Analysis of adaptive forward-backwarddiffusion flows with applications in imageprocessing

V B Surya Prasath1,3, José Miguel Urbano2 andDmitry Vorotnikov2

1Department of Computer Science, University of Missouri-Columbia, MO65211, USA2CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra,Portugal

E-mail: prasaths@missouri.edu, jmurb@mat.uc.pt and mitvorot@mat.uc.pt

Received 5 March 2015, revised 2 August 2015Accepted for publication 26 August 2015Published 24 September 2015

AbstractThe nonlinear diffusion model introduced by Perona and Malik (1990 IEEETrans. Pattern Anal. Mach. Intell. 12 629–39) is well suited to preserve salientedges while restoring noisy images. This model overcomes well-known edgesmearing effects of the heat equation by using a gradient dependent diffusionfunction. Despite providing better denoizing results, the analysis of the PMscheme is difficult due to the forward-backward nature of the diffusion flow.We study a related adaptive forward-backward diffusion equation which uses amollified inverse gradient term engrafted in the diffusion term of a generalnonlinear parabolic equation. We prove a series of existence, uniqueness andregularity results for viscosity, weak and dissipative solutions for such for-ward-backward diffusion flows. In particular, we introduce a novel functionalframework for wellposedness of flows of total variation type. A set of syn-thetic and real image processing examples are used to illustrate the propertiesand advantages of the proposed adaptive forward-backward diffusion flows.

Keywords: anisotropic diffusion, regularization, image restoration, forwardbackward diffusion, wellposedness, total variation flow

Inverse Problems

Inverse Problems 31 (2015) 105008 (30pp) doi:10.1088/0266-5611/31/10/105008

3 Author to whom any correspondence should be addressed.

0266-5611/15/105008+30$33.00 © 2015 IOP Publishing Ltd Printed in the UK 1

1. Introduction

The nonlinear diffusion model, introduced in image processing by Perona and Malik [55],involves solving the following initial-boundary value problem

�

O

ss

� � � 8 q

ss

� s8 q

� 8

⎧

⎨⎪⎪

⎩⎪⎪

u x tt

u x t u x t T

u x tT

u x u x

,div , , in 0, ,

,0 on 0, ,

, 0 in ,

1

2

0

( )( )( ) ∣ ( )∣ ( ) ( )( ) ( )

( ) ( )

( )

where �8 lu :0 is the observed (noisy) image, �8 � 2 is a bounded domain withLipschitz boundary. The function � � �l� �: 0 0 is non-increasing such that � �0 1( ) and

� �l�d slim 0.s ( ) Note if � ws 1( ) then we recover the heat equation. The diffusioncoefficient function �( · ) in equation (1) is an edge indicator function that reduces the amountof diffusion near edges and behaves locally as inverse heat equation. The original choices of�( · ) by Perona and Malik [55] are

� �� � ��

⎜ ⎟⎛⎝

⎞⎠s

sK

ss K

exp ,1

1, 21 2 2 2

( ) ( ) ( )

where K > 0 is a tunable parameter also known as the contrast parameter [57].Despite impressive numerical results obtained in image processing using the Perona–

Malik equation (1) with � �� ,1 it was shown [43, 44] to be an ill-posed PDE due to thedegenerate behavior for large gradients. Thereby, although existence of infinitely manysolutions in a relaxed setup involving Young measures was established in [21, 77], manyauthors have been looking for regularizations of the Perona–Malik equations which inherit itsusefulness in image restoration but have better mathematical behavior. One of the pioneeringworks of this kind [17] replaced the magnitude of the image gradient � �s u used in thediffusivity functions by the spatially regularized gradient, � � �Ts G u , where G! denotesthe two- dimensional Gaussian kernel, QT T� �T

�G x x2 exp 21 2 2( ) ( ) ( ) and * denotes theconvolution operation. Thus, the Catté et al [17] Gaussian regularized anisotropic diffusionequation (GRADE for short) reads as

�ss

� � � �Tut

G u udiv . 32( )( ) ( )

This modification of equation (1) is sufficient to obtain the existence and uniqueness ofsolution to the initial-boundary value problem for GRADE in equation (3). However, thespace-invariant Gaussian smoothing inside the divergent term tends to push the edges awayfrom their original locations, see figure 1 for an illustration of this effect on a synthetic cornerimage. This effect, known as edge dislocation, can be detrimental to further image analysis.This can also be seen via the regularity of solution to GRADE, which belongs to a high orderSobolev space. Furthermore, the use of isotropic smoothing is against the principle ofanisotropic diffusion which aims to smooth homogeneous regions without affecting edges. Toremedy this, one can use time regularization instead of spatial regularization, or a related ideaof decoupling the diffusion coefficient into a separate evolutionary PDE [4, 11, 12, 53, 63].Another direction to improve the well-posedness of the Perona–Malik equation (1) is toapproximate the nonlinear diffusion coefficient �1 with less degenerate ones, see [34–36, 38].Fourth order regularizations were introduced and studied in a bulk of papers, for instance, in[13, 33, 39, 40]. We refer to the recent paper [37] for a nice and detailed overview on

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

2

anisotropic diffusions arising in image processing from the perspective of mathematicalanalysis.

Motivated by the correspondence between the variational and PDE methods for imagingproblems, which we discuss in the next section, in this paper we consider a Perona–Maliktype PDE with the generic diffusion function inspired by the stationary nonlinear regular-ization approach. Engrafting a mollified gradient within the adaptive diffusion function weobtain a general forward-backward diffusion PDE. We prove a series of existence, uniquenessand regularity results for viscosity, weak, strong and dissipative solutions for a wide class ofthe proposed generalized forward-backward diffusion models. Experimental results on syn-thetic, noisy standard test, and biomedical images are provided to illustrate different diffusionschemes considered here.

One of the highlights of the paper is the introduction of the concept of partial variation,which enables us to define and employ the Banach space of functions of bounded partialvariation. Our approach appears to be relevant in the context of evolutionary problems whichinvolve singular diffusion of 1-Laplacian kind or gradients of linear growth functionals, andholds promise for wide applicability.

The rest of the paper is organized as follows. Section 2 is devoted to the modellingissues. Section 3 examines the conditions needed for wellposedness of the proposed reg-ularization-inspired forward-backward PDE in various scenarios. In section 4, we providenumerical experiments to prove the effectiveness of the proposed multi-scale scheme as wellas examples for various cases.

Figure 1. Spatial regularization in the diffusion coefficient alters discontinuities in agiven image. (a) Original synthetic image of size q31 31, a square (gray value = 160)at the bottom right corner with uniform background (gray value = 219). (b) Inputimage obtained by adding Gaussian noise !n = 30 to the original image. This noisyimage is used as the initial value u0 for the nonlinear PDEs with �1 diffusion coefficientand K = 20. Results of PMADE (1) with 20 iterations in (left) image (right) surfaceformat (c), and GRADE (3) with 20 iterations in (left) image (right) surface format (d).The intersection of red dotted lines indicate the exact corner location of the square. (Forinterpretation of color in this figure, the reader is referred to the web version of thisarticle.)

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

3

2. Preliminary observations and the proposed model

The motivation for the Perona–Malik nonlinear diffusivity is that inside the regions where themagnitude of the gradient of u is weak, equation (1) acts like a heat equation, resulting inisotropic smoothing, whereas near the edges, where the magnitude of the gradient is large, thediffusion is ‘stopped’ and the edges are preserved.

To see the underlying details, we split the divergence term in equation (1),

� � �� � � � � a � � � �u u u u u u u u u u u u udiv 2 2 .

4

x xx y yy x y xy xx yy2 2 2 2 2( ) ( )( ) ( ) ( )∣ ∣ ∣ ∣ ∣ ∣ ( )

( )Considering the tangent , and normal & directions of the isophote lines, and setting

� � �� � as s s s2 , 5( ) ( ) ( ) ( )we infer

� � �,, &&� � � � � �u u u u u udiv . 62 2 2( )( ) ( ) ( )∣ ∣ ∣ ∣ ∣ ∣ ( )

We thus see that the Perona–Malik diffusion (1) is the sum of the tangential diffusionweighted by the function �( · ) plus the normal (transverse) diffusion weighted by thefunction � ,( · ) resp. Since smoothing with edge preservation is of paramount importance inimage processing, it is desirable to smooth more in the tangent direction than in the normaldirection. This can be translated to the condition

��

��

- -a�

ld ld

ss

lim 0, or equivalently, lims s

s12

. 7s s

( )( )

( )( ) ( )

For example, in the case of the power growth functions

� xs sq( )the above limit gives that

- �q12

. 8( )

Note that for the original diffusion function �2 in (2) we have � �s 02( ) if s < K2, implyingforward diffusion in the regions where the gradient magnitude of the image function is lessthan K, whereas � �s 02( ) if s > K2, yielding backward diffusion in the area where absolutevalues of the gradient are larger than K. The same is true for �1 with the threshold value� K2 21 . Thus, the PDE (1) promotes combined forward-backward diffusion flow, seefigure 2 for a comparison with the heat equation. The Perona–Malik anisotropic diffusionequation (PMADE for short) thus balances forward and backward diffusion regimes using atunable K, the contrast parameter [48, 57]. These two competing requirements constitute acommon theme in many PDE based image restoration models [16, 18, 19, 76]. Moreover,from equation (6) we see that equation (1) is a time dependent nonlinear diffusion equationwith preferential smoothing in the tangential direction , than normal & to edges. Thisproperty has been exploited in image processing and in particular in edge preserving imagerestoration [9].

The PDE models of Perona–Malik type have strong connections to variational energyminimization problems and this fact is exploited by many to design various diffusion func-tions [9, 60]. Following [71], consider the next minimization problem for image restoration:

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

4

¨ ¨C

B G� � � �8 8

E u u x u x x u x xmin2

d d . 9u

02( )( ) ( ) ( ) (∣ ( )∣) ( )

Here " � 0 is regularization parameter, # � 0 fidelity parameter, and � �G l �: is an evenfunction, which is called the regularization function. The a priori constraint on the solution isrepresented by the regularizing term G �u ,( ) and the shape of the regularization determinesthe qualitative properties of solutions [52]. The formal gradient flow associated with thefunctional E(u) is given by,

BG

Css

�a ��

� � �⎛⎝⎜

⎞⎠⎟

ut

uu

u u udiv . 100(∣ ∣)∣ ∣ ( ) ( )

Figure 2. Diffusion process for a simple synthetic image. (a) Original synthetic imageof size 31 ! 31, a square (2 ! 2, gray value = 1) at the center with uniform background(gray value = 0). (b) Input image obtained by adding Gaussian noise !n = 30 to theoriginal image. This noisy image is used as the initial value u0. (c) Diffusion coefficient�1 in (2), with K = 20. This acts as a discontinuity detector and stops the diffusionspread across edges. (d) Flux function � � �u u .1( ) · (e) Result of heat equation with20 iterations in (left) image (right) surface format. (f) Result of PMADE equation (1)with 20 iterations in (left) image (right) surface format. The white dotted lines indicatethe influence region at the center. (For interpretation of color in this figure, the reader isreferred to the web version of this article.)

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

5

We recall two primary choices used widely as regularization functions in various imageprocessing tasks.

• f(s) = s2: this corresponds to the classical Tikhonov regularization method [70]. In thiscase the Euler–Lagrange equation (written with artificial time evolution, see equation (10))is,

B Css

� % � �ut

u u u ,0( )

which is an isotropic diffusion equation and hence does not preserve edges, seefigure 2(e). This heat flow provides a linear scale space and has been widely used invarious computer vision tasks such as feature point detection and object identifica-tion [65].

• f (s) = s: to reduce the smoothing when the magnitude of the gradient is high, Rudin et al[66] introduced total variation (TV) based scheme by setting f (s) = s. In this case theEuler–Lagrange equation is written as (see equation (10)),

�

C

C

ss

���

� �

ss

��

� �� �

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

ut

uu

u u

ut

u

uu u

div , or

div , 11

0

20

∣ ∣ ( )

∣ ∣( ) ( )

where � � 0 is a small number added to avoid numerical instabilities in discreteimplementations. In [18], the existence and uniqueness of the TV minimization is provedin the space of functions of bounded variation (BV), and the corresponding gradient flowis treated in [7], see also [5, 6, 15, 27, 31, 45] and remark 10 below. But this global TVmodel suffers from staircasing and blocky effects in the restored image [52]. Also, sharpcorners will be rounded and thin features are removed under this regularization model. Tosee this, let DB be the indicator function of ��B N a bounded set with Lipschitzboundary. Then the TV term in the regularization functional (9), ¨ D� B is the perimeterof the set B. This shows that TV regularization penalizes edge lengths of an image. Notethat this TV diffusion PDE (11) is a borderline case of anisotropic diffusion PMADE inequation (1) with � � �s s ,1 2( ) a singular diffusion model.

Figure 3. Well known regularization functions and their qualitative shapes: (a)regularization (b) flux (c) diffusion coefficients. Convex functions: Tikhonov, totalvariation (TV). Non-convex functions: Perona–Malik regularizations f1-PM1, f2-PM2see equation (12). Normalized to [0, 1] for visualization.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

6

It is easy to see that the equation (10) almost coincides with the Perona–Malik aniso-tropic PDE (1) for �Ga �s s s,2( ) ( ) see [54], and the difference between the two equations isthe lower order term coming from the data fidelity in the regularization functional (9). Foredge preservation we need to work with functions f with at most linear growth at infinity, see(8). For example, the Perona–Malik diffusion coefficients (2), up to multiplicative constants,correspond to the following non-convex regularization functions (see figure 3(a), denoted asPM1 and PM2),

G G� � � ��

� � ��⎜ ⎟ ⎜ ⎟

⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟u x

u xK

u xu xK

1 exp , log 1 .

12

1

2

2

2

(∣ ( )∣) ∣ ( )∣ (∣ ( )∣) ∣ ( )∣

( )Figure 4 shows an experimental analysis of PMADE (1) against TV PDE (11) on a

synthetic image which consist of various circles with constant pixel values. This piecewiseconstant image represents a near ideal scenario and both the PMADE (T = 20, 100) and TV-PDE (T = 100, 200) results indicate over-smoothing and staircasing artifacts.

Several studies [56, 60–64, 68] have introduced spatially adaptive regularization func-tions to reduce staircasing/blocky artifacts created by the classical TV and PMADE schemes.Such adaptive methods can be written as an energy minimization of the following form (see,e.g., [68]),

¨ ¨C B� � � �8 8

E u u x u x x x u x xmin d d , 13u

02( )( ) ( ) ( ) ( )∣ ( )∣ ( )

where the function B ( · ) self adjusts itself according to an estimate of edge information fromeach pixel. Since we want to reduce the regularization/smoothing effect of (9) near edges,hence "(x) is chosen to be inversely proportional to the likelihood of the presence of an edge.For example, the original function proposed in [68] is,

Figure 4. Advantages and disadvantages of TV-PDE and PMADE (with �1 as thediffusion coefficient and K = 20) models on a synthetic piecewise constant Circlesimage. Noisy image is obtained by adding Gaussian noise !n = 30. i ii iii iv : weshow in each sub-figure (i) gray-scale image (ii) surface (pixel values as z values) (iii)level lines (only top 4 level lines are shown for clarity), and (iv) contour maps tohighlight jaggedness of level lines and staircasing artifacts. Better viewed online andzoomed in.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

7

��B �

� � ��

Tx

G u x1

, 0. 140

( ) ( ) ( )

The term in the denominator provide an estimate of edges from the input image u0 at scale !using the Gaussian kernel G! filtered gradients. Introduction of such a spatially adaptiveparameter, which self adjusts according to the smoothed gradient of the image, reduces theTV flow in homogenous regions thereby alleviating the staircasing problem. In [20], theexistence and uniqueness of the functional satisfying (13) is proved under the weighted TVnorm. An edge indicator function of the form (14) can also be introduced directly in the PDEof the form (10). For example, using it in the PMADE (1), we write adaptive PMADE as

�

�ss

��

� � ��

T

⎛⎝⎜⎜

⎞⎠⎟⎟u x t

t

u x t

G u xu x t

,div

,, . 15

2

0

( )( ) ∣ ( )∣( ) ( ) ( )

It is advantageous to use the current estimate image u in the edge indicator "(x) in (15)instead of the initial noise image u0, that is

�

�ss

��

� � ��

T

⎛⎝⎜⎜

⎞⎠⎟⎟u x t

t

u x t

G u x tu x t

,div

,

,, . 16

2( )( ) ∣ ( )∣( ) ( ) ( )

See figure 5 for an illustration of using adaptive weight function in the final restoration resultson a synthetic image with multiple objects. As can be seen using an adaptive edge indicatorkeeps the edges through higher iterations. Moreover, integration of two scales (�ucorresponds to scale ! = 0 and � �TG u to scale !) in one term, see equation (17) below,can regularize the boundaries of the level set of u0 while at the same time keeping morefeatures. Numerical experiments will support our claims about the advantage of interaction oftwo scales, see section 4. Also we use the nonlinear function f (see equation (9)) to control

Figure 5. Using updated edge indicator function results in better final restoration inPMADE (with �1 diffusion coefficient and K = 20). Shown here are the final restorationresults at iterations T = 20, 100 for non-adaptive PMADE (15) and adaptive PMADE(16). i ii iii iv : we show in each sub-figure (i) gray-scale image (ii) edge map (heatcolormap) (iii) close-up gray-scale image and (iv) close-up edge map. Note that thesmoothness parameter ò = 10!6, is used in this example, see equation (14). Betterviewed online and zoomed in.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

8

the growth adaptively as mentioned. Thus the general multi-scale minimization problem nowreads as

�¨ ¨CG

� � ��

� � �T8 8E u u x u x x

u xG u x

xmin d d . 17u

02( )( ) ( ) ( ) ( ( ))

( ) ( )

A well known method to prove the existence and uniqueness of minimizer to this problem(17) is to obtain the lower semicontinuity of the functional E using the properties ofregularizing function f.

Remark 1. The functional (17) with f("u) replaced by � �TG u ,2 and with additionalquadratic regularization term ( ¨I �

8u xd2 ), which is related to robust Geman–Mclure

model [30], was considered in [41]. The arguments in [41] can also be extended to the generalminimization problem (17).

Motivated by the regularization functional (17) and previous discussions, we considerhere a general forward-backward PDE of the following form4,

Kss

�� �

� � �T

⎛⎝⎜⎜

⎞⎠⎟⎟u

t

x u u

K g G udiv

,

1. 18y

( )( ∣ ∣)

( )

Here Ky is the partial derivative of K x y,( ) with respect to the second variable � �y u . Thevariational problem could involve explicit dependence on the function u in the regularizationterm, G G� �x u x u x, , ,( ( ) ( )) with the corresponding changes in the evolutionary problem,but we will not study this general case in this article. For the existence and uniqueness of asolution u, we need additional assumptions on j which will be discussed at the end of thissection and in section 3. Observe also that, in the x- and u-independent case, f and j arerelated through G Ka � as s s .( ) ( )

Remark 2. The fidelity parameter # in (17) can be tuned to fit the the problem at hand[32, 64], i.e.,

¨ C� � �8

D u u x u x u x x p, d , 1, 2.p0 0( )( ) ( ) ( ) ( )

The major questions we are now concerned with are:

(1) What are the conditions on j to obtain existence of solutions for the PDE (18)?(2) What are the admissible inverse mollifier g functions?

The answer to the first question depends on the answer to the second. Let us brieflydiscuss this issue; more details will be given in the next section. If g is merely continuous,then we admit power growth, e.g., K �x y y, ,p( ) � � �dp1 , and logarithmic growthK �x y y, ln .( ) If g is locally Lipschitz, then we need a sort of strong parabolicity conditioninvolving j. If the derivative of g is sub-linear near zero (that is, g may be of order s p, p � 2,for small s), then K x y,( ) enjoys a wide range of possibilities with minor restrictions such ascoercivity and weak parabolicity.

4 We will omit the image fidelity term for brevity, since this lower order term does not cause any special difficultiesin the mathematical analysis of the model.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

9

3. Existence of various types of solutions

Here we study the equation

Kss

�� �

� � �

⎛⎝⎜

⎞⎠⎟

ut

x u u

K g G udiv

,

1, 19y ( ∣ ∣)

(∣ ∣) ( )

which is slightly more general than equation (18) in the sense that we admit arbitrary spacedimension n and generic convolution kernels G.

Throughout this section, we employ Einstein!s summation convention. The inner productin � ,n ��n , is denoted by a dot. The symbols "C E; ,( ) "C E; ,w ( ) "L E;2 ( ) etc denote thespaces of continuous, weakly continuous, quadratically integrable etc functions on an interval" �� with values in a Banach space E. We recall that a function " lu E: is weaklycontinuous if for any linear continuous functional g on E the function " �lg u :( ( · )) iscontinuous.

We are going to introduce three different concepts of generalized solution toequation (19): viscosity, weak and dissipative. The relation between different kinds ofsolution is not an issue here, since our goal is to construct at least one kind of solution for thewidest possible range of assumptions on j. The interrelation question is very delicate, but it istrue that a strong, regular solution, if it exists, is also a viscosity, dissipative, or weak solution.Moreover, in these circumstances, no other viscosity or dissipative solution may exist, but aweak solution might. On the other hand, if there are no strong solutions, then these classes ofsolutions intersect but do not coincide, e.g., there might exist weak solutions which areneither viscosity nor dissipative. Under additional assumptions, one can prove that a weaksolution is also dissipative. Roughly speaking, dissipative is the weakest notion of the three,and viscosity is the strongest. Nonetheless, in the logarithmic case, only weak solutions areproven to exist, and for the infinity-Laplacian flow (which however does not fit into ourframework here) we only know that viscosity solutions exist.

3.1. Viscosity solutions

We denote

K E K� �a x p x p x pp p

p, , , , 20ij y ij yy

i j( ) ( ∣ ∣) ( ∣ ∣) ∣ ∣ ( )

��

h qKg q1

1. 21( ) (∣ ∣)

( )

Here $ij is Kronecker!s delta, and ��x p q, , .n In this subsection we consider the case ofspatially periodic boundary conditions [2] for equation (19). Namely, we assume that there isan orthogonal basis {bi} in �n so that

�� � � �u x u x b x i n, , , , 1 ,..., . 22in( )( · ) · ( )

The problem is complemented with the initial condition

�u x u x0, , 230( ) ( ) ( )

where ��x ,n and u0 is Lipschitz and satisfies (22). Of course, j (and thus a) should alsosatisfy the same spatial periodicity restriction (with respect to x).

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

10

We also make the following assumptions:

K K K, , are continuous and bounded functions, 24y yx yxx ( )

K K v, are continuous for y 0, and satisfy, 25yy yyx ( )

�K K� �

� ly x y x ysup lim , , 0, 26

x yyy yyx

0n( )∣ ∣ ( ) ( ) ( )

. �Y Y Y Y Ys

s� �

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥a x p C mod

a x px

k n x p,,

, 1 ,..., , , , , 27ij i jk ij

i jn( ) ( ) ( )

� � �� � �d dh W h W G W, , . 28n n n1 213( ) ( ) ( ) ( )

Here and below C stands for a generic positive constant, which can take different values indifferent lines. The operator mod (see its exact definition in [64]) maps any symmetric matrixto its suitably defined ‘positive-semidefinite part’. Observe also that if

.

- -

� �d a �

a � aa �a

�

dg W g s g s O s

g s C g s g sg s

sC g s

0, , 0, ,

1 , 1 , 29

loc,2

3 2 2

( ) ( ) ∣ ( )∣ ( )

∣ ( )∣ ( ( )) ∣ ( )∣ ∣ ( )∣ ( ( )) ( )

then the required conditions for h are satisfied.

Definition 1. A function u from the space

� ��q d dC T L T W0, 0, , 30n n1( )( ) ( )[ ] ( )is a viscosity sub-/supersolution to (19), (22) and (23) if, for any �G � qC T0, n2 ([ ] ) andany point �� qt x T, 0, n

0 0( ) ( ] of local maximum/minimum of the function u ! f, one has

- .G K G Gss

�� �

� � �

⎛⎝⎜

⎞⎠⎟t

x

K g G udiv

,

10 0, 31y ( ∣ ∣)

(∣ ∣) ( )

and equalities (22), (23) hold in the classical sense. A viscosity solution is a function which isboth a subsolution and a supersolution.

Theorem 1. (i) Problems (19), (22) and (23) has a viscosity solution in class (30) for everypositive T. Moreover,

- -� �

u u t x uinf , sup . 320 0n n( ) ( )

(ii) Assume that

- �� � �a x p a z p C x z x z p, , , , , . 33ij

n( )( ) ( ) ∣ ∣ ( )

Here is the square root of a positive-semidefinite symmetric matrix [42]. Then the solutionis unique. Moreover, for any two viscosity solutions u and v to (19), (22), the followingestimate holds

-� �

� ' �u t v t t u vsup , , sup 0, 0, 34n n

∣ ( · ) ( · )∣ ( ) ∣ ( · ) ( · )∣ ( )

with some non-decreasing continuous scalar function ' dependent on u and v.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

11

Proof. (Sketch) We follow the strategy of [64, section 3], where we studied a relatedproblem, and refer to it for further details. Note first that (32) is a direct consequence of thedefinition of viscosity solution: e.g., to get the second inequality, one can put f = $t, to derivethat the function u(t, x) ! $t attains its global maximum at t = 0, and to let E l �0.

Now, we need to formally establish a Bernstein estimate for � �usup .n Consider theformal parabolic operator

$

K

K

K

K

�ss

� � � �s

s s

� � �s �

sss

� � � �ss

� � ��

�ss

� � � � �s�s

�ss

� � � � �s�s

�

�ss

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

th u G a x u

x x

h u Ga x u

pu

xh u G x u

x

h u Gx u u u

u x

h u G uG

xx u

x

h u G uG

x

x u u u

u x

,

,,

,

,

,. 35

iji j

ij

lx x

lyx

l

yyx x x

l

iy

i

i

yy x x

l

2

i j l

i l i

l i

( ) ( )

( ) ( ) ( ) ( ∣ ∣)

( )( ∣ ∣)

∣ ∣

( ) · ( ∣ ∣)

( ) ·( ∣ ∣)

∣ ∣ ( )

Fix T. Differentiating (19) with respect to each xk, k = 1,..., n, multiplying by u2 ,xk and addingthe results, we get

$

K

K

K

K

K

� �� � � �

� � � � �s�s

�

� � �s �

s

� � � � �s�s

�

� � � �

�s

s s� � �

ss s

�s

s s�

� � � � �s �s s

�

� � � � �s�s

�

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

u h u G a x u u u

h u G uG

xa x u u u

h u Ga x u

xu u

h u G uG

xx u u u

h u G x u u u

hq q

u G uG

x xu

Gx x

x u u u

h u G uG

x xx u u u

h u G uG

xx u u u

2 ,

2 ,

2,

2 ,

2 ,

2 ,

2 ,

2 , .

36

ij x x x x

kij x x x

ij

kx x x

kyx x x

yx x x x

j l i j k ly x x

i ky x x

iyx x x

2

2 2 2

2

k i k j

i j k

i j k

i i k

i k i k

i k

i k

k i k

( )∣ ∣ ( ) ( )

( ) · ( )

( ) ( )

( ) · ( ∣ ∣)

( ) ( ∣ ∣)

( ) ( ∣ ∣)

( ) · ( ∣ ∣)

( ) · ( ∣ ∣)( )

Using [64, lemma 1], it possible to get rid of the second-order terms in the right-handside, ending up with

$ -� � �u C u1 , 372 2( ) ( )∣ ∣ ∣ ∣ ( )

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

12

which by maximum principle implies

-�u C. 382∣ ∣ ( )

We can now approximate our problem by well-posed uniformly parabolic problems inthe sense of [47, chapter 5], namely, we need to separate h and the eigenvalues of a(x, p) awayfrom zero, and to make a(x, p) diagonal for large p , simultaneously keeping all involvedconstants uniformly bounded and the constant from (27) bounded away from zero. We obtainthe required viscosity solution by passing to the limit in the viscosity sense, employing thegeneral consistency/stability results from the viscosity solution theory [22] and the Arzelà-Ascoli compactness provided by the Bernstein estimate, see [2, 3, 10, 64] with similarconsiderations. The uniqueness of solutions follows from the stability inequality (34), whichmay be shown by mimicking the proofs of similar bounds in [2, 3, 64]. ,

3.2. Dissipative solutions

The concept of dissipative solution (see [23, 29, 50, 51, 63, 73, 75] and an illustrativediscussion in [72]) allows us to significantly relax the assumptions on j and g with respect tothe viscosity solution case.

In this subsection we use Neumann!s boundary condition. The Dirichlet boundaryconditions can also be treated with some technical adjustments. Let ٠be a bounded opendomain in � ,n ��n , with a regular boundary s8. We thus consider (19), (23) to be coupledwith

Oss

� � s8u

x0, . 39( )

The symbol & &· will stand for the Euclidean norm in L2(Ω). The corresponding scalarproduct will be denoted by parentheses , .( · · ) We will also use this notation for dualitybetween Lp(Ω) and 8�L .p p 1( )

We assume that for every natural number N, the functions K K, y are continuous andbounded on Ω ! (1/N, N). The parabolicity conditions (24)–(27) are replaced by a weakerone:

. �K K� � � 8 �x p p x p p p p x p p, , 0, , , 0 . 40y yn

1 1 2 2 1 2 1 2( )( ) ( ) · ( ) ⧹{ } ( )

We also put weaker assumptions on g, h and G. Namely, h is merely needed to beLipschitz, which holds, e.g., if g is non-negative, locally Lipschitz and -a �g C g1 ,2( )whereas G should be of class �W .n

21( )

We point out that G # "u means the convolution � �G u,˜ where u is an appropriatelinear and continuous extension5 of u onto �n which may depend on the boundary condition(see [17]).

Introduce the following formal expression

K' �

� �

� � �v t x

x v t x v t x

K g G v t x,

, , ,

1 ,. 41y( )( )

( ∣ ( )∣) ( )(∣ ∣( )) ( )

5 The simplest possible extension is letting u to be zero outside of Ω. Another option is to use Hestenes–Seeley-likeextensions [1] which conserve the Sobolev class of u.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

13

Definition 2. Let � 8u L .0 2 ( ) A function u from the class

�� d 8 d 8u C L L W0, ; 0, ; 42w 2 1 11( )([ ) ( )) ( ) ( )

is called a dissipative solution to problem (19), (23), (39) if, for all regular6 functions�d q 8 lv: 0,[ ) satisfying the Neumann boundary condition (39) and all non-negative

moments of time t, one has

-

¨

H

H

� �

� ' � � � �s

s��

& & & &⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

u t v t u v

v s u s v sv s

su s v s s

, 0,

2 , , ,,

, , d ,

43

t

tt s

20

2

0

( ) ( · ) ( · )

( ( ( · )) ( ) ( · )) ( · ) ( ) ( · )( )

where % is a certain constant7 depending on g, G, j and v (in particular, %=1 provided wg 0).

Usual argument [72] shows that these dissipative solutions possess the weak-stronguniqueness property (any regular solution is a unique dissipative solution).

Theorem 2. Assume

K � �dl�d �8

x y ylim inf , , 44y x

y ( ) ( )

K �l �8

x y ylim sup , 0. 45y x

y0

( ) ( )

Assume also that either we have strong parabolicity, namely,

. �

K K� �

� � 8 �

x p p x p p p p

C p p x p p

, ,

, , , 0 , 46

y y

n

1 1 2 2 1 2

1 22

1 2

( )( ) ( ) · ( )

⧹{ } ( )

or better regularity of h and G,

� �� �dh W G W, . 47n n222( ) ( ) ( )

Let � 8u L .0 2 ( ) Then there exists a dissipative solution to problems (19), (23) and (39).

Remark 3. In the case when (47) but not (46) holds, the test functions for (43) shouldadditionally satisfy the condition

K � � � �d 8dx v v L Ldiv , 0, ; , 48y 2( ) ( )( ∣ ∣) ( ) ( )

which, by the way, automatically holds provided j is more regular, e.g., satisfies theassumptions (24)–(26) of the previous subsection.

Remark 4. The TV flow [7] corresponds to the case wg 0, K �x y y, ln ,( ) so it satisfies(40) but is ruled out by (44) and (45). We will consider this particular form of j (with genericg) in the next subsection (see remark 10). The existence of dissipative solutions for the TVflow is an open problem. We however believe that the hypotheses of theorem 2 may besignificantly weakened.

6 Here ‘regular’ means that v and "v are uniformly bounded and sufficiently smooth, and � vv 0 a.e. ind q 80, .( )

7 The exact expression for % follows from the proof below.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

14

Proof of theorem 2. To begin with, we formally derive some a priori bounds and inequality(43) for the solutions to problem (19), (23), (39). Firstly, we formally take the L2(Ω)-scalarproduct of (19) with 2 u(t), and integrate by parts:

K�

� �

� � �� �& &

⎛⎝⎜

⎞⎠⎟t

ux u u

K g G uu

dd

2 ,

1, 0. 49y2

( ∣ ∣)(∣ ∣) ( )

Since the second term is non-negative due to (40) and (45), (49) a priori implies that

-�dd& & & &u u . 50L L0, ; 0

2( )( )

Thus,

- - -� � �& && & & &G u G u C u C. 51∣ ∣ ˜ ( )Consider the scalar function K: � �8y x y yinf , ,x y

2( ) ( ) y > 0, Ψ(0) = 0. From (49) and (51)we can conclude that

-¨ ¨ : ��d

8& &u x t C ud d . 52

00(∣ ∣) ( )

The function Ψ(y) is non-negative, continuous (for y = 0 this follows from (45), for positive yit can be derived from the compactness of 8) and satisfies the condition

: � �dl�d y ylim .y ( ) By the Vallée–Poussin criterion [26], "u a priori belongs to acertain weakly compact set in L T L0, ;1 1( ) for any T > 0.

Fix a regular test function �d q 8 lv : 0,[ ) satisfying the Neumann boundarycondition (39). Adding (19) with the identity

K�

ss

� �� �

� � �� ' �

ss

⎛⎝⎜

⎞⎠⎟

vt

x v v

K g G vv

vt

div,

1div ,y ( ∣ ∣)

(∣ ∣) ( )

which can be understood, e.g., in the sense of distributions, and formally multiplying by�u t v t2[ ( ) ( )] in L2(Ω), we find

K K

K

� �� � � � �

� � �� �

� � � � � � � � � �

� ' � � � �ss

�

& &⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

tu v

x u u x v v

K g G uu v

h G v h G u x v v u v

v u vvt

u v

dd

2, ,

1,

2 , ,

2 , 2 , . 53

y y

y

2

( )

( ∣ ∣) ( ∣ ∣)(∣ ∣) ( )

[ ( ) ( )] ( ∣ ∣) ( )

( ( ) ) ( )

If (46) holds, then, due to (51), (45) and boundedness of Ω, we have

-- -

K

K K

� � � � � � � � �

�� � � � �

� � �� �

� � � � � � � � �

� � � � � � � �d d

& & & & & & & &

& && && & & & & &

⎛⎝⎜

⎞⎠⎟

h G v h G u x v v u v

x u u x v v

K g G uu v

C h G u v u v C u v

C G u v u v C u v C u v

2 , ,

2, ,

1,

, 54

y

y y

L L L 12

12 2

1

( )[ ( ) ( )] ( ∣ ∣) ( )( ∣ ∣) ( ∣ ∣)

(∣ ∣) ( )

( ) ( ) ( )( ) ( ) ( )

where C1 is the doubled constant from (46).

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

15

If (47) and (48) hold, then, by virtue of (40), we get

-

-

-

-

K

K K

K

K

K

K

K

K

K

� � � � � � � � �

�� � � � �

� � �� �

� � � � � � � �

� � � � � � � � �

� � � �s�s

�

� � � �s�s

� �ss

�

� � � � � �

� � � � � � � �s�s

� �ss

�

� � � �s�s

� � �ss

�

� � � � �

� � � � � � � � �

�

d& & & && &

& && && && &

& && && & & && & & & & && &

& &

⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎡⎣ ⎤⎦ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎛⎝⎜

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎞⎠⎟

⎡⎣ ⎤⎦⎛⎝⎜

⎡⎣ ⎤⎦⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎡⎣ ⎤⎦

h G v h G u x v v u v

x u u x v v

K g G uu v

h G u h G v x v v u v

h G u h G v x v v u v

h G uG

xu

h G vG

xv x v

vx

u v

C G u v x v v u v

h G u h G vG

xu x v

vx

u v

h G vG

xu v x v

vx

u v

C G x v v u v u v

C G u G u v u v C G u v u v

C u v

2 , ,

2, ,

1,

2 div , ,

2 div , ,

2

, ,

div ,

2 , ,

2 , ,

div ,

.

55

y

y y

y

y

i

iy

i

L y

iy

i

iy

i

y

W W

2

21

21

( )( )

( )

( ) ( )( ) ( )

( )

( )

( ) ( )

( )

[ ( ) ( )] ( ∣ ∣) ( )( ∣ ∣) ( ∣ ∣)

(∣ ∣) ( )

˜ ˜ ( ∣ ∣)

˜ ˜ ( ∣ ∣)

˜ ˜

˜ ˜ ( ∣ ∣)

( ˜ ˜) ( ∣ ∣)

˜ ˜ ˜ ( ∣ ∣)

˜ ( ˜ ˜) ( ∣ ∣)

( ∣ ∣) ˜ ˜˜ ˜ ˜ ˜ ˜

( )

Note that this C can be set to be zero when wg 0.Thus, in both cases, there is % > 0 such that

- H� � � ' � � � �ss

�& & & &⎛⎝⎜

⎞⎠⎟t

u v u v v u vvt

u vdd

ln 2 , 2 , . 562 2( ) ( ( ) ) ( )

By Gronwall!s lemma, we infer (43).We recall the following abstract observation [69, 78]. Assume that we have two Hilbert

spaces, �X Y , with continuous embedding operator li X Y: , and i(X) is dense in Y. Theadjoint operator * * *li Y X: is continuous and, since i(X) is dense in Y, one-to-one. Since iis one-to-one, * *i Y( ) is dense in *X , and one may identify *Y with a dense subspace of *X .Due to the Riesz representation theorem, one may also identify Y with *Y . We arrive at thechain of inclusions:

* *� w �X Y Y X . 57( )Both embeddings here are dense and continuous. Observe that in this situation, for� �f Y u X, , their scalar product in Y coincides with the value of the functional f from *X

on the element �u X:

�f u f u, , . 58Y( ) ( )Such triples (X, Y, X*) are called Lions triples.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

16

We will work with the Lions triple *8 8 8H L H, ,m m2( ( ) ( ) ( ) ( )) where � �m 1 n

2is a

fixed number. Denote by A the Riesz bijection between the spaces Hm and *Hm( ) (which arenot identified). We will employ the Sobolev embedding �H C ,m 1 which is compact.

Consider the following approximate problem, where the first equality is understood in thesense of the space (Hm)*, whereas the second one is in the sense of the space L2:

�� � � ��ut

Q u Au u udd

0, . 59t 0 0( ) ∣ ( )

The operator *lQ H H: ,m m( ) which respects the Neumann boundary condition, isdetermined by the duality

K�

� �

� � �� � �

⎛⎝⎜

⎞⎠⎟Q u w

u u

K g G uw w H,

,

1, , .y m( )

( · ∣ ∣)(∣ ∣)

We do not use a notation for partial time derivative since we treat (59) as an ODE in a Banachspace.

The operator Q is bounded and continuous from *C to Hm1 ( ) . Note that there is no loss ofcontinuity as � lu 0 due to (45). Therefore, *lQ H H: m m( ) is a compact operator. Thisgives opportunity to secure existence of solutions to (59) in the class

*� ��d �d �dL H W H C L0, ; 0, ; 0, ; 60m mb2 2

12( )( ) ( ) ( )[ ) ( )

by an application of the Leray–Schauder degree theory (a systematic approach to parabolicproblems of kind (59) may be found, e.g., in [78]).

Repeating the arguments above, we deduce, for a fixed test function v, that theapproximate solutions satisfy the inequality

�

- H� � � ' � � �

�ss

� � �

& & & &

⎛⎝⎜

⎞⎠⎟

tu v u v v u v

vt

u v Au u v

dd

ln 2 ,

2 , 2 , . 61

2 2( ) ( ( ) )

( )

An application of Cauchy!s inequality yields

-� �Au u v Av v2 ,12

, .

Consequently, the approximate solutions satisfy inequality (43) up to a term of order �O .( )Due to the observations above, without loss of generality the approximate solutions

converge weakly in L T W0, ;1 11( ) and weakly-* in dL T L0, ; 2( ) as � l 0 for any T > 0. This

is enough for passing to the limit in inequality (43), since its left-hand side is the only termwhich is nonlinear in u, but this term is lower-semicontinuous (we refer to [51, 72] fordetailed passages to the limit in some dissipative solution inequalities). ,

3.3. Weak and strong solutions

For j of power and logarithmic growth, we can show existence of weak solutions. Moreover,in the first case the solutions are locally Lipschitz and their gradients are Hölder continuous.We maintain Ω to be a bounded open domain in �n with a regular boundary. We assume that

�d l �dg : 0, 0,[ ) [ ) is continuous8, and ��G W .n22 ( )

8 Thus, for weak solutions, g is not needed to be locally Lipschitz.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

17

Firstly, let K � �x y c y, ,p1

1( ) p > 1, c1 > 0. To simplify the presentation, in the sequelwe assume that c1 = 1. We keep the Neumann boundary condition, but generalization to theDirichlet case is straightforward.

Definition 3. A function u from the class

*� � �� 8 8 8�⎡⎣ ⎤⎦u C T L L T W W T L W0, ; 0, ; 0, ;

62

w p p p p p21

11

21( )( )( )[ ] ( ) ( ) ( )

( )is called a weak solution to problem (19), (39) if, for all � 8 q 8v L Wp2

1( ) ( ) and a.a.�t T0, ,( ) one has

�� � �

� � �� �

�⎛⎝⎜

⎞⎠⎟

ut

vp u u

K g G uv

dd

,1

1, 0. 63

p 2( )∣ ∣(∣ ∣) ( )

Theorem 3. LetK � �x y y, ,p 1( ) p > 1, � 8u L .0 2 ( ) There exists a weak solution u to (19),(39) satisfying (23).

Proof. (Sketch) Although the operator

�� �

� � �

�⎛⎝⎜

⎞⎠⎟

u uK g G u

div1

p 2∣ ∣(∣ ∣)

is not monotone, it is still possible to adapt the Minty–Browder technique to prove theorem 3.The key point is to pass to the limit. Let uk{ } be a sequence of solutions to (59) with� �� l 0.k Since .Au u, 0,k k every uk satisfies the a priori estimates (50), (51) and (52)(with T instead of �d). We have to prove that its limit u (in the weak-* topology ofdL T L0, ; 2( )) is a solution. Estimates (50), (51) and (52) imply that the solutions uk belong

to a uniformly bounded set in the space (62). Owing to [67, corollary 4], without loss ofgenerality we may assume that lu uk in *8C T W0, ; ,2

1([ ] [ ( )] ) so the initial condition ispreserved by the limit. One can check that the extension operators mentioned in the previoussubsection are continuous from *8W2

1[ ( )] to *�W .n21[ ( )] Therefore, lu uk˜ ˜ in

*�C T W0, ; .n21([ ] [ ( )] ) Thus, � � l � �G u G uk in 8dC T L0, ; .([ ] ( )) The operator

� � 8 l 8 � 8dG L W C: 21( ) ( ) ( ) is continuous. This implies that � � l � �G u G uk

in q 8C T0, .([ ] )) Due to the continuity of the Nemytskii operator on the space ofcontinuous of functions [46], we conclude that � � l � �h G u h G uk( ) ( ) uniformly on

q 8T0, .[ ] Then we can proceed similarly to the classical monotonicity argument[25, 28, 49] but with necessary changes. ,

Remark 5. In a similar way, theorem 3 may be generalized onto the case of more generalK x y,( ) with growth as l dy and decay as ly 0 of order �y .p 1

We next obtain the local Lipschitz-regularity of a weak solution, as well as the localHölder continuity of its gradient.

Theorem 4. Assume that h is Lipschitz and u is a weak solution to (19), (23) and (39), whichis locally bounded, together with its gradient. Then there exists B � 0, 1( ) such that for anycompact set # � q 8T0,( ) there is M > 0 so that

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

18

#-� � � � �u t x u t x M x x t t t x t x, , , , , , 641 1 2 2 1 2 1 2 1 1 2 2( )( ) ( ) ( ) ( ) ( )

and

#-� � � � � � �B

u t x u t x M x x t t t x t x, , , , , , .

65

1 1 2 2 1 2 1 2 1 1 2 2( )( ) ( ) ( ) ( )

( )

Remark 6. For the degenerate case �p 2 the behaviour of solutions is a purely local factand the local boundedness follows for any local weak solution. On the contrary, in thesingular case 1 < p < 2, it must be derived from global information and may require extraassumptions. Restricting the values of p to the range

�, 2n

n2

2( ) suffices though. Note that forapplications in imaging n = 2 and no extra assumption is needed.

Remark 7. The constant M is determined by the parabolic distance from # to the parabolicboundary of q 8T0,( ) and by the supremum of u and �u on #, see [24, chapter IX]. Theconstant " depends exclusively on p and dimension.

Proof of theorem 4. Equation (19) with K � �x y y, p 1( ) can be written in the form

ss

� � ��ut

h t x u udiv , , 66p 2( )˜( ) ∣ ∣ ( )

where � � � �h p h G u1 .˜ ( ) ( ) Similarly to the proof of theorem 3, one shows that G # "uis continuous on q 8T0, .[ ] Therefore � � � �h p h G u1˜ ( ) ( ) is continuous and boundedfrom below by a positive constant. Moreover, since h is Lipschitz, the partial derivatives

� � � � � ��p h G u u1hx

Gxi i( )( ) ( ) · ˜˜

are bounded. Thus, the structure conditions of [24,chapter VIII, pp 217–18] are fulfilled. The local regularity of the solution now follows fromthe general results of [24].

Concerning the optimal Lipschitz regularity of the weak solution we invoke the results of[14], with p constant. Indeed, condition (7) on page 912 of [14] holds since h is boundedabove and below by positive constants and Lipschitz in space. Moreover, h is Höldercontinuous in time because the same holds for the nonlocal term � �G u, due to the fact thatu is Hölder continuous in time up to the lateral boundary s8 (see [24, chapter III]). ,

Now we treat the logarithmic growth case. Again, just to simplify the presentation, wemerely consider K �x y y, ln .( ) We restrict ourselves to the Neumann boundary condition.Let% be the Banach space of finite signed Radon measures on (0, T) ! Ω. It is the dual ofthe space C0((0, T) ! Ω) (the space of continuous functions on (0, T) ! Ω that vanish at theboundary of this cylinder). For %�v , and ' � q 8C T0, ,([ ] ) Φ � 0, we define theweighted partial variation of v as

%� -

Z�Z Z

'� q8 '

qd

v vPV sup , div . 67C T

C0, ; :n

0

0( ) ( )

( )( ) ∣ ∣

Observe that if � 8v L T W0, ; ,1 11( ( )) then PVΦ(v) is equal to

¨ ¨ ' �8

t x v t x x t, , d d .T

0( ) ( )

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

19

The partial variation of v is

�v vPV PV . 681( ) ( ) ( )Define the ‘bounded partial variation space’ BPV as

% %� � � �d& & & &v v v PV v .BPV{ }≔ ( )Owing to lower semicontinuity of suprema, we have

Proposition 1. For any non-negative function ' � q 8C T0,([ ] ) and a weakly-*converging (in%) sequence �v BPV,m{ } one has

-'l�d

'v vPV lim infPV . 69m

m( ) ( ) ( )

Using (69) with ' w 1, we can prove

Proposition 2. BPV is a Banach space.

Definition 4. A function u from the class

*� � �� 8 8d⎡⎣ ⎤⎦u C T L W T L W0, ; BPV 0, ; 70w 2

12 1

1( )( )( ) ( ) ( )

is called a weak solution to problems (19), (39) and (23) withK �x y y, ln( ) and � 8u L0 2 ( )if

(i) there exists �� q 8dz L T0, ; ,n(( ) ) -q8d& &z 1,L T0,(( ) ) so that for all

�� 8v L W ,2 11( )

� � � � �ut

v h G u z vdd

, , 0, 71( ( ) ) ( )

a.e. on (0, T);(ii) for all �� 8 8w L T W W T L0, ; 0, ; ,1 1

111

2( ( )) ( ( )) one has

-

¨

¨

� � �

� � � � � � �

� �& &

& & & & & &

⎜ ⎟⎛⎝

⎞⎠u T w T

ws

s u s s PV u

u w w T w h G u s z s w s s

2dd

, d 2

0 0 2 , d ;

72

T

h G u

T

2

0

02 2 2

0

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( ( )) ( ) ( ))( )

( )

(iii) (23) holds in the space L2(Ω).

Remark 8. The motivation for this definition is the following one. Consider, formally, a pair(u, z) of sufficiently smooth functions satisfying (71), (72), (23). Then

-

¨

¨ ¨

¨ ¨

��

� � � � �

�

⎜ ⎟

⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

u ws

s u w s s

ws

s u s s h G u s u s s

ws

s w s sus

s w s s

2d

d, d

2dd

, d 2 , d

2dd

, d 2dd

, d . 73

T

T T

T T

0

0 0

0 0

( ) ( ) ( )( )

( ) ( ) ( ( ( )) ∣ ( )∣)

( ) ( ) ( ) ( ) ( )

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

20

Hence,

-¨ ¨� � � � ⎜ ⎟⎛⎝

⎞⎠h G u s u s s

us

s u s s, ddd

, d . 74T T

0 0( ( ( )) ∣ ( )∣) ( ) ( ) ( )

Therefore,

-¨ ¨� � � � � �h G u s u s s h G u s z s u s s, d , d . 75T T

0 0( ( ( )) ∣ ( )∣) ( ( ( )) ( ) ( )) ( )

On the other hand,

.� � � � � �h G u u h G u z u. 76( )∣ ∣ ( ) · ( )All this can be true if and only if

� � �u z u, 77∣ ∣ · ( )i.e.

K���

� � �zuu

u u, . 78y∣ ∣ ( · ∣ ∣) ( )

Substituting expression (78) for z into (71) and integrating by parts, we infer

¨

¨

K

KO

ss

� � � � �

� � � �ss

8

s8

�

⎡⎣⎢

⎤⎦⎥

ut

h G u x u u v x

h G u u vu

d

div , d

, . 79

y

yn 1

( )( ) ( ∣ ∣)

( ) ( · ∣ ∣) ( )

Testing (79) by any v compactly supported in Ω, we deduce (19). Consequently, bothintegrals in (79) are identically zero. By arbitrariness of v, �n 1-a.e. in $ Ω it holds

KO

� � �ss

�h G u uu

, 0. 80y( ) ( · ∣ ∣) ( )

Since both h and jy are non-vanishing, the Neumann boundary condition (39) is satisfied.

Theorem 5. LetK �x y y, ln ,( ) � 8u L .0 2 ( ) There exists a weak solution to (19), (39), (23).

Proof. Consider the approximate problem

��� � � ��ut

Q u Au u udd

0, . 81t 0 0( ) ∣ ( )

The notation and functional framework are the same as in the previous subsection, see theproof of theorem 2. The operator *� lQ H H: m m( ) is determined by the duality

�� ��

� � � � �� � �

⎛⎝⎜

⎞⎠⎟Q u w

uu K g G u

w w H,1

, , .m( ) ( ∣ ∣)( (∣ ∣))The existence of solutions to (81) in class (60) is similar to the existence of solutions to (59)in the previous subsection. Note that, since condition (45) is violated in the logarithmic case,we do not work directly with (59). However, the a priori bounds (50), (51) and

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

21

�-¨ ¨

�� �8

uu

C 82T

0

2∣ ∣∣ ∣ ( )

(an analogue of (52)) still hold, with C independent of ò. Estimate (82) easily yields

-¨ ¨ �8

u C. 83T

0∣ ∣ ( )

Let uk{ } be a sequence of solutions to (81) with with � �� l 0.k Set

��

�� �

zu

u.k

k

k k

Fixing a sufficiently smooth function �q 8 lw T: 0, ,[ ] testing (81) with 2(uk!w), andintegrating in time, we deduce

�

¨ ¨

¨

¨

� � � � � �

� � � � � � � �

� � �

& &

& & & & & &

⎜ ⎟⎛⎝

⎞⎠u t w t

ws

s u s s h G u s z s u s s

u w w t w h G u s z s w s s

Au s u s w s s t T

2dd

, d 2 , d

0 0 2 , d

2 , d , 0, ,

84

k

t

k

t

k k k

t

k k

k

t

k k

2

0 0

02 2 2

0

0

( )( )

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) [ ]( )

whence by Cauchy!s inequality

��

�

-

¨ ¨

¨

¨ ¨

� � � � � �

� � � � � � �

� � ��

� �� �

& &

& & & & & &

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

u t w tws

s u s s h G u s u s s

u w w t w h G u s z s w s s

h G u su s

us Aw s w s s t T

2dd

, d 2 , d

0 0 2 , d

2 , d2

, d , 0, .

85

k

t

k

t

k k

t

k k

k

t

kk

k k

kt

2

0 0

02 2 2

0

0 0

( )( )

( )( )

( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) [ ]( )

Let us prove that the limit � l�du z u z, lim , ,k k k( ) ( ) which we without loss of generalityassume to exist in the weak-* topology of qd d dL T L L T L0, ; 0, ; ,2( ) ( ) is a weaksolution. Estimates (50), (51) and (83) imply that the solutions uk belong to a uniformlybounded set in the space (62) with p = 1. Similarly to the proof of theorem 3, passing to afurther subsequence if necessary, we have that lu uk in *8C T W0, ; ,2

1([ ] [ ( )] ) and� � l � �h G u h G uk( ) ( ) in q 8C T0, .([ ] ) We can pass to the limit in (23) and in the

fourth term of the right-hand side of (85). We can also pass to the limit in the first term ofinequality (85) without affecting its sign, since this term is lower-semicontinuous (see [51, 72]and the proof of theorem 2). We employ proposition 1, triangle inequality and the aboveuniform convergence of � �h G uk( ) in order to pass to the limit in the third term of (85).The remaining terms of (85) are either linear, or constants, or of order �O ,k( ) so the passage tothe limit is now straightforward, and we get (72).

Testing (81) by a sufficiently smooth �q 8 lv T: 0, ,[ ] we derive

�� � � � � �ut

v h G u z v Av udd

, , , 0. 86kk k k k( )( ) ( )

Passing to the distributional limit, we obtain (71). Finally, by density, the test functions v andw in (71) and (72) can be taken from the spaces indicated in definition 4. ,

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

22

Remark 9. In view of lack of monotonicity/accretivity, uniqueness of weak solutions intheorems 3 and 5 is an open problem.

Remark 10. The TV flow equation ( wg 0, K �x y y, ln( ) ) was studied in [7] viamonotonicity/accretivity arguments such as the Crandall-Liggett theory. Since the operator

�� �

� � �

�⎛⎝⎜

⎞⎠⎟

u uK g G u

div1

871∣ ∣

(∣ ∣) ( )

is not accretive in any sense (for gv const), we had to use another approach. The case wg 0is not excluded from our results: we have obtained existence of some weak solutions for theTV flow (with Neumann boundary condition). However, the advantage of the solutions uobtained in [7] is that they belong to the space of functions of BV 8BV( ) for a.a. t, and it isthus possible to employ Anzellotti!s approach [8] to give more direct meaning to theNeumann boundary condition (39). We conjecture that a theory of that kind (traces, Greenformula, etc) may be developed, for instance, in the space �q 8L T0, BPV,2 (( ) ) but forthe purposes of this paper we opted to treat the boundary condition (39) in a more implicitway presented in remark 8.

4. Experimental results and discussions

In what follows, we provide some experimental results using adaptive forward-backwarddiffusion flows in image restoration. All the images are rescaled to 0, 1[ ] for visualization andthe PDE (18) is discretized using standard finite difference scheme via additive operatorsplitting (AOS) [74] with time step U � 0.2, and grid size one. The discretized version of thePDE equation (18) is utilized out using the following unconditionally stable semi-implicitscheme. Let h be the grid size, U � 0 the time step andUij

t be the pixel value u(i, j) at iterationt. The AOS discretization in 1-D with matrix-vector notation is given by,

U� �� �⎡⎣ ⎤⎦U A U U1 ,t t t1 1( )where �A U a Ut

ijt( ) [ ( )] with

� �&

� �

&�

��

��

��

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

a Uh

j

hj i

2, ,

2, ,

0, otherwise.

ijt

it

jt

i

k

it

kt

2

2i

( ) ≔

Here &i is the set of the two neighbors of location i (boundary locations have only oneneighbor) The values �i are the discrete values obtained by evaluating the overall diffusionfunction, � K� � � � �Tx u K g G u, 1y ( ) ( ) at location i. The Gaussian convolution inthe inverse mollification is carried out by the discrete version with rotationally symmetricGaussian lowpass filter of size 5 ! 5 with variance 2. For n-D images the modified semi-implicit scheme (AOS) is written as,

� U� ��

�

�⎡⎣ ⎤⎦Un

n A U U1

1 . 88t

l

n

lt t1

1

1( ) ( )

The matrix �A al ijl ij( ) corresponds to derivatives along the lth coordinate axis. Note that thisinvolves solving a linear system where the system matrix is tridiagonal and diagonally

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

23

dominant, see [59, 74] for more details. To avoid the directional (x–y axis) smoothing bias weadapted a multi-direction based modification [58], and to be fair the same approach was usedto compare all the PDEs here. We used a non-optimized MATLAB implementation on a2.3 GHz Intel Core i7, 8 GB 1600MHz DDR3 Mac Book pro Laptop. The pre-smoothingparameter ! = 1 in (18) is fixed for additive Gaussian noise level of standard deviation!n = 30 used here. Figure 6 shows the synthetic Shapes and Brain images (noise-free/groundtruth) and its corresponding noisy versions which are utilized in our experiments.

4.1. Effect of inverse mollification, contrast parameter

We first consider the effect of inverse mollification function:

�� � �T

M xK g G u x

1

189( )( )

( )( )

in edge detection under noise for different choices of the power growth in� � � � �T Tg G u G u p( ) and K values. Note that the overall diffusion coefficient

acts like an edge detector within PDE based image restoration thereby guiding the smoothingprocess in and around edges. Figure 7 shows the computed inverse mollification function M(x) in (89) as images for different values of K and power growth in g ( · ) for the noisy Shapesimage (noise standard deviation !n = 30, see figure 6(a) right). The noisy image is shown infigure 6(a) right. Higher growth in functions � � � � �T Tg G u G u ,p( ) p > 2 retainsnoisy edges and similarly lower K as well. Moreover, we see that the contrast parameter Kcontrols the density of edges and can be chosen adaptively [57].

4.2. Effect of different exponent values in power growth diffusion

Next, we compare restoration results using PDE equation (18) with and without inversemollification function (89). We consider the power growthK � � �x u u,y

p( ) for differentexponent values p = 1, 2, 3, 4, 5 as the diffusion function. Figure 8(a) shows restoration ofnoisy Brain image (noise standard deviation !n = 30, see figure 6(b) right) without inversemollification function, i.e. taking wg 0, and figure 8(b) with

� � � � �T Tg G u G u .2( ) As can be seen, taking the non-trivial inverse mollifierfunction preserves strong edges in the final result when compared to the smoother results forhigher exponent p values.

Figure 6. Synthetic Shapes and real Brain images used in our experiments. Noise-free(ground-truth) images (left), and noisy images obtained by adding Gaussian noise levelT � 30n (right).

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

24

4.3. GRADE versus our approach

Finally, we provide a comparison of Catté et al [17] GRADE (3) to illustrate qualitativedifferences in restoration with our inverse mollification term based PDE (18), with

Figure 7. Inverse mollification function � � �TKg G u1 1( ( )) with respect tog ( · ) and contrast parameter K computed using noisy synthetic Shapes image (noisestandard deviation !n = 30, see figure 6(a) right). We show [0,1] normalized

� � �TKg G u1 1( ( )) as an image when: (a) � � � � �T Tg G u G u p( ) forp = 1, 2, 3, 4, 5 with K = 10!4

fixed, and (b) � � � � � �K 10 , 10 , 10 , 10 , 101 2 3 4 5 with� � � � �T Tg G u G u 2( ) fixed. Better viewed online and zoomed in.

Figure 8. Inverse mollification function when combined with power growth numeratorcan stop diffusion across edges. Solution of the PDE (18) on noisy synthetic Brainimage (noise standard deviation !n = 30, see figure 6(b) right) with power growthK � � �x u u, ,y

p( ) p = 1, 2, 3, 4, 5 (left to right) without inverse mollification (a)wg 0, and with (b) � � � � �T Tg G u G u .2( ) In both cases we used K = 10!4

and terminal time 100. It is clear visually that the inverse mollification has an effect inkeeping homogenous regions separated by strong edges and avoids leakage.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

25

� � � � �T Tg G u G u ,2( ) and set K = 10!4. In our PDE (18), we consider three casesfor the diffusion function: Perona–Malik non-convex regularization functions (f1, f2 in (12)),and TV regularization f(s) = s. Remember that the corresponding j in (18) may be recoveredfrom the relation K G� ax y y y, .y ( ) ( ) In figure 9 we show restoration results for the noisyShapes image (noise standard deviation !n = 30, see figure 6(a) right) corresponding todiffusion coefficients Ky from the Perona–Malik non-convex regularizations (PM1 for f1,

Figure 9. GRADE versus our approach on noisy synthetic Shapes image (noisestandard deviation !n = 30, see figure 6(a) right). Results with: (a) GRADE [17], andour adaptive PDE with main diffusion function (b) PM1 f1 in (12), (c) PM2 f2 in (12)(d) TV. In each row (i ii iii iv): we show (i) final denoized results, (ii) contours fromfinal denoized results, (iii) close-up of the contour map, and (iv) close-up surface. InPM1, PM2 we used K = 10!4, with terminal time T = 50 and in TV terminaltime T = 200.

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

26

PM2 for f2) and the TV regularization jy(y) = 1/y (the corresponding PDE is ss

utplus the

diffusive term (87) equals zero; for technical reasons, we actually employ�K � �y y1y

2( ) with ò = 10!6). As can be seen, by comparing the contour images,GRADE tends to smooth and displace the level lines of resultant image whereas our adaptiveschemes obtain better preservation of level lines in both Perona–Malik and TV diffusion.Table 1 lists well-known error metrics in the image processing literature for comparingGRADE against our adaptive PDE methods and supports visual comparison results thatadaptive schemes are better at retaining structures while removing noise. Deeper quantitativeanalysis of the numerical examples is deferred to an upcoming work and we refer to [60] foran earlier attempt in this direction with a strictly convex regularization.

Acknowledgments

JMU and DV partially supported by FCT project PTDC/MAT-CAL/0749/2012 and byCMUC, funded by the European Regional Development Fund, through the program COM-PETE, and by the Portuguese Government, through the FCT, under the project PEst-C/MAT/UI0324/2011. Part of this work was done while the first author VBSP was visiting theInstitute for Pure and Applied Mathematics (IPAM), University of California, Los Angeles,CA, USA. The authors thank José Alberto Iglesias Martínez (Computational Science Center,University of Vienna, Austria) for discussions related to the regularization-diffusion flows.Authors thank the anonymous referees and the handling editor for their comments whichhelped to improve the presentation and content of the paper.

References

[1] Adams R 1975 Sobolev spaces Pure and Applied Mathematics vol 65 (New York: Academic)[2] Alvarez L and Esclarín J 1997 Image quantization using reaction-diffusion equations SIAM J.

Appl. Math. 57 153–75[3] Alvarez L, Lions P L and Morel J-M 1992 Image selective smoothing and edge detection by

nonlinear diffusion II SIAM J. Numer. Anal. 29 845–66[4] Amann H 2007 Time-delayed Perona–Malik type problems Acta Math. Univ. Comen. LXXVI

15–38

Table 1. Error metrics comparison for PDE based smoothing results on noisy syntheticShapes image obtained with noise standard deviation T � 30.n Higher improved signalto noise ratio (ISNR), signal to noise ratio (SNR), peak signal to noise ratio (PSNR),mean structural similarity (MSSIM) indicate better restoration whereas lower meansquared error (MSE), root mean square error (RMSE), maximum absolute error (MAE),maximum absolute difference (MAX) indicate better restoration.

Metric Noisy GRADE Our-PM1 Our-PM2 Our-TV

ISNR 0 1.1120 4.6094 8.4478 5.824SNR 16.1925 17.3044 24.3237 24.6402 31.5589PSNR 18.5571 19.6691 26.6883 27.0049 33.9235MSSIM 0.3333 0.7177 0.8776 0.8529 0.9591

MSE 906.4984 701.7305 139.395 129.5954 26.3467RMSE 30.1081 26.4902 11.8066 11.384 5.1329MAE 24.0084 14.7066 5.5361 6.2528 2.2002MAX 132.8668 208.9502 150.7193 159.5855 126.144

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

27

[5] Andreu F, Ballester C, Caselles V and Mazón J M 2001 The Dirichlet problem for the totalvariation flow J. Funct. Anal. 180 347–403

[6] Andreu F, Caselles V, Díaz J I and Mazón J M 2002 Some qualitative properties for the totalvariation flow J. Funct. Anal. 188 516–47

[7] Andreu-Vaillo F, Caselles V and Mazón J M 2004 Parabolic quasilinear equations minimizinglinear growth functionals Progress in Mathematics vol 223 (Basel: Birkhäuser)

[8] Anzellotti G 1983 Pairings between measures and bounded functions and compensatedcompactness Ann. Mat. Pura ed Appl. 135 293–318

[9] Aubert G and Kornprobst P 2006 Mathematical Problems in Image Processing: PartialDifferential Equation and Calculus of Variations (New York, USA: Springer)

[10] Barcelos C A Z and Chen Y 2000 Heat flows and related minimization problem in imagerestoration Comput. Math. Appl. 39 81–97

[11] Belahmidi A 2005 Solvability of a coupled system arising in image and signal processing Afr.Diaspora J. Math. 3 45–61

[12] Belahmidi A and Chambolle A 2005 Time-delay regularization of anisotropic diffusion and imageprocessing Math. Modelling Numer. Anal. 39 231–51

[13] Bellettini G and Fusco G 2008 The Γ-limit and the related gradient flow for singular perturbationfunctionals of Perona–Malik type Trans. Am. Math. Soc. 360 4929–87

[14] Bögelein V and Duzaar F 2012 Hölder estimates for parabolic p(x, t)-Laplacian systems Math.Ann. 354 907–38

[15] Bonforte M and Figalli A 2012 Total variation flow and sign fast diffusion in one dimensionJ. Differ. Equ. 252 4455–80

[16] Brook A, Kimmel R and Sochen N 2003 Variational restoration and edge detection for colorimages J. Math. Imaging Vis. 18 247–68

[17] Catte V, Lions P L, Morel J-M and Coll T 1992 Image selective smoothing and edge detection bynonlinear diffusion SIAM J. Numer. Anal. 29 182–93

[18] Chambolle A and Lions P L 1997 Image recovery via total variation minimization and relatedproblems Numer. Math. 76 167–88

[19] Chen Y, Levine S and Rao M 2006 Variable exponent, linear growth functionals in imagerestoration SIAM J. Appl. Math. 66 1383–406

[20] Chen Y and Wunderli T 2002 Adaptive total variation for image restoration in BV space J. Math.Anal. Appl. 272 117–37

[21] Chen Y and Zhang K W 2006 Young measure solutions of the two-dimensional Perona–Malikequation in image processing Commun. Pure Appl. Anal. 5 617–37

[22] Crandall M G, Ishii H and Lions P-L 1992 User!s guide to viscosity solutions of second orderpartial differential equations Am. Math. Soc. Bull. New Ser. 27 1–67

[23] de Lellis C and Székelyhidi L Jr 2010 On admissibility criteria for weak solutions of the Eulerequations Arch. Ration. Mech. Anal. 195 225–60

[24] DiBenedetto E 1993 Degenerate parabolic equations Universitext (New York: Springer)[25] Evans L C 2010 Partial differential equations Graduate Studies in Mathematics vol 19 2nd edn

(Providence, RI: American Mathematical Society)[26] Feireisl E and Pra!ák D 2010 Asymptotic Behavior of Dynamical Systems in Fluid Mechanics

(AIMS Series on Applied Mathematics vol 4) (Springfield, MO: American Institute ofMathematical Sciences (AIMS))

[27] Fukui T and Giga Y 1996 Motion of a graph by nonsmooth weighted curvature Proc. First WorldCongress of Nonlinear Analysis vol 1 pp 47–56

[28] Gajewski H, Gröger K and Zacharias K 1974 Nichtlineare operatorgleichungen undoperatordifferentialgleichungen Mathematische Lehrbücher und Monographien: II. Abteilung,Mathematische Monographien, Band 38 (Berlin: Akademie)

[29] Gamba I M, Panferov V and Villani C 2009 Upper Maxwellian bounds for the spatiallyhomogeneous Boltzmann equation Arch. Ration. Mech. Anal. 194 253–82

[30] Geman S and McClure D 1987 Statistical methods for tomographic image reconstruction Proc. 46-th Session of the ISI, Bulletin of the ISI vol 52 pp 22–26

[31] Giga M-H and Giga Y 1998 Evolving graphs by singular weighted curvature Arch. Ration. Mech.Anal. 141 117–98

[32] Gilboa G, Sochen N and Zeevi Y Y 2006 Variational denoising of partly textured images byspatially varying constraints IEEE Trans. Image Process. 15 2281–9

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

28

[33] Greer J B and Bertozzi A L 2004 Traveling wave solutions of fourth order PDEs for imageprocessing SIAM J. Math. Anal. 36 38–68

[34] Guidotti P 2009 A new nonlocal nonlinear diffusion of image processing J. Differ. Equ. 2464731–42

[35] Guidotti P 2010 A new well-posed nonlinear nonlocal diffusion Nonlinear Anal. Theory, MethodsAppl. 72 4625–37

[36] Guidotti P 2012 A backward-forward regularization of the Perona–Malik equation J. Differ. Equ.252 3226–44

[37] Guidotti P 2015 Anisotropic diffusions of image processing from Perona–Malik on AdvancedStudies in Pure Mathematics to appear

[38] Guidotti P and Lambers J 2009 Two new nonlinear nonlocal diffusions for noise reductionJ. Math. Imaging Vis. 33 25–37

[39] Guidotti P and Longo K 2011 Two enhanced fourth order diffusion models for image denoisingJ. Math. Imaging Vis. 40 188–98

[40] Guidotti P and Longo K 2011 Well-posedness for a class of fourth order diffusions for imageprocessing Nonlinear Differ. Equ. Appl. NoDEA 18 407–25

[41] Harjulehto P, Latvala V and Toivanen O 2013 A variant of the Geman–Mcclure model for imagerestoration J. Math. Anal. Appl. 399 676–81

[42] Horn R A and Johnson C R 1985 Matrix Analysis (Cambridge: Cambridge University Press)[43] Kawohl B and Kutev N 1998 Maximum and comparison principle for one-dimensional anisotropic

diffusion Math. Ann. 311 107–23[44] Kichenassamy S 1997 The Perona–Malik paradox SIAM J. Appl. Math. 57 1328–42[45] Kielak K, Mucha P B and Rybka P 2013 Almost classical solutions to the total variation flow

J. Evol. Equ. 13 21–49[46] Krasnosel!skii M A 1964 Topological Methods in the Theory of Nonlinear Integral Equations ed

J Burlak (New York: Pergamon) (translated by A H Armstrong)[47] Lady!enskaja O A, Solonnikov V A and Ural’ceva N N 1967 Linear and quasilinear equations of

parabolic type Translations of Mathematical Monographs vol 23 (Providence, RI: AmericanMathematical Society) (translated from the Russian by S Smith)

[48] Li X and Chen T 1994 Nonlinear diffusion with multiple edginess thresholds Pattern Recognit. 271029–37

[49] Lions J-L 1969 Quelques Méthodes de Résolution des Problèmes aux Limites Non linéaires (Paris:Dunod)

[50] Lions P-L 1994 Compactness in Boltzmann!s equation via Fourier integral operators andapplications: I, II J. Math. Kyoto Univ. 34 391–27 429–61

[51] Lions P-L 1996 Mathematical Topics in Fluid Mechanics (Oxford Lecture Series in Mathematicsand its Applications vol 1) (New York: Oxford University Press)

[52] Nikolova M 2004 Weakly constrained minimization: application to the estimation of images andsignals involving constant regions J. Math. Imaging Vis. 21 155–75

[53] Nitzberg M and Shiota T 1992 Nonlinear image filtering with edge and corner enhancement IEEETrans. Pattern Anal. Mach. Intell. 14 826–33

[54] Nordstrom K N 1990 Biased anisotropic diffusion: a unified regularization and diffusion approachto edge detection Image Vis. Comput. 8 318–27

[55] Perona P and Malik J 1990 Scale-space and edge detection using anisotropic diffusion IEEETrans. Pattern Anal. Mach. Intell. 12 629–39

[56] Prasath V B S 2011 A well-posed multiscale regularization scheme for digital image denoising Int.J. Appl. Math. Comput. Sci. 21 769–77

[57] Prasath V B S and Delhibabu R 2014 Automatic contrast parameter estimation in anisotropicdiffusion for image restoration Communications in Computer and Information Science(Analysis of Images, Social Networks, and Texts) (Yekaterinburg, Russia, 2014) ed D I Ignatovet al vol 436 (Berlin: Springer) pp 14–18

[58] Prasath V B S and Moreno J C 2013 Feature preserving anisotropic diffusion for image restoration4th National Conf. on Computer Vision, Pattern Recognition, Image Processing and Graphics(NCVPRIPG 2013) (India, December 2013)

[59] Prasath V B S and Moreno J C 2015 On convergent finite difference schemes for variational-PDEbased image processing Appl. Math. Comput. to appear

[60] Prasath V B S and Singh A 2010 A hybrid convex variational model for image restoration Appl.Math. Comput. 215 3655–64

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

29

[61] Prasath V B S and Singh A 2010 Well-posed inhomogeneous nonlinear diffusion scheme fordigital image denoising J. Appl. Math. 2010 763847

[62] Prasath V B S and Singh A 2012 An adaptive anisotropic diffusion scheme for image restorationand selective smoothing Int. J. Image Graph. 12 18

[63] Prasath V B S and Vorotnikov D 2014 On a system of adaptive coupled PDEs for imagerestoration J. Math. Imaging Vis. 48 35–52

[64] Prasath V B S and Vorotnikov D 2014 Weighted and well-balanced anisotropic diffusion schemefor image denoising and restoration Nonlinear Anal.: Real World Appl. 17 33–46

[65] Prasath V B S, Vorotnikov D, Pelapur R, Jose S, Seetharaman G and Palaniappan K 2015Multiscale Tikhonov-total variation image restoration using spatially varying edge coherenceexponent IEEE Trans. Image Process. submitted

[66] Rudin L, Osher S and Fatemi E 1992 Nonlinear total variation based noise removal algorithmsPhysica D 60 259–68

[67] Simon J 1987 Compact sets in the space Lp(0,T; B) Ann. Mat. Pura ed Appl. Ser. Quarta 14665–96

[68] Strong D and Chan T 2003 Edge-preserving and scale-dependent properties of total variationregularization Inverse Problems 19 165–87

[69] Temam R 1979 Navier–Stokes equations Studies in Mathematics and its Applications vol 2(Amsterdam: North-Holland) (revised edition) (Theory and numerical analysis, with anappendix by F Thomasset)

[70] Tikhonov A N and Arsenin V Y 1997 Solutions of Ill-posed Problems (New York: Wiley)[71] Vese L 2001 A study in the BV space of a denoising-deblurring variational problem Appl. Math.

Optim. 44 131–61[72] Vorotnikov D 2012 Global generalized solutions for Maxwell-alpha and Euler-alpha equations

Nonlinearity 25 309–27[73] Vorotnikov D A 2008 Dissipative solutions for equations of viscoelastic diffusion in polymers

J. Math. Anal. Appl. 339 876–88[74] Weickert J, Romeny B M H and Viergever M A 1998 Efficient and reliable schemes for nonlinear

diffusion filtering IEEE Trans. Image Process. 7 398–410[75] Wu J 2000 Analytic results related to magneto-hydrodynamic turbulence Physica D 136 353–72[76] You Y-L, Xu W, Tannenbaum A and Kaveh M 1996 Behavioral analysis of anisotropic diffusion

in image processing IEEE Trans. Image Process. 5 1539–53[77] Zhang K 2006 Existence of infinitely many solutions for the one-dimensional Perona–Malik

model Calculus Variations Partial Diff. Equ. 26 171–99[78] Zvyagin V G and Vorotnikov D A 2008 Topological approximation methods for evolutionary

problems of nonlinear hydrodynamics (de Gruyter Series in Nonlinear Analysis andApplications) vol 12 (Berlin: de Gruyter & Co)

Inverse Problems 31 (2015) 105008 V B Surya Prasath et al

30