â— Â— ax^~^ for a = |Vn| and x G [0, 1]. Then since P P a > 1, g{x) Â— a{l Â— x^ ^) Â— a{Â— Â— ^x^) > a{+ ^^xP -xP-^). P 1 J) Â— 1 Letting g{x) = a{H x^ Â— x^^^) we easily see that g is strictly decreasing on [0, 1] with ^(0) = 1/p and ^(1) = 0. Now let /(/r) = ^(1 Â— /r). Where |Vu| < 1 there holds iPi{Vu) Â— <^c(Vh) Â— V(u Â— h) Vh > c|V(m Â— 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 Â— > oo we get yjo(Vn) cpoi'^h) V(u h) V/i |Vh|2-P > c|V(n Â— /i)p For the singular measure we see / |D-Â«| > / DÂ‘u--^+f |D*Â«|(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 Â— 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^Â’Â°Â‘ 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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