EFFICIENT DESIGN OF A LOW REDUNDANT DISCRETE SHEARLET TRANSFORM Bart Goossens, Jan Aelterman, Hiêp Luong, Aleksandra Pižurica and Wilfried Philips Ghent University, Department of Telecommunications and Information Processing (TELIN-IPI-IBBT) Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium [email protected] ABSTRACT Shearlet Transform (DST), and we show that a design is possible with a redundancy factor as low as 2.6, while the Recently, there has been a huge interest in multiresolution number of orientation subbandscan be arbitrarily high and representations that also perform a multidirectional analy- while offering highly desirable properties such as shift in- sis. The Shearlet transform provides both a multiresolu- variance and self-invertibility of the transform. This com- tion analysis (such as the wavelet transform), and at the bination of properties is very difficult to achieve with ex- same time an optimally sparse image-independent repre- isting x-let transforms. Further, we put special attention sentation for images containing edges. Existing discrete on design choices and we point out possibilities for fur- implementations of the Shearlet transform have mainly fo- ther improvement. Results are given to demonstrate the cused on specific applications, such as edge detection or improved characteristics of this transform. denoising, and were not designed with a low redundancy The remainder of this paper is as follows: in Sec- in mind (the redundancy factor is typically larger than the tion 2 we give an overview the Continuous Shearlet Trans- number of orientation subbands in the finest scale). In this form (CST) and its properties that are of importance to paper, we present a novel design of a Discrete Shearlet our work. In Section 3 we explain a number of existing Transform, that can have a redundancy factor of 2.6, inde- DST implementations and their advantages or drawbacks. pendent of the number of orientation subbands, and that The novel DST implementation is presented in Section 4. has many interesting properties, such as shift-invariance Results and a discussion are given in Section 5. Finally, and self-invertability. This transform can be used in a Section 6 concludes this paper. wide range of applications. Experiments are provided to show the improved characteristics of the transform. 2. THE CONTINUOUS SHEARLET TRANSFORM 1. INTRODUCTION 2.1. Shearlet basis functions It is well known that while traditional separable multi- The CST is a multiresolution transform with basis func- dimensional wavelets are efficient for approximating im- tions well localized in space, frequency and orientation. ages with point-wise singularities, they are not very ef- Let ψj,k,l(x) denote the shearlet basis functions (or in the fective for line singularities. For this reason, there has remainder simply called shearlets), then the CST of an im- 2 recently been a lot of interest in multiresolution represen- age f(x) ∈ L2(R ) is defined by [29,30]: tations that have basis elements with a much better di- l x l x x rectional selectivity, i.e. that also perform a multidirec- [SHψf] (j, k, )= f( )ψj,k,l( − )d (1) ˆR2 tional analysis. To name a few: steerable pyramids [1, 2 2], dual-tree complex wavelets [3–7], steerable complex where j ∈ R, k ∈ R and l ∈ R denote the scale, orien- wavelets [8], Marr-like wavelet pyramids [9], 2-D (log) tation and the spatial location, respectively. The idea be- Gabor transforms [10,11], contourlets [12–14], ridgelets hind the Continuous Shearlet Transform (CST) is to com- [15,16], wedgelets [17,18], bandelets [19], brushlets [20], bine geometry and multiscale analysis [27]. Shearlets are curvelets [21, 22], phaselets [23], directionlets [24] and formed by dilating, shearing and translating a mother shear- R2 surfacelets [25]. let function ψ ∈ L2( ), as follows: One of the most recent siblings in this family of repre- j/2 k j ψ l(x)= |det A| ψ B A x − l (2) sentations is the Shearlet transform [26–30], that provides j,k, the mathematical rigidness of a traditional multiresolution where A and B are invertible 2×2 matrices, with det B = j/2 analysis (such as the wavelet transform), and that is at the 1. The normalization factor |det A| has been chosen l same time an optimally sparse, image-independent rep- such that the norm kψk2 = kψj,k,lk2 for all j, k, . The resentation for images containing edges. In this work, basis functions are subject to a composite dilation Aj and we further investigate the implementation of the Discrete geometrical transform Bk. For the shearlet analysis, we will use the following transform matrices: B. Goossens is supported in part by Special Research Fund (BOF) 4 0 1 1 of Ghent University, Belgium. A. Pižurica is a postdoctoral researcher A = and B = . (3) of the Fund for the Scientific Research in Flanders (FWO) Belgium. 0 2 0 1 ω y C2 ω ωx minC (a) (b) C1 C3 1 Figure 1. Geometric transformations used by the Shearlet transform (a) anisotropic dilation (matrix A). (b) shear C2 (matrix B). Here, A is an anisotropic scaling matrix (in the x-direction, the scaling is twice the scaling in the y-direction) and B is a geometric shear matrix. These transforms are illustrated Figure 2. Partitioning of the 2-D frequency plane into two in Figure 1. cones (C1 and C2) and a square (C3) at the origin. The shearlet mother function is a composite wavelet that satisfies appropriate admissibility conditions [29], and more equal treatment of the horizontal and vertical direc- that is defined in the Fourier transform domain as: tions, the frequency plane is split into two cones (for the high frequency band) and a square at the origin (for the ωy Ψ(ω)=Ψ1 (ωx)Ψ2 (4) low frequency band), as shown in Figure 2 [26]: ωx 2 C1 = (ωx,ωy) ∈ R | |ωx|≥ ω0, |ωy| ≤ |ωx| with ω = [ωx ωy], Ψ1(ωx) the Fourier transform of a R2 wavelet function and Ψ2(ωy) a compactly supported bump C2 = (ωx,ωy) ∈ | |ωy|≥ ω0, |ωy| > |ωx| function: 2 C3 = (ωx,ωy) ∈ R | |ωx| <ω0, |ωy| <ω0 Ψ2(ωy)=0 ⇔ ωy ∈/ [−1, 1]. (5) with ω0 the maximal frequency of the the center square Note that by this condition, the mother shearlet function is C3. This square is added to be able to construct a shearlet bandlimited in a diagonal band of the 2-D frequency spec- tight frame [26,27]. To treat horizontal and vertical fre- trum. Because the basis functions are obtained through quencies equally, in cone C2, the x- and y-componentsfor shears and dilations of the mother shearlet function, this x need to be switched before applying geometric trans- bandlimited property also directly controls the directional forms. This comes down to using the following dilation sensitivity of the basis functions. To see this, let us inves- and shear matrices in both cones: tigate the effect of a shear operationon the mother shearlet 4 0 1 1 function. For the shear transform in (3), we have: 1 A = , B = 1 0 2 1 0 1 Bk ωy Ψ ω =Ψ1 (ωx)Ψ2 k − (6) A 2 0 B 1 0 ω 2 = , 2 = . x 0 4 1 1 which means that a shear operation results in a shift in Consequently, the horizontal cone is dilated horizontally the argument of Ψ (ω /ω ), hence the orientation of the 2 y x by factor 4 per scale, while the vertical cone is dilated basis function is controlled by the parameter k (see Fig- vertically by factor 4. In the following, we make the dis- ure 3b). Similarly, the anisotropic scaling leads to: tinction between both cones explicit by assigning different j −j −j ωy shearlet basis functions to each cone d =1, 2: Ψ A ω =Ψ1 4 ωx Ψ2 2 . (7) ωx (d) x A j/2 BkAj x l ψj,k,l( )= |det d| ψ d d − (8) Here we see that changing the scale parameter j results in a scaling in the argument of the wavelet Ψ1, but it also Analogously to the wavelet transform [31], it is natural to affects the support of the bump function Ψ2. More con- discretize the scale, orientation and position indices. In cretely, when the scale parameter is increased by , the 1 the remainder, we will therefore restrict j, k, l to discrete bandwidth of the shearlet is halved (hence the shearlet has (integer) values. The resulting frequency tiling is illus- a finer directional selectivity). trated in Figure 3a. 2.2. Shearlets on the cone 2.3. Tight frames of shearlets So far, we considered shear operations in the vertical di- Next, we want to represent an arbitrary function f ∈ rection and anisotropic dilation, with a larger scaling fac- 2 L2(R ) by a set of projections of this function onto the tor in the x-direction than in the y-direction. To obtain a (d) shearlet basis elements, f, ψj,k,l . The family of func- 1Here, we rely on the fact that the Fourier transform D E of a geometrically transformed function f(Ax) is given by tions ¡ −1 |det A| F {f} A−T ω , with F {f} the Fourier transform of (1) x (2) x Z Z l Z2 f. ψj,k,l( ), ψj,k,l( )|j ∈ , k ∈ , ∈ , j ≥ 0 n o ω ω y y ω y 2j Ψ2 ω x 4j ω y ω ω Ψ2 ω + k x x x Ψ1(ωx) (a) (b) Figure 3. (a) Frequency tiling of the Shearlet transform in trapezoidal shaped tiles (wedges) [26]. (b) Individual compo- nents Ψ1(ωx) and Ψ2(ωy/ωx) of the Fourier transform of the shearlet mother function and the selection of orientations by the parameter k.