An Adaptive Variational Model for Image

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is given in a variational formulation with adaptive regularization norms for both ... decompose a given degraded image I into a component u which is the 'true' im-.
An Adaptive Variational Model for Image Decomposition Stacey E. Levine

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Department of Mathematics and Computer Science 440 College Hall, Duquesne University Pittsburgh, PA, 15282 phone: (412) 396-5489; fax (412) 396-5197 [email protected]

Abstract. We propose a new model for image decomposition which separates an image into a cartoon, consisting only of geometric objects, and an oscillatory component, consisting of textures and noise. The model is given in a variational formulation with adaptive regularization norms for both the cartoon and texture part. The energy for the cartoon interpolates between total variation regularization and isotropic smoothing, while the energy for the textures interpolates between Meyer’s G norm and the H −1 norm. These energies are dual in the sense of the LegendreFenchel transform and their adaptive behavior preserves key features such as object boundaries and textures while avoiding staircasing in what should be smooth regions. Existence and uniqueness of a solution is established and experimental results demonstrate the effectiveness of the model for both grayscale and color image decomposition.

1

Introduction

One of the fundamental objectives in image processing is to extract useful information from an image. This often requires decomposing an image into meaningful parts. A classic example is the image denoising problem in which the goal is to decompose a given degraded image I into a component u which is the ’true’ image and a component v containing ’noise’. These quantities are usually related by I = u + v. Most denoising methods aim to only recover u, the ’true’ image since in this setting the residual v is thought to contain no meaningful information. Traditionally, variational approaches for image denoising search for a solution u in BV , the set of functions of bounded variation. This is the natural space for modeling ’cartoon’ type images, since elements of BV consist of smoothly varying regions with sharp object boundaries. However, Gousseau and Morel [15] showed that real images which contain natural oscillations or textures cannot be modeled in BV. To overcome this problem, Meyer [18] formulated the image decomposition problem as follows. Given a degraded image, I, decompose I ?

This work was supported in part by NSF Grant DMS 0505729

into a ’cartoon’ part u containing only geometric objects, and an oscillatory part v containing noise and textures. I, u, and v should be still be related by the equation I = u + v. However, when solving this problem, the goal is to simultaneously recover both u and v, assuming they both contain meaningful information. Several different approaches have been used to determine such a ’cartoon + texture’ decomposition. One class of models is based on statistical methods or wavelet techniques, e.g. [7, 12, 17, 18]. Meyer’s work [18] has inspired several variational formulations for solving this problem, e.g. [3, 4, 16, 20, 22]; this is the approach we take here. In this work we propose a new variational model for image decomposition which separates an image into a cartoon and texture component. The main feature of the proposed model is that the regularization norms for both the cartoon and texture part are adaptive to the local image information and remain in duality. At locations with high gradient (likely edges or textures in I), total variation based regularization is used for the cartoon u and a minimization energy which favors highly oscillatory functions is used for v. At locations with low gradient (likely homogeneous regions in I), isotropic smoothing is used to prevent staircasing for u and a more regularizing energy is used for v. More precisely, for Ω ⊂ R2 and 1 ≤ q < ∞ denote the Sobolev spaces W 1,q (Ω) = {u ∈ Lq (Ω) | ∇u ∈ Lq (Ω)} and H 1 (Ω) = W 1,2 (Ω). Then W −1,p (Ω) is the dual space of W01,q (Ω) (functions in W 1,q (Ω) that vanish on the boundary of Ω) where 1q + p1 = 1. Using this notation, the proposed model behaves as follows. Depending on the strength of the gradient, the minimization energy for the cartoon interpolates between the total variation semi-norm (≈ W 1,1 norm) and the W 1,2 = H 1 norm while that for the oscillating component simultaneously interpolates between their dual norms in the spaces W −1,∞ and W −1,2 = H −1 . The duality between the energies for u and v enables the model to correctly self-adjust when handling key features in the image, specifically, object boundaries, textures and noise. This also prevents false edge generation (or ’staircasing’) in the cartoon part yielding a truly piecewise smooth image. The paper is organized as follows. Section 2 provides a brief survey of related variational decomposition models. In section 3, the proposed model is described. Section 4 contains the numerical implementation for both grayscale and color images, and section 5 contains the experimental results. Section 6 will have the concluding remarks. The appendix contains the proof of existence and uniqueness of a solution.

2

Total Variation (TV) Based Image Decomposition Models

The ‘cartoon + texture’ decomposition problem considered here for two dimensional grayscale images is formulated as follows: Given an image I : Ω ⊂ R 2 → R,

find a decomposition I = u + v where u : Ω → R is piecewise smooth consisting only of geometric objects and their boundaries and v : Ω → R contains only oscillating features such as textures and noise. The space most widely accepted for modeling piecewise smooth ’cartoon’ typeR images u is the set of functions of bounded variation, BV (Ω) = {u ∈ L1 (Ω) | Ω |Du| < ∞} [14], where the total variation semi-norm of u is Z Z |Du| := sup{ u(x)div(ξ(x)) | ξ ∈ Cc1 (Ω; R2 ), ||ξ||L∞ (Ω) ≤ 1}. Ω

Ω

Rudin-Osher-Fatemi [21] proposed the now classic total variation (TV) denoising model which can be formulated as follows: Z min 2 |Du| + λ||v||2L2 (Ω) (1) I=u+v,u∈BV ×L (Ω)

Ω

The fundamental strength of TV based denoising is its ability to keep sharp boundaries in the cartoon u while removing noise. However, if the desired goal is to create a ’cartoon + texture” decomposition, the L2 norm may not be most appropriate minimization energy for the oscillatory component v. While u, the part of the image containing geometric structures, can still be modeled in BV, the oscillating patterns in v should be modeled in a space which lies somewhere between BV and L2 [15]. In addition, Meyer showed there are simple cases in which (1) will not create a true u + v decomposition. For example, if I ∈ BV is the characteristic function of a region with smooth boundary, it would be reasonable to assume the (1) should yield the decomposition u = I and v = 0. However, this is not always the case for λ > 0 [18]. The problem does not necessarily stem from assuming v ∈ L2 (Ω), but from using the L2 norm as the minimization energy for v. Meyer proposed several alternate norms for v which allow for more oscillations. The one this work will focus on is that of the Banach space G(R2 ) = {f = divξ | ξ ∈ L∞ (R2 , R2 )}, with norm q ||v||G(R2 ) = inf{|| |ξ| ||L∞ (R2 ,R2 ) | ξ ∈ L∞ (R2 , R2 ), |ξ| = ξ12 + ξ22 }

where ξ = (ξ1 , ξ2 ). Very oscillatory signals have small G norm (see [18] for examples), so this energy will preserve both textures and noise. Meyer modified (1) using the G(R2 )-norm for v and proposed the following model: Z 1 inf |Du| + ||v||2G(R2 ) (2) 2λ (u,v)∈BV ×G(R2 ), R2 I=u+v

The notion of duality in the decomposition u + v exists, as G(R 2 ) is precisely the space W −1,∞ (R2 ), the dual of W01,1 (R2 ) which is very close to the space BV (R2 ). The duality also gives some insight into the presence of the divergence operator in the definition of G(R2 ), since the gradient and the divergence are dual operators.

Aubert and Aujol [2] studied a natural analogue of Meyer’s model (2) on a bounded domain Ω ⊂ R2 . In order to do this, they replaced G(R2 ) with the space G(Ω) := {v ∈ L2 (Ω) | v = divξ, ξ ∈ L∞ (Ω, R2 ), ξ · N = 0 on ∂Ω}

(3)

which is contained in W −1,∞ (Ω). Here, N is the unit outward normal to the boundary of Ω, ∂Ω. Remark 1. If an image I ∈ L2 (Ω) is decomposed using a u + v model where u ∈ BV (Ω) ⊂ L2 (Ω), then regardless of the minimization energy imposed on v, it must be that v ∈ L2 (Ω). So (2) still yields a u + v decomposition with v ∈ L2 (Ω). However, the u + v decomposition obtained by (2) is different than that obtained by (1) (see [2, 16, 18, 22] for examples). Vese and Osher [22, 23] proposed the first numerical implementation of (2). Their method decomposes an image into a cartoon part, u, and further separates the textures into two components, ξ1 and ξ2 , which model the oscillating patterns in the horizontal and vertical directions respectively. They proposed the model Z inf{ |Du| + λ||I − (u + divξ)||2L2 (Ω) + µ|| |ξ| ||Lp (Ω) | (4) Ω

(u, ξ) ∈ BV (Ω) × Lp (Ω, R2 ))} p where ξ = (ξ1 , ξ2 ), |ξ| = ξ12 + ξ22 , λ, µ > 0 and p ≥ 1 are fixed, and Ω ⊂ R2 is a bounded open set. Their model differs from (2) in the second and third terms. The second term is a data term enforcing fidelity with the initial data, I, that is, it forces I ≈ u + v = u + divξ. The third term is an approximation of the energy for v in Meyer’s model in the sense that inf ξ || |ξ| ||Lp → ||v||G as p → ∞. Following the the success of the Vese-Osher model, these two authors along with Sol´e [20] proposed a higher order denoising method based on (4) for the case p = 2. Recently, Le and Vese [16] proposed another decomposition in which the textures are modeled in div(BM O), a slightly larger space than G, thus allowing more oscillatory components. An alternate discrete approximation of the G-norm was proposed by Aujol et.al. [3, 4] based on Chambolle’s duality based projection method for implementing the discrete TV model [6]. All of the above mentioned models have demonstrated that variational decomposition methods are computationally stable, efficient and effective. Furthermore, they are based on mathematically sound foundations. In [2–4, 16] the authors prove the existence of a solution, as well as analyze the nature of the solutions. Current variational decomposition models have focused on developing more appropriate norms for modeling oscillating features v which are in some sense dual to the TV semi-norm. In all of the above mentioned models, the TV seminorm is used to model the cartoon u. However, the cartoon part still exhibits staircasing when using the TV energy for u (see figure 1). In this paper, we investigate different energies for both the cartoon u ∈ BV and oscillating component v ∈ L2 which show less evidence of staircasing in u while still preserving sharp object boundaries in u and textures in v.

Fig. 1. Image decomposition using the TV-based model (4) with µ = .001 and λ = .025. First Column: Initial Images I; Second Column: texture part v; Third Column: cartoon part u; Fourth Column: edge maps (18) of above cartoon images u. Both images were run for 250 iterations.

3

Image Decomposition via Adaptive Regularization

3.1

Adaptive Cartoon Reconstruction

Variational methods with cost functionals that interpolate between total variation based W 1,1 smoothing and isotropic W 1,2 = H 1 smoothing overcome the problems of staircasing in cartoon image reconstruction while still preserving sharp object boundaries. Several denoising models using this kind of interpolation have been studied (e.g. [5, 8, 11] and references there-in). In [11] the following adaptive model was proposed Z φ(x, ∇u) + λ||v||2L2 (Ω) . min 2 (5) I=u+v,u∈BV ∩L (Ω)

where

Z

φ(x, ∇u) = Ω

(

Ω

1 q(x) , q(x) |∇u| q(x)²−²q(x) |∇u| − , q(x)

e and q(x) = q(|∇I(x)|) satisfies

e =2 lim q(|∇I|)

e |∇I|→0

lim

e |∇I|→∞

|∇u| < ² |∇u| ≥ ²

,

e =α>1 q(|∇I|)

(6)

(7)

e is monotonically decreasing and q(|∇I|)

(here Ie is a smoothed version of I). For example, one could choose q(x) = 1 +

1 1 + k|∇Gσ ∗ I|2

(8)

Fig. 2. Image decomposition using the proposed model (13) with µ = .1 and λ = .05. First Column: Initial Images I; Second Column: texture part v; Third Column: cartoon part u; Fourth Column: edge maps (18) of above cartoon images u. Both images were run for 250 iterations.

¡ ¢ where Gσ = σ1 exp −|x|2 /4σ 2 is the Gaussian filter and σ, k > 0 are fixed. The main feature of this model is that the type of regularization depends on the local image information. At likely edges, or regions with sufficiently high gradient, TV based regularization is used. In likely homogeneous regions where the gradient is very low, q(x) ≈ 2 and isotropic smoothing is used. At all other locations, φ self adjusts to use the appropriate combination of TV and isotropic smoothing. Furthermore, the functional (6) is both convex and lower semi-continuous. In particular, when the threshold, ², is set to 1 we have that ( 1 Z q(x) , |∇u| < 1 q(x) |∇u| φ(x, ∇u) = , q(x)−1 |∇u| − q(x) , |∇u| ≥ 1 Ω ¶ Z µ 1 p(x) = sup −udivξ − |ξ| dx (9) p(x) |ξ|≤1,ξ∈C 1 (Ω,R2 ) Ω 1 1 + p(x) = 1 for all x ∈ Ω. This leads to a mathematically sound model where q(x) which is established in [11].

3.2

New Model: Adaptive Cartoon + Texture Decomposition

The relationship (9) R gives insight into the decomposition model proposed in this paper. Since Ω φ(x, ∇u) is proper, convex, and lower-semicontinuous, it demonstrates that the functionals Z Z 1 φ(x, ∇u) and Ψ (ξ) := |ξ|p(x) (10) Φ(u) := Ω p(x) Ω

are essentially conjugate in the sense of the Legendre-Fenchel transform [13], e where φ(x, ∇u) is defined in (6), 1 < α ≤ q(x) = q(|∇I(x)|) ≤ 2 satisfies (7), and 1 1 + = 1 for all x ∈ Ω. (11) q(x) p(x) Based on the notion of duality between the cartoon and oscillatory components, we define a functional F : BV (Ω) × L2 (Ω, R2 ) → R by F (u, ξ) := Φ(u) + λ||I − u − divξ||2L2 (Ω) + µΨ (ξ)

(12)

for I ∈ L2 (Ω) and propose the following image decomposition model: Z Z 2 Idx} udx = inf 2 {F (u, ξ) | divξ ∈ L (Ω) and 2 (u,ξ)∈BV (Ω)×L (Ω,R )

Ω

(13)

Ω

using definitions (10)-(12) with λ, µ > 0. Remark 2. We mention the key features of the model (12)-(13). 1. The adaptive nature of the model is exploited by the conjugacy of the variable exponents (11). At locations with large gradient such as edges, textures, and noise, Meyer’s model (1) should be used since these are the main features used to distinguish between geometric objects and oscillating patterns. In this case, T V (≈ W 1,1 ) regularization for the cartoon u retains sharp object boundaries. Furthermore, W −1,∞ regularization for the oscillatory component keeps only smaller scale patterns in v. The proposed model (12)e → ∞, (13) has this feature since as |∇I| e → 1 and p(|∇I|) e → ∞. q(|∇I|)

On the other hand, isotropic smoothing should be used in likely homogeneous regions in order to reduce the effects of staircasing (see figure 2). In this case, H 1 smoothing should be used for the cartoon and H −1 for the oscillatory component. The H −1 norm was proposed and analyzed in [20] for modeling the residual v in order to solve the image denoising problem. The authors demonstrate that it successfully contributes to a smoother cartoon image. e → 0, Again, the conjugacy of the exponents yields this behavior since as |∇ I| e → 2 and p(|∇I|) e → 2. q(|∇I|)

1 2. The coefficient p(x) in Ψ (ξ) (see (10)) not only preserves the notion of duality between the energies for the cartoon, u, and oscillations v = divξ, but it also provides a natural scaling for the oscillations. At highly oscillatory features, 1 p(x) will be very small, thus allowing more oscillations to be preserved in ξ. α Since 1 < α ≤ q(x) ≤ 2 we have that 2 ≤ p(x) ≤ α−1 < ∞ so the coefficient is never zero. 3. The fidelity term λ||I − (u + divξ)||2L2 ensures that I ≈ u + divξ = u + v as in (4).

R R 4. The constraints divξ ∈ L2 (Ω) and Ω udx = Ω Idx in (13) arise naturally in this decomposition problem. Since I ∈ L2 (Ω), it is necessary that v = divξ ∈ L2 (Ω). Then the texture component v = divξ lies in the space {v ∈ L2 (Ω) | v = divξ with ξ ∈ L2 (Ω, R2 )} which is slightly bigger than G(Ω) (see (3)). This may allow for more R osciludx = latory features to be captured in v = divξ. The requirement that Ω R R vdx ≈ 0, a natural assumption for oscillating patterns. Idx forces Ω Ω Furthermore, it ensures that the model is well-posed. 5. The model (12)-(13) is well-posed. The proof of existence and uniqueness can be found in the appendix. We refer the reader to new related work in staircase-reducing decomposition models using higher order energies for the cartoon in [10, 9].

4

Numerical Implementation

We implemented (12)-(13) by solving it’s corresponding Euler-Lagrange equations ³ ´ 1 div |∇u|qˆ(x)−2 ∇u (14) u = I − div(ξ1 , ξ2 ) + 2λ q p(x)−2 ∂ µξ1 ξ12 + ξ22 (15) = 2λ (u − I + div(ξ1 , ξ2 )) ∂x q p(x)−2 ∂ µξ2 ξ12 + ξ22 (16) = 2λ (u − I + div(ξ1 , ξ2 )) ∂y where ξ = (ξ1 , ξ2 ) and ( q(x), qˆ(x) = 1,

|∇u| < ², 1 1 + = 1 for all x ∈ Ω and q(x) p(x) |∇u| ≥ ²

(17)

1 Ix with q(x) chosen as in (8). Our initial data is u(x, 0) = I(x), ξ1 (x, 0) = − 2λ |∇I| I

y 1 and ξ2 (x, 0) = − 2λ |∇I| , and the natural Neumann boundary conditions are used. To avoid dependency on the threshold ², this value was automatically updated to ensure that at least 75% of the pixels were using adaptive smoothing, qˆ(x) = q(x). Division by zero ³is avoided in the diffusion ´ term by approximating p ¡ ¢ qˆ(x)−2 qˆ(x)−2 2 2 |∇u| + δ ∇u for small δ > 0. In the div |∇u| ∇u with div case qˆ(x) ≡ 1, this approximation yields the correct solution as δ → 0 [1]. Our experimental results found this approximation is also stable for general 1 ≤ qˆ(x) ≤ 2. We approximated (14)-(16) using a semi-implicit scheme with central differences. We obtained similar results using a minmod approximation [21] for the diffusion term and believe other difference schemes should work as well.

4.1

Parameters

We tested the proposed model for different values of λ and µ. In (12)-(13), the parameter λ controls the level of fidelity between the initial image I and it’s decomposition u + v. The parameter µ controls the level of oscillatory features retained in v. These parameters play the same role as in the models (2) and (4). Meyer [18] showed that as µ decreases, more oscillatory features are kept 1 reduced the effect of µ on in v. We found that the adaptive coefficient p(x) the v component in the proposed model. For both parameters, values less than one yielded optimal results. The parameter k in the exponent q(x) determines the interpolation between the W 1,1 and H 1 , and the W −1,∞ and H −1 norms respectively. All of our images ranged in intensity from 0 to 255, and we found that a value of k = .05 worked well in each case. The threshold, ², in (17) is set so that 75% of the pixels use adaptive smoothing, qˆ(x) = q(x), and the others use TV-based regularization, qˆ(x) ≡ 1.

Fig. 3. Top Row: Decomposition using the proposed model (12)-(13) with µ = .001 and λ = .001. Bottom Row: Decomposition using the TV-based model (4) with µ = .001 and λ = .001; First Column: Initial Images I; Second Column: texture part v; Third Column: cartoon part u; Fourth Column: edge maps (18) of above cartoon images u. Both images were run for 200 iterations.

4.2

Comparison with TV-based decomposition model (4)

We chose to compare our model with (4) as the representative TV-based decomposition model since it was straightforward to implement, yields very good results, and both inspired and is most closely related to (12)-(13). When comparing with (4), we used the implementation proposed by the authors [22] and follow their example of setting the exponent on the oscillatory component to

p = 1. They reported finding similar results for 1 ≤ p ≤ 10 and when p is too large, only smaller scale textures were preserved. The parameters λ and µ were chosen so that both models yielded optimal results and v had roughly the same standard deviation for both (4) and (12)-(13).

4.3

Color image decomposition

A simple modification of the proposed model (12)-(13) yields similar decomposition results for color images, I : Ω ⊂ R2 → R3 . Our main goal in this work was to verify that the adaptive nature of the model would be preserved when applied to color images. Therefore, for our experiments, we used a straightforward channel by channel processing. Specifically, we decomposed the color image into it’s RGB components, processed each channel separately using (14)-(16), and then recombined to get the final results. There are other, potentially more optimal, color image representations that will further enhance the decomposition, however we found this one to be easy to implement and yielded results comparable to that for grayscale images I : Ω ⊂ R2 → R. We used a similar modification for the TV based model (4).

5 5.1

Experimental Results Grayscale Image Decomposition

Figures 1-2 provide a series of comparisons of the proposed model and (4). We first tested the TV image decomposition model (4) on two textured images (figure 1) in an effort to examine the nature of the cartoon image. The TV-based model successfully decomposes the image, however, false edges still appear in u in what should represent smoothly varying regions. This indicates that the cartoon u has undergone staircasing, visible in the cartoons’ edge maps. All edge maps in our examples were obtained by applying a standard gradient based edge detector 1 1 + .05|∇Gσ ∗ u|2

(18)

In figure 2, the same series of images were obtained using the proposed model (12)-(13). The proposed model also successfully decomposes the image and on closer examination of the cartoon image u, it is evident that the geometric regions remain piecewise smooth. This is further demonstrated by examining the corresponding edge maps. It is also interesting to note that while the textures are accurately separated into v, small details such as corners are well preserved in the cartoon. Figures 3 and 4 contains several more examples demonstrating the effectiveness of the proposed model (12)-(13) in reducing the effect of staircasing in grayscale image decomposition.

Fig. 4. Top Row: Decomposition using the proposed model (12)-(13) with µ = .1 and λ = .005 for 300 iterations. Bottom Row: Decomposition using the TV-based model (4) with µ = .001 and λ = .005 for 400 iterations; First Column: Initial Images I; Second Column: texture part v; Third Column: cartoon part u; Fourth Column: edge maps (18) of above cartoon images u.

5.2

Color Image Decomposition

When decomposing color images, we separated the image into it’s RGB components, processed each channel separately using (4) or (12)-(13) respectively, and then recombined to get the final results. The decompositions in figure 5 demonstrate that both (4) and (12)-(13) can be easily and successfully modified for decomposing color images. The presence or absence of staircasing for both models was similar to that for grayscale image decomposition. The staircasing effect for the cartoon u is still reduced using the adaptive method (12)-(13). Furthermore, the oscillating features are well preserved in v while the boundaries of objects such as the baboon’s eye or the rock under the lizard’s foot remain sharp in u. In figures 6 and 7 we give an example where the proposed model can identify regions which are obstructed, but not necessarily segmented by textures. Figure 6 contains two multi-spectral ASTER (Advanced Spaceborne Thermal Spaceborne Emission and Reflectance Radiometer) images which have both been the site of several wildfires. Identification of the fire scar boundaries is difficult to obtain with spectral information information alone given continuously changing properties such as those of the vegetation, atmosphere, etc. [19]. Gradient based edge detection would be insufficient in detecting the boundaries of the target regions given the presence of textures such as road patterns and topography. Furthermore, the textures retain the same oscillatory features both inside and outside the scars, so a proper segmentation could not be based solely on texture analysis. However, we found that performing an appropriate decomposition

Fig. 5. Top Row: original images, Second Row: texture part v, Third Row: cartoon part u, Fourth Row: edge maps (18) of cartoon u. The bottom three rows were decomposed using: 1st and 3rd Columns: proposed model (12)-(13) with µ = .001, λ = .005 for 100 and 75 iterations respectively, 2nd and 4th Columns: TV-based decomposition model (4) with µ = .001, λ = .005 for 200 and 75 iterations respectively.

will leave the target regions intact in the cartoon while removing the textures obstructing their boundaries. We compare the results of the adaptive and TV based decomposition models applied to both ASTER images. Both models preserve the textures in v as well as the fidelity of the scar edges in u. As in the previous examples, the adaptive model (12)-(13) retains piecewise smooth regions with clear boundaries in u with less evidence of staircasing both inside and outside the target regions.

6

Concluding Remarks

We presented a new model for image decomposition which separates an image into a cartoon and texture component. The model automatically adapts to the

Fig. 6. Top Row: left: ASTER image of 4 fire scars in the Phoenix, Arizona valley (numbered areas indicate burned regions); right: ASTER image of a remote region near Ayers Rock in Australia (yellow indicates burned regions).

local image information while preserving duality between the cartoon and oscillatory component. Sharp object boundaries are preserved while staircasing is avoided in the cartoon, while oscillating features are simultaneously preserved in the texture component. This model is mathematically sound (see appendix) and is successful in decomposing both grayscale and color images. Acknowledgements: We thank Michael Ramsey and Tamara Misner at the University of Pittsburgh for providing the LANDSAT images in Figure 6.

Appendix: Existence and uniqueness of a solution to (12)-(13) R Lemma 1. The functional Φ(u) = Ω φ(x, ∇u), where φ is defined in (6), is convex and lower-semicontinuous on L1 (Ω); that is, if uj , u ∈ BV (Ω) satisfy uj → u in L1 (Ω) as j → ∞ then Φ(u) ≤ lim inf Φ(uj ). j→∞

(19)

Proof. See [11] R 1 |ξ|p(x) dx Lemma 2. The functional Ψ : L2 (Ω, R2 ) → R defined by Ψ (ξ) := Ω p(x) (2 ≤ p(x) ≤ β < ∞ for x ∈ Ω) is convex and lower-semicontinuous on L1 (Ω, R2 ); that is, if ξj , ξ ∈ L2 (Ω, R2 ) satisfy ξj → ξ in L1 (Ω, R2 ) as n → ∞, then Ψ (ξ) ≤ lim inf Ψ (ξj ). j→∞

Proof. Straightforward computations give us that Z 1 r·ξ− Ψ (ξ) = sup |r|q(x) dx 2 q(x) r∈C(Ω,R ) Ω

Fig. 7. First Row: texture part v, Second Row: cartoon part u, Third Row: edge maps (18) of cartoon u. The images were decomposed using: First and Third Columns: proposed model (12)-(13) with µ = .001 and λ = .005 for 100 and 75 iterations respectively, Second and Fourth Columns: TV-based decomposition model (4) with µ = .001 and λ = .005 for 200 iterations each.

1 1 where q(x) + p(x) = 1 for all x ∈ Ω. Therefore, Ψ is the pointwise supremum of a family of continuous affine functions on L1 (Ω, R2 ) and is thus convex and lower-semicontinuous.

To prove existence and uniqueness of a solution to (12)-(13), we follow a similar argument to that used in [2, 3, 16]. ˆ ∈ BV (Ω) × Theorem 1. Let I ∈ L2 (Ω). u, ξ) R Then there exists a solution (ˆ L2 (Ω, R2 ) of (12)-(13). If Ω Idx 6= 0, then the solution is unique.

Proof. Let (un , ξn ) be a minimizing sequence for F (u, ξ). Then there exists a constant C > 0 such that |un |BV (Ω) ≤ C ||I − un −

divξn ||2L2 (Ω)

(20) ≤C

(21)

||ξn ||L2 (Ω) ≤ C (22) R The Poincare-Wirtinger inequality (||un − Ω un ||L2 (Ω) ≤ |un |BV (Ω) ) and the R R fact that Ω un dx = Ω Idx, give us that ||un ||L2 (Ω) ≤ C.

(23)

By (20) and (23), there exists a function u ˆ ∈ BV (Ω) such that un → u ˆ strongly in L1 (Ω) 2

un * u ˆ weakly in L (Ω) Z Z Idx. u ˆdx = and Ω

(24) (25) (26)

Ω

For each n, let vn = divξn . Then vn ∈ L2 (Ω) and by (21) and (23) we also have that ||vn ||L2 (Ω) ≤ C. Therefore, there exists vˆ ∈ L2 (Ω) such that vn * vˆ in L2 (Ω) weak

(27)

and by (22), there exists ξˆ ∈ L2 (Ω) such that ξn * ξˆ in L2 (Ω) weak.

(28)

Therefore, for φ ∈ Cc∞ (Ω) we have that Z Z Z Z Z n→∞ ˆ ˆ divξφdx ξ · ∇φdx = ξn · ∇φdx −→ − divξn φdx = − vn φdx = Ω Ω Ω Ω Ω Z Z n→∞ and vn φdx −→ vˆφdx. Ω

Ω

So vˆ = divξˆ in the sense of distribution, but since vˆ ∈ L2 (Ω), vˆ = divξˆ a.e.

(29)

From (24), (27), (28), lemmas 1 and 2, and the weak lower-semicontinuity of ˆ ≤ lim inf n→∞ F (un , ξn ) = inf F (u, ξ). Since the L2 norm, we have that F (ˆ u, ξ) ˆ the minimizer (ˆ u, ξ) satisfies (26) and (29), existence of a solution to (12)-(13) is established. R Finally, we claim that if Ω Idx 6= 0 then the minimizer is unique. We observe that ||I − u − divξ||2L2 (Ω) is strictly convex on BV (Ω) × L2 (Ω, R2 ) except along ˆ is a solution of (12)the direction (u + tu, divξ − tu), t ∈ R. However, if (ˆ u, ξ) u + Rtˆ u, ξ) cannot be for any ξ ∈ L2 (Ω). If it were, then R(13), thenRfor t 6= 0, (ˆ u ˆdx = Ω Idx = (t + 1) Ω u ˆdx which is impossible. Ω

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