Integro-Differential Equations Based on (BV, L 1 ) Image Decomposition

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1 SIAM J IMAGING SCIENCES Vol 4, No, pp 3 32 c 2 Society for Industrial and Applied Mathematics Integro-Differential Equations Based on (BV, L Image Decomposition Prashant Athavale and Eitan Tadmor Abstract A novel approach for multiscale image processing based on integro-differential equations (IDEs was proposed in [E Tadmor and P Athavale, Inverse Probl Imaging, 3 (29, pp 693 7] These IDEs, which stem naturally from multiscale (BV,L 2 hierarchical decompositions, yield inverse scale representations of images in the sense that the BV -dual norms of their residuals are inversely proportional to the scaling parameters Motivated by the fact that (BV,L decomposition is more suitable for extracting local scale-space features than (BV,L 2, we introduce here the IDEs which arise from multiscale (BV,L hierarchical decompositions We study several variants of this (BV,L -based IDE model, depending on modifications to the curvature term Key words hierarchical image decompositions, total variation, denoising, deblurring, inverse scale, variational problem, integro-differential equation, energy decomposition AMS subject classifications 26B3, 65C2, 68U, 92C5 DOI 37/79554 Introduction In [32] we introduced a class of integro-differential equations (IDEs to model images, based on (BV,L 2 hierarchical decompositions [33, 34] In this paper we study integro-differential equations arising from hierarchical (BV,L image decomposition As in [32], we work here with grayscale images which can be realized as a graph of a discrete function f :Ω R 2 R The values of this function, f(x, denote the intensity of the image at the discrete points x Ω: the function f attains its maximum value at the brightest spots in the image and its minimum value of zero at the darkest spots The graph of an image consists of discrete pixels, which, for mathematical analysis, is postulated as an L 2 (Ω function The image f, which is obtained by any image capturing device, may be blurred and may contain some additive noise η Thus, the observed image, f, could be written as f = KU+η, whereu is the clean image sought without blurring and noise and K : L 2 (Ω L 2 (Ω is some blurring operator The problem of obtaining the clean image U from the observed image f is an ill-posed problem of image restoration, which can be addressed by several inverse problem solvers We mention in this context variational techniques using Tikhonov-like regularization, PDE-based methods, filtering, stochastic modeling, and wavelet-based techniques that were developed for solving these image processing problems [5, 9, 4, 5, 7, 8, 2, 2, 24, 25, 29, 3, 35] Received by the editors May 7, 2; accepted for publication (in revised form January 3, 2; published electronically March 7, 2 This research was supported in part by NSF grants DMS and DMS Institute of Pure and Applied Mathematics, University of California, Los Angeles, CA 995 (prashant@math uclaedu Department of Mathematics, Institute for Physical Science & Technology and Center of Scientific Computation and Mathematical Modeling (CSCAMM, University of Maryland, College Park, MD 2742 (tadmor@cscammumd edu 3

2 (BV, L INTEGRO-DIFFERENTIAL EQUATIONS FOR IMAGES 3 Image restoration leads to image decomposition For example, any denoising of an observed image f results in the decomposing of type f = U λ + η λ,whereη λ is interpreted as a noise by the denoising method Here, λ is an algorithm-specific scaling parameter: inthecaseof Gaussian smoothing, for example, the variance of the Gaussian kernel may serve as such a scaling parameter Small-scale features, categorized as noise, are then forced into η λ, resulting in a larger-scale version, U λ, of the original image f PDE-based, variational, and hierarchical decompositions Two main approaches for multiscale representations of images are PDE-based methods and variational methods The basic PDE-based method is the heat equation, u t =Δu, where,startingwiththegiven image as initial condition, u( := f, it produces a multiscale representation, {u(t} t>,forthe image f The heat equation removes noise through blurring, but being an isotropic diffusion it also blurs edges which define main features in the image This drawback can be removed by using instead the Perona Malik [29] nonlinear equation based on nonisotropic diffusion, u t =div(g( u u The equation is ill-posed, however, and as already noted by these authors, the Perona Malik diffusion can lead to false detection of edges in the presence of noise To resolve this issue, Catté etal[] proposed using a modified nonisotropic and nonlocal diffusive amplitude of the form g( G σ u ; this class of methods was studied in [9, 36] All these methods give rise to a forward multiscale representation of the image f, in the sense that u(t begins with the finest scale, u( := f, and then progressively blurs into coarser representations of f as t The class of variational methods is a widely used alternative to PDE-based methods in image processing Methods such as Mumford Shah segmentation [25, 26] and Rudin Osher Fatemi (ROF decomposition [3] fall under a general category of Tikhonov regularization [35, 3] Here one attempts to find a close approximation to a function f Y,inaspace X Y, which is an appropriate space adapted to measure edges and textures sought in u This leads to the following minimization problem: { u X + λ v Y } f = u λ + v λ, [u λ,v λ ]:=arginf f=u+v The term u X is a regularizing term, and u λ + v λ is a multiscale decomposition of f which varies with the positive scaling parameter, λ For references on this class of u + v methods we refer the reader to [] In the case of the ROF model [3], for example, edges are sought in the space of bounded variations, X = BV (Ω and f Y = L 2 ; see, eg, [6] This yields the (BV,L 2 -decomposition: { u BV + λ v 2 L 2 }, ( f = u λ + v λ, [u λ,v λ ]:=arginf f=u+v where u BV := Ω u is the BV -seminorm For small values of λ, the minimizer u λ is a large-scale image which consists of only main features and prominent edges in f As λ becomes larger, u λ captures the small-scale details of f The point of view advocated in [33] is to replace the fixed scaling parameter λ by a varying sequence of inverse-scale parameters, λ <λ 2 <, which dictate the scaling of the iterative refinement step, (2a [u λj+,v λj+ ]:= arginf v λj =u+v { u BV + λ j+ v 2 L 2 },

3 32 PRASHANT ATHAVALE AND EITAN TADMOR which in turn generates a multiscale (BV,L 2 hierarchical representation, (2b f j u λj 2 A novel (BV, L -based integro-differential equation (IDE In [32] we introduced the integro-differential equation t (2 u(x, s ds = f(x+ ( u(x, t 2λ(t div u, = u(x, t n Ω This novel IDE was based on a hierarchical (BV,L 2 decomposition (2 which was introduced by Tadmor, Nezzar, and Vese [33, 34], from which it inherits many important properties Once we were dealing with the framework of IDEs, several extensions of (2 wereproposedin[32] to deal with denoising and deblurring of images, which were not strictly associated with any variational problem The ROF decomposition ( usesthel 2 -fidelity term f u 2 L Analternativevariational problem based on an L -fidelity term, f u L, was proposed earlier by Alliney [4] 2 and was studied extensively by Chan, Esedoḡlu, and Nikolova (see [2, 27, 3]: { } (22 f = u λ + v λ, [u λ,v λ ]:=arginf u BV + λ v L f=u+v The resulting (BV,L minimization differs from the (BV,L 2 modelinseveralimportant aspects which have attracted considerable attention in recent years; see [, 2, 3, 6] We shall mention two such aspects The (BV,L minimization is contrast invariant, asopposedto the (BV,L 2 minimization Chan and Esedoḡlu [2] alsoshowedthemorelocal geometric aspects of the (BV,L minimization In particular, recall that if supp(f B R (, then both the (BV,L andthe(bv,l 2 minimizers vanish, namely, u λ andv λ = f for small λ s below the minimal threshold λ<λ L R On the other hand, if the f is a characteristic function χ Σ, then the (BV,L minimizer of (22 admits a maximal threshold, λ H, such that for all λ>λ H we have u λ = f and v λ, in contrast to the leaking phenomenon with the (BV,L 2 -based ROF model, where v λ R [22] 2 Quantization To proceed with the derivation of a (BV,L -based IDE, we first need to address a proper quantization of the image intensity If we let τ denote the small intensity quanta, then we can rescale the coarse image in (22, u λ,inτ-units With this in mind, we decompose the given image f using the (BV,L scheme, starting with initial scale λ : f = u λ τ + v λ, u λ := arginf u { u BV + λ τ f uτ L } The term div ( u(,t, which often arises in image processing, is the curvature of isolevel curves of function u(,t[8, 7] In the present context, it arises as the first variation of the BV -norm, u BV,whichin practice is replaced by its regularized form, eg, u BV = u 2 L + ε2, thus removing the ambiguity when u = and the possible lack of uniqueness in the corresponding (BV,L minimization in (22 [, 3]

4 (BV, L INTEGRO-DIFFERENTIAL EQUATIONS FOR IMAGES 33 The residual image v λ := f u λ τ can be further decomposed into a smaller scale with λ >λ : { v λ = u λ τ + v λ, u λ := arginf u BV + λ } u τ v λ uτ L We can continue this process for λ <λ <λ 2 < : v λk = u λk τ + v λk, u λk := arginf u { u BV + λ } k τ v λ k uτ L A telescoping sum of the refinement step v λk v λk = u λk τ yields the following hierarchical (BV,L decomposition of f: f = u λ τ + v λ = u λ τ + u λ τ + v λ = = u λ τ + u λ τ + + u λn τ + v λn Remark 2 We note that in order to begin with only the main features of f, the initial scale needs to be small but not too small: to capture a nontrivial minimizer, λ needs to be larger than a minimal threshold, which is quantified in terms of the dual -norm specified in (27 below[34, eq (25], [22]; ie, (23 λ >λ L := / sgn(f This yields a hierarchical (BV,L multiscale image decomposition, (24 f = N k=k u λk τ + v λn, where k signals the initial scale λ k to yield the first nontrivial minimizer u λ Consider the Nth step in the (BV,L scheme, ( u λn =arginf u + λ N v λn uτ u Ω τ Ω The Euler Lagrange equation associated with this minimizer reads (with the usual understanding of a regularized curvature term as (25 sgn (u λn τ v λn = ( uλn div λ N u λn From (24 we have N v λn = f u λk τ, k=k

5 34 PRASHANT ATHAVALE AND EITAN TADMOR and using this above expression in (25 we find N sgn u λk τ f k=k = λ N div ( uλn u λn Letting τ, we arrive at the following (BV,L IDE: (26 sgn u(x, s ds f(x = ( u(x, t s=t λ(t div u, =, t t, u(x, t n Ω subject to the initial condition u(,t = The scaling function λ(t is any monotone increasing function at our disposal We discuss the role of this function in the next section 22 On the scaling function λ(t It is argued in [22] that the dual norm, w, ϕ ϕ (27 w := sup L 2, ϕ = ϕ(xdx, ϕ BV ϕ BV Ω Ω is a proper norm to measure texture (equivalently, one can characterize this dual norm as w := inf g { g L :divg = w w} To understand the critical role of the scaling function λ(t intheidemodel(26 and its relationship with the star-norm, we first prove the following useful lemma Lemma 2 For u BV,letκ u be the curvature term, κ u := div ( u, whenever u Then κ u = Proof For ϕ BV we have the following: ( (28 (κ u,ϕ L 2 = div ( u u,ϕ L 2 ϕ BV Thus, we have κ u Letting ϕ = u in (28, we obtain κ u,u L 2 = u BV and the result follows The following theorem is a direct consequence of this lemma Theorem 22 Consider the IDE model (26 sgn (U(x, t f(x = λ(t div ( u(x, t u(x, t t, U(x, t := u(x, s ds t Then, the size of the residual, V (,t:=f t t u(,s ds, is dictated by the scaling function λ(t: sgn(v (,t = λ(t The star-norm measures oscillations [22] Thus, the above equality could be interpreted as saying that the oscillations in the residual image V (,t are inversely proportional to the scaling function λ(t In particular, the oscillations in the residual function diminish in a precise manner controlled by increasing λ(t

6 (BV, L INTEGRO-DIFFERENTIAL EQUATIONS FOR IMAGES 35 Remark 22 Recall that t in Theorem 22 is determined as the minimal effective scale in the sense that (23 holds; ie, λ(t >λ L =/ sgn(f One does not need, however, to have a priori knowledge on the amount of star-norm oscillation in f, sinceu(,t as long as λ(t is underestimated, ie, λ<λ L Eventually, the increasing λ(t will pick up the minimal effective scale 23 L -energy decomposition The increase of λ(t is directly linked to the energy of the underlying image For the (BV,L 2 -based IDE (2, we proved in [32] thel 2 -energy decomposition t s=t λ(s u(,s BV ds + V (,t 2 L = f 2 2 L 2 For the (BV,L -based IDE we have the corresponding L -energy decomposition Proposition 23 Consider the IDE (26 and let V (,t denote the residual, V (,t := t t u(,s ds f Then we have the following L -energy decomposition: t t λ(s u(,s BV ds + V (,t L = f L Proof We denote by U(,t:= t t u(,s ds the cumulative image so that the IDE (26 takes the form ( u(x, t λ(t div = sgn(u(x, t f(x u(x, t Integrating the above against u d dt (U f, we get λ(t u(,t BV = sgn(u(,t f d U(,t f dx Ω dt( = d sgn(u(,t f(u(,t f dx = d U(,t f dx, dt dt Ω and the result follows by time integration 3 IDEs for images: Numerical results and extensions 3 Comparing the (BV, L -and(bv, L 2 -based IDEs The multiscale image representation generated by the IDE (26 is shown in Figure It is instructive to look at the image representation produced by the (BV,L -based IDE (26 withthe(bv,l 2 -based IDE (2 In the first row of Figure 2 we see the multiscale representation using the new (26 Note that the larger blocks appear before the small blocks, irrespective of their intensity levels We observe that the appearance of these blocks is abrupt, as opposed to the results obtained with (2 shown in the second row of Figure 2, where these blocks appear gradually and the multiscale representation produced by (2 depends on the intensity levels rather than the size of the blocks This feature makes (BV,L IDE(26 more suitable for multiscale representation than (BV,L 2 IDE(2 The difference between the (BV,L -based IDE and the (BV,L 2 -based IDE is demonstrated clearly in Figure 3 We note that, for the same time, the (BV,L IDE gives better Ω

7 36 PRASHANT ATHAVALE AND EITAN TADMOR t = t = t =3 t =25 Figure The images U(t = t u(,s ds of the IDE (26 at t =,, 3, 25 Here, λ(t = 2t t = t =3 t =5 t =7 t = t =4 t =6 t = Figure 2 First row: the images U(t = t u(,s ds of the (BV,L -based IDE (26 at t =, 3, 5, 7 Here, λ(t =5 2 t Second row: the images U(t = t u(,s ds of the (BV,L2 -based IDE (2 at t =, 4, 6, Here, λ(t = 2 t results at the edges than the (BV,L 2 IDE The(BV,L IDE is seen to be contrast invariant and depends only on the scale of the blocks, whereas the (BV,L 2 IDEisnotcontrast invariant, as expected 32 IDE with filtered diffusion Recall that one of the drawbacks in using the heat equation for denoising is that it results in an isotropic diffusion The Perona Malik model removes this drawback by introducing a diffusion controlling function that controls the diffusion near prominent edges in a given image We propose a similar modification to our IDE model, seeking u(x, t :Ω R + R such that (3a sgn u(x, s dx f(x = g( G ( σ u(x, t u(x, t div, t > t, s=t λ(t u(x, t

8 (BV, L INTEGRO-DIFFERENTIAL EQUATIONS FOR IMAGES 37 (a (b (c Figure 3 In (a, (b, and (c we see the mesh views of the original image from Figure 2, the result U(t = t u(,s ds for the (BV,L2 -based IDE (2 at t =, and the image U(t = t u(,s ds for the (BV,L -based IDE (26, alsoatt =, respectively subject to normal boundary conditions u n Ω = As before, the IDE (3a is activated after the minimal time scale, t t, such that u(,t is the first slice of f to capture nontrivial features at that scale Similar to the Perona Malik-type models, we can choose the prefactor function g so that it vanishes at infinity to control the diffusion at prominent edges in the image Thus, the function g acts here as a high-pass filter which retains prominent edges in the image t u(x, s ds without diffusing them As choices for such a g-filter, Figure 5 displays the results of the modified IDE (3a with (3b g(s = +(s/β 2 Here, the constant β determines the extent to which edges are preserved: for small β s, relevant edges are preserved, whereas for large β s, they are diffused Detailed discussion of the numerical scheme for the filtered diffusion model (3 is given in section 4 Comparing the results of the filtered IDE (3a showninfigure5, we observe that edges, which are diffused by the basic IDE (26 as depicted in Figure 4, are preserved in Figure 5 As inthecaseof(2 (see[32], this phenomenon is more apparent for smaller values of t due to the fact that as t increases, U(,t in both models approaches f and, consequently, suffers from less diffusion of the edges The usefulness of the filtered diffusion IDE model becomes apparent when certain edges are required in the scale space for smaller values of t For example, in Figure 4, the edges are blurred for smaller values of t with the standard IDE (26, but with the filtered diffusion IDE (3a we retain relevant edges, as shown in Figure 5 We remark that this edge preserving filtered diffusion could be used in landmark-based hierarchical image registration algorithms [23, 28] 33 IDE with tangential smoothing The approach of using the diffusion controlling function works well with natural images with moderate gradients With other images, however, such as those which often arise in computer vision and industrial applications, the boundaries of their internal objects are marked with large, sharp gradients, for example, characteristic function χ D,whereD Ω In such cases, we can choose to smooth only in the tangential

9 38 PRASHANT ATHAVALE AND EITAN TADMOR t =4 t =42 t =44 t =6 Figure 4 The images U(t = t u(,s ds of the IDE (26 at t =4, 42, 44, 6 Here, λ(t = 2t t =2 t =38 t =4 t =6 Figure 5 The images U(t = t u(,s ds of the IDE (3a at t =2, 38, 4, 6 Here, λ(t = 2t direction to the boundaries of the objects, eg, [5] To this end, write Δu := u TT + u NN, where u TT and u NN are the tangential and normal diffusion components, ie, ( u u u TT =Δu u NN = u div, u NN = u u, 2 u u u If we restrict the diffusion in our IDE model to tangential directions, we obtain modified IDEs with tangential smoothing, (32 sgn u(x, s ds f(x = ( u(x, t u u(x, t div, =, t λ(t u(x, t n Ω and with tangential smoothing and filtering, (33 sgn u(x, s ds f(x t = g( G ( σ u(x, t u(x, t u(x, t div λ(t u(x, t As before, u :Ω R + R evolves in inverse scale space using the conditions u(, and u n Ω = Numerical experiments are shown in Figures 6 and 7 Comparethe(BV,L IDE results (26 shown in Figure 6 with the tangential smoothing (33 shown in Figure 7: the point here is that the tangential diffusion model preserves the edges, while denoising the rest of the image at a much faster rate than in the standard IDE model

10 (BV, L INTEGRO-DIFFERENTIAL EQUATIONS FOR IMAGES 39 f at t = t = t =4 t =8 Figure 6 A given noisy image f and the IDE images t u(,s ds of (26 at t =, 4, 8 Here, the scaling function is λ(t = 2 t Most of the noise is present at scale t =8 f at t = t = t =4 t =8 Figure 7 A given noisy image f and the IDE images t u(,s ds of (33 at t =, 4, 8 Here, the scaling function is λ(t = 2 t We observe that at t =8the noise is suppressed and at the same time the boundary of the circle is kept intact 4 Appendix: Numerical discretizations In this appendix we describe the numerical implementation of (26 and its variants First let us concentrate on the basic (BV,L IDE model (26, rewritten here for convenience (for simplicity we set the initial scale at λ(t at t =, (4 sgn u(x, s ds f(x = ( u(x, t λ(t div u(x, t In practice we approximate the BV -seminorm u BV with Ω u 2 + ɛ 2, which leads to the following IDE: ( (42 sgn u(x, s ds f(x = λ(t div u(x, t ɛ 2 + u(x, t 2 In our numerical experiments we used the regularization parameter ɛ = 3 Then, as usual, U(t := t u(x, s ds is the exact solution Let Δt bethetimestep,andletun+ denote the corresponding computed solution at t n+ =(n +Δt: U n+ = U n + W n+, W n+ Wi,j n+ := u n+ i,j Δt, where u n+ i,j u n+ (ih, jh is the approximate solution of the IDE at grid point (ih, jh We introduce α n := U n f, and we compute the increment W n+ by fixed-point iterations {ω k }

11 3 PRASHANT ATHAVALE AND EITAN TADMOR With this, the IDE (42 is discretized at t = t n+ : (43 U n i,j + ω k+ i,j + + = f i,j ωi+,j k ωk+ i,j ωi,j k+ ωi,j k ε 2 +(D +x ω ki,j 2 +(D y ω ki,j 2 ε 2 +(D x ωi,j k 2 +(D y ωi,j k 2 ωi,j+ k ωk+ i,j ωi,j k+ ωi,j k ε 2 +(D x ω ki,j 2 +(D +y ω ki,j 2 ε 2 +(D x ωi,j k 2 +(D y ωi,j k 2 α n λ (n+ h 2 α n λ (n+ h 2 The nonlinear system (43 is solved using Jacobi iterations, which leads to the fixed-point iterations for computing ω k+ : (44a i,j = λ(n+ h 2 (f i,j Ui,j n +αn (c E ωi+,j k + c W ωi,j k + c Sωi,j+ k + c N ωi,j k λ (n+ h 2 + α n (c E + c W + c S + c N ω k+ Here, λ (n+ = λ(t n+ are the discrete scaling parameters, and c E,c W,c S,c N are the discrete coefficients c E := c S := ε 2 +(D +x ω ki,j 2 +(D y ω ki,j 2, c W := ε 2 +(D x ω ki,j 2 +(D +y ω ki,j 2, c N := ε 2 +(D x ω ki,j 2 +(D y ω ki,j 2, ε 2 +(D x ω ki,j 2 +(D y ω ki,j 2 In the computations above we set h = To minimize the grid effects, we alternate the directions in which the above iterations are carried out, starting at the top-left corner position (, Fixing i =,wevaryj =toj max (east-south direction, initiating the next iteration at the top-right corner, and so on The fixed-point iterations (44a yield ω k k W n+ u n+ Δt, and we can then update the computed image U, (44b U n+ = U n + W n+ Next, we consider the filtered IDE (3a, which is rewritten here for convenience as u(x, s ds f(x = g( G ( σ u(x, t u(x, t div λ(t u(x, t The only difference here is the additional diffusion controlling function g( G σ u(x, t, where G σ is the two-dimensional Gaussian smoothing with standard deviation σ The function g(s = with β = 5 is used in our numerical experiments We approximate +(s/β 2 ( Gσ g( G σ u(x, t g ωn i,j, Δt and the expression on the right enters into the right-hand side of (43 Weendupwiththe same discrete IDE scheme (44 withλ (n λ (n/ g( G σ ω n i,j /Δt

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