Analysis of simultaneous inpainting and geometric separation based on sparse decomposition
AANALYSIS OF SIMULTANEOUS INPAINTING AND GEOMETRICSEPARATION BASED ON SPARSE DECOMPOSITION
VAN TIEP DO , , RON LEVIE , , GITTA KUTYNIOK , Abstract.
Natural images are often the superposition of various parts of different geometriccharacteristics. For instance, an image might be a mixture of cartoon and texture structures. Inaddition, images are often given with missing data. In this paper we develop a method for simul-taneously decomposing an image to its two underlying parts and inpainting the missing data. Ourseparation–inpainting method is based on an l minimization approach, using two dictionaries,each sparsifying one of the image parts but not the other. We introduce a comprehensive conver-gence analysis of our method, in a general setting, utilizing the concepts of joint concentration,clustered sparsity, and cluster coherence. As the main application of our theory, we consider theproblem of separating and inpainting an image to a cartoon and texture parts. Introduction
A digital image typically has two or more distinct constituents, e.g., it might contain a cartoon andtexture components. A key question is whether we can separate the image into its two components.This task is of interest in many applications, such as compression and restoration [5, 28, 57]. Theseparation problem is underdetermined and seems to be impossible to stably solve. However, if wehave prior knowledge about the types of geometric components underlying the image, the separationtask is possible, as shown in previous papers [4, 23, 27, 32, 36]. Some approaches commonly used inthis context for image separation are variational methods, e.g., [28, 37] and PDE-based separationmethods [37].More recently, compressed sensing approaches showed that l minimization can stably and pre-cisely solve this problem both theoretically [10, 26, 31, 35, 40, 42] and empirically [4, 27]. Thecore idea is to use multiple dictionaries, each sparsely representing one geometric part of the image.As opposed to thresholding approaches like [22], which are harder to analyze, the l minimizationapproach comes with strong theoretical results. Another classical problem in imaging science is torestore corrupted or missing parts of images, namely image inpainting . Images are often damageddue to different factors including improper storage, chemical processing, or losses of image dataduring transmission. The problem of image inpainting is of interest in numerous applications, fromthe restoration of scratched photos and corrupted images, to the removal of selected objects. Im-age inpainting is an ill-posed problem, so some prior information on the missing part is requiredhere as well. One approach that has been effectively applied to this problem is total variational[44, 48, 50, 51]. Variational approaches work well on piecewise smooth images, but generally per-form poorly on images that contain a superposition of cartoon and texture. On the other hand,local statistical analysis and prediction have been shown to perform well at inpainting the texturecontent [46, 49]. In our analysis, we focus on a compressed sensing driven approach, based on theprior information that the components of the image are sparsely represented by two representationsystems.For an illustration of texture and cartoon parts of an image, we present in Figure 1 a corruptedphoto with missing part and region covered by texture. Department of Mathematics, Technische Universität Berlin, 10623 Berlin, Germany Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334Nguyen Trai, Thanh Xuan, Hanoi Department of Mathematics, Ludwig-Maximilians-Universität München, Munich, Germany
E-mail address : [email protected], [email protected], [email protected] . a r X i v : . [ m a t h . F A ] S e p VAN TIEP DO, RON LEVIE, GITTA KUTYNIOK
Figure 1.
A corrupted photo, where the noise is a texture part together withmissing stripes, and the clean image is the cartoon part.1.1.
Separation and inpainting through cluster coherence.
The problem of inpainting andseparation of cartoon from texture was proposed in various papers, for instance [24, 33, 45]. Thegeneral approach is to find two dictionaries, each sparsely representing one image part, and notsparsely representing the other. For example, texture is sparsely represented by a Gabor system,and cartoon by curvelets/shearlets. Then, in the separation and inpainting algorithm, the image isdecomposed to a sparse representation based on the combined Gabor-curvelet system, using somecompressed sensing approach. However, in the above-mentioned papers, there is no theoreticalanalysis of the success of the proposed methods. In this paper, we consider a similar setting to theabove papers, and give a full approximation analysis of the method, with convergence guarantees.Our analysis is based on the theoretical machinery of joint concentration and cluster coherence.These notions were used in the past for analyzing separation problems and inpainting problems,but never (to the best of or knowledge) for the simultaneous problem. Given two dictionaries,both joint concentration and cluster coherence are definitions that quantify the ability of eachdictionary to sparsely represent signals of one type, but not signals of another type. Some papersuse joint concentration to do inpainting [19, 25], some to do separation [10, 13]. Proving theoreticalresults is typically easier when using joint concentration. However, checking joint concentrationon dictionaries in practice is difficult. Thus the notion of cluster coherence was introduced in [10],which is strictly stronger than joint concentration, and makes checking the assumptions of the theoryeasier in practical examples. Papers like [10, 13, 19, 25] use the notions of cluster coherence eitherfor separation or for inpainting, but not simultaneously.In this paper, we modify and extend the definitions of joint concentration to accommodate asimultaneous analysis of separation and inpainting. We then further extend the notion of clustercoherence to fit the modified definitions of joint concentration. With the new general definitions, weconsider the problem of inpainting and separating texture from cartoon. We show that the universalshearlet systems [25] and Gabor systems [1] satisfy the cluster coherence requirement, thus provingthe success of our inapinting–separation method.1.2.
Our contribution.
We summarize our main contribution as follows. • We modify the joint concentration and cluster coherence definitions to accommodate thesimultaneous separation-inpainting problem (Section 3). • Using the modified notions of joint concentration and cluster coherence we prove that the l sparse decomposition method successfully inpaints and separates the two parts of an imagein a generic setting (Theorem 3.8). • We use the general theory to inpaint and separate cartoon from texture in images. Here, thesparse decomposition method is based on a Gabor basis, sparsifying texture, and a universalshearlet dictionary, sparsifying cartoons (Section 5). The two systems are shown to satisfythe cluster coherence property, thus proving the success of the inpainting-separation cartoonfrom texture method (Theorem 6.4). ANALYSIS OF SIMULTANEOUS SEPARATION AND INPAINTING2.
Simultaneous inpainting and separation approach
In this section we summarize our general theory of simultaneous inpainting and separation.2.1.
The separation problem.
The task is to extract the two components C and T from theobserved image f , where we assume that f = C + T . (1)Here, only f is given, and the components C and T are unknown to us.To solve this underdetermined problem, we assume that each component can be sparsely rep-resented by some dictionary, that cannot sparsely represent the other component. In our theory,dictionaries are modeled as frames [38]. Definition 2.1.
Let I be a discrete index set. A sequence Φ = { φ i } i ∈ I in a separable Hilbert space H is called a frame for H if there exist constants < A ≤ B < ∞ such that A (cid:107) f (cid:107) ≤ (cid:88) i ∈ I |(cid:104) f, φ i (cid:105)| ≤ B (cid:107) f (cid:107) , ∀ f ∈ H , where A, B are called lower and upper frame bound. If A and B can be chosen to be equal, we callit ( A − ) tight frame . If A=B=1, { φ i } i ∈ I is called a Parseval frame .In our context, the Hilbert space H is the space of signals/images, and Φ is the dictionary. Byabusing notion, we also denote by Φ the synthesis operator Φ : l ( I ) → H , Φ( { a i } i ∈ I ) = (cid:88) i ∈ I a i φ i , which synthesizes an image f ∈ H from given coefficients { a i } i ∈ I . We denote by Φ (cid:63) the analysisoperator Φ (cid:63) : H → l ( I ) , Φ (cid:63) ( f ) = ( (cid:104) f, φ i (cid:105) ) i ∈ I . The analysis operator is interpreted as the transform that computed the different dictionary coeffi-cients of images f .2.2. The inpainting problem.
To model the inpainting problem, we suppose that there is somemissing data in the image f . Given the Hilbert space H , we assume that H = H K ⊕ H M , where thesubspaces H K , H M denote the known part and missing part respectively. Let P M and P K denotethe orthogonal projection of H upon these two subspaces, respectively. In the inpainting problem,we assume that we are only given the image in the known part P K f , and the goal is to reconstruct f . The inpainting problem is solved by considering a dictionary { φ i } i ∈ I that sparsely represents f using a known subset of indices Λ ⊂ I , but does not sparsely represent P M f using the indices Λ .This idea is formalized in Definition 3.2.2.3. The simultaneous separation–inpainting problem.
In the simultaneous problem, the goalis to extract C and T , satisfying (1), given only the image in the known part P K f . In our approach,we consider two Parseval frames Φ and Φ in H that sparsely represent their respective component C and T , but do not sparsely represent the other component. This is formalized through Definitions 3.3and 3.1 for the joint concentration approach, and 3.3 and 3.5 for the cluster coherence approach. Wemoreover suppose that the index subsets Λ and Λ of Φ and Φ respectively can sparsely represent C and T , but cannot sparsely represent P M T and P M C . This is formalized through Definitions 3.3and 3.2 for the joint concentration approach, and 3.3 and 3.5 for the cluster coherence approach.We consider the following algorithm, for simultaneously inpaint and separate (INP-SEP) geomet-ric components, based on l minimization. Algorithm (INP-SEP)INPUT: corrupted signal P K f ∈ H K , two Parseval frames { Φ } i ∈ I and { Φ } j ∈ J .COMPUTE: f (cid:63) = ( C (cid:63) , T (cid:63) ) where ( C (cid:63) , T (cid:63) ) = arg min x ,x (cid:107) Φ ∗ x (cid:107) + (cid:107) Φ ∗ x (cid:107) , s.t P K ( x + x ) = P K ( f ) . (2)OUTPUT: recovered components C (cid:63) , T (cid:63) . VAN TIEP DO, RON LEVIE, GITTA KUTYNIOKIn Section 3, we provide a theoretical analysis for the success of algorithm (2).3.
General separation and inpainting theory
In this section, we introduce a theory in which we can prove the success of Algorithm (INP-SEP)for a general separation and inpainting problem.3.1.
Joint concentration.
We propose a sufficient condition for the success of the (INP-SEP)algorithm, based on the notion of joint concentration. Joint concentration was first introduced in[10]. We present two slightly different notion of joint concentration, modified for our needs.
Definition 3.1.
Let Φ , Φ be two Parseval frames. Given two sets of coefficients Λ , Λ , definethe mixed joint concentration κ = κ (Λ , Λ ) by κ = κ (Λ , Λ ) = sup x,y ∈H (cid:107) Λ Φ ∗ x (cid:107) + (cid:107) Λ Φ ∗ y (cid:107) (cid:107) Φ ∗ y (cid:107) + (cid:107) Φ ∗ x (cid:107) . (3)Given an l normalized signal f , it is common to interpret the l norm of f as a measure of spread(opposite of sparsity). Thus, the mixed joint concentration with respect to the coefficient sets Λ and Λ quantifies the extent to which signals can have most of their energy supported and wellspread in Λ j while being sparsely concentrated in the other frame Φ k , for j (cid:54) = k . Bounding κ fromabove is one of the conditions that ensure that Algorithm (INP-SEP) succeeds in the separationtask. Next, we introduce another joint concentration notion which will be used to prove the successof the inpainting method. Definition 3.2.
Let Φ , Φ be two Parseval frames. Given two sets of coefficients Λ , Λ , definethe joint concentration of the missing part κ = κ (Λ , Λ ) by κ = κ (Λ , Λ ) = sup x,y ∈H ; P K x = P K y (cid:107) Λ Φ ∗ ( x − y ) (cid:107) + (cid:107) Λ Φ ∗ ( x − y ) (cid:107) (cid:107) Φ ∗ x (cid:107) + (cid:107) Φ ∗ y (cid:107) . The joint concentration of the missing part quantifies the extent to which signals which coincideon the known part can have most of the energy of their difference supported and well spread in Λ , Λ while begin sparse in Φ and Φ . The joint concentration κ is thus used to encode thegeometric relation between the missing part H M and expansions in Φ and Φ . Bounding κ fromabove is another one of the conditions that ensure the success of Algorithm (INP-SEP).3.2. Recovery guarantee through joint concentration.
The sufficient condition for recoveryis based on finding sets of coefficients Λ and Λ for which the joint concentrations are small, and Λ and Λ capture most of the energy of the ground truth separated components C and T . For thiswe, recall the following definition from [19]. Definition 3.3.
Fix δ > . Given a Hilbert space H with a Parseval frame Φ , f ∈ H is δ − relativelysparse in Φ with respect to Λ if (cid:107) Λ c Φ (cid:63) f (cid:107) ≤ δ, where Λ c denotes X \ Λ . We now prove our first separation result.
Proposition 3.4.
For δ , δ > , fix δ = δ + δ , and suppose that f ∈ H can be decomposed as f = C + T so that each component C , T is δ , δ − relatively sparse in Φ and Φ with respect to Λ and Λ , respectively. Let ( C (cid:63) , T (cid:63) ) solve (INP-SEP) . If we have κ + κ < , then (cid:107)C (cid:63) − C(cid:107) + (cid:107)T (cid:63) − T (cid:107) ≤ δ − κ + κ ) . (4) Proof.
First, we set κ = κ (Λ , Λ ) := sup P K u = P K v (cid:107) Λ Φ ∗ u (cid:107) + (cid:107) Λ Φ ∗ v (cid:107) (cid:107) Φ ∗ u (cid:107) + (cid:107) Φ ∗ v (cid:107) , and x = C (cid:63) − C , y = T − T (cid:63) . Since Φ and Φ are two Parseval frames, thus (cid:107)C (cid:63) − C(cid:107) + (cid:107)T (cid:63) − T (cid:107) = (cid:107) Φ ∗ ( C (cid:63) − C ) (cid:107) + (cid:107) Φ ∗ ( T (cid:63) − T ) (cid:107) ≤ (cid:107) Φ ∗ ( C (cid:63) − C ) (cid:107) + (cid:107) Φ ∗ ( T (cid:63) − T ) (cid:107) = (cid:107) Φ ∗ ( x ) (cid:107) + (cid:107) Φ ∗ ( y ) (cid:107) := S. ANALYSIS OF SIMULTANEOUS SEPARATION AND INPAINTINGNow we invoke the relation P K ( C (cid:63) + T (cid:63) ) = P K ( f ) = P K ( C + T ) and hence P K ( x ) = P K ( y ) . By thedefinition of κ, we have S = (cid:107) Λ Φ ∗ x (cid:107) + (cid:107) Λ Φ ∗ y (cid:107) + (cid:107) Λ c Φ ∗ ( C (cid:63) − C ) (cid:107) + (cid:107) Λ c Φ ∗ ( T (cid:63) − T ) (cid:107) ≤ κS + (cid:107) Λ c Φ ∗ C (cid:63) (cid:107) + (cid:107) Λ c Φ ∗ C(cid:107) + (cid:107) Λ c Φ ∗ T (cid:63) (cid:107) + (cid:107) Λ c Φ ∗ T (cid:107) ≤ κS + (cid:107) Λ c Φ ∗ C (cid:63) (cid:107) + (cid:107) Λ c Φ ∗ T (cid:63) (cid:107) + δ = κS + δ + (cid:107) Φ ∗ C (cid:63) (cid:107) + (cid:107) Φ ∗ T (cid:63) (cid:107) − (cid:107) Λ Φ ∗ C (cid:63) (cid:107) − (cid:107) Λ Φ ∗ T (cid:63) (cid:107) . We note that ( C (cid:63) , T (cid:63) ) is a minimizer of (INP-SEP). Thus (cid:107) Φ ∗ C (cid:63) (cid:107) + (cid:107) Φ ∗ T (cid:63) (cid:107) ≤ (cid:107) Φ ∗ C(cid:107) + (cid:107) Φ ∗ T (cid:107) . Therefore, S ≤ κS + δ + (cid:107) Φ ∗ C(cid:107) + (cid:107) Φ ∗ T (cid:107) − (cid:107) Λ Φ ∗ C (cid:63) (cid:107) − (cid:107) Λ Φ ∗ T (cid:63) (cid:107) ≤ κS + δ + (cid:107) Φ ∗ C(cid:107) + (cid:107) Φ ∗ T (cid:107) + (cid:107) Λ Φ ∗ x (cid:107) + (cid:107) Λ Φ ∗ y (cid:107) − (cid:107) Λ Φ ∗ C(cid:107) −(cid:107) Λ Φ ∗ T (cid:107) ≤ κS + 2 δ + κS = 2 κS + 2 δ. Thus , S ≤ δ − κ . The last inequality comes from the fact that (cid:107) Λ Φ ∗ x (cid:107) + (cid:107) Λ Φ ∗ y (cid:107) (cid:107) Φ ∗ x (cid:107) + (cid:107) Φ ∗ y (cid:107) ≤ (cid:107) Λ Φ ∗ y (cid:107) + (cid:107) Λ Φ ∗ x (cid:107) (cid:107) Φ ∗ x (cid:107) + (cid:107) Φ ∗ y (cid:107) + (cid:107) Λ Φ ∗ ( x − y ) (cid:107) + (cid:107) Λ Φ ∗ ( x − y ) (cid:107) (cid:107) Φ ∗ x (cid:107) + (cid:107) Φ ∗ y (cid:107) . This leads to κ ≤ κ + κ . Finally, we obtain (cid:107)C (cid:63) − C(cid:107) + (cid:107)T (cid:63) − T (cid:107) ≤ δ − κ + κ ) . (cid:3) Recovery guarantee through cluster coherence.
Deriving bounds for joint concentrationscan be difficult in practice. Our goal in this subsection is to replace the joint concentrations by easierto compute terms. For that, we recall the notion of cluster coherence. We then prove that jointconcentrations can be bounded by cluster coherence terms. Deriving bounds to cluster coherenceterms is generally easier than bounding joint concentrations. The following definition is taken from[10].
Definition 3.5.
Given two Parseval frames Φ = (Φ i ) i ∈ I and Φ = (Φ j ) j ∈ J . Then the clustercoherence µ c (Λ , Φ ; Φ ) of Φ and Φ with respect to the index set Λ ⊂ I is defined by µ c (Λ , Φ ; Φ ) = max j ∈ J (cid:88) i ∈ Λ |(cid:104) φ i , φ j (cid:105)| in case this maximum exists.We allow applying a projection P M on one or both of the frames in Definition 3.5. For example,similarly to [19], we use the notation µ c (Λ , P M Φ ; P M Φ ) to denote µ c (Λ , P M Φ ; P M Φ ) = max j ∈ J (cid:88) i ∈ Λ |(cid:104) P M φ i , P M φ j (cid:105)| . Next, we present two lemmas which bound the joint concentrations by corresponding clustercoherences.
Lemma 3.6.
We have κ (Λ , Λ ) ≤ max { µ c (Λ , Φ ; Φ ) , µ c (Λ , Φ ; Φ ) } . VAN TIEP DO, RON LEVIE, GITTA KUTYNIOK
Proof.
For x, y ∈ H , we set α = Φ ∗ y, α = Φ ∗ x . Then invoking the fact that Φ and Φ are twoParseval frames, hence x = Φ Φ ∗ x = Φ α and y = Φ Φ ∗ y = Φ α , we have (cid:107) Λ Φ ∗ x (cid:107) + (cid:107) Λ Φ ∗ y (cid:107) = (cid:107) Λ Φ ∗ Φ α (cid:107) + (cid:107) Λ Φ ∗ Φ α (cid:107) ≤ (cid:88) i ∈ Λ (cid:16) (cid:88) j |(cid:104) Φ i , Φ j (cid:105)|| α j | (cid:17) + (cid:88) j ∈ Λ (cid:16) (cid:88) i |(cid:104) Φ i , Φ j (cid:105)|| α i | (cid:17) = (cid:88) j (cid:16) (cid:88) i ∈ Λ |(cid:104) Φ i , Φ j (cid:105)| (cid:17) | α j | + (cid:88) i (cid:16) (cid:88) j ∈ Λ |(cid:104) Φ i , Φ j (cid:105)| (cid:17) | α i |≤ µ c (Λ , Φ ; Φ ) (cid:107) α (cid:107) + µ c (Λ , Φ ; Φ ) (cid:107) α (cid:107) ≤ max { µ c (Λ , Φ ; Φ ) , µ c (Λ , Φ ; Φ ) } ( (cid:107) α (cid:107) + (cid:107) α (cid:107) )= max { µ c (Λ , Φ ; Φ ) , µ c (Λ , Φ ; Φ ) } ( (cid:107) Φ ∗ x (cid:107) + (cid:107) Φ ∗ y (cid:107) ) . Thus, we obtain κ (Λ , Λ ) ≤ max { µ c (Λ , Φ ; Φ ) , µ c (Λ , Φ ; Φ ) } . (cid:3) Lemma 3.7.
We have κ (Λ , Λ ) ≤ max { µ c (Λ , P M Φ ; P M Φ ) + µ c (Λ , P M Φ ; P M Φ ) ,µ c (Λ , P M Φ ; P M Φ ) + µ c (Λ , P M Φ ; P M Φ ) } = max { µ c (Λ , P M Φ ; Φ ) + µ c (Λ , P M Φ ; Φ ) ,µ c (Λ , P M Φ ; Φ ) + µ c (Λ , P M Φ ; Φ ) } . Proof.
For each h ∈ H M and x ∈ H , set α = Φ ∗ ( x + h ) and β = Φ ∗ x then we have h = x + h − x = Φ Φ ∗ ( x + h ) − Φ Φ ∗ ( x )= Φ α − Φ β. Note that P K is an orthogonal projection, and hence h = P ∗ M P M h = P ∗ M P M Φ α − P ∗ M P M Φ β. Therefore, we obtain (cid:107) Λ Φ ∗ h (cid:107) = (cid:107) Λ Φ ∗ P ∗ M P M Φ α − Λ Φ ∗ P ∗ M P M Φ β (cid:107) = (cid:107) Λ ( P M Φ ) ∗ ( P M Φ ) α − Λ ( P M Φ ) ∗ ( P M Φ ) β (cid:107) ≤ (cid:107) Λ ( P M Φ ) ∗ ( P M Φ ) α (cid:107) + (cid:107) Λ ( P M Φ ) ∗ ( P M Φ ) β (cid:107) ≤ (cid:88) i ∈ Λ (cid:16) (cid:88) j |(cid:104) P M Φ i , P M Φ j (cid:105)|| α j | (cid:17) + (cid:88) i ∈ Λ (cid:16) (cid:88) j |(cid:104) P M Φ i , P M Φ j (cid:105)|| β j | (cid:17) ≤ (cid:88) j (cid:16) (cid:88) i ∈ Λ |(cid:104) P M Φ i , P M Φ j (cid:105)| (cid:17) | α j | + (cid:88) j (cid:16) (cid:88) i ∈ Λ |(cid:104) P M Φ i , P M Φ j (cid:105)| (cid:17) | β j |≤ µ c (Λ , P M Φ ; P M Φ ) (cid:107) α (cid:107) + µ c (Λ , P M Φ ; P M Φ ) (cid:107) β (cid:107) = µ c (Λ , P M Φ ; P M Φ ) (cid:107) Φ ∗ ( x + h ) (cid:107) + µ c (Λ , P M Φ ; P M Φ ) (cid:107) Φ ∗ ( x ) (cid:107) . Similarly, we have (cid:107) Λ Φ ∗ h (cid:107) ≤ µ c (Λ , P M Φ ; P M Φ ) (cid:107) Φ ∗ ( x + h ) (cid:107) + µ c (Λ , P M Φ ; P M Φ ) (cid:107) Φ ∗ ( x ) (cid:107) . This leads to (cid:107) Λ Φ ∗ h (cid:107) + (cid:107) Λ Φ ∗ h (cid:107) ≤ max { µ c (Λ , P M Φ ; P M Φ ) + µ c (Λ , P M Φ ; P M Φ ) ,µ c (Λ , P M Φ ; P M Φ ) + µ c (Λ , P M Φ ; P M Φ ) } ( (cid:107) Φ ∗ ( x + h ) (cid:107) + (cid:107) Φ ∗ ( x ) (cid:107) ) . ANALYSIS OF SIMULTANEOUS SEPARATION AND INPAINTINGFinally, κ (Λ , Λ ) ≤ max { µ c (Λ , P M Φ ; P M Φ ) + µ c (Λ , P M Φ ; P M Φ ) ,µ c (Λ , P M Φ ; P M Φ ) + µ c (Λ , P M Φ ; P M Φ ) } = max { µ c (Λ , P M Φ ; Φ ) + µ c (Λ , P M Φ ; Φ ) ,µ c (Λ , P M Φ ; Φ ) + µ c (Λ , P M Φ ; Φ ) } . (cid:3) We can now formulate a general guarantee for the success of Algorithm (INP-SEP), based oncluster coherence.
Theorem 3.8.
Let Φ , Φ be two Parseval frames for a Hilbert space H . For δ , δ > , fix δ = δ + δ , and suppose that f ∈ H can be decomposed as f = C + T so that each component C , T is δ , δ − relatively sparse in Φ and Φ with respect to Λ , Λ respectively. Let ( C (cid:63) , T (cid:63) ) solve (INP-SEP) . If we have κ + κ < , then (cid:107)C (cid:63) − C(cid:107) + (cid:107)T (cid:63) − T (cid:107) ≤ δ − µ c , (5) where µ c = max { µ c (Λ , P M Φ ; Φ ) + µ c (Λ , P M Φ ; Φ ) , µ c (Λ , P M Φ ; Φ ) + µ c (Λ , P M Φ ; Φ ) } + max { µ c (Λ , Φ ; Φ ) , µ c (Λ , Φ ; Φ ) } .Proof. The bound (5) holds as a consequence of Lemmas 3.6, 3.7 and Proposition 3.4. (cid:3)
Let us interpret this estimate. The relative sparsity δ and the cluster coherence µ c depend on thegeometric sets of indices Λ , Λ , which are not used at all in Algorithm (INP-SEP). Hence, Λ , Λ are analytic tools that we can choose arbitrarily, and are used only for deriving theoretical bounds.If we choose very large Λ , Λ , we get very small relative sparsities δ , δ , but we might loose controlover the cluster coherence µ c . Therefore, choosing appropriate Λ and Λ is an important step whenapplying our theory in concrete examples.4. Mathematical models of texture and cartoon
We now focus on the specific problem of inpainting and separating texture from cartoon. In thissection we define our models of texture and cartoon parts, and of the missing part.4.1.
Model of texture.
One of the earliest qualitative texture description that corresponds tohuman visual perception is: coarseness, contrast, directionality, line-likeness, regularity, roughness[41]. From a mathematical modeling point of view, many definitions of texture were proposed inthe past, for instance [13, 30, 39, 43]. Our model for texture is inspired by [13], and based on anexpansion of Gabor frame elements. To motivate our definition of texture we offer the followingdiscussion. A very restrictive definition of texture would be a periodical signal with a short period.The Fourier transform of a periodical signal is a signal supported on a delta train on a regular gridin the frequency domain. To relax the hard periodicity condition, suppose that we allow to perturbthe locations of the points of the regular frequency grid. This would result in a signal supportedon a sparse set of Fourier coefficients, which is exactly our definition of texture. Such a definitionindeed produces images that look like a repeating pattern, but not in a strictly periodical fashion(see Figure 2 for example).Before formally defining texture, we recall the
Schwartz functions or the rapidly decreasing func-tions S ( R ) := (cid:110) f ∈ C ∞ ( R ) | ∀ K, N ∈ N : sup x ∈ R (1 + | x | ) − N/ (cid:88) | α |≤ K | D α f ( x ) | < ∞ (cid:111) . (6)We define the Fourier transform and inverse Fourier transform for f, F ∈ S ( R ) by ˆ f ( ξ ) = F [ f ]( ξ ) = (cid:90) R f ( x ) e πix T ξ dx, ˇ F ( x ) = F − [ F ]( x ) = (cid:90) R F ( ξ ) e πiξ T x dx, VAN TIEP DO, RON LEVIE, GITTA KUTYNIOKwhich can be extended to a well define Fourier transform and inverse Fourier transform for functionsin L ( R ) (cf. [56]). Using a window function g : R → R which localizes a texture patch in thespatial domain, we now introduce our model for texture. Definition 4.1.
Let g ∈ L ( R ) be a window with ˆ g ∈ C ∞ ( R ) , and frequency support supp ˆ g ∈ [ − , satisfying the partition of unity condition (cid:88) n ∈ Z | ˆ g ( ξ + n ) | = 1 , ξ ∈ R . (7)For s > , we define the L − normalized scaled version of g by g s ( x ) = s · g ( sx ) . Let I T ⊆ Z , be asubset of Fourier elements. A texture is defined by T s ( x ) = (cid:88) n ∈ I T d n g s ( x ) e πix T sn , (8)where ( d n ) n ∈ Z denotes a sequence of complex numbers.In Subsection 6.3 we add a restriction on the size of the index set I T . For now, we just mentionthat I T is a small/sparse set in some sense. We also remark that by Definition 4.1, we have ˆ g s ( ξ ) = s − · ˆ g ( s − ξ ) , supp ˆ g s ⊆ [ − s, s ] and the partition of unity condition now reads (cid:88) n ∈ Z | ˆ g s ( ξ + sn ) | = s − , ξ ∈ R . (9) Figure 2.
A texture sample produced by randomly choosing a sparse set ofFourier coefficients with random values, with the rest of the Fourier coefficients setto zero.4.2.
The local cartoon patch.
In [52, 53], cartoon functions are defined as C = f + f · B τ , (10)where f , f ∈ L ( R ) ∩ C β ( R ) , β ∈ (0 , + ∞ ) , with compact support and B τ denotes the interior ofa closed, non-intersecting curve τ in C β ( R ) . In the separation algorithm of cartoon and texture,we analyze the input image using a Gabor frame and the so-called universal shearlet frame (seeSubsection 5). Both of these frames analyze images on local patches. Thus, for the sake of simplicity,we also reduce our model of the cartoon part to a local cartoon model. This is done in two steps.Cartoon images are locally close to piecewise constant functions with discontinuity along an edgecurve, which is locally close to a line. We thus consider the windowed step function w S ( x ) = (cid:40) w ( x ) x ≤ x > , (11)where w ∈ C ∞ ( R ) is a weighted function satisfying w (cid:54)≡ , ≤ w ( u ) ≤ and supp w ⊂ [ − ρ, ρ ] . Inour work, the cartoon part is analyzed using universal shearlets, which is a shear invariant systemup to the cone adaptation. Since shearing is a way of changing orientation, our analysis also appliesto the more general case where the discontinuity in (11) is along a general line in any orientation. ANALYSIS OF SIMULTANEOUS SEPARATION AND INPAINTINGWe note that it is possible to extend our results from the local cartoon model (11) to the globalcartoon model (10) by using a tubular neighborhood argument as shown in [13].The cartoon part (11) can be seen as a tempered distribution acting on
Schwartz functions by (cid:104) w S , f (cid:105) = (cid:90) ρ − ρ w ( x ) (cid:90) −∞ f ( x , x ) dx dx , f ∈ S ( R ) . (12)The following lemma is used for computing the Fourier transform of w S . Lemma 4.2.
We have (cid:90) −∞ e − πiωx dx = 12 (cid:104) δ ( ω ) + iπω (cid:105) . By Lemma 4.2, we can compute the Fourier transform of w S by (cid:104) (cid:100) w S , f (cid:105) = (cid:104) w S , ˆ f (cid:105) = (cid:90) ρρ w ( x ) (cid:90) −∞ (cid:16) (cid:90) R f ( ξ ) e − πi ( ξ x + ξ x ) dξ (cid:17) dx dx = (cid:90) R ˆ w ( ξ ) (cid:16) (cid:90) −∞ e − πiξ x dx (cid:17) f ( ξ ) dξ = (cid:90) R ˆ w ( ξ ) (cid:16) δ ( ξ ) + iπξ (cid:17) f ( ξ ) dξ, f ∈ L ( R ) . (13)By (13) , we obtain (cid:100) w S ( ξ ) = ˆ w ( ξ ) (cid:16) δ ( ξ ) + iπξ (cid:17) . Now, we modify the local cartoon part to get an image in L ( R ) . We note that we are interestedin the local behaviour of w S about ( x , x ) = 0 . Thus, the “DC part” of (cid:100) w S is not of interest tous, as it models some global asymptotic behaviour of the patch in R . We thus filter our the band | ξ | < r from (cid:100) w S ( ξ , ξ ) for some arbitrarily small r > , to obtain (cid:99) w S = {| ξ |≥ r } (cid:100) w S . (14)The main change incurred by this filtering of w S is a “translation along the y axis w S (cid:55)→ w S − DC ”,which does not affect the analysis via shearlet frames. This filtering changes the behaviour of w S about ( x , x ) = 0 negligibly, since r is small, and retains the quality of w S being approximatelypiecewise constant locally with a line discontinuity (see Figure 3). Henceforth, we call the localcartoon part w S simply a cartoon part. Figure 3.
A local cartoon patch.4.3.
The missing part.
In our analysis, the shape of the missing region is chosen to be M h = { x = ( x , x ) ∈ R | | x | ≤ h } , for h > , (15)and the orthogonal projection associated with this missing part is P M = M h . Figure 4.
The missing part (grey) at scale j .Note that this models a local and axis aligned missing part, corresponding to the local model oftexture (8) and the local and axis aligned cartoon model (14). When combining and re-orientingthe local patches, we can obtain many missing stripes at various orientations. Moreover, in practice,the missing part can be any domain contained in M h of (15).One practical example where the missing part is of the form (15) is seismic data, where the imageis commonly incomplete due to missing or faulty sensor or land development causing white strips[16, 17]. 5. Sparsifying systems for texture and cartoon
In this section, we choose frames that sparsely represent the texture and cartoon parts. Torepresent texture, it is clear from Definition 4.1 that a Gabor frame is a natural choice. For thecartoon part, among popular sparsifying representation systems such as wavelets, curvelets, andshearlets, it was shown in [15] that shearlets outperform not only wavelets, but also most otherdirectional sparsifying systems for inpainting larger gap size [19, 25].We hence choose the following sparse representation system. • Gabor frame - a tight frame with time-frequency balanced elements • Universal shearlet frame - a directional tight frame.5.1.
Gabor frame.
Gabor frames are defined as follows.
Definition 5.1.
Denote for x, y ∈ R and g ∈ L ( R ) g ( x, y )( t ) = g ( t − x ) e πit T y . For a, b > we call the collection of functions { g ( ma, nb ) | m, n ∈ Z } a Gabor system . Such asystem is called a
Gabor frame if it forms a frame.For our Gabor system, we use a window ˜ g that satisfies the assumptions of the window g of texture(Definition 4.1). We note that ˜ g and g may be different in general. However, the analysis for ˜ g (cid:54) = g is almost identical to the analysis in case g = ˜ g . Without loss of generality, we assume henceforththat both Definitions 4.1 and 5.1 are based on the same window g , satisfying the assumptions ofDefinition 4.1.For each scaling factor s > , we consider the Gabor tight frame G s = { ( g s ) λ ( x ) } λ . This frameis represented in the frequency domain as (ˆ g ) λ ( ξ ) = ˆ g s ( ξ − sn ) e πiξ T m s , (16)where λ = ( m, n ) is the spatial and frequency position and the parameter s denotes the band-size.This system constitutes a tight frame for L ( R ) , see [11] for more details.1 ANALYSIS OF SIMULTANEOUS SEPARATION AND INPAINTING5.2. The universal-scaling shearlet frame.
Shearlets were first introduced in [6] as a represen-tation system extending the wavelet framework. They are a directional representation system withsimilar optimal approximation properties as curvelets, but with the advantage of allowing for a uni-fied treatment of the continuum and digital domains. Extending the shearlet system, α -shearlets,first introduced in [20], can be regarded as a parametrized family ranging from wavelets to shear-lets. In [52, 53] it was shown that α − shearlets provide optimally sparse approximations for cartoonimages defined as piecewise C /α − functions, separated by a C /α singularity curve. A further ex-tension is universal shearlets [25], which is motivated by [29], with the aim at constructing a typeof α − shearlets that forms a Parseval frame. In our setting we choose universal shearlets as thesparsifying system of the cartoon part, since they form a Parseval frame, a necessary condition inour theory.Let φ be a function in S ( R ) satisfying ≤ ˆ φ ( u ) ≤ for u ∈ R , ˆ φ ( u ) = 1 for u ∈ [ − / , / and supp ˆ φ ⊂ [ − / , / . Define the low pass function Φ( ξ ) and the corona scaling functions for j ∈ N and ξ = ( ξ , ξ ) ∈ R , ˆΦ( ξ ) := ˆ φ ( ξ ) ˆ φ ( ξ ) ,W ( ξ ) := (cid:113) ˆΦ (2 − ξ ) − ˆΦ ( ξ ) , W j ( ξ ) := W (2 − j ξ ) . (17) Figure 5.
Frequency tiling of a cone-adapted shearlet.It is easy to see that we have the partition of unity property ˆΦ ( ξ ) + (cid:88) j ≥ W j ( ξ ) = 1 , ξ ∈ R . (18)Next, we use a bump-like function v ∈ C ∞ ( R ) to produce the directional scaling feature of thesystem. Suppose supp ( v ) ⊂ [ − , and | v ( u − | + | v ( u ) | + | v ( u + 1) | = 1 for u ∈ [ − , . Definethe horizontal frequency cone and the vertical frequency cone C (h) := (cid:110) ( ξ , ξ ) ∈ R | (cid:12)(cid:12)(cid:12) ξ ξ (cid:12)(cid:12)(cid:12) ≤ (cid:111) (19)and C (v) := (cid:110) ( ξ , ξ ) ∈ R | (cid:12)(cid:12)(cid:12) ξ ξ (cid:12)(cid:12)(cid:12) ≤ (cid:111) . (20)Define the cone functions V (h) , V (v) by V (h) ( ξ ) := v (cid:16) ξ ξ (cid:17) , V (v) ( ξ ) := v (cid:16) ξ ξ (cid:17) . (21)The shearing and scaling matrix are defined by A α, (h) := (cid:20)
00 2 α (cid:21) , S (h) := (cid:20) (cid:21) , (22)2 VAN TIEP DO, RON LEVIE, GITTA KUTYNIOK A α, (v) := (cid:20) α
00 2 (cid:21) , S (v) := (cid:20) (cid:21) , (23)where α ∈ ( −∞ , is the scaling parameter . The following two definitions are taken from [25]. Definition 5.2.
Let Φ , W, v be defined as above. For α ∈ ( −∞ , , k ∈ Z , we define(1) Coarse scaling functions : ψ − ,k ( x ) := Φ( x − k ) , k ∈ Z , x ∈ R . (2) Interior shearlets : let j ∈ N , l ∈ Z such that | l | < (2 − α ) j , k ∈ Z and ( ι ) ∈ { (h) , (v) } .Then we define ψ α, ( ι ) j,l,k ( x ) by its Fourier transform ˆ ψ α, ( ι ) j,l,k ( ξ ) := 2 − (2+ α ) j/ W j ( ξ ) V ( ι ) (cid:16) ξ T A − jα, ( ι ) S − l ( ι ) (cid:17) e − πiξ T A − jα, ( ι ) S − l ( ι ) k , ξ ∈ R . (3) Boundary shearlets : let j ∈ N , and l = ±(cid:100) (2 − α ) j (cid:101) we define ˆ ψ α, (b) j,l,k ( ξ ) := − (2+ α ) j/ − / W j ( ξ ) V (h) (cid:16) ξ T A − jα, (h) S − l (h) (cid:17) e − πiξ T A − jα, (h) S − l (h) k , ξ ∈ C (h) , − (2+ α ) j/ − / W j ( ξ ) V (v) (cid:16) ξ T A − jα, (v) S − l (v) (cid:17) e − πiξ T A − jα, (v) S − l (v) k , ξ ∈ C (v) , and in the case j = 0 , l = ± , we define ˆ ψ α, (b)0 ,l,k ( ξ ) := W ( ξ ) V (h) (cid:16) ξ T S − l (h) (cid:17) e − πiξ T k , ξ ∈ C (h) W ( ξ ) V (v) (cid:16) ξ T S − l (v) (cid:17) e − πiξ T k , ξ ∈ C (v) . Definition 5.3.
Let ( α j ) j ∈ N ⊂ R be a scaling sequence , i.e, α j ∈ Z j := (cid:110) mj | m ∈ Z , m ≤ j − (cid:111) . (24)We define the associated universal-scaling shearlet system , or shorter, universal shearlet system , by Ψ = SH ( φ, v, ( α j ) j ) := SH Low ( φ ) ∪ SH Int ( φ, v, ( α j ) j ) ∪ SH Bound ( φ, v, ( α j ) j ) , where SH Low ( φ ) := { ψ − ,k ( x ) | k ∈ Z } SH Int ( φ, v, ( α j ) j ) := (cid:110) ψ α j , ( ι ) j,l,k ( x ) | j ∈ N , l ∈ Z , | l | < (2 − α j ) j , k ∈ Z , ( ι ) ∈ { (h) , (v) } (cid:111) SH Bound ( φ, v, ( α j ) j ) := (cid:110) ψ α j , (b) j,l,k ( x ) | j ∈ N , l ∈ Z , | l | = ± (2 − α j ) j , k ∈ Z (cid:111) . In [25] it was shown that { Ψ } η , η = ( j, l, k ; α j , ( ι )) constitutes a Parseval frame for L ( R ) . Theuniversal shearlet system is based on discrete α j ∈ Z j in order for the set of sheared cone functionsto exactly cover the horizontal and vertical frequency cones. We note that for real α ∈ ( −∞ , wecan choose ( α j ) j that provide the best possible approximation of α , i.e, α j := arg min ˜ α j ∈ Z j | ˜ α j − α | , j ≥ . (25)This implies that α j j = Θ(2 αj ) and lim α j = α as j → ∞ . Indeed, we can show that α j = (cid:98) jα + 0 . (cid:99) j . (26)For our model, we consider universal shearlets with an arbitrary scaling sequence ( α j ) j ∈ N chosenas in (26) with α ∈ (0 , .6. Inpainting and separation of cartoon and texture
In this section we formulate our main result on inpainting and separation of cartoon and texture.3 ANALYSIS OF SIMULTANEOUS SEPARATION AND INPAINTING6.1.
Multi-scale separation and inpainting.
In our analysis, the separation problem is analyzedin a multi-scale setting. What we show is that the inpainting and separation method is more accuratefor smaller scale components of the image, as long as the size of the missing part gets smaller inscale. This is the same approach taken by [25] for the problem of inpainting.Consider the window function W defined in (17). We construct a family of frequency filters F j with the Fourier representation ˆ F j ( ξ ) := W j ( ξ ) = W ( ξ/ j ) , ∀ j ≥ , ξ ∈ R , (27)and in the case j = 0 ˆ F ( ξ ) := Φ( ξ ) , ξ ∈ R . Notice that W j is compactly supported in the corona A j := [ − j − , j − ] \ [ − j − , j − ] , ∀ j ≥ . (28)We now consider the texture and cartoon parts at different scales by filtering them with F j . Forthe patch size of texture, we typically use a fixed size s for all scales j . However, we also allow s to depend on the scale j to make the setting more flexible. We hence consider a variable s j , andassociated with s j the domain A s,j , ∀ j ≥ , defined by A s,j = { ξ ∈ R : s j ξ ∈ A j } . (29)We split the components C and T s into pieces of different scales C j = F j (cid:63) C and T s,j = F j (cid:63) T s , where T s is the texture of patch size s = s j . The total signal at scale j is defined to be f j = C j + T s,j . As mentioned above, we typically pick a constant s j = s . In the constant s case, the pieces f j can be filtered directly from f , namely, f j = F j (cid:63) f . We moreover have by (18) f = (cid:88) j f j . (30)In the general case, the Fourier transform ˆ f j of f j is supported in the annulus with inner radius j − and outer radius j − .Now, at each scale j we consider the simultaneous inpainting and separation problem of extracting C j and T s,j from (1 − P j ) f j = (1 − P j ) C j + (1 − P j ) T s,j . (31)As in [25], we assume that at scale j the size of the missing part h j depends on j . Denote by P j thecharacteristic function of the missing part at scale j , i.e, P j = {| x |≤ h j } . (32)After inpainting and decomposing each scale component (1 − P j ) f j into texture and cartoon com-ponents T s,j and C j , the total components T s and C can be reconstructed similarly to (30) , in case s is constant. Since generally s = s j depends on j , we use different Gabor frames to solve (31) fordifferent scales. We denote by G j the Gabor tight frame associated with s = s j , i.e, G j := G s j = { ( g s j ) ( m,n ) ( x ) } ( m,n ) ∈ Z . (33)In the analysis of the next subsections, we need the following representation of P j in the frequencydomain. Lemma 6.1.
We have ˆ P j ( ξ ) = 2 h j sinc(2 h j ξ ) δ ( ξ ) . Proof.
By definition, we have ˆ P j ( ξ ) = (cid:90) R P j ( x ) e − πix T ξ dx = (cid:16) (cid:90) h j − h j e − πix ξ dx (cid:17)(cid:16) (cid:90) R e − πix ξ dx (cid:17) = 2 h j sinc(2 h j ξ ) δ ( ξ ) . (cid:3) Balancing the texture and cartoon parts.
It is important to avoid the trivial case whereone of the parts, cartoon or texture, is much larger than the other. This would lead to the trivialseparation outcome in which the whole image is taken as the estimation of the larger part, and thesmaller part is estimated as zero. We thus suppose that the filtered components C j and T s,j havecomparable magnitudes at each scale.Consider the sub-band components w S j = F j ∗ w S , (34)and T s,j = F j ∗ T s , (35)where w S is defined in (14) and texture is in Definition 4.1 . The following claim formulates the energy balancing condition . Claim 6.2.
Consider the cartoon patch and texture with sub-bands defined in (34), (35), respec-tively. We have (cid:107) w S j (cid:107) ∼ − j , j → ∞ , and (cid:107)T s,j (cid:107) ∼ (cid:88) n ∈ Z ∩A s,j | d n | . Therefore, energy balance is achieved for c − j ≤ (cid:88) n ∈ I T ∩A s,j | d n | ≤ c − j , c , c > . (36) Proof.
We present the following sketch of the proof. For W sufficient nice we have (cid:107) w S j (cid:107) = (cid:90) ξ ∈ R , | ξ |≥ r | ˆ w ( ξ ) | · W (2 − j ξ ) · (cid:12)(cid:12)(cid:12) δ ( ξ ) + iπξ (cid:12)(cid:12)(cid:12) dξ ∼ (cid:90) ξ ∈A j , | ξ |≥ r ξ · | ˆ w ( ξ ) | dξ ∼ (cid:90) ξ ∈A j , | ξ |≥ j − ξ | ˆ w ( ξ ) | dξ + (cid:90) ξ ∈A j , j − ≥| ξ |≥ r ξ | ˆ w ( ξ ) | dξ . Note that by (28), A j = [ − j − , j − ] \ [ − j − , j − ] . Thus, { ξ ∈ A j , j − ≥ | ξ | ≥ r } ⊂ { ξ ∈A j , | ξ | ≥ j − } . We now use the rapid decay of ˆ w sup | ξ |≥ j − | ˆ w ( ξ ) | ≤ C N (cid:104)| j |(cid:105) − N , ∀ N ∈ N . This leads to (cid:90) ξ ∈A j , j − ≥| ξ |≥ r ξ | ˆ w ( ξ ) | dξ ≤ C N · j r · j (cid:104)| j |(cid:105) − N , ∀ N ∈ N . (37)In addition, (cid:90) ξ ∈A j , | ξ |≥ j − ξ | ˆ w ( ξ ) | dξ ∼ C · − j . (38)Combining (37) and (38), we finally obtain (cid:107) w S j (cid:107) ∼ − j . (39)On the other hand, we have T s = (cid:88) n ∈ I T d n g s j ( x ) e πi (cid:104) s j n,x (cid:105) . This leads to ˆ T s = (cid:88) n ∈ I T d n ˆ g s j ( ξ − s j n ) . (40)5 ANALYSIS OF SIMULTANEOUS SEPARATION AND INPAINTINGNow, using the change of variable ω = ξs j we get (cid:107)T s,j (cid:107) = (cid:107) ˆ T s,j (cid:107) = (cid:88) n, ˜ n ∈ I T (cid:90) d n d ˜ n W ( ξ/ j )ˆ g s j ( ξ − s j n )ˆ g s j ( ξ − s j ˜ n ) dξ = s j · (cid:88) | n − ˜ n |≤ n, ˜ n ∈ I T (cid:90) d n d ˜ n W ( s j ω/ j )ˆ g s j ( s j ( ω − n ))ˆ g s j ( s j ( ω − ˜ n )) dω = (cid:88) | n − ˜ n |≤ n, ˜ n ∈ I T (cid:90) d n d ˜ n W ( s j ω/ j )ˆ g ( ω − n )ˆ g ( ω − ˜ n ) dω. Note that for each n , there exist a finite number of ˜ n ’s independent of j satisfying | n − ˜ n | ≤ , and since W is sufficiently nice, (cid:90) W ( s j ω/ j )ˆ g ( ω − n )ˆ g ( ω − ˜ n ) dω ∼ (cid:90) A s,j ˆ g ( ω − n )ˆ g ( ω − ˜ n ) dω. Thus, we obtain (cid:107)T s,j (cid:107) ∼ (cid:88) | n − ˜ n |≤ n ∈ I T ∩A s,j (cid:90) A s,j d n d ˜ n ˆ g ( ω − n )ˆ g ( ω − ˜ n ) dω ∼ (cid:88) n ∈ I T ∩A s,j (cid:90) A s,j | d n | | ˆ g ( ω − n ) | dω. Up to a small set of coefficients, we assume that the support of ˆ g is always entirely contained in A s,j . Hence, (cid:107)T s,j (cid:107) ∼ (cid:88) n ∈ I T ∩A s,j (cid:90) R | d n | | ˆ g ( ω − n ) | dω = (cid:88) n ∈ I T ∩A s,j | d n | (cid:90) R | ˆ g ( ω ) | dω ∼ (cid:88) n ∈ I T ∩A s,j | d n | . (41)Combining (39) and (41), we finish the proof. (cid:3) For later analysis, we also need an energy estimate of the cartoon patch in the missing region.We use the following useful notation in the whole paper (cid:104)| x |(cid:105) = (1 + | x | ) / . (42) Lemma 6.3.
For h j = o (2 − α j j ) , α j ∈ (0 , , and ω (0) (cid:54) = 0 , there exists a constant C > such that (cid:107) P j w S j (cid:107) ≥ C · h j − j (43) Proof.
By Plancherel’s theorem and Lemma 6.1, we have (cid:107) P j w S j (cid:107) = (cid:90) ξ ∈ R | ξ |≥ r (cid:12)(cid:12)(cid:12) (cid:90) R ˆ P j ( t ) (cid:100) w S j ( ξ − t ) dt (cid:12)(cid:12)(cid:12) dξ = (cid:90) ξ ∈ R | ξ |≥ r (cid:12)(cid:12)(cid:12) (cid:90) R h j sinc(2 h j t ) ˆ w ( ξ − t ) 1( πξ ) W j ( ξ − t , ξ ) dt (cid:12)(cid:12)(cid:12) dξ = 4 h j π (cid:90) ξ ∈ R | ξ |≥ r ξ (cid:12)(cid:12)(cid:12) (cid:90) R sinc(2 h j t ) ˆ w ( ξ − t ) W j ( ξ − t , ξ ) dt (cid:12)(cid:12)(cid:12) dξ. (44)Next, recalling the definition of W j ( ξ ) in (17), we have W j ( ξ ) = 1 , ∀ ξ such that ξ ∈ [ − j − , j − ] \ [ − j − , j − ] . (45)6 VAN TIEP DO, RON LEVIE, GITTA KUTYNIOKFor ξ in the corona-shape { ξ ∈ R | ξ ∈ [ − j − , j − ] \ [ − j − , j − ] } , if ξ is small then ξ must be large. More accurately, for ξ with | ξ | ≤ , we are guaranteed to have ξ ∈ [ − j − , j − ] \ [ − j − , j − ] in case ξ ∈ (2 j − , j − ) . Combining this observation with (44), the followingholds (cid:107) P j ω S j (cid:107) ≥ Ch j (cid:90) j − j − ξ (cid:90) − (cid:12)(cid:12)(cid:12) (cid:90) R sinc(2 h j t )ˆ ω ( ξ − t ) W j ( ξ − t , ξ ) dt (cid:12)(cid:12)(cid:12) dξ dξ . (46)Next, we study the term I j ( ξ ) := (cid:12)(cid:12)(cid:12) (cid:82) R sinc(2 h j t )ˆ ω ( ξ − t ) W j ( ξ − t , ξ ) dt (cid:12)(cid:12)(cid:12) . By the triangleinequality, we have I j ( ξ ) ≥ (cid:12)(cid:12)(cid:12) (cid:90) | t |≤ αjj − sinc(2 h j t )ˆ ω ( ξ − t ) W j ( ξ − t , ξ ) dt (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) (cid:90) | t |≥ αjj − sinc(2 h j t )ˆ ω ( ξ − t ) W j ( ξ − t , ξ ) dt (cid:12)(cid:12)(cid:12) . (47)We now use the rapid decay of ˆ w to obtain sup | t |≥ αjj | ˆ w ( ξ − t ) | ≤ C N (cid:104)| α j j |(cid:105) − N , ∀ ξ ∈ [ τ, τ ] , ∀ N ∈ N . (48)This leads to (cid:12)(cid:12)(cid:12) (cid:90) | t |≥ αjj sinc(2 h j t )ˆ ω ( ξ − t ) W j ( ξ − t , ξ ) dt (cid:12)(cid:12)(cid:12) ≤ C (cid:48) N (cid:104)| α j j |(cid:105) − N . (49)For the other term, we observe that for fine scale j , we have ( ξ − t , ξ ) ∈ [ − j − , j − ] \ [ − j − , j − ] for ξ ∈ [ − , , | t | ≤ α j j − , α j ∈ (0 , and ξ ∈ (2 j − , j − ) . Combining thiswith (45), we derive W j ( ξ − t , ξ ) = 1 . Thus, we obtain (cid:90) | t |≤ αjj − sinc(2 h j t )ˆ ω ( ξ − t ) W j ( ξ − t , ξ ) dt = (cid:90) | t |≤ αjj − sinc(2 h j t )ˆ ω ( ξ − t ) dt . We now prove that there exists a constant C (cid:48) > such that for sufficiently large j , we have (cid:12)(cid:12)(cid:12) (cid:90) | t |≤ αjj − sinc(2 h j t )ˆ ω ( ξ − t ) dt (cid:12)(cid:12)(cid:12) ≥ C (cid:48) , ∀ ξ ∈ [ − , . (50)By triangle inequality, we obtain (cid:12)(cid:12)(cid:12) (cid:90) | t |≤ αjj − sinc(2 h j t )ˆ ω ( ξ − t ) dt (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) (cid:90) | t |≤ αjj − ˆ w ( ξ − t ) dt (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) (cid:90) | t |≤ αjj − (cid:16) sinc(2 h j t ) − (cid:17) ˆ w ( ξ − t ) dt (cid:12)(cid:12)(cid:12) . (51)For the first term, the following holds for ∀ ξ ∈ [ − , (cid:12)(cid:12)(cid:12) (cid:90) | t |≤ αjj − ˆ w ( ξ − t ) dt (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) (cid:90) R ˆ w ( t ) dt (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) (cid:90) | t |≥ αjj − ˆ w ( ξ − t ) dt (cid:12)(cid:12)(cid:12) by (48) ≈ | w (0) | = const > . (52)For the second term of the right hand side of (51), by the assumption h j α j j j → + ∞ −−−−→ , we have (cid:12)(cid:12)(cid:12) (cid:90) | t |≤ αjj − (sinc(2 h j t ) −
1) ˆ w ( ξ − t ) dt (cid:12)(cid:12)(cid:12) ≤ sup | t |≤ αjj − | sinc(2 h j t ) − | (cid:90) R | ˆ w ( t | dt ≤ C · sup | t |≤ αjj − | sinc(2 h j t ) − | j → + ∞ −−−−→ . (53)Thus, by (51), (53) and (52), we conclude the proof of (50).7 ANALYSIS OF SIMULTANEOUS SEPARATION AND INPAINTINGNow, by (47), (49), and (50), we have I j ( ξ ) ≥ C (cid:48) = const. Combining this with (46), we finallyobtain (cid:107) P j ω S j (cid:107) ≥ Ch j (cid:16) (cid:90) j − j − ξ dξ (cid:17)(cid:16) (cid:90) − C (cid:48) dξ (cid:17) = C · h j − j , which concludes the proof. (cid:3) Sparsity assumptions on texture.
Next, we restrict the definition of texture by boundingthe size of the index set I T of Definition 4.1. Define the domain M j,l, ( ι ) := (cid:104) s − j · S − l ( ι ) · ( supp ˆ ψ α j , ( ι ) j, , + B (0 , (cid:105) ∩ Z , (54)where A + B := { a + b | a ∈ A, b ∈ B } is the Minkowski sum, and multiplication of a set ofvectors by a matrix is done elementwise. We assume that at every scale j , the number of non-zeroGabor elements with the same position, generating T s,j , is not too large. More accurately, denote | A | = { n ∈ Z ∩ A } , and define the neighborhood set I ± T of the index set I T by I ± T = { n (cid:48) ∈ Z | ∃ n ∈ I T : | n (cid:48) − n | ≤ } . (55)What we assume is that at every scale j , for all l ∈ Z satisfying | l | ≤ (2 − α j ) j we have | I T ∩ M j,l, ( ι ) | ≤ (2 − α j − (cid:15) ) j/ = o (2 (2 − α j ) j/ ) , ∀ ( ι ) ∈ { (h) , (v) , (b) } , (56)and | I ± T ∩ A s,j | ≤ α j j s j as j → ∞ . (57) Figure 6.
Left: Interaction between a cluster of Gabor elements (grey smallsquares) and a shearlet (green) in the frequency domain; Right: Some shearletelements (green) associated with Λ ,j in the spatial domain for l = 0 .6.4. Cluster sets for texture and cartoon.
We define the cluster for texture at scale j to be Λ ,j := (cid:16) Z ∩ B (0 , M j ) (cid:17) × (cid:16) I ± T ∩ A s,j (cid:17) , (58)where M j := 2 (cid:15)j/ and B (0 , r ) denotes the closed l ball around the origin in R . The term M j = 2 (cid:15)j/ controls the trade-off between the relative sparsity and cluster coherence of the Gaborsystems.To represent the cartoon part, consider a universal shearlet system with α j from (25), where α ∈ (0 , is a global constant. To define the cluster sets, we fix a constant (cid:15) satisfying < (cid:15) < − α . (59)8 VAN TIEP DO, RON LEVIE, GITTA KUTYNIOKSince α j satisfies lim j →∞ α j = α , we have < (cid:15) < − α j , (60)at fine enough scales j . For the cartoon model w S j we define the set of significant coefficients ofuniversal shearlet system by Λ ± ,j := Λ ,j − ∪ Λ ,j ∪ Λ ,j +1 , ∀ j ≥ , (61)where Λ ,j := (cid:110) ( j, l, k ; α j , v) | | l | ≤ , k = ( k , k ) ∈ Z , | k − lk | ≤ (cid:15)j (cid:111) . The separation and inpainting theorem for cartoon and texture.
We now presentour main result. In the following theorem, we prove the success of separation and inpaining viaAlgorithm (INP-SEP). Since we inpaint a band of width h j for each scale j , it is important to showthat the inpainting error is asymptotically smaller than the energy that typical cartoon and textureparts have in the missing band. For the cartoon part, we can prove that the relative reconstructionerror, restricted to the missing part, goes to zero as j → ∞ . For texture, we note that it is possiblefor T s,j to be close to zero in the missing part, and thus the relative error restricted to the missingpart need not go to zero in the general case. However, generic texture parts are not close to zero inthe missing band, and typically (cid:107) P j T s,j (cid:107) ∝ h j (cid:107)T s,j (cid:107) ∝ h j − j . We thus consider for texture a relative error of the form (cid:107) P j T ∗ j − P j T s,j (cid:107) h j − j Theorem 6.4.
Consider the cartoon patch and texture with w S j and T s,j defined in (34) and (35)respectively. Suppose that the energy matching (36) holds. Suppose that the index set of the texture T s,j satisfies (56) and (57) . Then, for < h j = o (2 − ( α j + (cid:15) ) j ) with α j ∈ (0 , , lim inf α j > and (cid:15) satisfying (60), the recovery error provided by Algorithm (INP-SEP) decays rapidly and we haveasymptotically perfect simultaneous separation and inpainting. Namely, for all N ∈ N , (cid:107)C ∗ j − w S j (cid:107) + (cid:107)T ∗ j − T s,j (cid:107) (cid:107) w S j (cid:107) + (cid:107)T s,j (cid:107) = o (2 − Nj ) → , j → ∞ , (62) where ( C ∗ j , T ∗ j ) is the solution of (INP-SEP) and ( C j , T s,j ) are ground truth components. In addition,if ω (0) (cid:54) = 0 , we have asymptotically accurate relative reconstruction error in the missing part (cid:107) P j C ∗ j − P j ω S j (cid:107) (cid:107) P j ω S j (cid:107) j → + ∞ −−−−→ and (cid:107) P j T ∗ j − P j T s,j (cid:107) h j − j j → + ∞ −−−−→ . (63)We postpone the proof of Theorem 6.4 to Section 10, after we discuss preliminary material.Note that in Theorem 6.4 there is no direct restriction on the texture patch sizes s − j , and thetheorem works even when the texture patch at each scale is smaller than the scale − j . However,the most useful case of Theorem 6.4, which appropriately models the relation between texture andcartoon, is when s j = s is constant for all j . This is formulated in the following corollary. Corollary 6.5.
Consider the conditions of Theorem 6.4, and suppose that the texture part T s,j isbased on a constant s j = s . Then, we have asymptotically perfect separation and inpainting, i.e, for ∀ N ∈ N , (cid:107)C j − C j (cid:107) + (cid:107)T j − T s,j (cid:107) (cid:107)C j (cid:107) + (cid:107)T s,j (cid:107) = o (2 − Nj ) → , j → ∞ . (64) If in addition ω (0) (cid:54) = 0 , we have (cid:107) P j C ∗ j − P j ω S j (cid:107) (cid:107) P j ω S j (cid:107) j → + ∞ −−−−→ and (cid:107) P j T ∗ j − P j T s,j (cid:107) h j − j j → + ∞ −−−−→ . (65)9 ANALYSIS OF SIMULTANEOUS SEPARATION AND INPAINTING7. Extensions and future directions
In this section, we present potential extensions and future directions of our approach.(1)
Global cartoon model . Using the technique introduced in [10] and [13], we can localize aglobal cartoon part C j by a partition of unity ( w Q ) Q ∈Q . Denoting C j,Q = C j · w Q , we have (cid:88) Q C j,Q = C j . In this approach, using a tubular neighborhood theorem ([10, Sect. 6] and [13]), we applya diffeomorphism Φ Q to each piece C j,Q to straighten out the local curve discontinuity. Bycombining the different neighborhoods, we can derive a convergence results for a globalcartoon part.(2) Other types of components . Our theoretical analysis holds for general representationsystems which form Parseval frames. Thus, our general technique can be applied to problemsof image separation and image ipainting of other types of image parts.(3)
Noisy case . In future work we will extend our theory to the case where the image containsnoise.(4)
More components . In our analysis we consider the case where there are two components.In future work we will extend our theoretical guarantee for separation and inpainting ofmore than two geometric components.In the rest of the paper we prove Theorem 6.4. For that, in section 8 we study the relativesparsity of texture and cartoon, and in Section 9 we study the cluster coherence of the sparsifyingsystems. 8.
Relative sparsity of texture and cartoon
In this section, we bound the relative sparsity of texture with respect to shearlet frame (Subsection8.2) and texture with respect to Gabor frame (Subsection 8.3).8.1.
Decay estimates of shearlet and Gabor elements.
First, we provide decay estimates ofshearlet and Gabor elements.
Lemma 8.1.
Consider the shearlet frame Ψ of Definition 5.3 and the Gabor frame of scale j, G j defined in (33). For any arbitrary integer N = 1 , , . . . there exists a constant C N depending on j such that the following estimates holdi) | ( g s j ) m,n ( x ) | ≤ C N · s j · (cid:104)| s j x + m |(cid:105) − N (cid:104)| s j x + m |(cid:105) − N , ii) | ψ α j , (v) j,l,k ( x ) | ≤ C N · (2+ α j ) j/ · (cid:104)| α j j x − k |(cid:105) − N (cid:104)| j x + l α j j x − k |(cid:105) − N , | ψ α j , (h) j,l,k ( x ) | ≤ C N · (2+ α j ) j/ · (cid:104)| j x + l α j j x − k ) |(cid:105) − N (cid:104)| α j j x − k |(cid:105) − N , iii) |(cid:104) ( g s j ) m,n , ψ α j , ( ι ) j,l,k (cid:105)| ≤ C N · − (2 − α j ) j/ , ∀ ( ι ) ∈ { (v) , (h) , (b) } . Proof. i) By the change of variable ζ = s − j ξ − n , we have | ( g s j ) m,n ( x ) | = (cid:12)(cid:12)(cid:12) (cid:90) R (ˆ g s j ) m,n ( ξ ) e πiξ T x dξ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:90) R s − j ˆ g ( s − j ξ − n ) e πiξ T ( x + m sj ) dξ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) s j e πin T ( s j x + m ) (cid:90) R ˆ g ( ζ ) e πiζ T ( s j x + m ) dζ (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) s j (cid:90) R ˆ g ( ζ ) e πiζ T ( s j x + m ) dζ (cid:12)(cid:12)(cid:12) . We now apply integration by parts for N , N = 1 , , . . . , with respect to ζ , ζ , respectively, weobtain | ( g s j ) m,n ( x ) | = (cid:12)(cid:12)(cid:12) (cid:90) R s j ( s j x + m − N ∂ N ∂ζ N [ˆ g ( ζ )] e πiζ T ( s j x + m ) dζ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:90) R s j ( s j x + m − N ( s j x + m − N ∂ N + N ∂ζ N ∂ζ N [ˆ g ( ζ )] e πiζ T ( s j x + m ) dζ (cid:12)(cid:12)(cid:12) ≤ s j | s j x + m | − N | s j x + m | − N (cid:12)(cid:12)(cid:12) (cid:90) R ∂ N + N ∂ζ N ∂ζ N [ˆ g ( ζ )] dζ (cid:12)(cid:12)(cid:12) , | ( g s j ) m,n ( x ) | ≤ s j | s j x k + m k | − N k (cid:12)(cid:12)(cid:12) (cid:90) R ∂ N k ∂ζ N k k [ˆ g ( ζ )] dζ (cid:12)(cid:12)(cid:12) for k = 1 , . Here, the boundary terms vanish due to the compact support of (ˆ g s j ) m,n ( ζ ) . Thus, s − j (cid:16) | s j x + m | N + | s j x + m | N + | s j x + m | N | s j x + m | N (cid:17) | ( g s j ) m,n ( x ) | = s − j (1 + | s j x + m | N )(1 + | s j x + m | N ) | ( g s j ) m,n ( x ) |≤ (cid:12)(cid:12)(cid:12) (cid:90) R ˆ g ( ζ ) dζ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:90) R ∂ N ∂ζ N [ˆ g ( ζ )] dζ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:90) R ∂ N ∂ζ N [ˆ g ( ζ )] dζ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:90) R ∂ N + N ∂ζ N ∂ζ N [ˆ g ( ζ )] dζ (cid:12)(cid:12)(cid:12) . By the smoothness of ˆ g ( ζ ) and supp ˆ g = [ − , , there exists a constant C (cid:48) N ,N independent of j such that (cid:12)(cid:12)(cid:12) (cid:90) R ˆ g ( ζ ) dζ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:90) R ∂ N ∂ζ N [ˆ g ( ζ )] dζ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:90) R ∂ N ∂ζ N [ˆ g ( ζ )] dζ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:90) R ∂ N + N ∂ζ N ∂ζ N [ˆ g ( ζ )] dζ (cid:12)(cid:12)(cid:12) ≤ C (cid:48) N ,N . We obtain | ( g s j ) m,n ( x ) | ≤ C (cid:48) N ,N · s j ·
11 + | s j x + m | N
11 + | s j x + m | N . Moreover, for each N , N = 1 , , . . . there exists C (cid:48) N , C (cid:48) N such that (cid:104)| s j x + m |(cid:105) N = (1 + | s j x + m | ) N / ≤ C (cid:48) N (1 + | s j x + m | ) N , (cid:104)| s j x + m |(cid:105) N = (1 + | s j x + m | ) N / ≤ C (cid:48) N (1 + | s j x + m | ) N . Thus, | ( g s j ) m,n ( x ) | ≤ C N ,N · s j · (cid:104)| s j x + m |(cid:105) − N (cid:104)| s j x + m |(cid:105) − N . This proves the claim for any arbitrary integer N .ii) By the change of variable ζ T = ξ T A − jα j , ( ι ) S − l ( ι ) , we have | ψ α j , ( ι ) j,l,k ( x ) | = (cid:12)(cid:12)(cid:12) (cid:90) R ˆ ψ α j , ( ι ) j,l,k ( ξ ) e πiξ T x dξ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:90) R − (2+ α j ) j/ W j ( ξ ) V ( ι ) (cid:16) ξ T A − jα j , ( ι ) S − l ( ι ) (cid:17) e πiξ T ( x − A − jαj, ( ι ) S − l ( ι ) k ) dξ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:90) R (2+ α j ) j/ W j (( S l ( ι ) A jα j , ( ι ) ) T ζ ) V ( ι ) ( ζ ) e πiζ T ( S l ( ι ) A jαj, ( ι ) x − k ) dζ (cid:12)(cid:12)(cid:12) . Similarly to in (i) we apply integration by parts for N , N = 1 , , . . . with respect to ζ , ζ . Weobtain the decay estimate of universal shearlets | ψ α j , (v) j,l,k ( x ) | ≤ C N · (2+ α j ) j/ · (cid:104)| α j j x − k |(cid:105) − N (cid:104)| j x + l α j j x − k |(cid:105) − N , | ψ α j , (h) j,l,k ( x ) | ≤ C N · (2+ α j ) j/ · (cid:104)| j x + l α j j x − k ) |(cid:105) − N (cid:104)| α j j x − k |(cid:105) − N . iii) We consider three cases.Case 1: ( ι ) = (v) . Applying (i) and (ii) and the change of variables ( y , y ) = ( s j x , j x + l α j j x ) ,we obtain |(cid:104) ( g s j ) m,n , ψ α j , (v) j,l,k (cid:105)| = (cid:12)(cid:12)(cid:12) (cid:90) R ( g s j ) m,n ( x ) ψ α j , (v) j,l,k ( x ) dx (cid:12)(cid:12)(cid:12) ≤ C (cid:48) N · − (2 − α j ) j/ (cid:90) R (cid:104)| y + m |(cid:105) − N (cid:104)| y − k |(cid:105) − N dy dy ≤ C N · − (2 − α j ) j/ . Case 2: ( ι ) = (h) . Similarly, we can use (i), (ii) and the change of varibles ( y , y ) = (2 j x + l α j j x , s j x ) to obtain |(cid:104) ( g s j ) m,n , ψ α j , (v) j,l,k (cid:105)| ≤ C N · − (2 − α j ) j/ . Case 3: ( ι ) = (b) . By the definition of boundary shearlets, we can verify this estimate by combiningtwo cases above. (cid:3)
Lemma 8.2.
For each n ∈ N there exists a constant C N > such that (cid:90) R (cid:104)| z |(cid:105) − N (cid:104)| z + t |(cid:105) − N dz ≤ C N (cid:104)| t |(cid:105) − N , ∀ t ∈ R . Proof.
We have (cid:90) R (cid:104)| z |(cid:105) − N (cid:104)| z + t |(cid:105) − N dz = (cid:90) R max {(cid:104)| z |(cid:105) − N , (cid:104)| z + t |(cid:105) − N } · min {(cid:104)| z |(cid:105) − N , (cid:104)| z + t |(cid:105) − N } dz ≤ (cid:90) R (cid:16) (cid:104)| z |(cid:105) − N + (cid:104)| z + t |(cid:105) − N (cid:17) · (cid:104)| t/ |(cid:105) − N dz ≤ C N (cid:104)| t |(cid:105) − N . (cid:3) Cartoon patch.
For the sake of brevity, we use some indexing sets for universal shearlets ∆ := (cid:110) ( j, l, k ; α j , ( ι )) | j ≥ , | l | < (2 − α j )j , k ∈ Z , ( ι ) ∈ { (h) , (v) } (cid:111)(cid:91) (cid:110) ( j, l, k ; α j , (b)) | j ≥ , | l | = 2 (2 − α j )j , k ∈ Z (cid:111) , (66) ∆ j := { ( j (cid:48) , k, l ; α j , ( ι )) ∈ ∆ | j (cid:48) = j } , j ≥ , (67) ∆ ± j := ∆ j − ∪ ∆ j ∪ ∆ j +1 , (68)where ∆ − = ∅ . We now have the following result. Proposition 8.3.
Consider the shearlet frame Ψ of Definition 5.3 and the cartoon patch w S j asdefined in (34). We assume that lim inf j →∞ α j > . Then, we have the following decay estimate ofthe cluster approximate error δ ,j δ ,j := (cid:88) η ∈ ∆ ,η / ∈ Λ ± ,j |(cid:104) ψ α j (cid:48) , ( ι ) j (cid:48) ,l,k , w S j (cid:105)| = o (2 − Nj ) , ∀ N ∈ N , (69) where η = ( j (cid:48) , l, k ; α j (cid:48) , ( ι )) . The proof of this proposition roughly follows the lines of the Proposition 5.2 in [25]. For the sakeof brevity we denote t (v) = ( t (v)1 , t (v)2 ) := A − jα j , (v) S − l (v) k = (2 − α j j k , − j ( k − lk )) , (70) t (h) = ( t (h)1 , t (h)2 ) := A − jα j , (h) S − l (h) k = (2 − j ( k − lk ) , − α j j k ) . (71)The proof also relies on the following two lemmas. Lemma 8.4.
Consider the shearlet frame Ψ of Definition 5.3 and the cartoon patch with sub-band w S j defined in (34). Then, for α j ∈ (0 , , j (cid:48) ∈ { j − , j, j + 1 } , j ≥ , the following estimates holdfor arbitrary integers M ≥ i) If ( ι ) = (v) and | l | > , we have (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j (cid:48) , (v) j (cid:48) ,l,k (cid:105) (cid:12)(cid:12)(cid:12) ≤ C M · − (2+ α j (cid:48) ) j/ · (cid:104)| t (v)1 |(cid:105) − (cid:104)| t (v)2 |(cid:105) − (cid:104)| α j (cid:48) j (cid:48) |(cid:105) − M . ii) If ( ι ) = (h) , we have (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j (cid:48) , (h) j (cid:48) ,l,k (cid:105) (cid:12)(cid:12)(cid:12) ≤ C M · j (cid:48) r · (cid:104)| t (h)1 |(cid:105) − (cid:104)| t (h)2 |(cid:105) − (cid:104)| j (cid:48) |(cid:105) − M , where r is defined in (14).iii) If ( ι ) = (b) and | l | = 2 (2 − α j (cid:48) ) j (cid:48) , we have (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j (cid:48) , (b) j (cid:48) ,l,k (cid:105) (cid:12)(cid:12)(cid:12) = C M · − (2+ α j (cid:48) ) j (cid:48) / · (cid:104)| t (b)1 |(cid:105) − (cid:104)| t (b)2 |(cid:105) − (cid:104)| α j (cid:48) j (cid:48) |(cid:105) − M . Proof.
Without loss of generality, we prove for j (cid:48) = j . The other cases are treated similarly.(i) By the definition of w S j and Plancherel’s theorem, we obtain (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:104) (cid:99) w S j , ˆ ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:90) R ˆ w ( ξ ) 12 πξ W j ( ξ ) ˆ ψ α j , (v) j,l,k ( ξ ) (cid:12)(cid:12)(cid:12) = 12 π (cid:12)(cid:12)(cid:12) (cid:90) R e πit (v)2 ξ (cid:90) R ˆ w ( ξ ) 1 ξ W j ( ξ ) ˆ ψ α j , (v) j,l, ( ξ ) e πit (v)1 ξ dξ dξ (cid:12)(cid:12)(cid:12) = 12 π (cid:12)(cid:12)(cid:12) (cid:90) R e πit (v)2 ξ (cid:90) R ˆ w ( ξ ) τ j,l ( ξ ) e πit (v)1 ξ dξ dξ (cid:12)(cid:12)(cid:12) , where τ j,l ( ξ ) := ξ W j ( ξ ) ˆ ψ α j , (v) j,l, ( ξ ) .We now apply repeated integration by parts with respect to ξ i , i = 1 , , . We get (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) = 12 π (cid:12)(cid:12)(cid:12) (cid:90) R e πit (v)2 ξ (cid:90) R ∂ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] 1(2 πit (v)1 ) e πit (v)1 ξ dξ (cid:105) dξ (cid:12)(cid:12)(cid:12) ≤ | t (v)1 | (cid:12)(cid:12)(cid:12) (cid:90) R e πit (v)2 ξ (cid:90) R ∂ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] e πit (v)1 ξ dξ dξ (cid:12)(cid:12)(cid:12) ≤ | t (v)1 | − (cid:90) R (cid:90) R (cid:12)(cid:12)(cid:12) ∂ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) dξ dξ (72)and similarly (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) ≤ | t (v)2 | − (cid:90) R (cid:90) R (cid:12)(cid:12)(cid:12) ∂ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) dξ dξ (73) (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) ≤ | t (v)1 | − | t (v)2 | − (cid:90) R (cid:90) R (cid:12)(cid:12)(cid:12) ∂ ∂ξ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) dξ dξ . (74)Thus, | t (v)1 | (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) (72) ≤ (cid:90) R (cid:90) R (cid:12)(cid:12)(cid:12) ∂ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) dξ dξ | t (v)2 | (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) (73) ≤ (cid:90) R (cid:90) R (cid:12)(cid:12)(cid:12) ∂ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) dξ dξ | t (v)1 | | t (v)2 | (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) (74) ≤ (cid:90) R (cid:90) R (cid:12)(cid:12)(cid:12) ∂ ∂ξ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) dξ dξ . This leads to (1 + | t (v)1 | + | t (v)2 | + | t (v)1 | | t (v)2 | ) (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) = (cid:104)| t (v)1 |(cid:105) (cid:104)| t (v)2 |(cid:105) (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) ≤ (cid:90) R (cid:12)(cid:12)(cid:12) ˆ w ( ξ ) τ j,l ( ξ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ ∂ξ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) dξ (75) = (cid:82) R Γ( ξ ) dξ , where Γ( ξ ) := (cid:82) R (cid:12)(cid:12)(cid:12) ˆ w ( ξ ) τ j,l ( ξ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ ∂ξ ∂ξ [ ˆ w ( ξ ) τ j,l ( ξ )] (cid:12)(cid:12)(cid:12) dξ . Next, we bound Γ . By the definition of the universal shearlets, ˆ ψ α j , (v) j,l,k has compact support in the trapezoidal region supp( ˆ ψ α j , (v) j,l,k ) = (cid:110) ξ ∈ R | ξ ∈ [ − j − , j − ] \ [ − j − , j − ] , (cid:12)(cid:12)(cid:12) ξ ξ − l − (2 − α j ) j (cid:12)(cid:12)(cid:12)(cid:111) . (76)This implies that for any ξ ∈ supp τ j,l we have ( l − ( α j − j ≤ ξ ξ ≤ ( l + 1)2 ( α j − j and j − ≤ | ξ | ≤ j − . Using the assumption | l | > follows that there exist constants C , C > such that ξ ∈ I j,l := [ − C l α j j , − C l α j j ] ∪ [ C l α j j , C l α j j ] . τ j,l ( ξ ) is zero for ξ < j − , we obtain (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂τ j,l ∂ξ ( ξ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∂∂ξ (cid:104) ξ W j ( ξ ) v (cid:16) (2 − α j ) j ξ ξ − l (cid:17)(cid:105)(cid:12)(cid:12)(cid:12) ≤ j − (cid:104) j W ( ξ j , ξ j ) ∂W∂ξ ( ξ j , ξ j ) v (cid:16) (2 − α j ) j ξ ξ − l (cid:17) +2 (2 − α j ) j ξ ∂v∂ξ (cid:16) (2 − α j ) j ξ ξ − l (cid:17) W ( ξ j , ξ j ) (cid:105) ≤ j − (cid:16) C − j + C (2 − α j ) j | ξ | (cid:17) <α j ≤ ≤ C − j . (77)Furthermore, we have (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂τ j,l ∂ξ ( ξ ) (cid:12)(cid:12)(cid:12) = 2 (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂ σ j,l ξ ∂ξ (cid:12)(cid:12)(cid:12) , where σ j,l := W j ( ξ ) v (cid:16) (2 − α j ) j ξ ξ − l (cid:17) . Thus, in the support (76) we have (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂σ j,l ∂ξ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∂∂ξ (cid:104) W j ( ξ ) v (cid:16) (2 − α j ) j ξ ξ − l (cid:17)(cid:105)(cid:12)(cid:12)(cid:12) = 22 j W ( ξ j , ξ j ) ∂W∂ξ ( ξ j , ξ j ) v (cid:16) (2 − α j ) j ξ ξ − l (cid:17) − (2 − α j ) j ξ ξ ∂v∂ξ W ( ξ j , ξ j ) ≤ C − j + C (2 − α j ) j | ξ || ξ | α j ≤ ≤ C − α j j . Consequently, we get (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂τ j,l ∂ξ ( ξ ) (cid:12)(cid:12)(cid:12) = 2 (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂σ j,l ∂ξ ξ − σ j,l ( ξ ) ξ (cid:12)(cid:12)(cid:12) ≤ C − α j j j − + C (cid:48) j − ≤ C (cid:48)(cid:48) − j . (78)Similarly, we obtain (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂ τ j,l ∂ξ ( ξ ) (cid:12)(cid:12)(cid:12) ≤ C − j , (79)and (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂ τ j,l ∂ξ ( ξ ) (cid:12)(cid:12)(cid:12) ≤ C − j . (80)We now exploit the specific form of I j,l as well as the rapid decay of ˆ w to study ˆ w ( n ) for n = 0 , , .These lead to (see Figure 7 for illustration) (cid:107) ˆ w ( n ) (cid:107) L ( I j,l ) ≤ vol ( I j,l ) sup ξ ∈ I j,l | ˆ w ( n ) ( ξ ) | ≤ C M,n | l α j j |(cid:104)| l α j j |(cid:105) − ( M +1) ≤ C M,n (cid:104)| l α j j |(cid:105) − M l> ≤ C M,n (cid:104)| α j j |(cid:105) − M ∀ M ∈ N , ∀ n = 1 , , . . . (81)Combining (77), (78), (79), (80), (81), with (75), we have Γ( ξ ) ≤ C · − (2+ α j ) j/ · − j max (cid:110) (cid:107) ˆ w (cid:107) L ( I j,l ) , (cid:107) ∂ ˆ w∂ξ (cid:107) L ( I j,l ) , (cid:107) ∂ ˆ w∂ξ (cid:107) L ( I j,l ) (cid:111) (82) ≤ C M · − (2+ α j ) j/ · − j · (cid:104)| α j j |(cid:105) − M . (83)In addition, since supp Γ( ξ ) = [2 j − , j − ] , we finally obtain (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) ≤ C M · − (2+ α j ) j/ · (cid:104)| t (v)1 |(cid:105) − (cid:104)| t (v)2 |(cid:105) − (cid:104)| α j j |(cid:105) − M . (ii) This claim is similar to (i) but we need to modify the intervals I j,l to be [ − C j , − C j ] ∪ [ C j , C j ] , where C , C > . This leads to the additional terms (cid:104)| j |(cid:105) − M and r in (ii) whencomparing with (i). Next, we use r ≤ | ξ | ≤ j − instead of using j − ≤ | ξ | ≤ j − forthe support of τ j,l . Thus, redefining τ j,l ( ξ ) = ξ W j ( ξ ) ˆ ψ α j , (h) j,l, ( ξ ) , we modify (77) , (78) , (79) , (80) asfollows.4 VAN TIEP DO, RON LEVIE, GITTA KUTYNIOK (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂τ j,l ∂ξ ( ξ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∂∂ξ (cid:104) ξ W j ( ξ ) v (cid:16) (2 − α j ) j ξ ξ − l (cid:17)(cid:105)(cid:12)(cid:12)(cid:12) ≤ r (cid:12)(cid:12)(cid:12) j W ( ξ j , ξ j ) ∂W∂ξ ( ξ j , ξ j ) v (cid:16) (2 − α j ) j ξ ξ − l (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (2 − α j ) j ξ ∂v∂ξ W ( ξ j , ξ j ) (cid:12)(cid:12)(cid:12) ≤ r (cid:16) C − j + C (2 − α j ) j | ξ | (cid:17) <α j ≤ ≤ Cr · j , (84)Furthermore, we have (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂τ j,l ∂ξ ( ξ ) (cid:12)(cid:12)(cid:12) = 2 (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂ σ j,l ξ ∂ξ (cid:12)(cid:12)(cid:12) , where σ j,l := W j ( ξ ) v (cid:16) (2 − α j ) j ξ ξ − l (cid:17) . Thus, (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂σ j,l ∂ξ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∂∂ξ (cid:104) W j ( ξ ) v (cid:16) (2 − α j ) j ξ ξ − l (cid:17)(cid:105)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) j W ( ξ j , ξ j ) ∂W∂ξ ( ξ j , ξ j ) v (cid:16) (2 − α j ) j ξ ξ − l (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (2 − α j ) j ξ ∂v∂ξ W ( ξ j , ξ j ) (cid:12)(cid:12)(cid:12) ≤ C − j + C (2 − α j ) j | ξ | <α j ≤ ≤ C. Consequently, we have (cid:12)(cid:12)(cid:12) ∂τ j,l ∂ξ ( ξ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∂σ j,l ∂ξ ξ − σ j,l ( ξ ) ξ (cid:12)(cid:12)(cid:12) ≤ C (cid:48) r . (85)Similarly, we obtain (2+ α j ) j/ (cid:12)(cid:12)(cid:12) ∂ τ j,l ∂ξ ( ξ ) (cid:12)(cid:12)(cid:12) ≤ Cr · j , (86)and (cid:12)(cid:12)(cid:12) ∂ τ j,l ∂ξ ( ξ ) (cid:12)(cid:12)(cid:12) ≤ C (cid:48) r . (87)These bounds lead to Γ( ξ ) ≤ Cr · max (cid:110) (cid:107) ˆ w (cid:107) L ( I j,l ) , (cid:107) ∂ ˆ w∂ξ (cid:107) L ( I j,l ) , (cid:107) ∂ ˆ w∂ξ (cid:107) L ( I j,l ) (cid:111) . Thus, by the rapid decay of ˆ w , as before, (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) ≤ C M · j r · (cid:104)| t (v)1 |(cid:105) − (cid:104)| t (v)2 |(cid:105) − (cid:104)| j |(cid:105) − M . For an illustration, we refer to Figure 7. Finally, we obtain (iii). Recalling the definition of theshearlets, we can easily verify this estimate since boundary shearlets is just a particular case ofhorizontal and vertical shearlets. (cid:3)
Lemma 8.5.
Consider the shearlet frame Ψ of Definition 5.3 and the cartoon patch with sub-band w S j defined in (34). For j ∈ { j − , j, j + 1 } , j ≥ , and for any arbitrary N ≥ , there exists aconstant C N > such that (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j (cid:48) , (v) j (cid:48) ,l,k (cid:105) (cid:12)(cid:12)(cid:12) ≤ C N (2 − α j (cid:48) ) j (cid:48) / (cid:90) R ˜ w N,j (2 − α j (cid:48) j (cid:48) ( x + k )) (cid:104)| x |(cid:105) − N (cid:104)| lx + lk − k |(cid:105) − N dx , where ˜ w N,j (cid:48) := | w | (cid:63) (cid:104)| j (cid:48) [ · ] |(cid:105) − N , and (cid:104)| · |(cid:105) is defined from (42) .Proof. Without loss of generality, we can prove only for j (cid:48) = j. We first note that in supp ψ α j , (v) j,l,k wehave w S ≡ w S , where wS is defined in (14). We now consider the line distribution w L introducedin [25] by (cid:104) w L , f (cid:105) = (cid:90) ρ − ρ w ( x ) f ( x , dx , f ∈ S ( R ) , (88)5 ANALYSIS OF SIMULTANEOUS SEPARATION AND INPAINTINGwith Fourier transform (cid:104) (cid:99) w L , f (cid:105) = (cid:90) R ˆ w ( ξ ) f ( ξ ) dξ. (89)Next, we observe that (cid:104) w L , f (cid:105) = (cid:90) ρ − ρ w ( x ) f ( x , dx = (cid:90) ρ − ρ w ( x ) (cid:90) R f ( x , x ) δ ( x ) dx dx = (cid:90) (cid:90) R w ( x ) δ ( x ) f ( x , x ) dx dx = (cid:104) w ( x ) δ ( x ) , f (cid:105) , where δ ( x ) denotes the usual Dirac delta functional. Thus, w L ( x ) = w ( x ) δ ( x ) . (90)In addition, (cid:104) w S , f (cid:105) = (cid:90) ρ − ρ w ( x ) (cid:90) −∞ f ( x ) dx dx = (cid:90) ρ − ρ w ( x ) (cid:90) + ∞−∞ f ( x ) { ≥ x } (0) dx dx = (cid:90) ρ − ρ w ( x ) (cid:90) R f ( x ) (cid:16) (cid:90) R { y ≥ x } ( y ) δ ( y ) dy (cid:17) dx dx = (cid:90) ρ − ρ w ( x ) (cid:90) R f ( x ) (cid:16) (cid:90) + ∞ x δ ( y ) dy (cid:17) dx dx = (cid:68) (cid:90) + ∞ x w ( x ) δ ( y ) dy, f ( x ) (cid:69) . Hence, we obtain w S ( x ) = (cid:90) + ∞ x w L ( x , y ) dy. (91)Since integration commutes with convolution, we also have w S j = w S (cid:63) F j = (cid:90) + ∞ x w L j ( x , y ) dy. where w L j is defined by w L (cid:63) F j . In addition, by integration by parts with respect to variable x ,we obtain (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) = (cid:104) (cid:90) + ∞ x w L j ( x , y ) dy, ψ α j , (v) j,l,k (cid:105)(cid:105) = (cid:104) w L j ( x ) , (cid:90) + ∞ x ψ α j , (v) j,l,k ( x , y ) dy (cid:105) , where the boundary terms vanish due to the compact support of ψ α j , (v) j,l,k . Now we put Ξ α j , (v) j,l,k ( x ) := (cid:90) + ∞ x ψ α j , (v) j,l,k ( x , y ) dy, and note that ˆΞ α j , (v) j,l,k ( ξ ) = i πξ · ˆ ψ α j , (v) j,l,k ( ξ ) . Similarly to Lemma 8.1, which was based on ψ α j , (v) j,l,k , wecan prove a lemma for Ξ α j , (v) j,l,k , and prove the rapid decay property | Ξ α j , (v) j,l,k ( x ) | ≤ C (cid:48) N · j − · (2+ α j ) j/ · (cid:104)| α j j x − k |(cid:105) − N (cid:104)| l α j j x + 2 j x − k |(cid:105) − N ≤ C N · ( α j − j/ · (cid:104)| α j j x − k |(cid:105) − N (cid:104)| l α j j x + 2 j x − k |(cid:105) − N . (92)Now we can use the decay estimate of the line singularities w L j to obtain | w L j ( x ) | = | [ w L (cid:63) F j ]( x ) | = (cid:12)(cid:12)(cid:12) (cid:90) R w ( y ) F j ( x − ( y , dy (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) (cid:90) R | w ( y ) | j | ˇ W (2 j ( x − ( y , | dy (cid:12)(cid:12)(cid:12) ≤ (cid:90) R | w ( y ) | j C N (cid:104)| j x |(cid:105) − N (cid:104)| j ( y − x ) |(cid:105) − N dy = C N j (cid:104)| j x |(cid:105) − N [ | w (cid:63) (cid:104) j [ · ] |(cid:105) − N ]( x )= C N j (cid:104)| j x |(cid:105) − N ˜ w N,j ( x ) , ˜ w N,j ( x ) := [ | w (cid:63) (cid:104) j [ · ] |(cid:105) − N ]( x ) , x = ( x , x ) ∈ R . Combining this with the rapid decayproperties of Ξ α j , (v) j,l,k ( x ) from (92) and Lemma 8.2, we have (cid:12)(cid:12)(cid:12) (cid:104) w S j , ψ α j , (v) j,l,k (cid:105) (cid:12)(cid:12)(cid:12) ≤ C N (cid:90) R j (cid:104)| j x |(cid:105) − N ˜ w N,j ( x )2 ( α j +2) j/ (cid:104)| α j j x − k |(cid:105) − N ·(cid:104)| l α j j x + 2 j x − k |(cid:105) − N dx = C N (2 − α j ) j/ (cid:90) R (cid:104)| x |(cid:105) − N ˜ w N,j (2 − α j j ( x + k )) (cid:104)| x |(cid:105) − N (cid:104)| lx + lk + x − k |(cid:105) − N dx ≤ C N (2 − α j ) j/ (cid:90) R ˜ w N,j (2 − α j j ( x + k )) (cid:104)| x |(cid:105) − N (cid:104)| lx + lk − k |(cid:105) − N dx . (cid:3) Figure 7.
Interactions between horizontal shearlets, vertical shearlets and thesub-image of cartoon part w S j . Left: Frequency support of horizontal shearletsat scale j (grey), w S j (blue). Right: Frequency support of vertical shearlets with | l | > (brown), w S j (blue).Now we are ready to prove Proposition 8.3 Proof of Proposition 8.3.
By the definition of δ ,j and the support of the window function F j , onlyshearlet coefficients in ∆ ± j have nonzero inner products with wS j , so we have δ ,j = (cid:88) η ∈ ∆ ,η / ∈ Λ ± ,j |(cid:104) ψ α j (cid:48) , ( ι ) j (cid:48) ,l,k , w S j (cid:105)| = (cid:88) η ∈ ∆ ± j ,η / ∈ Λ ± ,j |(cid:104) ψ α j (cid:48) , ( ι ) j (cid:48) ,l,k , w S j (cid:105)| = (cid:88) k ∈ Z , | l |≤ | k − lk | > (cid:15)j (cid:48) |(cid:104) ψ α j (cid:48) , (v) j (cid:48) ,l,k , w S j (cid:105)| + (cid:88) k ∈ Z , | l | > |(cid:104) ψ α j (cid:48) , (v) j (cid:48) ,l,k , w S j (cid:105)| + (cid:88) k ∈ Z ,l ∈ Z |(cid:104) ψ α j (cid:48) , (h) j (cid:48) ,l,k , w S j (cid:105)| + (cid:88) k ∈ Z |(cid:104) ψ α j (cid:48) , (b) j (cid:48) , ± (2 − αj (cid:48) ) j (cid:48) ,k , w S j (cid:105)| = T + T + T + T , where T := (cid:88) k ∈ Z , | l |≤ | k − lk | > (cid:15)j (cid:48) |(cid:104) ψ α j (cid:48) , (v) j (cid:48) ,l,k , w S j (cid:105)| , T := (cid:88) k ∈ Z , | l | > |(cid:104) ψ α j (cid:48) , (v) j (cid:48) ,l,k , w S j (cid:105)| , T = (cid:88) k ∈ Z ,l ∈ Z |(cid:104) ψ α j (cid:48) , (h) j (cid:48) ,l,k , w S j (cid:105)| , T := (cid:88) k ∈ Z |(cid:104) ψ α j (cid:48) j (cid:48) , ± (2 − αj (cid:48) ) j (cid:48) ,k , w S j (cid:105)| . Without loss of generality we restrict to the scale index j (cid:48) = j . The respective arguments for theother cases j (cid:48) = j ± are similar.We start with an estimation of T . By Lemma 8.5, for N ≥ , we obtain T = (cid:88) k ∈ Z , | l |≤ | k − lk | > (cid:15)j |(cid:104) ψ α j , (v) j,l,k , w S j (cid:105)|≤ C N (2 − α j ) j/ (cid:88) k ∈ Z , | l |≤ | k − lk | > (cid:15)j (cid:90) R ˜ w N,j (2 − α j j ( x + k )) (cid:104)| x |(cid:105) − N (cid:104)| lx + lk − k |(cid:105) − N dx = C N (2 − α j ) j/ (cid:88) k ∈ Z , | l |≤ | k | > (cid:15)j (cid:90) R ˜ w N,j (2 − α j j ( x + k )) (cid:104)| x |(cid:105) − N (cid:104)| lx + k |(cid:105) − N dx = C N (2 − α j ) j/ (cid:88) k ∈ Z , | l |≤ | k | > (cid:15)j (cid:90) R (cid:16) (cid:88) k ∈ Z ˜ w N,j (2 − α j j ( x + k )) (cid:17) (cid:104)| x |(cid:105) − N (cid:104)| lx + k |(cid:105) − N dx . Furthermore, we have (cid:88) k ∈ Z ˜ w N,j (2 − α j j ( x + k )) = (cid:88) k ∈ Z (cid:90) R | w ( y ) |(cid:104)| j ( y − − α j j ( x + k )) |(cid:105) − N dy = (cid:88) k ∈ Z (cid:90) R | w ( y ) |(cid:104)| (2 − α j ) j ( k + x − α j j y ) |(cid:105) − N dy α j ≤ ≤ (cid:90) R | w ( y ) | (cid:16) (cid:88) k ∈ Z (cid:104)| k + x − α j j y |(cid:105) − N (cid:17) dy ≤ C (cid:48) (cid:90) R | w ( y ) | (cid:16) (cid:90) R (cid:104)| t + x − α j j y |(cid:105) − N dt (cid:17) dy = C (cid:48) (cid:90) R | w ( y ) | (cid:16) (cid:90) R (cid:104)| t |(cid:105) − N dt (cid:17) dy ≤ C N . This implies T ≤ C N (2 − α j ) j/ (cid:88) k ∈ Z , | l |≤ | k | > (cid:15)j (cid:48) (cid:90) R (cid:104)| x |(cid:105) − N (cid:104)| lx + k |(cid:105) − N dx . Lemma . ≤ C N (2 − α j ) j/ (cid:88) k ∈ Z , | k | > (cid:15)j (cid:48) (cid:104)| k |(cid:105) − N dx . ≤ C N (2 − α j ) j/ (cid:90) t> (cid:15)j (cid:104)| t |(cid:105) − N ≤ C N (2 − α j ) j/ − ( N − (cid:15)j . We now bound the decay rate for the term T . First we fix j ≥ and | l | > . For N ≥ , weobtain (cid:88) k ∈ Z (cid:104)| t (v)1 |(cid:105) − | t (v)2 | − N = (cid:88) k ∈ Z (cid:104)| − α J j k |(cid:105) − (cid:104)| − j ( k − lk ) |(cid:105) − ≤ C (cid:48) (2+ α j )2 j (cid:90) R (cid:90) R (cid:104)| x |(cid:105) − (cid:104)| x |(cid:105) − dx dx ≤ C (2+ α j )2 j . (93)8 VAN TIEP DO, RON LEVIE, GITTA KUTYNIOKNow, by Lemma 8.4, we have T = (cid:88) k ∈ Z < | l |≤ (2 − αj ) j |(cid:104) ψ α j , (v) j,l,k , w S j (cid:105)|≤ C M (cid:88) k ∈ Z < | l |≤ (2 − αj ) j − (2+ α j ) j/ · (cid:104)| t (v)1 |(cid:105) − (cid:104)| t (v)2 |(cid:105) − (cid:104)| α j j |(cid:105) − M ≤ C M · (2 − α j ) j/ · (cid:104)| α j j |(cid:105) − M (cid:88) k ∈ Z (cid:104)| t (v)1 |(cid:105) − (cid:104)| t (v)2 |(cid:105) − ≤ C (cid:48) M · (2 − α j ) j/ · (2+ α j )2 j · (cid:104)| α j j |(cid:105) − M . Using the assumption lim inf j →∞ α j > and choosing M sufficiently large we obtain the desireddecay rates. By using (ii) and (iii) of the Lemma 8.4 the estimates for T and T are done similarly.Here, note that r · o (2 − Nj ) = o (2 − Nj ) for fixed r > . For an illustration of Lemmas 8.4, 8.5, andthe main proposition, we refer to Figure 7. (cid:3) Texture.
We now provide a relative sparsity error (Definition 3.3 ) for the texture part.
Proposition 8.6.
Consider the Gabor frame of scale j, G j , defined in (33), and the texture definedin Definition 4.1. Then, for every N ∈ N the sequence ( δ ,j ) j ∈ N decays rapidly in the sense δ ,j := (cid:88) λ/ ∈ B (0 ,M j ) × I ± T |(cid:104)T s,j , ( g s j ) λ (cid:105)| = o (2 − Nj ) . (94) Proof.
First, note that ˆ T s = (cid:90) R (cid:88) n ∈ I T d n g s j ( x ) e − πi (cid:104) ξ − s j n,x (cid:105) dx = (cid:88) n ∈ I T d n ˆ g s j ( ξ − s j n ) . Using the support condition of ˆ g , denoting λ = ( ˜ m, ˜ n ) , we obtain |(cid:104)T s,j , ( g s j ) λ (cid:105)| = |(cid:104) ˆ T s,j , (ˆ g s j ) λ (cid:105)| = (cid:12)(cid:12)(cid:12) (cid:88) n ∈ I T d n (cid:90) W j ( ξ )ˆ g s j ( ξ − s j n ) · ˆ g s j ( ξ − s j ˜ n ) e − πi (cid:104) ˜ m,ξ (cid:105) sj dξ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:88) n ∈ I T , | n − ˜ n |≤ d n (cid:90) W j ( ξ )ˆ g s j ( ξ − s j n ) · ˆ g s j ( ξ − s j ˜ n ) e − πi (cid:104) ˜ m,ξ (cid:105) sj dξ (cid:12)(cid:12)(cid:12) . Indeed, |(cid:104)T s,j , ( g s j ) λ (cid:105)| = |(cid:104) ˆ T s,j , (ˆ g s j ) λ (cid:105)| = 0 , for ˜ n / ∈ I ± T ∩ A s,j . Moreover, we have { ( ˜ m, ˜ n ) / ∈ B (0 , M ) × I ± T } = { ( ˜ m, ˜ n ) ∈ Z × Z , | ˜ m | > M j } ∪ { ( ˜ m, ˜ n ) ∈ Z × Z , ˜ n / ∈ I ± T } . Thus, since the Gabor coefficients of T s,j are zero for frequencies outside I ± T , δ ,j = (cid:88) λ/ ∈ B (0 ,M j ) × I ± T |(cid:104)T s,j , ( g s j ) λ (cid:105)| = (cid:88) | ˜ m |≥ M j , ˜ n ∈ I ± T ∩A s,j |(cid:104)T s,j , ( g s j ) λ (cid:105)| = (cid:88) | ˜ m |≥ M j , ˜ n ∈ I ± T ∩A s,j (cid:12)(cid:12)(cid:12) (cid:90) R (cid:88) n ∈ I T , | n − ˜ n |≤ d n ( g s j ) (0 ,n ) ( x )( g s j ) ( ˜ m, ˜ n ) ( x ) dx (cid:12)(cid:12)(cid:12) ≤ C (cid:88) | ˜ m |≥ M j , ˜ n ∈ I ± T ∩A s,j | d ˜ n | (cid:90) R | ( g s j ) (0 , ˜ n ) ( x ) | · | ( g s j ) ( ˜ m, ˜ n ) ( x ) | dx, where d ˜ n = max {| d n | | n ∈ I T , | n − ˜ n | ≤ } . Note that the last inequality holds since for each ˜ n ,there exist a finite number of n satisfying | n − ˜ n | ≤ . Moreover, by (56), there are o (2 (2 − α j ) j/ ) points of I ± T in each support of a shearlet and there are o (2 (2 − α j ) j ) shearlets in A j , so { n ∈ I ± T ∩ A s,j } ≤ { n ∈ I T ∩ A s,j } = o (2 (2 − α j ) j/ ) · (2 − α j ) j = o (2 (2 − α j )3 j/ ) . (cid:88) n ∈ I T ∩A s,j | d n | ≤ c − j , which implies d n ≤ √ c , ∀ n ∈ I T ∩ A s,j , we have δ ,j ≤ C · o (2 (2 − α j )3 j/ ) (cid:88) | ˜ m | >M j (cid:90) R | ( g s j ) (0 , ˜ n ) ( x ) | · | ( g s j ) ˜ m, ˜ n ( x ) | dx. Lemma . ≤ C N · o (2 (2 − α j )3 j/ ) (cid:90) R (cid:88) | ˜ m | >M j s j · (cid:104)| s j x |(cid:105) − N (cid:104)| s j x |(cid:105) − N (cid:104)| s j x + ˜ m |(cid:105) − N ·(cid:104)| s j x + ˜ m |(cid:105) − N dx. By Lemma 8.2, we now obtain δ ,j ≤ C N · o (2 (2 − α j )3 j/ ) · (cid:88) | ˜ m | >M j (cid:104)| ˜ m |(cid:105) − N (cid:104)| ˜ m |(cid:105) − N ≤ C N · o (2 (2 − α j )3 j/ ) · (cid:90) R (cid:90) M j (cid:104) t |(cid:105) − N (cid:104)| t |(cid:105) − N dt dt M j =2 (cid:15)j/ ≤ C N · o (2 (2 − α j )3 j/ ) · − ( N − (cid:15)j/ . This proves the claim since we can choose an arbitrarily large N ∈ N . (cid:3) Cluster coherence for cartoon and texture
In the previous section we proved relative sparsity for the texture and cartoon parts. For thesuccess of separation and inpainting, by Theorem 3.8, we need to prove that the cluster coherenceterms are less than / when summed together. To guarantee this asymptotically, we prove thatthe cluster coherence terms decay to zero when the scale goes to infinity.9.1. Cluster coherence of the un-projected frames.
We now analyze the cluster coherence ofthe un-projected frames µ c (Λ ± ,j , Ψ ; G j ) and µ c (Λ ,j , G j ; Ψ ) . Proposition 9.1.
Consider the shearlet frame Ψ of Definition 5.3 and the Gabor frame of scale j, G j , defined in (33). We have µ c (Λ ± ,j , Ψ ; G j ) → , j → ∞ . where Λ ± ,j is defined in (61).Proof. Let us recall the definition of the cluster of shearlets Λ ,j = (cid:110) ( j, l, k ; α j , (v)) | | l | ≤ , k ∈ Z , | k − lk | ≤ (cid:15)j (cid:111) , and Λ ± ,j = Λ ,j − ∪ Λ ,j ∪ Λ ,j +1 as defined in (61). Without loss of generality, we prove only µ c (Λ ,j , Ψ ; G j ) → as j → ∞ , and note that summing over j (cid:48) = j − , j, j + 1 does not change theasymptotics. By the definition of the cluster coherence, we have µ c (Λ ,j , Ψ ; G j ) = max ( m,n ) (cid:88) η ∈ Λ ,j |(cid:104) ψ α j , ( ι ) j,l,k , ( g s j ) m,n (cid:105)| , where η = ( j, l, k ; α j , ( ι )) . Suppose that the maximum is attained for ¯ m, ¯ n ∈ Z (the maximumexists, since both frames are translation invariant, and the shearlet elements at scale j are compactlysupported in frequency). Applying the change of variables ( y , y ) = ( s j x , j x + l α j j x − lk ) µ c (Λ ,j , Ψ ; G j ) = (cid:88) η ∈ Λ ,j |(cid:104) ψ α j , ( ι ) j,l,k , ( g s j ) ¯ m, ¯ n (cid:105)|≤ (cid:88) | l |≤ ,k ∈ Z | k − lk |≤ (cid:15)j (cid:90) R C N · s j · (2+ α j ) j/ · (cid:104)| α j j x − k |(cid:105) − N (cid:104)| j x + l α j j x − k |(cid:105) − N ··(cid:104)| s j x + m |(cid:105) − N (cid:104)| s j x + m |(cid:105) − N dx dx . = (cid:88) | l |≤ ,k ∈ Z | k − lk |≤ (cid:15)j (cid:90) R C N · − (2 − α j ) j/ · (cid:104)| α j j s j y − k |(cid:105) − N (cid:104)| y + lk − k |(cid:105) − N (cid:104)| y + m |(cid:105) − N ·(cid:104)| s j j y − l − (2 − α j ) j y + s j − j lk + m |(cid:105) − N dy dy ≤ (cid:88) | l |≤ ,k ∈ Z | k |≤ (cid:15)j (cid:90) R C N · − (2 − α j ) j/ (cid:104)| α j j s j y − k |(cid:105) − N (cid:104)| y − k |(cid:105) − N (cid:104)| y + m |(cid:105) − N dy dy = (cid:88) | l |≤ ,k ∈ Z | k |≤ (cid:15)j (cid:90) R C N − (2 − α j ) j/ (cid:16) (cid:88) k ∈ Z (cid:104)| α j j s j y − k |(cid:105) − N (cid:17) (cid:104)| y − k |(cid:105) − N ·(cid:104)| y + m |(cid:105) − N dy dy ≤ C N · − (2 − α j ) j/ · (cid:15)j = C N · − (2 − α j − (cid:15) ) j/ j → + ∞ −−−−→ , where (2 − α j − (cid:15) ) > (cid:15) at each scale j by (60). (cid:3) Next, we prove the following cluster coherence decays.
Proposition 9.2.
Consider the shearlet frame Ψ of Definition 5.3 and the Gabor frame of scale j, G j , defined in (33). We have µ c (Λ ,j , G j ; Ψ ) → , j → ∞ , where Λ ,j is defined in (58).Proof. Suppose that the maximum in the mutual coherence is attained at ¯ l ∈ Z , ¯ k ∈ Z and ( ι ) ∈{ (h) , (v) , (b) } (there is a maximum since the systems are translation invariant and the inner productgoes to zero as n goes to infinity). For each N = 1 , . . . using (56) and Lemma 8.1 (iii), we have µ c (Λ ,j , G j ; Ψ ) = (cid:88) m ∈ B (0 ,M j ) ,n ∈ I ± T |(cid:104) ( g s j ) m,n , ψ α j ,ιj, ¯ l, ¯ k (cid:105)|≤ C N · − (2 − α j ) j/ { m ∈ B (0 , M j ) } · { n ∈ I ± T ∩ M j,l, ( ι ) }≤ C N · − (2 − α j ) j/ · M j · (2 − α j − (cid:15) ) j/ M j =2 (cid:15)j/ = C N · − (cid:15)j/ j → + ∞ −−−−→ . (cid:3) Cluster coherence of the projected frames.
In this subsection we compute the clustercoherence terms corresponding to the missing stripe. To be able to inpaint the missing part, thefollowing propositions rely of the fact that the gap size h j at each scale j is smaller than the essentiallength of the shearlet elements in this scale. For illustration, see Figure 8.In [25] the problem of inpainting a missing strip from a line singularity via shearlet frames wasstudied. As part of their construction, they prove the following proposition. Proposition 9.3. ( [25], Prop. 5.6)
Consider the shearlet frame Ψ of Definition 5.3. For h j = o (2 ( α j + (cid:15) ) j ) with α j ∈ (0 , , lim inf α j > and (cid:15) satisfying (60), we have µ c (Λ ± ,j , P j Ψ ; Ψ ) → , j → ∞ , where Λ ± ,j is defined in (61), P j is defined in (32). Figure 8.
Left: Frequency support of a shearlet (green) and Gabor elements(grey small squares). Right: Missing part (light brown), essential spatial supportof a shearlet (green) and a cluster of Gabor elements (grey). The gap width doesnot exceed the essential length of the shearlet.Next, we prove the following result.
Proposition 9.4.
Consider the the Gabor frame of scale j, G j defined in (33). Suppose that h j = o (2 − α j j ) with α j ∈ (0 , , lim inf α j > , and I T satisfies (56) and (57) . Then we have µ c (Λ ,j , P j G j ; G j ) → , j → ∞ , where Λ ,j is defined in (58) and P j is defined in (32). For an illustration of this Proposition, we refer to Figure 8 (grey part). Next, we prove thefollowing result.
Proof.
First, from the definition of the cluster, we may assume that the maximum is attained for ( m (cid:48) , n (cid:48) ) ∈ Λ ,j . Namely, we have µ c (Λ ,j , P j G j ; G j ) = (cid:88) ( m,n ) ∈ M j × ( I ± T ∩A s,j ) |(cid:104) P j ( g s j ) m,n , ( g s j ) m (cid:48) ,n (cid:48) (cid:105)| = (cid:88) ( m,n ) ∈ M j × ( I ± T ∩A s,j ) (cid:12)(cid:12)(cid:12) (cid:90) h j − h j (cid:90) R ( g s j ) m,n ( x )( g s j ) m (cid:48) ,n (cid:48) ( x ) dx (cid:12)(cid:12)(cid:12) ≤ (cid:88) ( m,n ) ∈ M j × ( I ± T ∩A s,j ) (cid:90) h j − h j (cid:90) R | ( g s j ) m,n ( x ) || ( g s j ) m (cid:48) ,n (cid:48) ( x ) | dx. (95)By Lemma 8.1, we obtain the following decay estimate of ( g s j ) m,n ( x ) for any N ∈ N | ( g s j ) m,n ( x ) | ≤ C N · s j · (cid:104)| s j x + m |(cid:105) − N (cid:104)| s j x + m |(cid:105) − N . y = s j x , we have (cid:90) h j − h j (cid:90) R | ( g s j ) m,n ( x ) || ( g s j ) m (cid:48) ,n (cid:48) ( x ) | dx ≤ (cid:90) h j − h j (cid:90) R C N s j (cid:104)| s j x + m |(cid:105) − N (cid:104)| s j x + m |(cid:105) − N (cid:104)| s j x + m (cid:48) |(cid:105) − N ·(cid:104)| s j x + m (cid:48) |(cid:105) − N dx = (cid:90) h j s j − h j s j (cid:90) R C N (cid:104)| y + m |(cid:105) − N (cid:104)| y + m |(cid:105) − N (cid:104)| y + m (cid:48) |(cid:105) − N (cid:104)| y + m (cid:48) |(cid:105) − N dy. (96)Next, (cid:88) m (cid:104)| y + m |(cid:105) − N (cid:104)| y + m |(cid:105) − N ≤ C (cid:48) N , and (cid:90) R (cid:104)| y + m (cid:48) |(cid:105) − N dy ≤ C (cid:48)(cid:48) N . (97)Thus, by (57) and (95) combined with (96) and (97) , we obtain µ c (Λ ,j , P j G j ; G j ) ≤ (cid:88) n ∈ I ± T ∩A s,j C N · h j s j ≤ C N · α j j s j h j s j = C N · α j j h j j → + ∞ −−−−→ . (cid:3) Proposition 9.5.
Consider the shearlet frame Ψ of Definition 5.3 and the Gabor frame of scale j, G j , defined in (33). Assuming that I T satisfies (56) and (57) , we have µ c (Λ ,j , P j G j , Ψ ) → , j → ∞ , where Λ ,j is defined in (58) and P j is defined in (32) .Proof. The maximum of the cluster coherence is attained for some ( j, ¯ l, ¯ k ; α j , ( ι )) ∈ ∆ j . Thus, µ c (Λ ,j , P j G j , Ψ ) = (cid:88) ( m,n ) ∈ M j × ( I ± T ∩A s,j ) |(cid:104) P j ( g s j ) m,n , ψ α j , ( ι ) j, ¯ l, ¯ k (cid:105)| = (cid:88) ( m,n ) ∈ M j × ( I ± T ∩A s,j ) (cid:12)(cid:12)(cid:12) (cid:90) h j − h j (cid:90) R ( g s j ) m,n ( x ) ψ α j , ( ι ) j, ¯ l, ¯ k ( x ) dx (cid:12)(cid:12)(cid:12) ≤ (cid:88) ( m,n ) ∈ M j × ( I ± T ∩A s,j ) (cid:90) h j − h j (cid:90) R | ( g s j ) m,n ( x ) || ψ α j , ( ι ) j, ¯ l, ¯ k ( x ) | dx. Now we consider three cases1) Case ( ι ) = (v) . For each N = 1 , , . . . , by Lemma 8.1, we derive the following decay estimatesof ( g s j ) m,n ( x ) and ψ α j , ( ι ) j, ¯ l, ¯ k ( x ) | ( g s j ) m,n ( x ) | ≤ C N · s j · (cid:104)| s j x + m |(cid:105) − N (cid:104)| s j x + m |(cid:105) − N , | ψ α j , (v) j, ¯ l, ¯ k ( x ) | ≤ C N · (2+ α ) j/ · (cid:104)| α j j x − ¯ k |(cid:105) − N (cid:104)| j x + ¯ l α j j x − ¯ k |(cid:105) − N . y = S ¯ l (v) A jα j , (v) x = (2 α j j x , j x + ¯ l α j j x ) , we have (cid:90) h j − h j (cid:90) R | ( g s j ) m,n ( x ) || ψ α j , (v) j, ¯ l, ¯ k ( x ) | dx ≤ C N · − (2+ α j ) j/ · s j (cid:90) αjj h j − αjj h j (cid:90) R (cid:104)| s j − α j j y + m |(cid:105) − N (cid:104)| s j − j ( y − ¯ ly ) + m |(cid:105) − N ·(cid:104)| y − ¯ k |(cid:105) − N (cid:104)| y − ¯ k |(cid:105) − N dy dy . ≤ C N · − (2+ α j ) j/ · s j (cid:90) R (cid:90) R (cid:104)| s j − α j j y + m |(cid:105) − N (cid:104)| s j − j ( y − ¯ ly ) + m |(cid:105) − N ·(cid:104)| y − ¯ k |(cid:105) − N (cid:104)| y − ¯ k |(cid:105) − N dy dy . (98)Furthermore, (cid:88) m (cid:104)| s j − α j j y + m |(cid:105) − N (cid:104)| s j − j ( y − ¯ ly ) + m |(cid:105) − N ≤ C (cid:48) N (99)and (cid:90) R (cid:104)| y − ¯ k |(cid:105) − N dy ≤ C (cid:48)(cid:48) N , (cid:90) R (cid:104)| y − ¯ k |(cid:105) − N dy ≤ C (cid:48)(cid:48)(cid:48) N . (100)Thus, by (98), (99), (100) and (57), we obtain µ c (Λ ,j , P j G j ; Ψ ) ≤ (cid:88) n ∈ I ± T ∩A s,j C N · − (2+ α j ) j/ · s j ≤ C N · − (2+ α j ) j/ · s j · α j j s j = C N · − (2 − α j ) j/ j → + ∞ −−−−→ .
2) Case ( ι ) = (h) . Similarly, by the change of variable y = S ¯ l (h) A jα j , (h) x = (2 j x +¯ l α j j x , α j j x ) ,we have (2+ α j ) j/ · s − j · (cid:90) h j − h j (cid:90) R | ( g s j ) m,n ( x ) || ψ α j , (h) j, ¯ l, ¯ k ( x ) | dx ≤ C N (cid:90) R (cid:104)| s j − j ( y − ¯ ly ) + m |(cid:105) − N (cid:104)| s j − α j j y + m |(cid:105) − N (cid:104)| y − ¯ k |(cid:105) − N (cid:104)| y − ¯ k |(cid:105) − N dy. By using similar argument as in case ( ι ) = (v) , we finally obtain µ c (Λ ,j , P j G j ; Ψ ) ≤ (cid:88) n ∈ I ± T ∩A s,j C N · − (2+ α j ) j/ · s j (57) ≤ C N · − (2 − α j ) j/ j → + ∞ −−−−→ .
1) Case ( ι ) = (b) . Recalling the definition of boundary shearlets, the decay estimate can be donesimilarly. (cid:3)
Proposition 9.6.
Consider the shearlet frame Ψ of Definition 5.3 and the Gabor frame of scale j, G j , defined in (33). Suppose that s j ≤ α j j , h j = o (2 α j j ) , and α j ∈ (0 , , lim inf α j > . Thenwe have µ c (Λ ± ,j , P j Ψ ; G j ) → , j → ∞ , where Λ ± ,j is defined in (61) and P j is defined in (32). Proof.
Without loss of generality, we prove only µ c (Λ ,j , P j Ψ ; G j ) → , j → ∞ . First, we assumethat the maximum in the definition of the cluster is attained for some ( ¯ m, ¯ n ) ∈ Z × Z . We have µ c (Λ ,j , P j Ψ ; G j ) = (cid:88) ( j,l,k ; α j ,ι ) ∈ Λ ,j |(cid:104) P j ψ α j , ( ι ) j,l,k , ( g s j ) ¯ m, ¯ n (cid:105)| = (cid:88) | l |≤ ,k ∈ Z | k − lk |≤ (cid:15)j (cid:12)(cid:12)(cid:12) (cid:90) h j − h j (cid:90) R ψ α j , (v) j,l,k ( x )( g s j ) ¯ m, ¯ n ( x ) dx (cid:12)(cid:12)(cid:12) ≤ (cid:88) | l |≤ ,k ∈ Z | k − lk |≤ (cid:15)j (cid:90) h j − h j (cid:90) R | ψ α j , (v) j,l,k ( x ) || ( g s j ) ¯ m, ¯ n ( x ) | dx | . For each N = 1 , , . . . , by Lemma 8.1, we derive the following decay estimates of ψ α j , (v) j,l,k ( x ) and ( g s j ) ¯ m, ¯ n ( x ) | ψ α j , (v) j,l,k ( x ) | ≤ C N · (2+ α ) j/ · (cid:104)| α j j x − k |(cid:105) − N (cid:104)| j x + l α j j x − k |(cid:105) − N , | ( g s j ) ¯ m, ¯ n ( x ) | ≤ C N · s j · (cid:104)| s j x + ¯ m |(cid:105) − N (cid:104)| s j x + ¯ m |(cid:105) − N . Thus, by changing variable y = S l (v) A jα j , (v) x = (2 α j j x , j x + l α j j x ) , we have (cid:90) h j − h j (cid:90) R | ψ α j , (v) j,l,k ( x ) | · | ( g s j ) ¯ m, ¯ n ( x ) | dx ≤ C N · − (2+ α j ) j/ · s j · (cid:90) αjj h j − αjj h j (cid:90) R (cid:104)| y − k |(cid:105) − N (cid:104)| y − k |(cid:105) − N (cid:104)| s j − α j j y + ¯ m |(cid:105) − N · (cid:104)| s j − j ( y − ly ) + ¯ m |(cid:105) − N dy dy . (101)Next, (cid:104)| s j − α j j y + ¯ m |(cid:105) − N (cid:104)| s j − j ( y − ly ) + ¯ m |(cid:105) − N ≤ (102)and (cid:90) R (cid:16) (cid:88) | l |≤ ,k ∈ Z | k − lk |≤ (cid:15)j (cid:104)| y − k |(cid:105) − N (cid:104)| y − k |(cid:105) − N (cid:17) dy = (cid:88) | l |≤ ,k ,k ∈ Z | k |≤ (cid:15)j (cid:104)| y − k |(cid:105) − N · (cid:16) (cid:90) R (cid:104)| y − k − lk |(cid:105) − N dy (cid:17) ≤ C N · (cid:15)j · (cid:90) R (cid:104)| t − y |(cid:105) − N dt ≤ C (cid:48) N (cid:15)j . (103)Thus, by (101), (102), (103), we obtain µ c (Λ ,j , P j Ψ ; G j ) ≤ C (cid:48) N · − (2+ α j ) j/ · s j · α j j h j · (cid:15)js j ≤ αjj ≤ C (cid:48) N · − (2 − α j − (cid:15) ) j/ · α j j h j j → + ∞ −−−−→ . (cid:3) Proof of theorem 6.4
We are now ready to present the proof of Theorem 6.4.
Proof.
By Theorem 3.8, we have (cid:107) P j C ∗ j − P j w S j (cid:107) + (cid:107) P j T ∗ j − P j T s,j (cid:107) ≤ δ j − µ c,j , (104)where δ j = δ ,j + δ ,j and µ c,j = max { µ c (Λ ± ,j , P j Ψ ; Ψ ) + µ c (Λ ,j , P j G j ; Ψ ) , µ c (Λ ,j , P j G j ; G j ) + µ c (Λ ± ,j , P j Ψ ; G j ) } + max { µ c (Λ ± ,j , Φ ; G j ) , µ c (Λ , G j ; Ψ ) } . Moreover, it follows from Propositions 8.3 and 8.6 that δ j = δ + δ = o (2 − Nj ) , ∀ N ∈ N . (105)For the other term, the following estimate holds as a consequence of Propositions 9.1, 9.2, 9.3, 9.4,9.5 and 9.6 µ c,j −→ , j → + ∞ . (106)Combining (104), (105), and (106), we get (cid:107)C ∗ j − w S j (cid:107) + (cid:107)T ∗ j − T s,j (cid:107) = o (2 − Nj ) , ∀ N ∈ N . This proves the first claim of the theorem.The second claim is obtained since the following estimate holds for any arbitrarily large number N ∈ N (cid:107) P j C ∗ j − P j w S j (cid:107) + (cid:107) P j T ∗ j − P j T s,j (cid:107) ≤ (cid:107)C ∗ j − C j (cid:107) + (cid:107)T ∗ j − T s,j (cid:107) ≤ − Nj . (107)By Lemma 6.3 and (107), for N ∈ N chosen sufficiently large, we obtain (cid:107) P j C ∗ j − P j w S j (cid:107) (cid:107) P j w S j (cid:107) ≤ o (2 − Nj ) h j − j j → + ∞ −−−−→ , and (cid:107) P j T ∗ j − P j T s,j (cid:107) h j − j = o (2 − Nj ) h j − j j → + ∞ −−−−→ , which concludes the proof. (cid:3) References [1] Ole Christensen,
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