Construction of Compactly Supported Shearlet Frames

CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES

PISAMAI KITTIPOOM, GITTA KUTYNIOK, AND WANG-Q LIM

Abstract. Shearlet tight frames have been extensively studied during the last years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital setting. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial. In this paper, we focus on cone-adapted shearlet systems which – accounting for stability questions – are associated with a general irregular set of parameters. We first derive sufficient conditions for such cone-adapted irregular shearlet systems to form a frame and provide explicit estimates for their frame bounds. Secondly, exploring these results and using specifically designed wavelet scaling functions and filters, we construct a family of cone-adapted shearlet frames consisting of compactly supported shearlets. For this family, we derive estimates for the ratio of their frame bounds and prove that they provide optimally sparse approximations of cartoon-like images. Finally, we discuss an implementation strategy for the shearlet transform based on compactly supported shearlets. We further show that an optimally sparse N-term approximation of an M M cartoon-like image can be derived with asymptotic computational cost O(M 2(1+τ)) for× some positive constant τ depending on M and N.

1. Introduction Over the last 20 years wavelets have established themselves as a key methodology for efficiently representing signals or operators with applications ranging from more theoretical tasks such as adaptive schemes for solving elliptic partial differential equations to more practically tasks such as data compression. It is fair to say that nowadays wavelet theory can be regarded as an essential area in applied mathematics. A major breakthrough in this field was achieved through Daubechies in 1988 by introducing compactly supported wavelet orthonormal bases with favorable vanishing moment and support properties [5]. Recently, it was proven by Candés and Donoho in [2] that wavelets do not perform optimally when representing and analyzing anisotropic features in multivariate data such as singularities concentrated on lower dimensional embedded manifolds, where their focus was on the 2-dimensional situation. This observation initiated a flurry of activity to design a

1991 Mathematics Subject Classification. Primary: 42C40; Secondary: 42C15, 65T60, 65T99, 94A08. Key words and phrases. Curvilinear discontinuities, edges, filters, nonlinear approximation, optimal sparsity, scaling functions, shearlets, wavelets. P.K. acknowledges support from the Faculty of Science, Prince of Songkla University. G.K. and W.-Q L. would like to thank Wolfgang Dahmen, Chunyan Huang, Demetrio Labate, Christoph Schwab, and Gerrit Welper for various discussions on related topics. We also thank the anonymous referees for valuable comments and suggestions. G.K. and W.-Q L. acknowledge support from DFG Grant SPP-1324, KU 1446/13-1. 1 2 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM representation system which provides optimally sparse approximations of so-called cartoon- like images while having as favorable theoretical and computational properties as wavelet systems. We would like to exemplarily mention the first continuous system based on parabolic scaling introduced in [25] for the study of Fourier Integral Operators, followed by the introduction of curvelets in [2], and contourlets in [24, 7]. A particularly interesting study is also [11], in which negative results on the existence of orthonormal bases of ‘very directional’ wavelets were derived. In 2005, shearlets were developed by Labate, Weiss, and two of the authors in [22] (see also [14]) as the first directional representation system which allows a unified treatment of the continuum and digital world similar to wavelets, while providing (almost) optimally sparse approximations within a cartoon-like model [15] – ‘almost’ in the sense of an additional log-factor which is customarily regarded as negligible. Aiming for sparse approximations as well as robust expansions, it is often desirable to extend the well-known concept of a basis. This is achieved by the notion of a frame, which 2 is a family of functions ϕλ λ∈Λ in a Hilbert space such that, for all f , A f H 2 2{ } H ∈ H ≤ λ∈Λ f,ϕλ B f H, where 0 < A and B < are called frame bounds. Of particular interest| are |tight≤ frames , for which A = B, or even∞ Parseval frames with A = B = 1. Several constructions of discrete shearlet frames are already known to date, see [14, 18, 4, 23, 17]. However, taking applications into account, spatial localization of the analyzing elements of the encoding system is of uttermost importance both for a precise detection of geometric features as well as for a fast decomposition algorithm. But, one must admit that none of the previous approaches encompasses this crucial case. Hence the main goal of this paper is to provide a comprehensive analysis of cone-adapted – the variant of shearlet systems feasible for applications (see Section 2.1.1)– discrete shearlet frames encompassing in particular compactly supported shearlet generators. Our contribution is two-fold: We firstly provide sufficient conditions for a cone-adapted irregular shearlet system to form a frame for L2(R2) with explicit estimates for the ratio of the associated frame bounds; and secondly, based on these results, we introduce a class of cone-adapted compactly supported shearlet frames, which are even shown to provide (almost) optimally sparse approximations of cartoon-like images, alongside with estimates for the ratio of the associated frame bounds. Finally, we discuss an algorithmic realization of the shearlet transform based on compactly supported shearlets, which is spatial domain based with upsampling. We further show that it can provide (almost) optimally sparse N-term approximations of cartoon-like images with computational cost O(M 2(1+τ)) for an M M input image, τ depending on the size of input image M and the accuracy of approximation× N. We should mention that when M N 4, τ = 1/2 and a naive algorithm based on convolutions realized directly by multiplication≈ in the Fourier domain would have complexity O(M 5/2 log M) in this case. However, in practice, the size of the associated constant C together with the log M-factor might lead to CM 5/2 log M M 2(1+τ), and, much more important, this naive algorithm does not provide the (almost) optimally≈ sparse approximation rate we aim for.

1.1. Shearlet Systems. Referring to the detailed introduction of shearlet systems in Sec- tion 2, we allow us here to just brieﬂy mention those main ideas and results, which are crucial for an intuitive understanding. CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 3

Shearlet systems are designed to efficiently encode anisotropic features. In order to achieve optimal sparsity, shearlets are scaled according to a parabolic scaling law, thereby exhibiting a spatial footprint of size 2−j times 2−j/2. They parameterize directions by slope encoded in a shear matrix, since choosing shear rather than rotation is in fact decisive for a unified treatment of the continuum and digital setting. We refer for more details to [14, 19] for the continuum theory and [21, 12, 23] for the digital theory. Shearlet systems come in two ways: One class being generated by a unitary representation of the shearlet group and equipped with a particularly ‘nice’ mathematical structure, however causes a biasedness towards one axis, which makes it unattractive for applications; the other class being generated by a quite similar procedure however restricted to a horizontal and vertical cone in frequency domain, thereby ensuring an equal treatment of all directions. For both cases, the continuous shearlet systems are associated with a 4-dimensional parameter space consisting of a scale parameter measuring the resolution, a shear parameter measuring the orientation, and a translation parameter measuring the position of the shearlet. A sampling of this parameter space leads to discrete shearlet systems, and it is obvious that the possibilities for this are numerous. A canonical sampling approach – using dyadic sampling – leads to so-called regular shearlet systems. However, lately, questions from sampling theory and stability issues brought the study of irregular shearlet systems to the researcher’s attention. 1.2. Necessity of Compactly Supported Shearlets. Shearlet theory has already im- pacted various applications for which the sparse encoding or analysis of anisotropic features is crucial, and we refer to results on denoising [23], edge detection [16], or geometric separation [9, 10]. However, similar as in wavelet analysis, a significant improvement of the applicability of shearlets is possible, if compactly supported shearlet frames would be made available. Let us focus on imaging sciences and discuss the problem of edge detection as one exemplary application to illustrate the necessity of such a study. In computer vision, edges were detected as those features governing an image while sepa- rating smooth regions in between. Thus imaging tasks typically concern an especially careful handling of edges, for instance, avoiding to smoothen them during a denoising process. Therefore, when exploiting a decomposition in terms of a representation system for such tasks, a superior localization in spatial domain of the analyzing elements is in need. This would then allow a very precise focus on edges, thereby reducing or even avoiding various artifacts in the decomposition and analysis process. Secondly, due to the increased necessity to process very large data sets, fast decomposition algorithms are crucial. Learning from the experience with structured transforms such as the fast wavelet transform, it is conceiv- able that a representation system consisting of compactly supported elements will provide a significant gain in complexity. 1.3. Previous Work on Shearlet Constructions. Let us now discuss the history and the state-of-the-art of shearlet constructions. The first class of shearlets, which were shown to generate a tight frame, were band-limited with a wedge-like support in frequency domain specifically adapted to the shearing operation (see [22, 14]). This particular class of cone- adapted shearlet frames was already extensively explored, for instance, for analyzing sparse approximation properties of the associated shearlet frames [15]. 4 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

Shortly afterwards, a different avenue was undertaken in [18], where a first attempt was made to derive sufficient conditions for the existence of irregular shearlet frames, which, however, could not decouple from the previous focus on band-limited shearlet generators. In addition, this result was stated for shearlet frames which arose directly from a group representation. In some sense, this path was continued in [4], where again sufficient conditions for this class of irregular shearlet frames were studied. Let us also mention at this point that a more extensive study of these systems was performed in [17] with a focus on necessary conditions and a geometric analysis of the irregular parameter set. Now returning to the situation of cone-adapted shearlet frames, a particular interesting study was recently done in [23], where a fast decomposition algorithm for separable shearlet frames – which are non-tight – was introduced, however without analyzing frame or sparsity properties. We should mention that it was this work which gave us the intuition of taking a viewpoint of separability for the construction of compactly supported shearlet frames in this paper. The question of optimal sparsity of a large class of cone-adapted compactly supported shearlet frames was very recently solved by two of the authors in [20]. However, no sufficient conditions for or construction of cone-adapted compactly supported shearlet frames were studied so far. These are the questions we will attack and provide answers for in this paper.

1.4. Our Contribution. Our contribution in this paper is two-fold. Firstly, Theorem 3.4 provides sufficient conditions on the irregular set of parameters and the generating shearlet to form a cone-adapted shearlet frame alongside with estimates for the ratio of the frame bounds. This result greatly extends the result in [18] by, in particular, encompassing compactly supported shearlet generators. Secondly, exploring these results and using specifically designed wavelet scaling functions and filters, we explicitly construct a family of cone-adapted shearlet frames consisting of compactly supported shearlets with estimates for their frame bounds in two steps: For functions supported on the horizontal cone in frequency domain and then for functions in L2(R2) (see Theorems 4.7 and 4.9). Theorem 4.10 then proves that this family does even provide (almost) optimally sparse approximations of cartoon-like images.

1.5. Outline. In Section 2, we introduce the necessary definitions and notation for cone- adapted irregular shearlet systems. In particular, in Subsections 2.1 and 2.2 we present the viewpoint of deriving cone-adapted irregular shearlet systems from sampling the parameters of cone-adapted continuous shearlet systems for the first time. Subsection 2.3 is then concerned with introducing a large class of cone-adapted irregular shearlet systems, coined feasible shearlet systems, which will be the focus of this paper. Sufficient conditions for such feasible shearlet systems to form a frame and explicit estimates for their frame bounds are then discussed in Section 3. In Section 4, these results are explored to construct a family of cone-adapted regular compactly supported shearlet frames with estimates for the ratio of the associated frame bounds (Subsections 4.1 and 4.2). In Subsection 4.3, it is proven that this family does even provide (almost) optimally sparse approximations of cartoon-like images. Some proofs are deferred to Section 6. In section 5, we then comment on applicability of the constructed families of cone-adapted regular compactly supported shearlet frames. CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 5

2. Regular and Irregular Shearlet Systems 2.1. Cone-Adapted Shearlets. Shearlets are highly anisotropic representation systems, which optimally sparsify C2(R2)-functions apart from C2-discontinuity curves. Their main advantage over other proposed directional representation systems is the fact that they provide a unified treatment of the continuum and digital world. Today, shearlets are utilized for various applications, and we refer to [16, 9, 10] as some examples. Introduced in [14], discrete shearlet systems – shearlet systems for L2(R2) with discrete parameters – exist in two different variants: One coming directly from a group representation of a particular semi-direct product, the so-called shearlet group, and the other being adapted to a cone-like partitioning of the frequency domain. In fact, the second type of shearlet systems are the ones exhibiting the favorable property of treating the continuum and digital setting uniformly similar to wavelets, and are the ones relevant for applications. Hence those are the discrete shearlet systems we focus on in this paper. We now introduce cone-adapted discrete regular shearlet systems – previously also referred to as ‘shearlets on the cone’ – as a special sampling of the (cone-adapted) continuous shearlet systems, which were exploited in [19]. Since all previous papers consider discrete regular shearlet systems as sampling the parameters of continuous shearlet systems refer to the first type of shearlet systems arising from a group representation (see, e.g., [18, 4]), this is the first time this view is taken and hence is presented in a more elaborate manner. 2.1.1. (Cone-Adapted) Continuous Shearlet Systems. Shearlets are scaled according to a parabolic scaling law encoded in the parabolic scaling matrices Aa or Aã, a > 0, and exhibit directionality by parameterizing slope encoded in the shear matrices S , s R, defined by s ∈ a 0 √a 0 A = or A˜ = a 0 √a a 0 a and 1 s S = , s 0 1 respectively. To ensure an (almost) equal treatment of the different slopes, we partition the frequency plane into the following four cones – : C1 C4 2 (ξ1,ξ2) R : ξ1 1, ξ2/ξ1 1 : ι =1, { ∈ 2 ≥ | | ≤ } (ξ1,ξ2) R : ξ2 1, ξ1/ξ2 1 : ι =2, ι =  { ∈R2 ≥ | | ≤ } C  (ξ1,ξ2) : ξ1 1, ξ2/ξ1 1 : ι =3,  {(ξ ,ξ ) ∈ R2 : ξ ≤ −1, |ξ /ξ | ≤ 1} : ι =4, { 1 2 ∈ 2 ≤ − | 1 2| ≤ }  and a centered rectangle = (ξ ,ξ ) R2 : (ξ ,ξ ) < 1 . R { 1 2 ∈ 1 2 ∞ } For an illustration, we refer to Figure 1(a). The rectangle corresponds to the low frequency content of a signal, which is customarily represented by translationsR of some scaling function. Anisotropy now comes into play when encoding the high frequency content of a signal, which corresponds to the cones – , C1 C4 6 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

C2 = C C1 R

C3 C4

(a) (b)

Figure 1. (a) The cones 1 – 4 and the centered rectangle in frequency domain. (b) The tiling of theC frequencyC domain induced by discreteR shearlets with the support of one shearlet from Example 2.3 exemplarily highlighted.

where the cones 1 and 3 as well as 2 and 4 are treated separately as can be seen in the following definition.C C C C Definition 2.1. The (cone-adapted) continuous shearlet system (φ, ψ, ψ˜) generated by SH a scaling function φ L2(R2) and shearlets ψ, ψ˜ L2(R2) is defined by ∈ ∈ (φ, ψ, ψ˜)=Φ(φ) Ψ(ψ) Ψ(˜ ψ˜), SH ∪ ∪ where Φ(φ)= φ = φ( t): t R2 , { t − ∈ } − 3 −1 −1 1/2 1/2 2 Ψ(ψ)= ψ = a 4 ψ(A S ( t)) : a (0, 1], s [ (1 + a ), 1+ a ], t R , { a,s,t a s − ∈ ∈ − ∈ } and − 3 −1 −T 1/2 1/2 2 Ψ(˜ ψ˜)= ψ˜ = a 4 ψ˜(A˜ S ( t)) : a (0, 1], s [ (1 + a ), 1+ a ], t R , { a,s,t a s − ∈ ∈ − ∈ } where a indexes scale, s indexes shear, and t indexes translation. Setting S = (a, s, t): a (0, 1], s [ (1 + a1/2), 1+ a1/2], t R2 , cone { ∈ ∈ − ∈ } 2 2 3 the associated (Cone-Adapted) Continuous Shearlet Transform ˜f : R S C SHφ,ψ,ψ × cone → of some function f L2(R2) is given by ∈ ′ ˜f(t , (a, s, t), (ã, s,˜ t˜))=( f,φ ′ , f, ψ , f, ψ˜ ˜ ). SHφ,ψ,ψ t a,s,t a,˜ s,˜ t Notice that in [19] only special generators ψ were considered, whereas here we state the definition for all ψ L2(R2); see also the interesting extension [13]. ∈ 2.1.2. (Cone-Adapted) Discrete Shearlet Systems. A discretization of the shearlet parameters is typically achieved through sampling of the parameter set of (cone-adapted) continuous 2 2 shearlet systems, where R is sampled as c1Z and, for the horizontal cone 1 3, Scone is discretized as C ∪ C −j −j/2 j/2 j/2 2 (2 ,k2 ,S −j/2 A −j M m : j 0, k 2 ,..., 2 , m Z , (1) { k2 2 c ≥ ∈ {−⌈ ⌉ ⌈ ⌉} ∈ } CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 7 where, for c =(c ,c ) (R+)2, M denotes the sampling matrix 1 2 ∈ c c1 0 Mc = . 0 c2 For the vertical cone 2 4, Scone is discretized in a similar way, now using the sampling matrix C ∪ C c 0 M˜ = 2 . c 0 c 1 The anisotropic sampling of the position parameter is chosen to allow the flexibility to oversample only in the shearing direction, which will reduce the redundancy of the generated system. This will later become apparent in the ratio of the associated frame bounds. Also the range of values for k deserves a comment, since the two values 2j/2 1 and 2j/2 + 1 which arose from the sampling procedure were replaced by 2j/2 and− 2j/−2 . The reason for this is that 2j/2 is not an integer for odd scales j. Also,−⌈ for the⌉ classical⌈ almost-⌉ separable shearlet generator introduced in Example 2.3, the support of the Fourier transform j/2 j/2 of the associated shearlets does not anymore fall into 1 3 for k = 2 1 and k =2 +1 (if considering the horizontal cone). C ∪C − − This now gives the following discrete system. Definition 2.2. For some sampling vector c = (c ,c ) (R+)2, the (cone-adapted) regu- 1 2 ∈ lar discrete shearlet system (c; φ, ψ, ψ˜) generated by a scaling function φ L2(R2) and SH ∈ shearlets ψ, ψ˜ L2(R2) is defined by ∈ (c; φ, ψ, ψ˜)=Φ(c ,φ) Ψ(c, ψ) Ψ(˜ c, ψ˜), SH 1 ∪ ∪ where Φ(c ,φ)= φ = φ( c m): m Z2 , 1 { m − 1 ∈ } 3j/4 j/2 2 Ψ(c, ψ)= ψ =2 ψ(S A j M m): j 0, k 2 , m Z , { j,k,m −k 2 − c ≥ | | ≤ ⌈ ⌉ ∈ } and 3j/4 T j/2 2 Ψ(˜ c, ψ˜)= ψ˜ =2 ψ˜(S A˜ j M˜ m): j 0, k 2 , m Z . { j,k,m −k 2 − c ≥ | | ≤ ⌈ ⌉ ∈ } Setting j/2 2 Λcone = (j, k, m): j 0, k 2 , m Z , { ≥ | | ≤ ⌈ ⌉ ∈ } 2 2 the associated (Cone-Adapted) Regular Discrete Shearlet Transform ˜f : Z Λ SHφ,ψ,ψ × cone → C3 of some function f L2(R2) is given by ∈ ′ ˜f(m , (j, k, m), (˜j, k,˜ m˜ ))=( f,φ ′ , f, ψ , f, ψ˜˜ ˜ ). SHφ,ψ,ψ m j,k,m j,k,m˜ The reader should keep in mind that although not indicated by the notation, the functions φm, ψj,k,m, and ψ˜j,k,m all depend on the sampling constants c1,c2. Notice further that the particular sampling set (1) forces a change in the ordering of parabolic scaling and shearing, which will be mimicked later by the class of irregular parameters we will be considering. The definition itself shows that the shearlets ψj,k,m (also ψ˜j,k,m) live on anisotropic regions of −j − j size 2 2 2 (in an asymptotic sense due to the operation of the parabolic scaling matrix) at various× orientations in spatial domain and hence essentially on anisotropic regions of size j j 2 2 2 in frequency domain (see also Figure 1(b)). × 8 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

Example 2.3. The classical example of a generating shearlet is a function ψ L2(R2) satisfying ∈ ψˆ(ξ)= ψˆ(ξ ,ξ )= ψˆ (ξ ) ψˆ ( ξ2 ), 1 2 1 1 2 ξ1 where ψ L2(R) is a discrete wavelet, i.e., satisﬁes the discrete Calder´on condition given 1 ∈ by ψˆ (2−jω) 2 =1 for a.e. ω R, with ψˆ C∞(R) and supp ψˆ [ 5 , 1 ] [ 1 , 5 ], j∈Z | 1 | ∈ 1 ∈ 1 ⊆ − 4 − 4 ∪ 4 4 and ψ L2(R) is a ‘bump’ function, namely 1 ψˆ (ω + k) 2 = 1 for a.e. ω [ 1, 1], 2 ∈ k=−1 | 2 | ∈ − satisfying ψˆ C∞(R) and supp ψˆ [ 1, 1]. There are several choices of ψ and ψ 2 2 1 2 satisfying those∈ conditions, and we refer⊆ to−[14] for further details. The tiling of frequency domain given by this band-limited generator and choosing ψ˜ = ψ(x2, x1) is illustrated in Figure 1(b). For this particular choice, using an appropriate scaling function φ for , it was proven in [14, Thm. 3] that the associated (cone-adapted) discrete R regular shearlet system (1; φ, ψ, ψ˜) forms a Parseval frame for L2(R2). SH 2.2. Irregular Sampling of Shearlet Parameters. Questions arising from sampling theory and the inevitability of perturbations yield the necessity to study shearlet systems with arbitrary discrete sets of parameters. To derive a better understanding of this problem, we take the viewpoint of regarding the previously considered set of parameters

−j −j/2 j j 2 (2 ,k2 ,S −j/2 A −j M m : j 0, k 2 2 ,..., 2 2 , m Z { k2 2 c ≥ ∈ {−⌈ ⌉ ⌈ ⌉} ∈ } as a discrete subset of Scone (for 1 3 and similarly for 1 3 with Mc substituted by C ∪ C2 C ∪ C M˜ c) and for the low frequency part R likewise. Then, an arbitrary set of parameters is merely a diﬀerent sampling set, and questions of density conditions of such a sampling set or its geometric properties are lurking in the background. Some density-like properties will indeed come into play in this paper mixed with decay conditions on the shearlet generator. For an extensive analysis of the geometric properties of the sampling set yielding necessary conditions for shearlet frames directly arising from a group representation, we refer to [17]. Next we formally deﬁne (cone-adapted) irregular shearlet systems.

2 Definition 2.4. Let ∆ and Λ, Λ˜ be discrete subsets of R and Scone, respectively, and let φ L2(R2) as well as ψ, ψ˜ L2(R2). Then the (cone-adapted) irregular discrete shearlet ∈ ∈ system (∆, Λ, Λ;˜ φ, ψ, ψ˜) is defined by SH (∆, Λ, Λ;˜ φ, ψ, ψ˜) = Φ(∆,φ) Ψ(Λ, ψ) Ψ(˜ Λ˜, ψ˜), SH ∪ ∪ where Φ(∆,φ)= φ = φ( t): t ∆ , { t − ∈ } − 3 −1 −1 Ψ(Λ, ψ)= ψ = a 4 ψ(A S ( t)):(a, s, t) Λ , { a,s,t a s − ∈ } and − 3 −1 −T Ψ(˜ Λ˜, ψ˜)= ψ˜ = a 4 ψ˜(A˜ S ( t)):(a, s, t) Λ˜ . { a,s,t a s − ∈ } Then the associated (Cone-Adapted) Irregular Discrete Shearlet Transform ˜f : ∆ SHφ,ψ,ψ × Λ Λ˜ C3 of some function f L2(R2) is given by × → ∈ ′ ˜f(t , (a, s, t), (ã, s,˜ t˜))=( f,φ ′ , f, ψ , f, ψ˜ ˜ ). SHφ,ψ,ψ t a,s,t a,˜ s,˜ t CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 9

2.3. Class of Irregular Shearlet Systems. Since the low frequency part is already very well studied, and since both cone pairs 1 and 3 as well as 2 and 4 are treated similarly, we from now on restrict our attention toC the horizontalC coneC C := , C C1 ∪ C3 and are interested in frame properties of the system Ψ(Λ, ψ), when expanding functions in L2( )= f L2(R2) : supp fˆ . C { ∈ ⊆ C} We remark that it is suﬃcient to consider the function space L2( ), since in classical shearlet theory, the representing shearlets are orthogonally projected ontoC the cones (and, in particular, onto the cone under consideration). For clarity, we further introduce the notion C (Λ, ψ) := Ψ(Λ, ψ). SH We will now focus on a class of sets of parameters satisfying some weak conditions, which – loosely speaking – allows to view it as a perturbation of a regular sampling set from (1). We remark that, since one main motivation for considering irregular sets of parameters are stability questions, such a constraint seems very natural. Also, we impose weak decay conditions on the shearlet generator, which still include spatially compactly supported shearlets; our main objective.

2.3.1. Feasible Sets of Parameters. We now first discuss our hypotheses on the set of parameters. For this, let (a ,s , tc ): j 0,k K , m Z2 , K Z, c =(c ,c ) (R+)2, { j j,k j,k,m ≥ ∈ j ∈ } j ⊆ 1 2 ∈ be an arbitrary discrete set of parameters in Scone. + For the decreasing sequence aj j≥0 R to act as a scaling parameter, we require a certain growth condition, which{ we} phrase⊂ as follows: For each (0, 1), there shall exist some positive integer p such that ∈ a j+p for all j 0. (2) aj ≤ ≥ −j Obviously, this condition is satisfied for the customarily utilized sequence aj = 2 with p = 1 and =1/2. Further, we assume the sequence s R of shear parameters to be of the form { j,k}j,k ⊂ sj,k = sk√aj for each j 0, k Kj, ≥ ∈ which allows a change of the parabolic scaling and shear operators similar to the discretization of the (cone-adapted) continuous shearlet systems (see Definition 2.2). We also restrict to s a1/2 + 1 for all j 0, k K , (3) | k| ≤ j ≥ ∈ j and mimic the discretization in the regular case. From now on, we assume that 0 Kj for each j 0 and s = 0 for simplicity, and remark that all of the results in this paper∈ can ≥ 0 be easily extended to a general sequence of shear parameters sk without this assumption. Finally, we assume the sequence s to satisfy a density-like condition, which will ensure { k}k 10 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM that the shearing operation does not cause a ‘piling up’ of the essential supports of the shearlets. In particular, we require that

−γ sup min 1, (Ssk x)2 min 1, (Ssk x)1 C(γ) < for any γ > 1. (4) R2 x=(x1,x2)∈ Z { | |} { | | } ≤ ∞ k∈ c R2 The sequence tj,k,m j,k,m of translation parameters shall have a grid structure, which we impose by assuming{ } that⊂ tc = S A M m, for some c =(c ,c ) (R+)2. (5) j,k,m sj,k aj c 1 2 ∈ Summarizing, we get the following R+ R Definition 2.5. Let the sequences aj j≥0 , sj,k j≥0,k∈Kj , and the sequence c 2 { } ⊂ { } ⊂ 2 R tj,k,m j≥0,k∈Kj,m∈Z satisfy (2), (3)–(4), and (5), respectively. Then we call the sequences{ } ⊂ c 2 (aj,sj,k = sk√aj, t = Ss Aa Mcm): j 0,k Kj, m Z { j,k,m j,k j ≥ ∈ ∈ } a feasible set of parameters. 2.3.2. Feasibility Conditions for Generating Shearlet. We next impose the following decay condition on the generating shearlets ψ L2(R2): ∈ Definition 2.6. A function ψ L2(R2) is called a feasible shearlet, if there exist α>γ> 3, and q > q′ > 0, q>r> 0 such∈ that ψˆ(ξ ,ξ ) min 1, qξ α min 1, q′ξ −γ min 1, rξ −γ . (6) | 1 2 | ≤ { | 1| } { | 1| } { | 2| } At first sight it seems picky to choose three different constants q, q′, and r for the different ‘decay areas’ in frequency domain. However, this flexibility will become crucial for deriving estimates for the lower frame bound, and hence for analyzing the class of compactly supported shearlet frames we aim to introduce. The class of feasible shearlets certainly includes the classical example of generating shearlets (Example 2.3) as well as all L2-functions whose Fourier transform is of polynomial decay of a certain degree. In particular, it includes functions which are spatially compactly supported. Moreover, condition (6) is chosen in such a way that ψ exhibits an essential support in frequency domain somehow similar to the support of the classical shearlets defined in Example 2.3, but is adapted to separable shearlet frames which later on will serve as the structure for our explicitly constructed compactly supported shearlet frames. For an illustration we refer to Figure 2.

2.3.3. Class of Shearlet Systems. Concluding, in this paper we are concerned with a particular class of (cone-adapted) irregular shearlet systems deﬁned as follows. Deﬁnition 2.7. Let Λ = (a ,s , tc ): j 0,k K , m Z2 be a feasible set of { j j,k j,k,m ≥ ∈ j ∈ } parameters and let ψ L2(R2) be a feasible shearlet. Then we call ∈ 3 − 4 2 −1 Z (Λ, ψ)= ψj,k,m = aj ψ(S−sk Aa Mcm): j 0,k Kj, m SH { j − ≥ ∈ ∈ } a feasible shearlet system. CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 11

1 r

1 − r 1 1 q q′

Figure 2. Essential support of ψˆ in the frequency domain for a shearlet ψ satisfying the decay condition (6).| |

Notice that such a system depends in particular on the parameters ,α,γ,q,q′, and r as well as on the function C( ). We further remark the abuse of notation for ψj,k,m as compared to Deﬁnition 2.2 to keep the notation simple. However, it will always be made clear which interpretation is meant.

3. A General Sufficient Condition for Shearlet Frames In this section, we will state a very general sufficient condition for the existence of cone- adapted irregular shearlet frames. A similar condition can be found in [18], however for shearlet frames arising from a group representation. This result can not be directly carried −1/2 over, since the relation between the cone, the bound aj for the parameters sk, and the most likely not band-limited shearlet generator ψ needs careful handling. Also, the result in [18] does not include compactly supported shearlet generators. 3.1. Covering Properties. For stating and proving the result, we require a particular function, which in an adapted form already appeared in [18] and is moreover a generalization of the function β in [6, Sect. 3.3.2] for the wavelet situation, with ξ as an additional variable. Let (Λ, ψ) be a feasible shearlet system as defined in Definition 2.7. The main objective willSH now be the function Φ : R2 R defined by C × → ˆ T ˆ T Φ(ξ,ω)= ψ(S Aa ξ) ψ(S Aa ξ + ω) . (7) | sk j || sk j | j≥0 k∈Kj Notice that this function measures the extent to which the essential supports of the scaled and sheared versions of the shearlet generator overlaps. For later use, we also introduce the function Γ : R2 R defined by → Γ(ω) = ess sup Φ(ξ,ω), ξ∈C measuring the maximal extent to which these versions overlap for a fixed distance, as well as the values Linf = ess inf Φ(ξ, 0) and Lsup = ess sup Φ(ξ, 0), (8) ξ∈C ξ∈C 12 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM which relate to the classical discrete Calderón condition. Finally, we also require the values

R(c)= Γ M −1m Γ M −1m 1/2 , where c =(c ,c ) (R+)2. (9) c − c 1 2 ∈ m∈Z2\{0} −1 The function R(c) measures the average of the symmetrized function values Γ(Mc m). We now first turn our attention to the terms Lsup and R(c) and provide explicit estimates for those, which will later be used for estimates for frame bounds associated to a shearlet system. We start by estimating Lsup in Proposition 3.1. For any (0, 1), let p be such that it satisfies (2), let (Λ, ψ) be a ∈ SH feasible shearlet system, and let Lsup be defined as in (8). Then q q 1 1 L p C(2γ) log + + < . (10) sup ≤ r 1/ q′ 1 2α−1 1 2γ ∞ − − In the special case s = k, k Z, the following stronger estimate holds: k ∈ q 2 q 1 1 L p 2+ +1 log + + < . (11) sup ≤ r 2γ 1 1/ q′ 1 2α−1 1 2γ ∞ − − − One essential ingredient in the proof of this lemma is a result from [26], which we state here for the convenience of the reader.

+ + Lemma 3.2. [26, Lem. 2.1] Let (0, 1),p Z , and let aj j∈Z R be a decreasing sequence which satisﬁes (2). Then, for∈ all α>∈0 and t> 0, { } ⊂ 1 1 a−α p t−α , aα p tα . j ≤ 1 α j ≤ 1 α a ≥t a ≤t j − j − This now allows us to prove Proposition 3.1.

Proof of Proposition 3.1. Let ξ =(ξ ,ξ )T R2. By (6), we obtain 1 2 ∈ Φ(ξ, 0) 2α ′ −2γ 1/2 −2γ ess sup min 1, qajξ1 min 1, q ajξ1 min 1, r(aj ξ2 + skajξ1) (12). ≤ R2 { | | } { | | } { | | } ξ∈ j≥0 k Letting η1 = qξ1,

2α−1 ′ −1 −2γ q −1 Φ(ξ, 0) ess sup min 1, ajη1 min 1, q q ajη1 min 1, rq ajη1 ≤ R2 { | | } { } r (η1,ξ2)∈ j≥0 k∈Z min 1, ra1/2ξ + rq−1a η s −2γ . (13) { | j 2 j 1 k| } By (4),

q −1 1/2 −1 −2γ q min 1, rq ajη1 min 1, raj ξ2 + rq ajη1sk C(2γ), Z r { | |} { | | } ≤ r k∈ CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 13

Hence we can continue (13) by

q 2α−1 ′ −1 −2γ Φ(ξ, 0) C(2γ) ess sup min 1, ajη1 min 1, q q ajη1 ≤ r η1∈R { | | } { | | } j≥0 q 2α−1 q C(2γ) ess sup χ[0,1)( ajη1 ) ajη1 + χ[1, ′ )( ajη1 ) ≤ r η ∈R | | | | q | | 1 j≥0 q ′ −1 −2γ +χ[ ′ ,∞)( ajη1 ) q q ajη1 q | | | | q 2α−1 q C(2γ) ess sup ajη1 + χ[1, ′ ]( ajη1 ) ≤ r η ∈R | | q | | 1 j≥0 |ajη1|≤1 ′ −1 −2γ + q q ajη1 . ′ −1 | | |q q aj η1|≥1 The ﬁrst claim (10) now follows from Lemma 3.2, if we show that q q ess sup χ[1, ′ ]( ajη1 ) p log1/ ′ . (14) η ∈R q | | ≤ q 1 j≥0 To provide this, we ﬁrst use (2) to obtain

p−1 ∞

1 1 q 1 1 1 q χ[ ,( ) ′ ]( η1 )+ χ[( ) ,( ) ′ ]( η1 ) ≤ ar ar q | | µj ar ar+jp q | | r=0 j=1 p−1 ∞ r r = g0(η1)+ gj (η1) , (15) r=0 j=1 where r 1 1 1 q gj (η1)= χ[( ) ,( ) ′ ]( η1 ) for each r =0,...,p 1 and j 0. µj ar ar+jp q | | − ≥ r For each r = 0,...,p 1, let Nr be the number of scales j 0 such that supp (g0) supp (gr) = 0. N is the− smallest possible positive integer satisfying≥ | ∩ j | r 1 Nr+1 1 q 1 ′ , ar ≥ q ar q which implies N +1 log ′ . Thus r ≤ 1/ q p−1 ∞ q gr(η ) p log . j 1 ≤ 1/ q′ r=0 j=0 Combining this estimate with (15) proves (14). 14 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

Next assume that sk = k for all k Z. To derive an estimate for Lsup in this case, we ﬁrst split (12) into the cases k = 0 and k∈= 0 with η = qξ and η = ra ξ , which yields 1 1 2 j 2 Φ(ξ, 0) 2α−1 ′ −1 −2γ q −1 ess sup min 1, ajη1 min 1, q q ajη1 ess sup min 1, rq ajη1 ≤ η ∈R { | | } { | | } −1 r { | |} 1 j≥0 η2∈[0,|rq aj η1|] k=0 −1 −2γ 2α ′ −1 −2γ min 1, η2 + rq ajη1k + ess sup min 1, ajη1 min 1, q q ajη1 .(16) { | | } η ∈R { | | } { | | } 1 j≥0 Notice that

q −1 −1 −2γ q 2 ess sup min 1, rq ajη1 min 1, η2 + rq ajη1k 2+ −1 η2∈[0,|rq aj η1|] r { | |} { | | } ≤ r 2γ 1 k=0 − and min 1, a η 2α min 1, q′q−1a η −2γ min 1, a η 2α−1 min 1, q′q−1a η −2γ . { | j 1| } { | j 1| } ≤ { | j 1| } { | j 1| } Therefore, we can continue (16) by

q 2 2α−1 ′ −1 −2γ Φ(ξ, 0) 2+ +1 ess sup min 1, ajξ1 min 1, q q ajη1 . ≤ r 2γ 1 η ∈R { | | } { | | } 1 j≥0 − Now (11) follows from Lemma 3.2; the proposition is proved. The strengthened estimate in the special case of a shearing parameter sequence s with { k}k sk = k for k Z will become important when estimating the frame bounds of the concrete class of compactly∈ supported shearlet frames we will introduce in Section 4. The next result reveals the dependence of R(c) on the sampling matrix Mc and provides a useful estimate for such. For its proof, we refer to Subsection 6.1.2. Proposition 3.3. Let (Λ, ψ) be a feasible shearlet system with associated sampling vector + 2 SH ′ c = (c1,c2) (R ) satisfying c1 c2, and let R(c) be deﬁned as in (9). Then, for any γ , which satisﬁes∈ 1 <γ′ <γ 2, we≥ have − c R(c) T D (γ) + min 1 , 2 T D (γ γ′)+ T (D (γ)+ D (γ)), (17) ≤ 1 1 c 2 2 − 3 1 2 2 where γ q q 1 1 2c1 T1 = p C(γ) log 1 + + , r µ q′ 1 γ 1 α−γ q′ − − γ−γ′ q ′ q 1 1 1 1 2qc2 T2 = p C(γ ) 2 log 1 + ′ + ′ + + , r µ q′ 1 γ 1 α−γ 1 γ 1 α−γ q′r − − − − q 1 2c γ T = p C(γ) 1 , 3 r 1 γ q′ − and, for any h> 0, 1 4 1 1 4 1 D (h)=2 1+ + 1+ ,D (h)=6 1+ + 1+ . 1 h 1 h 1 h 2 2 h 1 h 1 h 2 − − − − − − CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 15

In particular, for any γ′, which satisﬁes 1 <γ′ <γ 2, there exist positive constants κ and − 1 κ2 (in particular, independent on c) such that

′ 2c γ 2qc γ−γ R(c) κ 1 + κ 2 . (18) ≤ 1 q′ 2 q′r 3.2. General Suﬃcient Condition. Retaining the notions introduced in the previous subsection, we can now formulate a general suﬃcient condition for the existence of cone-adapted irregular shearlet frames. We defer the proof to Subsection 6.1.1.

Theorem 3.4. Let (Λ, ψ) be a feasible shearlet system, let Linf , Lsup be deﬁned as in (8), and let R(c) be deﬁnedSH as in (9). If

R(c) < Linf , then (Λ, ψ) is a frame for L2( ) with frame bounds A and B satisfying SH C 1 1 [L R(c)] A B [L + R(c)]. det M inf − ≤ ≤ ≤ det M sup | c| | c| The role of the determinant of the sampling matrix Mc, which can also be regarded as the inverse of the Beurling density of the translation sampling grid in spatial domain, requires a careful examination. The ‘quality’ of a frame can be measured by the magnitude of the quotient of the frame bounds B/A, since the closer it is to 1, the better frame reconstruction algorithms perform. In this case, this quotient is bounded by B L + R(c) sup . (19) A ≤ L R(c) inf − We remark that in order to estimate the quotient B/A, it is suﬃcient to estimate R(c) and Linf in (19), since we can use Proposition 3.1 for Lsup. Lower bounds for Linf will in the sequel be typically denoted by L˜inf . We make the following observations:

(a) Notice that as c1 approaches 0 – allowed by (18) – the quotient B/A approaches Lsup/Linf . This is very intuitive, since as c1 0 the translation grid becomes denser, the frame property more likely, and it becomes→ merely a question depending on the values Linf and Lsup. (b) As we indicated in the previous remark, one can achieve better frame bounds by simply choosing a sufficiently small sampling constant c1 (= max c1,c2 ). However, this leads to a highly redundant system, which causes a significant{ co}mputational complexity for practical applications. Therefore, we might not want to choose c1 too small to avoid high redundancy, but instead seek a threshold, which gives the largest possible value for c1 and also c2. (c) Reasoning about the previous two observations it becomes apparent that the balance between the possible values of c1 and c2 is an optimization problem, which presumably is different depending on the application at hand. 16 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

4. Compactly Supported Shearlet Frames We must now brace ourselves for the considerably harder challenge to construct compactly supported (separable) shearlet frames with provably ‘reasonable’ frame bounds. However, this task seems, at least to us, much more rewarding as it shows the delicacies of exploiting the previously derived general sufficient frame conditions. The main idea will be to start with a maximally flat low pass filter and use this to build a separable shearlet generator, which is then shown to generate a shearlet frame with ‘good’ frame bounds – in the sense of a small ratio. The modeling of the shearlet mimics the classical Example 2.3 with a wavelet-like function in horizontal direction and a bump-like function in vertical direction (on C), while ensuring spatial compact support. This is first performed for L2( ), followed by a discussion of L2(R2). The new shearlet frame is finally shown to provide (almost)C optimally sparse approximations for cartoon-like images.

4.1. Shearlet Frame for L2( ). We start by constructing a compactly supported (separable) shearlet frame (Λ, ψ)C for L2( ). Our construction will ensure that (Λ, ψ) is a feasible shearlet system,SH which enablesC us to exploit our previous results, inSH particular, Proposition 3.1, Theorem 3.4, and Proposition 3.3 to derive estimates of its frame bounds. For our construction, we focus on regular discrete shearlet systems, i.e., we choose

−j j/2 j/2 2 + 2 Λ= (2 ,k2 ,S j/2 A j M m): j 0, k 2 , m Z with c =(c ,c ) (R ) , { k2 2 c ≥ | | ≤ ⌈ ⌉ ∈ } 1 2 ∈ hence the main objective will be to construct a compactly supported shearlet ψ, which satisfies the decay condition (6). Our first goal towards this aim is to define a particular univariate scaling function, which will become a main building block for the compactly supported shearlet ψ we aim to construct, and prove controllable estimates for its function values. For this, given K, L Z+, we ∈ first let m0 be the trigonometric polynomial with real coefficients which satisfies m0(0) = 1 and L−1 K 1+ n m (ξ ) 2 = (cos(πξ ))2K − (sin(πξ ))2n. (20) | 0 1 | 1 n 1 n=0 Notice that m0 can be obtained from (20) using spectral factorization (see [6]). The two 1 parameters K and L determine the level of flatness near 2 and 0, respectively. In particular, 2 1 if K = L, then m0(ξ1) is symmetric with respect to the point ξ1 = 4 , and in this case m0 generates the Daubechies| | scaling function [6]. The more general trigonometric polynomial m0 we are considering here is sometimes also referred to as the maximally flat low pass filter. 2 For an illustration of m0 we refer to Figure 3. The following result| states| some useful properties, which will become handy in the sequel. For the proof, we refer to Subsection 6.2.1. Lemma 4.1. Retaining the previously introduced notations, we have the following conditions. 2 (i) m0 is even. (ii) |m |2 is decreasing on (0, 1 ). | 0| 2 (iii) If L−1 1 , then m 2 is concave on (0, 1 ). K+L−2 ≥ 4 | 0| 6 CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 17

1.4 L>K L=K K>L 1.2

0.8

0.6

0.4

0.2

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Figure 3. Graph of m 2 with K > L (solid), K = L (dot), and K < L (dash). | 0|

The next step is to deﬁne the sought scaling function φ L2(R) by ∈ ∞ −j φˆ(ξ1)= m0(2 ξ1). (21) j=0 Our ﬁrst result investigating this function concerns a lower bound for the function φˆ 2 on [ 1 , 1 ]. | | − 6 6 Proposition 4.2. Retaining the previously introduced notations, if L−1 1 , then K+L−2 ≥ 4

J0−1 − 2 − 2 2 j −2 J0+2(1−|m ( 1 )|2) φˆ(ξ ) m e 0 6 χ (ξ ), ξ R, (22) | 1 | ≥ 0 6 [−1/6,1/6] 1 1 ∈ j=0 where J0 is chosen sufficiently large. Moreover, the function φ is compactly supported. 1 ˆ 2 Proof. First note, that it suffices to consider the interval [0, 6 ], since φ(ξ1) is even (see 1 | | 2 Lemma 4.1(i)). Now assume that ξ1 [0, 6 ]. Since, by Lemma 4.1(ii)+(iii), m0 is monotone 1 ∈ | | and concave on (0, 6 ), we have m (ξ ) 2 m (0) 2 6( m (0) 2 m ( 1 ) 2) ξ . | 0 1 | ≥| 0 | − | 0 | −| 0 6 | | 1| Since by construction m (0) = 1, this implies | 0 | m (ξ ) 2 1 6C ξ , | 0 1 | ≥ − 1| 1| where C =1 m ( 1 ) 2 > 0. 1 −| 0 6 | Let now J > 0 be chosen sufficiently large such that 2−J0 C 1 . Then, since 1 x e−2x 0 1 ≤ 2 − ≥ for x [0, 1 ], we can continue our estimation by ∈ 2 −J m (2−J0 ξ ) 2 e−12(2 0 C1)|ξ1|. | 0 1 | ≥ 18 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

By again using Lemma 4.1(ii), this ﬁnally implies

J0−1 ∞ ′ φˆ(ξ ) 2 = m (2−jξ ) 2 m (2−J0−j ξ ) 2 | 1 | | 0 1 | | 0 1 | j=0 ′ j=0 J0−1 ∞ ′ −j 2 −j −J m 2 e(−12C1)(2 0 ξ1) ≥ 0 6 j=0 ′ j=0 J0−1 −j 2 −J +2 2 −2 0 (6C1)ξ1 = m0 6 e j=0 J0−1 −j 2 −J +2 m 2 e−2 0 C1 . ≥ 0 6 j=0 Finally, by standard arguments (cf. [6]), the function φ is compactly supported. The proposition is proved. In addition, an upper, ‘mathematically controllable’ bound for the values of φˆ 2 will be required for the analysis of the sought compactly supported shearlet ψ. We start| with| some preparation for being able to state the precise estimate. For this, we need to introduce a trigonometric polynomialm ˜ 0 slightly diﬀering from the previously discussed m0. Letting K′ Z be such that 0

L−1 ′ ′ K n 2 K 1+ n K n max m˜ 0(ξ1) − ′ ′ =: C2. (23) ξ1∈[0,1] | | ≤ n K + n K + n n=0 This now enables us to state the upper estimate for m˜ 2 we aimed for. | 0| Proposition 4.4. Retaining the previously introduced notations, and letting the non-negative ′ ′ K−K 1 integers K,K , and L be chosen such that L 6, L +1 K 3L 2, and ′ , ≥ ≤ ≤ − K +L−1 ≥ 4 then, for J1 suﬃciently large,

J1−1 2 −j − 2 J1+1 φˆ(ξ ) 2 min 1,C 2πξ −2γ m˜ e2 Pn |h(n)||n| , ξ R, | 1 | ≤ 2 | 1| 0 2π 1 ∈ j=0 CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 19 where γ =(K K′) 1 log (C ) and h(n) being the Fourier coeﬃcients of m˜ 2. − − 2 2 2 | 0| 2 2 Proof. First, observe that φˆ(ξ1) 1 for all ξ1 R, since m0(ξ1) 1. Now ﬁx ξ1 R. If 2πξ 1, we use the| easy| conclusion≤ from∈ the proof of| Lemma| ≤ 4.3(i), ∈ | 1| ≤ J1−1 2 −j − 2 2 J1+1 P |h(n)||n| C m˜ e n 1, 2 0 2π ≥ j=0 to show that J1−1 2 −j − −2γ 2 2 J1+1 |h(n)||n| 2 min 1,C 2πξ m˜ e Pn =1 φˆ(ξ ) , 2 | 1| 0 2π ≥| 1 | j=0 which was claimed. Now assume that 2πξ1 1. First, we observe that | | ≥ ′ m (ξ ) 2 = cos(πξ )2(K−K ) m˜ (ξ ) 2 | 0 1 | 1 | 0 1 | and ∞ 2(K−K′) φˆ(ξ ) 2 = cos(π2−jξ ) m˜ (2−jξ ) 2. | 1 | 1 | 0 1 | j=0 ∞ −j sin(x) Using the classical formula j=1 cos(2 x)= x (see [6]), it follows that

′ ∞ 2(K−K ) ˆ 2 sin(2πξ1) −j 2 φ(ξ1) = m˜ 0(2 ξ1) . (24) | | 2πξ1 | | j=0 Since 2πξ 1, there exists a positive integer J such that 2J−1 2π ξ 2J . Fixing this | 1| ≥ ≤ | 1| ≤ J, by (24) and the deﬁnition of C2, we have J−2 ∞ 2 −2(K−K′) −j′ 2 −j 2 φˆ(ξ1) = 2πξ1 m˜ 0(2 ξ1) m˜ 0(2 ξ1) | | | | ′ | | | | j=0 j=J−1 J−2 ∞ ′ 2πξ −2(K−K ) 2log2(C2) m˜ (2−j(2−J+1ξ )) 2 ≤ | 1| | 0 1 | ′ j=0 j=0 ∞ ′ 2πξ −2(K−K ) 2(J−1) log2(C2) m˜ (2−j(2−J+1ξ )) 2. ≤ | 1| | 0 1 | j=0 J−1 J (J−1) log (C2) log 2(C2) −J+1 Since 2 2π ξ1 2 implies 2 2 2πξ1 , letting η1 = 2 ξ1, hence 1 η ≤1 , yields| | ≤ ≤ | | 2π ≤| 1| ≤ π ∞ ′ φˆ(ξ ) 2 2πξ −2(K−K )+log2(C2) sup m˜ (2−jη ) 2 | 1 | ≤ | 1| | 0 1 | η1∈[0,1/π] j=0 ∞ ′ C 2πξ −2(K−K )+log2(C2) sup m˜ (2−jη ) 2. ≤ 2 | 1| | 0 1 | η1∈[0,1/2π] j=0 20 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

By Lemma 4.3(i), this implies that, for all J1 > 0,

J1−1 −j 2 ∞ 2 2 φˆ(ξ ) 2 C 2πξ −2γ m˜ sup m˜ 2−j−J1 η , | 1 | ≤ 2 | 1| 0 2π 0 1 j=0 η1∈[0,1/2π] j=0 Since 2π P |h(n)||n| |ξ | m˜ (ξ ) 2 1+ 2π h(n) n ξ e( n ) 1 . | 0 1 | ≤ | || | | 1| ≤ n we can continue by concluding that

J1−1 2 −j − 2 J1+1 φˆ(ξ ) 2 C 2πξ −2γ m˜ e2 Pn |h(n)||n|. | 1 | ≤ 2 | 1| 0 2π j=0 The proposition is proved. −γ ′ 4K−L+1 We can conclude from this result that φˆ(ξ1) = O( ξ1 ) for 0 K (which is ′ 5 K−K 1 | ′| 1 | | ≤ ≤ equivalent to ′ ), where γ =(K K ) log (C ). Let us now take a closer look K +L−1 ≥ 4 − − 2 2 2 at the decay rate γ. This depends on K′ for fixed K and L, where K′ can be chosen to be an arbitrary integer satisfying 0 K′ < K. Now regarding the parameters K′ and K as ≤ control parameters for the low pass filter m0 given by L−1 ′ ′ K 1+ n m (ξ ) 2 = cos(πξ )2(K−K ) cos(πξ )2(K ) − (sin(πξ ))2n , | 0 1 | 1 1 n 1 n=0 we see that, for any choice of K′ satisfying 0 K′ < K, the same scaling function φ is ≤ generated. Therefore, for analyzing the decay of φˆ, it is sufficient to show that there exists some K′ such that γ is above a certain threshold; later we aim for γ > 3. The following lemma makes this consideration explicit by choosing K′ = 0. We will comment on this choice in more detail after we stated Proposition 4.6. 3L Lemma 4.5. Let m0 be the low pass filter defined in (20) with K 2 and L 2, and let ′ ≥ ≥ C2 and m˜ 0 be as in Lemma 4.3 with K =0. Then 2 2K−L/2−1 max m˜ 0(ξ1) 2 ξ1∈[0,1] | | ≤ and the constant γ from Proposition 4.4 satisfies. 1 1 L γ = K log (C ) > ( + 1). − 2 2 2 2 2 Finally, we have reached the stage, where we can introduce a compactly supported shearlet ψ L2(R2), which generates a (separable) feasible shearlet frame (Λ, ψ) with ‘good’ mathematical∈ controllable frame bounds. For illustrative purposes,SH some elements of the shearlet frame (Λ, ψ) are displayed in Figure 4. SH Z+ 3L Proposition 4.6. Let K, L be such that L 10 and 2 K 3L 2, let m0 be the associated low pass filter as∈ defined in (20), and≥ let φ be the associated≤ ≤ scaling− function as defined in (21). Further, we define the bandpass filter m1 by m (ξ ) 2 = m (ξ +1/2) 2, ξ R. | 1 1 | | 0 1 | 1 ∈ CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 21

Then the shearlet defined by ψˆ(ξ)= m (4ξ )φˆ(ξ )φˆ(2ξ ), ξ =(ξ ,ξ ) R2 1 1 1 2 1 2 ∈ satisfies the feasibility condition (6); in particular, for any 0 K′ 4K−L+1 , we have ≤ ≤ 5 ψˆ(ξ) min 1, qξ α min 1, q′ξ −γ min 1, rξ −γ , (25) | | ≤ { | 1| } { | 1| } { | 2| } where 1 α = K K′ and γ =(K K′) log (C ) (26) − − − 2 2 2 with C2 being defined in (23), and

1 − 2γ 1 J1−1 −j 2 ′ − 2(K−K ) ′ 2 2 J1+1 P |h(n)||n| ′ q =4πC , q =2π C m˜ e n , r =2q . (27) 2 2 0 2π j=0 Further, we have 2 J0−1 2 −j − 2 1 2 2 −2 J0+2(1−|m ( 1 )|2) 2 ψˆ(ξ) m ( ) m e 0 6 χ (ξ) > 0, ξ R , (28) | | ≥| 0 6 | 0 6 Ω ∈ j=0 R2 1 1 1 1 1 1 where Ω= ξ =(ξ1,ξ2) : ξ1 [ , ] [ , ], ξ2 [ , ] . { ∈ ∈ 12 6 ∪ − 12 − 6 ∈ − 12 12 } In Lemma 4.5, K′ was chosen as K′ = 0 to show that the shearlet ψ defined as in Proposition 4.6 satisfies the feasibility condition (6). We note that certainly also different values for K′ can be chosen to estimate the constants α and γ in (25) (by using (26)), which might lead to improved estimates for the frame bounds. We refer to Table 1 for such estimates.

Figure 4. Display of compactly supported shearlets ψ2,k,0,k = 2, 1, 0, 1, 2, in spatial domain. − −

Z+ 3L Now for K, L satisfying L 10 and 2 K 3L 2, Proposition 4.6 together with (11) implies that∈ L < . Moreover,≥ by Proposition≤ ≤ 3.3,− R(c) can be made arbitrarily sup ∞ small by choosing a sampling matrix Mc with suﬃciently small determinant. Finally, let the set Ω be deﬁned as in Proposition 4.6. From (28), we see that ψˆ(ξ) 2 L˜ χ (ξ), where | | ≥ inf Ω 2 J0−1 2 −j − 1 2 2 −2 J0+2(1−|m ( 1 )|2) L˜ := m m e 0 6 L . (29) inf 0 6 0 6 ≤ inf j=0 22 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

Since ∞ T A j S Ω= , 2 k C j=0 j/2 |k|≤⌈2 ⌉ it then follows that

T 2 T Φ(ξ, 0) = ψˆ(S A −j ) L˜ χ (S A −j ξ) L˜ on . | k 2 ξ | ≥ inf Ω k 2 ≥ inf C j,k j,k Thus, by Theorem 3.4, both the lower and upper frame bound exist with explicit estimates for the shearlet frame (Λ, ψ) for a sampling matrix Mc with suﬃciently small determinant. This directly implies theSH following main result. Theorem 4.7. Let

−j j/2 j/2 2 Λ= (2 ,k2 ,S j/2 A j M m): j 0, k 2 , m Z { k2 2 c ≥ | | ≤ ⌈ ⌉ ∈ } + be the regular sampling set of Scone. Further, let K, L Z be such that L 10 and 3L K 3L 2, and deﬁne a shearlet ψ L2(R2) by ∈ ≥ 2 ≤ ≤ − ∈ ψˆ(ξ)= m (4ξ )φˆ(ξ )φˆ(2ξ ), ξ =(ξ ,ξ ) R2, 1 1 1 2 1 2 ∈ where m0 is the low pass ﬁlter satisfying

L−1 K 1+ n m (ξ ) 2 = (cos(πξ ))2K − (sin(πξ ))2n, ξ R, | 0 1 | 1 n 1 1 ∈ n=0 m1 is the associated bandpass filter defined by m (ξ ) 2 = m (ξ +1/2) 2, ξ R, | 1 1 | | 0 1 | 1 ∈ and φ is the scaling function given by ∞ φˆ(ξ )= m (2−jξ ), ξ R. 1 0 1 1 ∈ j=0 Then there exists a sampling constant cˆ1 > 0 such that the shearlet system (Λ, ψ) forms 2 + 2 SH a frame for L ( ) for any sampling matrix Mc with c = (c1,c2) (R ) and c2 c1 cˆ1. Furthermore, theC corresponding frame bounds A and B satisfy ∈ ≤ ≤ 1 1 [L˜ R(c)] A B [L + R(c)], det(M ) inf − ≤ ≤ ≤ det(M ) sup | c | | c | if R(c) < L˜inf , where L˜inf is defined as in (29). To illustrate how this result can be applied, we discuss the derived estimates for the frame ′ bounds using the following particular choices for the free parameters K,K ,L,c1, and c2. However, we do not claim that this is the optimal choice. In fact, optimizing the estimate for the ratio of the frame bounds A/B is a highly delicate task in this situation. CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 23

′ ′ K L c1 c2 K B/A K L c1 c2 K B/A 39 18 1.00 0.40 (27,17) 37.1204 39 19 0.9 0.40 (27,18) 44.5359 39 18 1.00 0.30 (27,15) 32.0208 39 19 0.9 0.30 (27,16) 28.4307 39 18 1.00 0.25 (27,15) 31.9105 39 19 0.9 0.25 (27,15) 28.0983 39 18 1.00 0.15 (27,15) 31.9019 39 19 0.9 0.20 (27,15) 28.0699 39 18 1.00 0.10 (27,15) 31.9019 39 19 0.9 0.15 (27,15) 28.0683 (a) (b) Table 1. Some estimates for the ratio of the frame bounds for the compactly supported shearlet frame constructed in Theorem 4.7 for various choices of the ′ ′ parameters K,K , L and the sampling constants c1,c2. K is given in the form ( , ), where the ﬁrst component is used to estimate α and γ in (11) and the second component is used to estimate α and γ in (17) using (26).

Example 4.8. For our discussion, we decided upon the following choices for the various parameters necessary for exploiting Theorem 4.7 to derive an estimate for the ratio of the frame bounds: We may choose L˜inf as in (29) for the expression for the lower frame bound. The value 1 Lsup can be estimated using (11) with = 2 ,p =1. Further, for a given sampling matrix Mc + 2 with c = (c1,c2) (R ) and c1 c2 > 0, the value R(c) can be estimated using (17) with 1 ∈ ≥ ′ = 2 ,p = 1 from Proposition 3.3. For both those estimates, given positive integers K,K , ′ 4K−L+1 1 ′ and L such that for K 5 we choose = 2 , the values q, q ,r are chosen as in (27), and the values α,γ as in≤(26). Table 1 shows some estimates for the ratio of the frame bounds of (Λ, ψ) using Theorem ′ SH 4.7 for specific choices of the parameters K,K , and L with various sampling constants c1,c2. We wish to emphasize that the estimates (11) and (17) behave differently depending on α and ′ γ. One certain value of K minimizing the upper estimate (11) for Lsup does not necessarily minimize the upper bound (17) for R(c). In fact, among all possible values of K′ – that is, ′ 4K−L+1 ′ 0 K 5 – we choose K differently for (11) and (17) so that such different choices of≤K′ minimize≤ (11) and (17), respectively, which leads to significantly improved theoretical frame bounds. It should be pointed out that numerical experiments show much better results, but the presented ones are the estimates we are able to prove theoretically. Notice the flexibility in choosing K,K′, and L. A full blown optimization analysis might lead to an even better estimate for the ratio of the frame bounds of (Λ, ψ), but this is beyond the scope of this paper. SH

4.2. Shearlet Frame for L2(R2). We now aim to generate a frame for the whole space L2(R2) using the compactly supported (separable) shearlet frame introduced in the previous subsection. One classical way for achieving this is by orthogonally projecting each shearlet element in Fourier domain onto the cone , and proceeding likewise for the vertical cone. However, this procedure is quite counterproductiveC in the sense that it destroys most of the advantageous properties, e.g., compact support and regularity, of the elements of such 24 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM a shearlet frame, in particular, if the Fourier transform of the to be analyzed signal is not entirely supported in either the horizontal or the vertical cone. This problem can though be quite easily resolved by taking the union of two cone-adapted shearlet systems – one for the horizontal cone = 1 3 and the other for the vertical cone C C ∪ C 2 2 = 2 4. In fact, a shearlet frame (c; φ, ψ, ψ˜) (cf. Definition 2.2) for L (R ) can be CconstructedC ∪ C by the following SH Theorem 4.9. Let ψ L2(R2) be the shearlet with associated scaling function φ L2(R) ∈ ∈ both introduced in Theorem 4.7, and set θ(x1, x2) = φ(x1)φ(x2) and ψ˜(x1, x2) = ψ(x2, x1). Then the corresponding shearlet system (c; θ, ψ, ψ˜) forms a frame for L2(R2) for any SH sampling matrices M and M˜ with c =(c ,c ) (R+)2 and c c cˆ . c c 1 2 ∈ 2 ≤ 1 ≤ 1 Proof. In the sequel, the constants α,γ and q, q′,r will be those defined in (26) and (27), respectively. First note that the function Φ defined in (7) now becomes Φ : R2 R2 R defined by × → Φ(ξ,ω)= θˆ(ξ) θˆ(ξ + ω) + Φ (ξ,ω) + Φ (ξ,ω), (30) | || | 1 2 where

ˆ T − ˆ T − Φ1(ξ,ω)= ψ(Sk A2 j ξ) ψ(Sk A2 j ξ + ω) j≥0 |k|≤⌈2j/2⌉ and

˜ˆ ˜ − ˜ˆ ˜ − Φ2(ξ,ω)= ψ(SkA2 j ξ) ψ(SkA2 j ξ + ω) . j≥0 |k|≤⌈2j/2⌉ Also, R(c) deﬁned in (9), is now given by the term

1 1 −1 −1 −1 −1 −1 −1 1 Γ (c m)Γ ( c m) 2 + Γ (M m)Γ ( M m) 2 + (Γ (M˜ m)Γ ( M˜ m)) 2 , 0 1 0 − 1 1 c 1 − c 2 c 2 − c m=0 where

Γ0(ω) = ess sup θˆ(ξ) θˆ(ξ + ω) and Γi(ω) = ess sup Φi(ξ,ω) for i =1, 2. ξ∈R2 | || | ξ∈R2 In fact, Theorem 3.4 can be easily extended to this case so that we only need to estimate Linf , Lsup, and R(c) to derive finiteness of the estimate for the ratio of the frame bounds in (19). Using the function Φ(ξ,ω) from (30), Linf and Lsup are defined as in (8) and (9) – note that in this case, is replaced by R2 in the definition. C We start by estimating the first term θˆ(ξ) θˆ(ξ + ω) from (30). WLOG, we may assume that m = m = 0 for m Z2 0 . Then,| || by Proposition| 4.4, we have ∞ 1 ∈ \{ } θˆ(ξ) θˆ(ξ + c−1m) φˆ(ξ ) φˆ(ξ + c−1m) | || 1 | ≤ | 1 || 1 1 | min 1, q′ξ −γ min 1, q′ξ + q′c−1m −γ ≤ { | 1| } { | 1 1 1| } 2c γ 1 m −γ, (31) ≤ q′ ∞ provided that c−1q′m 4. Now choose c > 0 sufficiently small so that c−1q′ 4. 1 1 ≥ 1 1 ≥ CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 25

For the second and third term in (30), we can apply the estimates (11) and (18) from the proof of Proposition 3.3. In particular for the third term, we only need to switch the roles of the variables ξ1 and ξ2 to reach the same estimates as (11) and (18). Set now γ′′ = γ γ′ for an arbitrarily ﬁxed γ′ satisfying 1 < γ′ < γ 2. Then, by (31) and (18), we obtain− − ′′ 2c γ 2qc γ R(c) C 1 + C′ 2 . (32) ≤ q′ q′r Also, using (11), q 2 q 1 1 L 1+2 2+ +1 log + + . (33) sup ≤ r 2γ 1 2 q′ 1 21−2α 1 2−2γ − − − since θˆ(ξ) 1. This gives an upper bound of L . | | ≤ sup Finally, let us derive a lower bound of Linf . By (22) and (28), we have

θˆ(ξ) 2 > Cχ (ξ), ψˆ(ξ) 2 > Cχ (ξ), and ψ˜ˆ(ξ) 2 > Cχ (ξ), (34) | | Ω0 | | Ω1 | | Ω2 where Ω = ξ R2 : ξ 1 , 0 { ∈ ∞ ≤ 6 } Ω = ξ R2 : 1 < ξ < 1 , ξ < 1 , 1 { ∈ 12 | 1| 6 | 2| 12 } Ω = ξ R2 : 1 < ξ < 1 , ξ < 1 . 2 { ∈ 12 | 2| 6 | 1| 12 } Setting

˜ T Ω= Ω0 A2j SkΩ1 A2j Sk Ω2 , ∪ ∪ j,k j,k we observe that Ω = R2 and, by (34), Φ(ξ, 0) > L˜ χ (ξ) for some L˜ > 0. (35) inf Ω inf 2 This implies Φ(ξ, 0) > L˜inf > 0 on R , which yields the lower bound of Linf . Concluding, the estimates (32), (33), and (35) provide bounds for Linf , Lsup, and R(c) to derive an upper estimate for (L + R(c))/(L R(c)) in (19), and hence for A/B. For sup inf − this, it is in particular important to notice that we can choose a sampling matrix Mc with sufficiently small determinant such that L˜inf R(c) > 0 with L˜inf as defined in (35). This proves the existence of the frame bounds. − Finally, it is obvious that all functions in (Λ, Λ˜, ψ, ψ,˜ θ) are compactly supported in spatial domain, which finished the proof. SH

In the proof of Theorem 4.9, we see that our upper bounds of Lsup and R(c) are about twice as large as the upper bounds in the cone case and the lower bound L˜inf is about the same compared to the cone case. Also c can be chosen so that R(c) is suﬃciently small. This indicates that in this case, our estimate for the ratio of the frame bounds is about twice as large as the estimate for the ratio B/A in the cone case for suﬃciently small determinant of the sampling matrix Mc. 26 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

4.3. Sparse Approximation using Compactly Supported Shearlets. One essential performance criterion for a frame composed of anisotropic elements – besides the ratio of the frame bounds – is the approximation rate of curvilinear objects. This viewpoint arose due to the fact that edges are the most prominent features in images, hence representation systems should in particular provide sparse representations for those. To stand on solid ground, we first briefly recall the mathematical model of a cartoon-like image introduced in [2]. For ν > 0, the set ST AR2(ν) is defined to be the set of all B [0, 1]2 such that B is a translate of a set x R2 : x ρ(θ), x = ( x , θ) in polar coordinates⊂ ′′ { ∈ | | ≤ 2 | | } which satisfies sup ρ (θ) ν, ρ ρ0 < 1. Then, (ν) denotes the set of functions f L2(R2) of the form| | ≤ ≤ E ∈ f = f0 + f1χB, where f , f C2([0, 1]2) and B ST AR2(ν). 0 1 ∈ 0 ∈ Assume now that (c; φ, ψ, ψ˜) forms a frame for L2(R2), which for illustrative purposes SH for a moment we denote by (c; φ, ψ, ψ˜)=(σ ) , say. Is it well-known that a frame is SH i i∈I associated with a so-called canonical dual frame, which in this case we want to call (˜σi)i∈I , and which, for each f L2(R2), provides the decomposition ∈ f = f, σ σ˜ . i i i∈I A (non-linear) N-term approximation fN of f is then generated by selecting N indices from I, say IN , and setting f = f, σ σ˜ . N i i i∈I N Typically – and this will also be our choice here – one chooses those N indices which are associated with the N largest coefficients f, σ in magnitude. We would though like to i emphasize that in case (σi)i∈I is not a basis, this approximation is not necessarily the best N-term approximation. In [8], it was proven that the optimal approximation rate for such cartoon-like image models which can be achieved under some restrictions on the representation system as well as on the selection procedure of the approximating coefficients is f f 2 C N −2 as N , − N 2 ≤ →∞ where fN is the N-term approximation generated by the N largest coefficients in magnitude. In [20], two of the authors proved that a large class of shearlet frames – including, in particular, a significant set of compactly supported shearlet frames – provide optimally sparse approximations of such cartoon-like images. For the convenience of the reader, we state the result below. Notice that in [20], the result was proven for a isotropic sampling matrix, but the extension to our anisotropic sampling is immediate. Theorem 4.10 ([20]). Let c> 0, and let φ, ψ, ψ˜ L2(R2) be compactly supported. Suppose that, in addition, for all ξ =(ξ ,ξ ) R2, the shearlet∈ ψ satisfies 1 2 ∈ α −γ −γ (i) ψˆ(ξ) C min 1, ξ1 min 1, ξ1 min 1, ξ2 and | | ≤ { | | } −{γ | | } { | | } ∂ |ξ2| (ii) ψˆ(ξ) h(ξ1) 1+ , ∂ξ2 ≤| | |ξ1| CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 27 where α> 5, γ 4, h L1(R), and C is a constant, and suppose that the shearlet ψ˜ satisfies ≥ ∈ ˜ (i) and (ii) with the roles of ξ1 and ξ2 reversed. Further, suppose that (c; φ, ψ, ψ) forms a frame for L2(R2). SH Then, for any ν > 0, the shearlet frame (c; φ, ψ, ψ˜) provides (almost) optimally sparse approximations of functions f 2(ν), i.e.,SH there exists some C > 0 such that ∈E f f 2 C (log N)3 N −2 as N , − N 2 ≤ →∞ where fN is the nonlinear N-term approximation obtained by choosing the N largest shearlet coefficients of f. Using this result, we can prove that in fact the compactly supported shearlet frames we introduced in Theorem 4.7 even provide (almost) optimally sparse approximations of cartoon-like images, i.e., functions in 2(ν). Although the optimal rate is not achieved, the log-factor is typically considered negligibleE compared to the N −2-factor, wherefore the term ‘almost optimal’ has been adopted into the language. Theorem 4.11. Let ψ L2(R2) be the shearlet with associated scaling function φ L2(R) ∈ ∈ both introduced in Theorem 4.7 with L > 18, and set θ(x1, x2) = φ(x1)φ(x2) as well as ˜ ˜ ψ(x1, x2) = ψ(x2, x1). Then the compactly supported shearlet frame (c; θ, ψ, ψ) provides (almost) optimally sparse approximations of functions f 2(ν),SH i.e., there exists some C > 0 such that ∈ E f f 2 C (log N)3 N −2 as N , − N 2 ≤ →∞ where fN is the nonlinear N-term approximation obtained by choosing the N largest shearlet coefficients of f. Proof. The fact that (c; θ, ψ, ψ˜) is a compactly supported frame follows from Theorem 4.9. Hence it remainsSH to prove that the shearlet ψ satisfies conditions (i) and (ii) from Theorem 4.10, and the shearlet ψ˜ likewise. By Lemma 4.5 and Proposition 4.6, there exists a constant C such that ψˆ(ξ) C min 1, ξ K min 1, ξ −L/4−1/2 min 1, ξ −L/4−1/2 , | | ≤ { | 1| } { | 1| } { | 2| } where L/4+1/2 > 4 and K > 5. This already proves condition (i) of Theorem 4.10. To show condition (ii), we first observe that there exists a function ψ1 such that we can write ψˆ(ξ)= ψˆ1(ξ1)φˆ(2ξ2) so that ψˆ (ξ ) C min 1, ξ K min 1, ξ −L/4−1/2 and φˆ(ξ ) C min 1, ξ −L/4−1/2 . | 1 1 | ≤ { | 1| } { | 1| } | 2 | ≤ { | 2| } Hence it is sufficient to prove that ′ ′ (φˆ) (ξ ) C min 1, ξ −γ with γ′ 4. (36) | 2 | ≤ { | 2| } ≥ ′ ′ L 1 For this, we first choose a positive integer ρ > 0 such that 4 ρ < 4 2 < K, which we are allowed to do since L> 18. Then, by the definition of φ, ≤ − ′ ′ sin(2πξ ) ρ sin(2πξ ) K−ρ ∞ φˆ(ξ )= 2 2 m˜ (2−jξ ), 2 2πξ 2πξ 0 2 2 2 j=0 28 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM where m˜ (ξ ) 2 = L−1 K−1+n (sin(πξ ))2n. Now define S and φ˜ by | 0 2 | n=0 n 2 ′ ′ sin(2πξ ) ρ sin(2πξ ) K−ρ ∞ S(ξ )= 2 and φ˜ˆ(ξ )= 2 m˜ (2−jξ ), 2 2πξ 2 2πξ 0 2 2 2 j=0 which gives ′ ′ ˆ ˆ ′ (φˆ) (ξ2)= S (ξ2)φ˜(ξ2)+ S(ξ2)(φ˜) (ξ2). ′ Since max S′(ξ ) , S(ξ ) C ξ −ρ , it remains to show that both φ˜ˆ and (φ˜ˆ) are {| 2 | | 2 |} ≤ | 2| | | | | bounded. Notice that we may assume thatm ˜ 0(0) = 1 and φ˜ is compactly supported (see ˆ ˆ [6]). Therefore, boundedness of φ˜ implies that φ˜ and (φ˜)′ are bounded, which enables us to ˆ restrict our task further and it remains to check that φ˜ L1(R). This will now be proved ∈ by showing that, for all ξ2,

ˆ −η φ˜(ξ ) C min 1, ξ with η > 1, (37) | 2 | ≤ { | 2| } ˆ which obviously implies φ˜ L1(R). ∈ ˆ First, it is easy to show that φ˜ is bounded on [ 1, 1]. Therefore, if ξ 1, then (37) | | − | 2| ≤ holds. Now assume that ξ2 > 1. Then there exists a positive integer J > 0 such that J−1 J | | 2 2K−L/2−1 2 ξ2 2 , and, using the fact that Lemma 4.5 implies maxξ2 m˜ 0(ξ2) < 2 , we obtain≤ | | ≤ | | J−1 ∞ ˆ −(K−ρ′) −j−J φ˜(ξ2) C ξ2 max m˜ 0(ξ2) m˜ 0(2 ξ2) | | ≤ | | ξ2 | | | | j=0 j=0 ∞ ′ C ξ −(K−ρ −K+L/4+1/2) sup m˜ (2−jξ ) ≤ | 2| | 0 2 | ξ∈[−1,1] j=0 ′ C ξ −(L/4+1/2−ρ ). ≤ | 2| Hence, L/4+1/2 ρ′ > 1 and this proves (37), and hence (36). The theorem is proved. − We would like to make the reader aware of the fact that the constant C in Theorem 4.11 we derive from this proof depends strongly on the bound of the inverse of the lower frame bound detM A−1 | c| . ≤ L R(c) inf −

5. Implementation of Compactly Supported Shearlets The class of cone-adapted regular compactly supported shearlet frames, which we introduced and analyzed in the previous section allows an eﬃcient and exact digitalization of the associated transform. In the sequel we will ﬁrst discuss the redundancy of the associated digital system and argue that this can indeed be controlled. Secondly, a fast implementation will be discussed followed by a comparison with other directional transforms. CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 29

5.1. Controllable Redundancy. Let us first quantify the redundancy of the constructed compactly supported shearlet frames. For this, let f L2(R2). Without loss of generality, we can assume that f is a finite linear combination of∈ translates of a 2-dimensional scaling function θ at scale J as follows: 2J −1 2J −1 f(x)= d θ(2J x n). n − n =0 n =0 1 2 Let us now count the number of shearlet elements necessary to represent f. For this, we first consider elements from Ψ(ψ; c) for fixed scale j 0,...,J 1 . We observe that there j/2+1 j j/2 −1∈{ − } are 2 shearing indices k and 2 2 (c1c2) translates associated with the scaling 2j+1 matrix A2j and the sampling matrix Mc. Thus, we require 2 /(c1c2) shearlet elements from Ψ(ψ; c). Due to symmetry reasons, we require the same number of shearlet elements ˜ ˜ −2 from Ψ(ψ; c). Finally, about c1 translates of the scaling function θ in Φ(θ; c1) are necessary at the coarsest scale j = 0. Summarizing, the total number of necessary shearlet elements across all scales, and hence the number of necessary shearlet coefficients is about 4 J−1 4 22J +2 22j +1 = (38) c c c c 3 1 2 j=0 1 2 The redundancy of each shearlet frame can now be computed as the ratio of the number of coefficients dn and the number of the shearlet coefficients from (38). Letting J , we obtain as redundancy → ∞ 4 1 . 3 c1c2 Setting exemplarily c1 = 1 and c2 = 0.4, we have an asymptotic redundancy of 10/3 with theoretical estimates for the frame bounds given in Table 1. 5.2. Discrete Implementation. The shearlet transform associated with the class of cone- adapted regular compactly supported shearlet frames introduced in Section 4 has been im- plemented, and we would like to refer to ShearLab1 for a freely available implementation. Although the details of the implementation are presented in [23], for the convenience of the reader, we will now briefly review the main ideas. To construct the shearlet generator, let φ L2(R) be a compactly supported 1D scaling function satisfying ∈

φ(x1)= h(n1)√2φ(2x1 n1) Z − n1∈ for some ‘appropriately chosen’ filter h. An associated compactly supported 1D wavelet γ L2(R) can then be defined by ∈ γ(x1)= g(n1)√2φ(2x1 n1), Z − n1∈ where again g is an ‘appropriately chosen’ filter. The selected shearlet generator is then defined by the 2D separable wavelet function ψ(x1, x2) = γ(x1)φ(x2), and – specifying the

1ShearLab (Version 1.1) is available from http://www.shearlab.org. 30 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM term ‘appropriately chosen’ – we assume that h and g are chosen so that ψ satisfies condition (6). For the signal f L2(R2) to be analyzed, we assume that, for J > 0 fixed, f is of the form ∈ J J J f(x)= fJ (n)2 θ(2 x1 n1, 2 x2 n2). (39) Z2 − − n∈ Aiming towards computing the shearlet coefficients f, ψj,k,m for j = 0,...,J 1, we first observe that −

f, ψ = f(S −j/2 ( )), ψ ( )) . (40) j,k,m 2 k j,0,m For simplicity, we will from now on assume that j/2 is an integer; otherwise either j/2 or j/2 would need to be taken. Our observation (40) implies that the shearlet coefficients⌈ ⌉ ⌊f, ψ⌋ can be explicitly computed by applying the discrete separable wavelet transform j,k,m associated with the anisotropic sampling matrix A2j provided that f(S2−j/2k ) is contained in the scaling space V = 2J θ(2J n , 2J n ):(n , n ) Z2 . J { − 1 − 2 1 2 ∈ } Despite (39), this is in particular not the case, if the shear parameter 2−j/2k is non-integer. The reason for this failure is that the shear matrix S2−j/2k does not preserve the regular grid −J Z2 Z2 Z2 2 in VJ , i.e., S2−j/2k( ) = . In order to resolve this issue, consider the new scaling space V k = 2J+j/4θ (2J+j/2 n , 2J n ):(n , n ) Z2 , where θ (x)= θ(S x). J+j/2,J { k − 1 − 2 1 2 ∈ } k k k −J Z2 We remark that the scaling space VJ+j/2,J is obtained by refining the regular grid 2 along the x -axis by a factor of 2j/2. With this modification, the new grid 2−J−j/2Z 2−J Z 1 × is now invariant under the shear operator S2−j/2k. After a short calculation, this leads to the following representation of f(S −j/2 (x)) with scaling coefficients f˜ =(f ) j/2 h : 2 k J J ↑2 ∗1 j/2 ˜ J+j/4 J+j/2 J f(S2−j/2k(x)) = (fJ )(Skn)2 θk(2 x1 n1, 2 x2 n2), (41) Z2 − − n∈ where 2j/2 and are the 1D upsampling operator by a factor of 2j/2 and the 1D convolu- ↑ ∗1 tion operator along the x1-axis, respectively, and hj/2(n1) are the Fourier coefficients of the trigonometric polynomial

j/2−1 k −2πin12 ξ1 Hj/2(ξ1)= h(n1)e . (42) Z k=0 n1∈ Using (40), the shearlet coefficients f, ψ can then be explicitly computed by basically j,k,m applying the discrete separable wavelet transform associated with the sampling matrix A2j to the scaling coefficients in (41). To be more precise, for j1, j2 > 0, let the discrete separable j1 j2 wavelet transform Wj1,j2 associated with the scaling matrix diag(2 , 2 ) be defined by

j1 j2 Wj1,j2 (c)(n1, n2)= gj1 (m1 2 n1)hj2 (m2 2 n2)c(m1, m2) Z2 − − m∈ CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 31 for c ℓ(Z2), where h and g are the Fourier coefficients of the trigonometric polynomials ∈ j j Hj defined in (42) and Gj defined by j−2 k j−1 −2πi2 n1ξ1 −2πi2 n1ξ1 Gj(ξ1)= h(n1)e g(n1)e Z Z k=0 n1∈ n1∈ d for j > 0 fixed. Also, define the discrete shear operator S2−j/2k by d ˜ 2 Z2 S2−j/2k(fJ )(n)= (fJ (Sk ) Θk) 1 hj/2 (n) for fJ ℓ ( ), (43) ∗ ∗ ↓2j/2 ∈ 2 j/2 where Θk(n)= θk( ), θk( n) for n Z , hj/2(n1)= hj/2( n1) and 2 is 1D downsam- j/2 − ∈ − ↓ pling by a factor of 2 along the axis x1. Then the shearlet coefficients are explicitly given by d f, ψ = W S −j/2 (f ) (m), j,k,m J−j,J−j/2 2 k J The shearlet coefficients for the vertical cone can be computed similarly. d We would like to emphasize that employing the discrete shear operator S2−j/2k, which Z2 computes interpolated sample values on the sheared grid S−2−j/2k( ), is key to not only computing the shearlet coefficients exactly. It is even more the faithful digital equivalent to the shear operator S2−j/2k from continuum space, thereby deriving directionality faith- fully in the digital implementation. For illustration purposes, consider the 2D function Z2 fc = χ{x:x1=0}. Let then fd by the function on the digital grid , which is derived by 2 sampling fc, i.e., fd(n)= fc(n) for all n Z . Now consider the continuum sheared version ∈ 2 fc(S−1/4 ) of fc and its digital analog fc(S−1/4(n)), n Z again derived by mere sampling. 2 ∈ Since Z is not invariant under S−1/4, the function fc(S−1/4(n)) is aliased as illustrated in Figures 5(a) and (b). Deriving a digital version of fc(S−1/4 ) by application of the discrete d d shear operator S2−j/2k to fd instead, i.e., S−1/4(fd), effectively removes directional aliasing as visually evidenced by Figures 5(c) and (d).

(a) (b) (c) (d)

Figure 5. (a) Aliased image: fc(S−1/4n). (b) Aliased image: 2D-FFT of d fc(S−1/4(n)). (c) De-aliased image: S−1/4(fd)(n). (d) De-aliased image: 2D- d FFT of S−1/4(fd)(n).

5.3. Computational Complexity. Let us now comment on the computational complexity of this digital shearlet transform. The most time consuming step is the computation of the discrete shear operator in (43) for the finest scale j = J 1. This step requires 1D upsampling by a factor of 2j/2 followed by 1D convolution for each− direction associated with the shear 32 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM parameter k – the filter Θk in (43) can be precomputed for each shear parameter k. Letting L be the total number of directions at finest scale j = J 1 and M 2 the size of 2D input data, the computational complexity for computing the scaling− coefficients in (41) is O(2j/2L M 2). The complexity of the discrete separable wavelet transform associated with A2j for (40) requires O(M 2) operations, wherefore it is negligible. Since L = 2(2 2j/2 + 1), the total computational cost for this digital shearlet transform hence amounts to O(2log2(1/2(L/2−1))L M 2). (44) We wish to mention that the total computational cost depends on the number L of shear parameters at finest scale, and this total cost grows approximately by a factor of L2 as L is increased. It should though be emphasized that L can be chosen so that this shearlet transform is favorably comparable to other redundant directional transforms with respect to running time as well as performance. For instance, when this shearlet transform is imple- mented with 6 directions at finest scale, the constant factor 2log2(1/2(L/2−1)) in (44) equals 1 implying that the running time of this shearlet transform is only about 6 times slower than the discrete orthogonal wavelet transform, thereby remaining in the range of the running time of other directional transforms. For the inverse shearlet transform, an apparent problem is the fact that although the families of shearlet frames provide excellent spatial localization, dual frames need to be computed in order to reconstruct a given function from its shearlet coefficients. However, this can be overcome noticing that due to the small ratio of the frame bounds proven in Theorem 4.7 (see also Table 1), which are in fact numerically even much smaller (of the order of about 6), iterative schemes like the conjugate gradient method can be used very effectively, which typically requires only a few iterations. Implementations are also already available in ShearLab2. Before we finish this subsection, we analyze the asymptotic computational cost of the shearlet transform based on compactly supported shearlets, which yields (almost) optimally sparse N-term approximations of cartoon-like images f 2(ν) as stated in Theorem 4.11. ∈E Proposition 5.1. Let the shearlet frame (c; θ, ψ, ψ˜) be defined as in Theorem 4.9 with θ SH an orthogonal scaling function, and let (ψ˜ ) denote its dual frame. Further, let f 2(ν), λ λ ∈E M be a positive integer, and define J0 to be a scale satisfying M−1 M−1 P (f)(x)= f (m)2jθ(2jx m), x R2, J0 J0 − ∈ m =0 m =0 1 2 j 2 where, for each scale j, Pj denotes the orthogonal projection onto span θ(2 m): m Z j j { −Z ∈ } and fj(m) = f( ), 2 θ(2 m) . Let N > M be such that J := 2 log2 N and J J0. Then − ∈ ≤ f P (f), ψ ψ˜ 2 C (log N)3 N −2 as N , − J λ λ2 ≤ →∞ λ∈ΛN where ΛN are the shearlet indices corresponding to the N largest coefficients f, ψλ in magnitude. This coefficient sequence ( P (f), ψ ) can be computed from the M M digital J λ λ × 2ShearLab (Version 1.1) is available from http://www.shearlab.org. CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 33

2(1+2 log2 N/ log2 M) data fJ0 by the shearlet transform with total computational cost O(M ) as N . →∞ Proof. Let Λ denote the total shearlet indexing set, and notice that WLOG we may assume that λ =(j, k, m) ΛN implies j J; see, e.g., the proof of Theorem 4.10 in [20]. Then we have ∈ ≤ f P (f), ψ ψ˜ − J λ λ2 λ∈ΛN (f P (f)), ψ ψ˜ + P (f), ψ ψ˜ ≤ − J λ λ2 J λ λ2 λ∈Λ λ/∈ΛN (f P (f)), ψ ψ˜ + f, ψ ψ˜ + (f P (f)), ψ ψ˜ . ≤ − J λ λ2 λ λ2 − J λ λ2 λ∈Λ λ/∈ΛN λ/∈Λ Letting A and B be the lower and upper frame bounds of the shearlet frame (c; θ, ψ, ψ˜), respectively, by Theorem 4.11 we conclude that SH

3/2 −1 1+ √B f PJ (f), ψλ ψ˜λ 2 C (log N) N + f PJ (f) 2. (45) − ≤ √A − λ∈ΛN j j To estimate f PJ (f) 2, we let ϕj,m(x)=2 ϕ(2 x m) be orthogonal wavelets associated with θ, and for − each j > 0, set − Ω0 = m Z2 : supp(ϕ ) Γ = and Ω1 = m Z2 : supp(ϕ ) Γ= , j { ∈ j,m ∩ ∅} j { ∈ j,m ∩ ∅} where Γ is the discontinuity curve of f. Then we obtain

f P (f) 2 = f,ϕ 2 = f,ϕ 2 + f,ϕ 2 . − J 2 | j,m| | j,m| | j,m| j>J m∈Z2 j>J m∈Ω0 m∈Ω1 j j Since Ω0 C 2j and f,ϕ C 2−j, it follows that | j | ≤ | j,m| ≤ f P (f) 2 C 2j 2−2j + C N −2 C N −2. − J 2 ≤ ≤ j>J Hence, by (45), f P (f), ψ ψ˜ 2 C (log N)3 N −2, − J λ λ2 ≤ λ∈ΛN which proves the ﬁrst claim. The shearlet transform we described in this section computes the sequence of coeﬃcients

( PJ (f), ψλ )λ∈ΛN by the following two steps. First, compute the scaling coeﬃcients fJ from the M M input data f , which has complexity O(2αJ ) with α = (2 log M)/J. Secondly, × J0 2 apply the shearlet transform to compute PJ (f), ψλ for each λ ΛN . Since the size of fJ is O(22J ), by (44), the computational cost for this second step is∈O(23J ). Summarizing, the αJ (1+2/α) 2(1+2 log2 N/ log2 M) total cost of computing ( PJ (f), ψλ )λ∈ΛN is O((2 ) )= O(M ), which ﬁnishes the proof. 34 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

Proposition 5.1 indicates how large the number of directions L should be in order to achieve (almost) optimally sparse approximation of cartoon-like images by the shearlet transform. In fact, by (44), L needs to be chosen such that

2log2(1/2(L/2−1))L M 4 log2 N/ log2 M . ≈

5.4. Comparison with Other Directional Transforms. Let us also briefly compare this implementation with the digital curvelet transform [1] and the contourlet transform [3, 7, 24]. For the discrete shearlet transform, excellent spatial localization is achieved by using compactly supported shearlets. In addition, by utilization of a shear matrix instead of rotation, directionality is naturally adapted for the digital setting in the sense that the shear matrix preserves the structure of the integer grid, and a unified concept for the continuum and digital setting is achieved. However, due to use of a shear matrix, the discrete shearlet transform is not fully structured compared to other directional transforms based on a directional filter bank (c.f. [3, 11, 7, 24]) in the sense that a full tree structure across scales is not available requiring an additional computation step to compute scaling coefficients resampled on an anisotropically refined and sheared grid in (41). Another drawback is that these compactly supported shearlet systems do not form a tight frame and, accordingly, the synthesis process needs to be performed by iterative methods. Curvelets provide excellent frequency localization and directional selectivity by use of bandlimited functions whose Fourier transform is supported in directional wedges. However, due to the bandlimitedness, curvelets interact with an image on the whole domain, which often creates some artifacts as we can observe in Figure 6. Finally, contourlets could be compactly supported in spatial domain and this would allow excellent spatial localization like compactly supported shearlets. However, they are built from a directional filter bank, which guarantees directional selectivity only when IIR filters are used, which prohibits to employ compactly supported contourlets. This ob- struction is also indicated in [11]. Let us now be more specific about the choice of the generator in the implementation. In principle, we could use a wavelet function defined as in Theorem 4.9 to implement the shearlet transform, in which case its stability can be theoretically guaranteed with reasonable redundancy — see Table 1. However, this choice requires a relatively large size of support of shearlets in the spatial domain. Since this is not desirable, in particular, for denoising, the implementation uses other orthogonal wavelets like ’Symmlets’ and ’Coiflets’. We though emphasize that numerical experiments show that this choice still provides a fast convergence rate for approximating the inverse, and we refer to [23] for such numerical tests. Also, it is important to notice that the choice of the wavelet function employed for the implementation fulfills the conditions in Theorem 3.4, which shows that it can generate a shearlet frame. However, Theorem 4.9, which proves reasonable frame bounds, does not cover this choice. In this sense, there is still a gap between our theory and implementation. We though believe that the conditions in Theorem 4.9 can be significantly relaxed to cover more separable wavelet functions (even nonseparable ones) generating shearlet frames with reasonable frame bounds and relatively small size of support. CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 35

We included Figure 6 to illustrate the peak signal to noise ratio results for the three discussed transforms. For our computations, we used CurveLab available from http:// www.curvelet.org/ and the nonsubsampled contourlet transform available from www.ifp. illinois.edu/~minhdo/software/ for curvelets and contourlets, respectively. For each of transforms, we perform hard thresholding on the transform domain. We apply the shearlet transform with redundancy of 4 for the comparison with curvelets. For the comparison of contourlets and shearlets, we apply the shearlet transform to 4 translated images which leads to redundancy of 40. We wish to remark that the computing time of our shearlet transform is signiﬁcantly faster than the one of contourlets. The running time of both transforms was calculated on a desktop with 2.53 GHz and 1.98 of RAM.

6. Proofs 6.1. Proofs of Results from Section 3.

6.1.1. Proof of Theorem 3.4. Let f L2( ). Then, by deﬁnition of ψ , ∈ C j,k,m 2 f,ˆ ψˆ 2 j,k,mL (C) j≥0 k∈Kj m∈Z2 2 3/2 ˆ ˆ T 2πiξ,Aaj Ssk Mcm = aj f(ξ)ψ(Ssk Aaj ξ)e dξ . (46) Z2 j≥0 k∈Kj m∈ C Z2 1 1 2 We now ﬁrst decompose the sum over m . For this, set Ω = 2 , 2 . Then, by appropriate changes of variables, ∈ − 2 3/2 ˆ ˆ T 2πiξ,Aaj Ssk Mcm aj f(ξ)ψ(Ssk Aaj ξ)e dξ Z2 m∈ C −3/2 a 2 j ˆ −1 −T −1 −1 −T −1 ˆ −1 2πiξ,m = f A S M ξ χC A S M ξ ψ (M ξ)e dξ det(M ) aj sk c aj sk c c Z2 c m∈ | |R2 −3/2 a 2 j ˆ −1 −T −1 −1 −T −1 ˆ −1 2πiξ,m = f Aaj Ssk Mc ξ χC Aaj Ssk Mc ξ ψ (Mc ξ)e dξ 2 det(Mc) 2 Ω+ℓ m∈Z | | ℓ∈Z −3/2 aj −1 −T −1 −1 −T −1 = fˆ A S M (ξ + ℓ) χC A S M (ξ + ℓ) det(M ) aj sk c aj sk c Z2 c Ω Z2 m∈ | | ℓ∈ 2 ψˆ (M −1(ξ + ℓ))e2πiξ,m dξ c 36 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

(a) (b)

(e) (f)

Figure 6. Denoising of the Goldhill image (512 512) using shearlets, curvelets and contourlets. (a) Original Goldhill image.× (b) Noisy image, PSNR: 22.1dB. (c) Denoised with curvelets, PSNR: 28.65dB, redundancy: 4.8, computing time: 1.35 sec. (d) Denoised with shearlets (29.17dB, 4, 1.10sec.). (e) Denoised with contourlets, (29.92dB, 53, 384.55 sec.). (f) Denoised with shearlets, (30.15dB, 40, 10.08 sec.). Next, applying Plancherel’s theorem, 2 3/2 ˆ ˆ T 2πiξ,Aaj Ssk Mcm aj f(ξ)ψ(Ssk Aaj ξ)e dξ Z2 m∈ C −3/2 a 2 j ˆ −1 −T −1 −1 −T −1 ˆ −1 = f Aaj Ssk Mc (ξ + ℓ) χC Aaj Ssk Mc (ξ + ℓ) ψ (Mc (ξ + ℓ)) dξ det(Mc) Ω 2 | | ℓ∈Z CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 37

Resolving the absolute values yields,

2 3/2 ˆ ˆ T 2πiξ,Aaj Ssk Mcm aj f(ξ)ψ(Ssk Aaj ξ)e dξ Z2 m∈ C a−3/2 j −1 −T −1 −1 −T −1 −1 = fˆ A S M (ξ + ℓ) χC A S M (ξ + ℓ) ψˆ (M (ξ + ℓ)) det(M ) aj sk c aj sk c c c Ω Z2 | | m,ℓ∈ fˆ A−1S−T M −1(ξ + m) ψˆ M −1(ξ + m) dξ aj sk c c −3/2 a j −1 −T −1 −1 −T −1 −1 = fˆ A S M ξ χC A S M ξ ψˆ (M ξ) det(M ) aj sk c aj sk c c c Z2 Ω+ℓ | | ℓ∈ fˆ A−1S−T M −1(ξ + m ℓ) ψˆ M −1(ξ + m ℓ) dξ aj sk c − c − m∈Z2 a−3/2 j ˆ −1 −T −1 −1 −T −1 ˆ −1 −T −1 = f Aaj Ssk Mc ξ χC Aaj Ssk Mc ξ f Aaj Ssk Mc (ξ + m) det(M ) R2 c Z2 | | m∈ ψˆ (M −1ξ)ψˆ M −1(ξ + m) dξ. c c Combining with (46),

2 1 −1 −T −1 f,ˆ ψˆj,k,m 2 = fˆ(ξ) fˆ ξ + A S M m L (C) det(M ) aj sk c Z2 c Z2 j≥0 k∈Kj m∈ | | C j≥0 k∈Kj m∈ T T −1 ψˆ S Aa ξ ψˆ S Aa ξ + M m dξ sk j sk j c = T1 + T2, (47) where 2 2 1 ˆ ˆ T T1 = f (ξ) ψ Ssk Aaj ξ dξ det(Mc) | | j≥0 k∈Kj C and 1 ˆ ˆ −1 −T −1 T2 = f (ξ) f ξ + A S M m det(M ) aj sk c c j≥0 Z2 | | k∈Kj C m∈\{0} T T −1 ψˆ S Aa ξ ψˆ S Aa ξ + M m dξ. sk j sk j c With the same arguments in [6, Sect. 3.3.2], we can conclude that

1 ˆ 2 −1 −1 1/2 T2 f Γ Mc m Γ Mc m . (48) | | ≤ det(Mc) − | | m∈Z2\{0} 38 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

By (47) and (48), we ﬁnally obtain 2 Linf R(c) 2 Lsup+R(c) 2 − fˆ f,ˆ ψˆ 2 fˆ . det(M ) j,k,m L (C) det(M ) c ≤ Z2 ≤ c | | j≥0 k∈Kj m∈ | | 1 2 The introduction of the lower bound for Linf in the estimate fˆ [Linf R(c)] is | det(Mc)| − immediate. The theorem is proved.

6.1.2. Proof of Proposition 3.3. We start by estimating Γ(2ω1, 2ω2), and will use this later 2 to derive the claimed upper estimate for R(c). For each (ω1,ω2) R 0 , we ﬁrst split the sum over j as ∈ \{ }

Γ(2ω1, 2ω2) ˆ 1/2 ˆ 1/2 = ess sup ψ ajξ1,skajξ1 + aj ξ2 ψ ajξ1 +2ω1,skajξ1 + aj ξ2 +2ω2 ξ∈R2 j≥0 k∈Kj ˆ 1/2 ess sup + ψ ajξ1,skajξ1 + aj ξ2 ≤ ξ∈R2 Z {j:|aj ξ1|<ω∞} {j:|ajξ1|≥ω∞} k∈ ψˆ a ξ +2ω ,s a ξ + a1/2ξ +2ω j 1 1 k j 1 j 2 2 = ess sup(I1 + I2). (49) ξ∈R2

The next step consists in estimating I1 and I2. Before we delve in the estimations, we ﬁrst introduce some useful inequalities which will be used later. Recall that α>γ> 3, and q, q′,r are positive constants satisfying q > q′ > 0, q>r> 0. Further, let γ′′ = γ γ′ for an arbitrarily ﬁxed γ′ satisfying 1 <γ′ <γ 2. Then we have the following inequalities− for x, y, z R. The proofs are all elementary, and− hence we will skip them. ∈ min 1, qx α min 1, q′x −γ min 1, ry −γ { | | } { | | } { | | } min 1, qx α−γ min 1, q′x −γ min 1, (qx)−1ry −γ , (50) ≤ { | | } { | | } | | γ 1+ z ′′ ′′ ′ ′′ min 1, x −γ min 1, 2γ y −γ min 1, x −γ max 1, 1+ z γ , (51) { | | } x + y ≤ | | { | | } { | | } α−γ ′ −γ γ′′ ′ −γ′′ min 1, qx min 1, q x x (q ) , (52) { | | } { | | }| | ≤ and ′′ ′′ ′′ ′ min 1, qx α−γ min 1, q′x −γ x γ (q′)−γ min 1, qx α−γ+γ min 1, q′x −γ . (53) { | | } { | | }| | ≤ { | | } { | | } We start with I1. By applying (6) and (50), I min qa ξ α−γ, 1 min q′a ξ −γ, 1 min q(a ξ +2ω ) α−γ , 1 1 ≤ {| j 1| } {| j 1| } {| j 1 1 | } {j:|aj ξ1|≤ω∞} −γ ′ −γ r ξ2 min q (ajξ1 +2ω1) , 1 min sk + 1/2 , 1 {| | } Z q a ξ1 k∈ j CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 39

−γ γ r 2ω2 ξ2 2ω1 min + sk + 1/2 1+ , 1 . (54) q ajξ1 a ξ1 ajξ1 j We now distinguish two cases, namely ω ∞ = ω1 ajξ1 and ω ∞ = ω2 ajξ1 . | |≥| | | |≥| | Notice that these two cases indeed encompass all possible relations between ω and ξ1.

Case 1: We assume that ω ∞ = ω1 ajξ1 , hence ajξ1 +2ω1 ω1 . This implies −γ | |≥| | | |≥| | q′(a ξ +2ω ) q′ω −γ. Thus, continuing (54), | j 1 1 | ≤ ∞ I min qa ξ α−γ, 1 min q′a ξ −γ, 1 q′(a ξ +2ω ) −γ 1 ≤ {| j 1| } {| j 1| }| j 1 1 | {j:|aj ξ1|≤ω∞} −γ r ξ2 min sk + 1/2 , 1 Z q a ξ1 k∈ j q′ω −γ min qa ξ α−γ, 1 min q′a ξ −γ, 1 ≤ ∞ {| j 1| } {| j 1| } {j:|aj ξ1|≤ω∞} −γ q r r ξ2 min sk + 1/2 , 1 . r Z q q a ξ1 k∈ j By (4) and Lemma 3.2, ﬁnally conclude q r C(γ) α−γ ′ −γ I1 ′ γ min qajξ1 , 1 min q ajξ1 , 1 ≤ q ω ∞ {| | } {| | } j∈Z q r C(γ) q 1 1 p ′ γ log1/ ′ + γ + α−γ . (55) ≤ q ω ∞ q 1 1 − − Case 2: We now assume that ω = ω a ξ . Then (51) applied to (54) yields ∞ | 2|≥| j 1| ′′ I 2γ min qa ξ α−γ, 1 min q′a ξ −γ, 1 min q(a ξ +2ω ) α−γ , 1 1 ≤ {| j 1| } {| j 1| } {| j 1 1 | } {j:|ajξ1|≤ω∞} ′ −γ ′′ r ξ r −γ min q′(a ξ +2ω ) −γ , 1 min s + 2 , 1 {| j 1 1 | }  q k 1/2  q Z aj ξ1 k∈   ′′ ′′ 2ω −γ 2ω γ 2 max 1, 1+ 1 .   a ξ a ξ j 1 j 1 Applying now (4), we obtain q ′ ′′ γ r C(γ ) α−γ ′ −γ α−γ I 2 ′′ min qa ξ , 1 min q a ξ , 1 min q(a ξ +2ω ) , 1 1 ≤ r γ {| j 1| } {| j 1| } {| j 1 1 | } 2 q ω ∞ j≥0 γ′′ ′′ 2ω min q′(a ξ +2ω ) −γ , 1 a ξ γ max 1, 1+ 1 . (56) {| j 1 1 | }| j 1| a ξ j 1 40 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

We next split Case 2 further into the following two subcases. Subcase 2a: If1 1+ 2ω1 , then ≤| aj ξ1 | γ′′ ′′ 2ω ′′ a ξ γ max 1, 1+ 1 a ξ +2ω γ . | j 1| a ξ ≤| j 1 1| j 1 Hence, by Lemma 3.2 and exploring inequality (52), we can conclude from (56) that q ′ r C(γ ) q 1 1 I1 p q′r γ′′ log1/ ′ + γ + α−γ . (57) ≤ ω ∞ q 1 1 q − − Subcase 2b: If1 1+ 2ω1 , then, for all j 0, ≥| aj ξ1 | ≥ ′′ 2ω γ min q(a ξ +2ω ) α−γ , 1 min q′(a ξ +2ω ) −γ , 1 max 1, 1+ 1 1. {| j 1 1 | } {| j 1 1 | } a ξ ≤ j 1 Hence, by exploring inequality (53), we can conclude from (56) that q ′ r C(γ ) q 1 1 I1 p q′r γ′′ log1/ ′ + γ′ + α−γ+γ′′ . (58) ≤ ω ∞ q 1 1 q − − We next estimate I2. First, notice that the inequality (54) still holds for I2 with modified index set for j – we have j : ajξ1 > ω ∞ instead of j : ajξ1 ω ∞ in this case. Therefore, by (4), { | | } { | |≤ } −γ r ξ I min qa ξ α−γ , 1 min q′a ξ −γ, 1 min s + 2 , 1 2 ≤ {| j 1| } {| j 1| } q k 1/2 ∞ Z aj ξ1 {j:|aj ξ1|>ω } k∈ q C(γ) min qa ξ α−γ, 1 min q′a ξ −γ, 1 ≤ r {| j 1| } {| j 1| } {j:|aj ξ1|>ω∞} q C(γ) q′a ξ −γ ≤ r | j 1| {j:|aj ξ1|>ω∞} By Lemma 3.2, this estimation can be finalized to q r C(γ) 1 I2 p ′ γ γ . (59) ≤ q ω ∞ 1 − We are now ready to prove the claimed estimate for R(c) using (49). For this, we define the constants T1, T2, and T3 as in Proposition 3.3. Now define = m Z2 : m > m and ˜ = m Z2 : c−1 m >c−1 m . Q { ∈ | 1| | 2|} Q { ∈ 1 | 1| 2 | 2|} If c−1 m >c−1 m , then, by (55) and (59), 1 | 1| 2 | 2| Γ( M −1m) T m −γ + T m −γ for all m ˜. ± c ≤ 1 ∞ 3 ∞ ∈ Q −1 −1 If c1 m1 c2 m2 with m = 0, then, by (57), (58), and (59), | | ≤ | | ′′ Γ( M −1m) T m −γ + T m −γ for all m ˜c 0 . ± c ≤ 2 ∞ 3 ∞ ∈ Q \{ } CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 41

Therefore, we obtain 1/2 R(c) = Γ(M −1m)Γ( M −1m) c − c m∈Z2\{0}

′′ T m −γ + T m −γ + T m −γ + T m −γ (60) ≤  1 ∞ 3 ∞   2 ∞ 3 ∞  ˜ ˜c m∈Q m∈Q \{0}     Notice that, since ˜ , Q ⊂ Q m −γ m −γ. ≤ ˜ m∈Q m∈Q Also, we have ′′ c ′′ m −γ min 1 , 2 m −γ . ≤ c2 ˜c c m∈Q \{0} m∈Q\{0} Therefore, (60) can be continued by

c ′′ R(c) T m −γ + T m −γ + min 1 , 2 T m −γ . 3 ∞ 1 ∞ c 2 ≤ Z2 2 c m∈\{0} m∈Q m∈Q\{0} −γ To provide an explicit estimate for the upper bound of R(c), we compute m∈Q m ∞ and −γ c m as follows: m∈Q \{0} ∞ ∞ ∞ ∞ m −γ = 2 m −γ +4 m −γ ∞ | 1| | 1| m∈Q m =1 m =1 m =m +1 1 2 12 1 4 1 2 1+ + 1+ . ≤ γ 1 γ 1 γ 2 − − − and ∞ m −γ = m −γ +4 m −γ ∞ ∞ | 1| c m∈Q m =1 m∈Q\{0} 1 1 4 1 6 1+ + 1+ ≤ γ 1 γ 1 γ 2 − − − This completes the proof.

6.2. Proofs of Results from Section 4.1.

2 6.2.1. Proof of Lemma 4.1. First, it is obvious that the function m0 is even. Next, letting 2 | 2 | y = sin (πξ1), the values of the trigonometric polynomial m0(ξ1) can be expressed in the form | | L−1 K 1+ n m (ξ ) 2 = P (y), where P (y)=(1 y)K − yn. | 0 1 | − n n=0 42 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

We compute

P ′(y) L−1 K 1+ n L−1 K 1+ n = (1 y)K−1 K − yn + (1 y) n − yn−1 − − n − n n=0 n=1 L−1 K 1+ n L−1 K + n 1 L−2 K + n = (1 y)K−1 K − yn n − yn + (n + 1) yn − − n − n n +1 n=0 n=1 n=0 K + L 2 K + L 2 = (1 y)K−1 K − yL−1 (L 1) − yL−1 − − L 1 − − L 1 − − K + L 2 = (K + L 1) − yL−1(1 y)K−1. − − L 1 − − ′ 2 Hence, P (y) < 0 for y (0, 1), and this immediately implies that m0(ξ1) is decreasing on 1 ∈ | | (0, 2 ). The second derivative can be derived by P ′′(y) K + L 2 = (L 1)(K + L 1) − yL−2(1 y)K−1 +(K + L 1)(K 1) − − − L 1 − − − − K + L 2 − yL−1(1 y)K−2 L 1 − − K + L 2 K + L 2 L 1 = (K 1)(K + L 1) − − yL−2(1 y)K−2 y − . − − K 1 L 1 − − K + L 2 − − − ′′ L−1 Thus P (y) < 0 for 0

2 1 ∂ 2 1 L 1 2 2 m0(ξ1) < 0 for0 <ξ1 < arcsin − . ∂ξ1 | | π K + L 2 − Since L−1 1 implies K+L−2 ≥ 4 1 1 L 1 2 1 arcsin − , π K + L 2 ≥ 6 − concavity of m 2 on (0, 1 ) is proven. | 0| 6

2 6.2.2. Proof of Lemma 4.3. Letting y = sin (πξ1) for ξ1 [0, 1], the values of the trigonometric polynomial m˜ (ξ ) 2 can be expressed in the form∈ | 0 1 | L−1 ′ K 1+ n m˜ (ξ ) 2 = P˜(y), where P˜(y)=(1 y)K − yn. | 0 1 | − n n=0 If K′ = 0, P˜(y) is increasing on (0, 1), which proves one part of claim (i). CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 43

To prove the remaining part of claim (i), let us assume that K′ > 0. For y (0, 1), direct computation yields ∈ (P˜)′(y) L−2 ′ K 1+ n K L 2 = (1 y)K −1 (K K′) − yn (K′ + L 1) − − yL−1 − − n − − L 1 n=0 − L−2 ′ K 1+ n K L 2 = yL−1(1 y)K −1 (K K′) − yn−L+1 (K′ + L 1) − − . − − n − − L 1 n=0 − ′ Hence, (P˜)′(y) is a product of a decreasing function and of yL−1(1 y)K −1 > 0 on (0, 1). ′ − This implies that P˜(y) is increasing on (0, z0) as long as (P˜) (z0) 0. Obviously, the same 2 1 2 π ≥ ˜ ′ 1 is true for m˜ 0(ξ1) . Thus, since 4 = sin ( 6 ), it suﬃces to show that (P ) ( 4 ) 0. For this, we compute| | ≥

′ L−2 3 K −1 K K′ K 1+ n K + L 2 P˜′( 1 )=(K′ +L 1) 41−L − − 22L−22−2n − . 4 − 4 K′ + L 1 n − L 1 n=0 − − Since K 7 and K+L−2 4, the term in the bracket [ ] can be further estimated by ≥ L−1 ≤ K K′ 1 L−2 K 1+ n K + L 2 22L − − 2−2n − K′ + L 1 4 n − L 1 n=0 − − K K′ 1 K K(K + 1) K(K + 1)(K + 2) K + L 2 22L − 1+ + + − ≥ K′ + L 1 4 4 42 2 43 6 − L 1 − − K K′ K + L 2 22L − − ≥ K′ + L 1 − L 1 − − K K′ (K + L 2) ... K) K K′ =22L − − 22L − 22L−2 0, K′ + L 1 − (L 1)! ≥ K′ + L 1 − ≥ − − − K−K′ 1 where the last inequality follows from ′ . This completes the proof of claim (i). K +L−1 ≥ 4 To prove claim (ii), observe that, for 0 y 1, ≤ ≤ L−1 K 1+ n ′ P˜(y) − max (1 y)K yn | | ≤ n y∈[0,1] | − || | n=0 L−1 ′ K 1+ n K′ K n n = − . n K′ + n K′ + n n=0 Thus claim (ii) is proven. 6.2.3. Proof of Lemma 4.5. Without loss of generality we assume that L is even, since the ‘odd case’ can be proven similarly. Since K′ = 0 and by obvious rules for binomial coeﬃcients, L−1 K 1+ n K−1 K 1+ n K−1 K 1+ n C = − = − − 2 n n − n n=0 n=0 n=L 44 P.KITTIPOOM,G.KUTYNIOK,ANDW.-QLIM

K−1 K−1+n 2K−2 Utilizing the estimate n=0 n < 2 from [6], we have K−1 K 1+ n C < 22K−2 − (61) 2 − n n=L Now let a = K−1+n for n =0,...,L 1, and note that n n − an+1 K + n K + L 1 = − 2, an n +1 ≥ L ≥ which implies

aL + + aK−1 aL + aL+1 + + a 3L −1 ≥ 2 2L−2(2a )+2L−3(2a )+ +2L/2−1(2a ) ≥ 1 3 L−1 2L/2−1(a + + a ). ≥ 0 L−1 Thus, using (61), C < 22K−2 2L/2−1C , 2 − 2 and hence 2K−2−L/2+1 2K−L/2−1 C2 < 2 =2 . Thus, we have max m˜ (ξ ) 2 22K−L/2−1 R 0 1 ξ1∈ | | ≤ and L 2γ =2K log (C ) > +1, − 2 2 2 which is what was claimed.

6.2.4. Proof of Proposition 4.6. By (i) and (ii) in Lemma 4.1, we obtain m (4ξ ) 2 m ( 1 ) 2χ (ξ ). (62) | 1 1 | ≥| 0 6 | [1/12,1/6]∪[−1/12,−1/6] 1 On the other hand, by deﬁnition of m1, L−1 K 1+ n m (ξ ) 2 = (sin(πξ ))2K − (cos(πξ ))2n | 1 1 | 1 n 1 n=0 L−1 ′ ′ K 1+ n = (sin(πξ ))2(K−K )(sin(πξ ))2K − (cos(πξ ))2n 1 1 n 1 n=0 ′ L−1 ′ K n ′ K 1+ n K n πξ 2(K−K ) − ≤ | 1| n K′ + n K′ + n n=0 ′ C πξ 2(K−K ). ≤ 2| 1| Therefore, ′ m (ξ ) 2 min 1,C πξ 2(K−K ) (63) | 1 1 | ≤ { 2| 1| } CONSTRUCTION OF COMPACTLY SUPPORTED SHEARLET FRAMES 45 since m (ξ ) 2 1. Let now | 1 1 | ≤ − 1 J1−1 2 2γ ′ −j − 1/(2(K−K )) ′ 2 2 J1+1 P |h(n)||n| ′ q =4π(C ) , q =2π C m˜ e n , and r =2q , 2 2 0 2π j=0 ′ with C2 being deﬁned in (23). It is easy to check that q>r>q > 0. By Propositions 4.2 and 4.4,(62) and (63), we obtain (25) and (28). ′ 1 In (25), the decay rate γ is given by γ = K K 2 log2(C2). Hence, by Lemma 4.5, in the special case that K′ = 0 and L 10, we− have −γ > L/4+1/2 3. This implies the feasibility condition (6). ≥ ≥

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Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90112, Thailand E-mail address: [email protected] Institute of Mathematics, University of Osnabruck,¨ 49069 Osnabruck,¨ Germany E-mail address: [email protected] Institute of Mathematics, University of Osnabruck,¨ 49069 Osnabruck,¨ Germany E-mail address: [email protected]