Optimally Sparse Image Representations using Shearlets Glenn R. Easley Demetrio Labate Wang–Q Lim System Planning Corporation, Department of Mathematics, Department of Mathematics, 1000 Wilson Blvd North Carolina State University, Washington University, Arlington, VA 22209 USA Raleigh, NC 27695 USA St. Louis, Missouri 63130 USA Abstract— It is now widely acknowledged that traditional has a simple mathematical construction, extends naturally to wavelets are not very effective in dealing with multidimensional higher dimensions and can be associated to a multiresolution signals containing distributed discontinuities. This paper presents analysis [12]. In addition, as we will show in this paper, this a new discrete multiscale directional representation called the Discrete Shearlet Transform. This approach, which is based on approach is amenable to a fast algorithmic implementation and the shearlet transform previously developed by the authors and is very competitive for image denoising. their colaborators, combines the power of multiscale methods The paper is organized as follows. In Section II we introduce with a unique ability to capture the geometry of multidimensional a two-dimensional continuous transform called the Continuous data and is optimally efficient in representing images containing Shearlet Trasform, which is well suited to locate discontinu- edges. Numerical experiments demonstrate that the Discrete Shearlet Transform is very competitive in denoising applications ities along edges. In Section III we show that the Discrete both in terms of performance and computational efficiency. Shearlet Transform, obtained by discretizing the corresponding continuous transform, provides optimal representation for a I. INTRODUCTION large class of two-dimensional signals. In Section IV, we de- The most useful feature of the wavelet transform is the scribe the algorithmic implementation of the Discrete Shearlet ability to deal with signals containing isolated point sin- Transform and in Section V we describe some applications for gularities. This fact, together with the availability of fast image denoising. discrete implementations, explains the spectacular success of II. THE CONTINUOUS SHEARLET TRANSFORM wavelets in a variety of signal processing applications. Indeed, 2 if a one-dimensional signal s(t), smooth away from point An affine family generated by à 2 L (R) is a collection of discontinuities, is approximated using the best M-term wavelet functions of the form: expansion, then the rate of decay of the approximation error, ¡1=2 ¡1 fÃa;t(x) = a Ã(a x ¡ t): a > 0; t 2 Rg: as a function of M, is optimal. In fact, it is significantly better 2 than the corresponding Fourier approximation error [1], [2]. à is called a continuous wavelet if, for all f 2 L (R) Z Z However, despite their optimal approximation properties for 1 1 da one-dimensional signals, traditional wavelet methods do not f(x) = hf; Ãa;ti Ãa;t(x) dt : 0 ¡1 a perform as well with multidimensional data. Indeed, wavelets are very efficient in dealing with isolated point singularities The continuous wavelet transform of f is: only. In higher dimensions, other types of singularities are usu- Wf(a; t) = hf; Ãa;ti; ally present or even dominant. Images, for example, typically contain sharp transitions such as edges. Since edges interact and the discrete wavelet transform is obtained by dicretizing extensively with the elements of the wavelet basis, “many” Wf(a; t) on an appropriate set [2]. wavelet coefficients are needed to accurately represent these The natural way of extending the theory of the continuous objects. A number of recent results have shown that a much wavelet transform to higher dimensions is by considering the more efficient representation of multidimensional data is ob- affine system ¡1=2 ¡1 n tained by better exploiting their geometric regularities. These fÃM;t(x) = j det Mj Ã(M x ¡ t): M 2 G; t 2 R g; various methods include contourlets [3], [4], complex wavelets 2 n [5] brushlets [6], ridgelets [7], curvelets [8], bandelets [9] and where à 2 L (R ) and G is a subset of GLn(R), the group other schemes of “directional wavelets” [10]. of invertible n £ n matrices. Similarly to the one-dimensional 2 n The authors and their collaborators have recently intro- case, à is called a continuous wavelet if, for all f 2 L (R ) Z Z duced the shearlet representation [11], [12], which applies f(x) = hf; ÃM;ti ÃM;t(x) dt d¸(M); the framework of affine systems to capture very efficiently G Rn the geometry of multidimensional signals. As a result, this where ¸(M) is a measure on G, and approach provides optimal approximation properties for a large class of two-dimensional images. The shearlet representation Wf(M; t) = hf; ÃM;ti ^ ^ 1 b ^ 1 1 1 1 is the continuous wavelet transform of f [13]. with Ã1; Ã2 2 C (R), supp Ã1 ½ [¡ 2 ; ¡ 16 ] [ [ 16 ; 2 ] and ^ The traditional way of discretizing the continuous wavelet supp Ã2 ½ [¡1; 1]. In addition, we assume that f 2 L2(Rn) M 2 G; t 2 Rn transform of replaces X 1 with a discrete set Aj; j 2 Z; k 2 Zn. However, starting jÃ^ (2¡2j!)j2 = 1 for j!j ¸ ; (4) 1 8 from the continuous wavelet transform other types of discrete j¸0 transforms can be deduced. Indeed, the shearlet transform, and, for each j ¸ 0, which will be described in the next section, is a obtained by discretizing the 2-dimensional continuous wavelet transform 2Xj ¡1 ^ j 2 in a ‘non-traditional’ way. jÃ2(2 ! ¡ `)j = 1 for j!j · 1: (5) For à 2 L2(R2), consider the 2-dimensional affine system `=¡2j 1 ^(0) ¡ 2 ¡1 2 From these assumptions it follows that the functions à fÃast(x) = j det Masj Ã(Mas x ¡ t): t 2 R ;Mas 2 ¡g; j;`;k (1) (with j ¸ 0, ¡2j · ` · 2j ¡ 1, k 2 Z2) form a tiling of 1 »2 D0 = f(»1;»2): j»1j ¸ ; j j · 1g. This is illustrated in where ¡ is the 2–parameter dilation group 8 »1 µ p ¶ Figure 1(a). In very similar way, we construct a second set a a s + (1) ^(1) ¡ = fMas = p :(a; s) 2 R £ Rg: of discrete shearlets Ãj;`;k(x) such that the set fÃj;`;k : j ¸ 0 a j j 2 0; ¡2 · ` · 2 ¡ 1; k 2 Z g is a tiling of D1 = f(»1;»2): 1 »1 2 2 à j»2j ¸ ; j j · 1g (see Figure 1(a)). Finally, let ' 2 L (R ) We choose such that 8 »2 2 » be such that the set f'k(x) = '(x ¡ k): k 2 Z g is a tight ^ ^ ^ ^ 2 2 1 1 2 _ Ã(») = Ã(»1;»2) = Ã1(»1) Ã2( ); (2) frame for L ([¡ ; ] ) . We deduce the following result. » 16 16 1 Theorem 3.1 ([12]): The collection: ^ 1 where Ã1 is a continuous wavelet for which Ã1 2 C (R) (d) j j 2 ^ f'k;Ãj;`;k : j ¸ 0; ¡2 · ` · 2 ¡ 1; k 2 Z ; d = 0; 1g with supp Ã1 ½ [¡2; 1=2] [ [1=2; 2] and Ã2 is chosen so that ^ 1 ^ ^ Ã2 2 C (R), supp Ã2 ½ [¡1; 1], with Ã2 > 0 on (-1,1), and is a tight frame for L2(R2). kÃ2k = 1. Under these assumptions, à is a continuous wavelet This indicates that the decomposition is invertible and the + 2 [14] and for a 2 R , s 2 R, and t 2 R transformation is numerically well-conditioned. Sf(a; s; t) = hf; Ãasti »2 will be called the continuous shearlet transform of f 2 L2(R). The elements of the affine system, which we shall call con- tinuous shearlets, are oriented waveforms whose orientation is 6» 2j ? controlled by the shear parameter s. They become increasingly elongated at fine scales (as a ! 0). We refer to [14] for more »1 - details. » 22j III. DISCRETE SHEARLET TRANSFORM By sampling the Continuous Shearlet Transform Sf(a; s; t) (a) (b) on an appropriate discrete set we obtain a discrete transform which is able to better deal with distributed discontinuities. Fig. 1. (a) The tiling of the frequency plane induced by the shearlets. The Observe that the matrix Mas can be factored as tiling of D0 is illustrated in solid line, the tiling of D1 is in dashed line. (b) µ p ¶ µ ¶ µ ¶ The frequency support of a shearlet Ãj;`;k satisfies parabolic scaling. a a s 1 s a 0 p = p : 0 a 0 1 0 a Details about this construction can be found in [12] and ` j [15]. Let us summarize here the main mathematical properties Thus, it will be “discretized” as Mj` = B A , where µ ¶ µ ¶ of shearlets: 1 1 4 0 B = ;A = ² Shearlets are well localized. They are compactly sup- 0 1 0 2 ported in the frequency domain and have fast decay in are the shear matrix and the anisotropic dilation matrix, the spatial domain. ^ respectively. Hence, the discrete shearlets are the functions ² Shearlets satisfy parabolic scaling. Each element Ãj;`;k is of the form supported on a pair of trapezoids, each one contained in a j 2j 3j box of size approximately 2 £ 2 (see Figure 1(b)). In (0) 2 ` j Ãj;`;k(x) = 2 Ã(B A x ¡ k); (3) the spatial domain, each Ãj;`;k is essentially supported on a box of size 2¡j £ 2¡2j. Their supports become where µ ¶ increasingly thin as j ! 1. ^(0) ^(0) ^ ^ »2 ² Shearlets exhibit highly directional sensitivity. The ele- à (») = à (»1;»2) = Ã1(»1) Ã2 ; ^ »1 ments Ãj;`;k are oriented along lines with slope given by ¡j j ¡` 2 . As a consequence, the corresponding elements and thus, fd [n1; n2] are the discrete samples of a function ¡j j bj Ãj;`;k are oriented along lines with slope ` 2 .