Efficient Design of a Low Redundant Discrete Shearlet Transform

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

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. forms a tight frame (or Parseval frame) if there exist a pos- By comparing equation (2) to equation (13), it can be 2 itive constant A> 0 so that for all f ∈ L2(R ) the Parse- noted that the scaling function is more or less isotropic. val relationship holds: This behavior resembles the isotropy of the scaling functions in the 2-D discrete wavelet transform (DWT), with 2 (d) 2 the only difference that in the DWT, 2-D scaling functions f, ψj,k,l = A kfk2 (9) l D E are formed by a tensor products of one-dimensional scal- j,k,X,d ing functions, instead of being defined per cone. 2 This relationship implies that any function in L2(R ) can be expanded into a set of functions [31]: 2.4. Shearlets or curvelets? Shearlets are very similar to curvelets in the sense that −1 (d) (d) f = A f, ψj,k,l ψj,k,l (10) both perform a multiscale and multidirectional analysis. l D E j,k,X,d Each basis element has a frequency support that is con- j j For the shearlet functions, equation (9) amounts to a sim- tained in a rectangle of size proportional to 2 × 4 (or 4j ×2j ) in both transforms, which means that the length of ple design constraint for the functions Ψ1(ω) and Ψ2(ω): the frequency support is approximately the squared width −j 2 of the frequency support. This property is called par- Ψ1(4 ω) =1 for |ω|≥ ω0 (11) Xj≥0 abolic scaling, hence the frequency supports become in- creasingly thin as decreases [22,27]. Both transforms 2j −1 j j 2 have very similar asymptotic approximation properties: Ψ2(2 ω − k) =1 for |ω|≤ π (12) for images f(x) that are C2 everywhere except near edges, k=X−2j where f(x) is piecewise C2, the approximation error of a which practically means that the sum of the energies of reconstruction with the N-largest coefficients (fN (x)) in Ψ1(ω) and Ψ2(ω) for respectively scaled frequencies and the shearlet/curvelet expansion is given by [22,26]: shifted frequencies must be one. The Parseval relation- kf − f k2 ≤ B · N −2 (log N)3 , N → ∞ ship holds for the part of the frequencyplane that excludes N 2 the center square (see Figure 2), although this can be triv- with B a constant. Because this is the optimal approxi- ially extended to the complete frequency plane by adding mation rate for this type of functions [26], this property is a bandlimited scaling function [27]: often referred to as optimal sparsity. Still, therearea number of differences between shearlets and curvelets [27]: Φ˜ (ωx) |ωy| ≤ |ωx| <ω0 • Shearlets are generated by applying a family of op- Φ(ωx,ωy)= Φ˜ (ωy) |ωx| ≤ |ωy| <ω0 (13)  erators to a single function, while curvelet basis el- 0 else ements are not in the form of equation (2).  with Φ(˜ ω) a 1-D scaling function that satisfies: • Shearlets are normally associated to a fixed translation lattice, while curvelets are not. This is of im- 2 −j 2 portance for applications: when combining infor- Ψ1(4 ω) + Φ˜ (ω) =1 for |ω| <ω0 (14) mation from multiple scales and orientations (e.g. Xj≥0

to model inter- or intrascale dependencies), curvelet where x denotes the complex conjugate of x. techniques need to take into account that the trans- Even thoughthe scheme is computationallysimple and lation lattice is not fixed. shift-invariant, the redundancy factor is high due to the lack of downsampling operations in the decomposition. • In the constructionof the shearlet tight frame above, More specifically, the redundancy factor is: the number of orientations doubles at every scale, while in the curvelet frame, this number doubles at J every other scale. 1+ 2j+1 =2J+2 − 3, Xj=1 • Shearlets are associated to a multiresolution analysis, while curvelets are not. when choosing 2J+1 orientations for the first scale. For example, using scales gives redundancy factor ! In Perhaps the most primary advantage, that we want to point 3 29 out in this work, is that shearlets allow for a much less re- analogy to the undecimated DWT, we will call this trans- dundant sparse tight frame representation, while offering form the undecimated DST because of the lack of decimations. shift invariance. 3. EXISTING DISCRETE SHEARLET 3.2. Related existing implementations TRANSFORMS Recently, a number of related DST implementations have 3.1. Direct discretization been proposed. Easley et al. [27] propose a discrete implementation with one of the main applications in image The most straightforward way to design the DST is to ap- denoising. In their work, a Laplacian pyramid is followed (d) x by windowing filters in the Pseudo-Polar DFT domain. ply a direct discretization to the shearlet filters ψj,k,l( ). This approach is applied for example in [28]. First, the By including decimations in the Laplacian pyramid, the functions Ψ1(ω), Ψ2(ω) and Φ(˜ ω) are designed according redundancy of the transform is reduced. Because the re- to the constraints in equations (11)-(13) (see [27] for the dundancy factor per scale of the transform increases lin- design details). Next, the shearlet filters can be expressed early with the number of orientations for that scale, the in the discrete time Fourier transform domain as: overall redundancy factor is still high. Finally, we remark that the Laplacian pyramid representation of [32] that is ω (1) ω 3j/2 −j −j y used in [27] is not a tight frame in its standard form, how- Ψj,k,0( ) = 2 Ψ1(4 ωx)Ψ2 2 − k ωx ever a tight frame can be constructed by using orthogonal (2) ω 3j/2 −j −j ωx pyramid filters [33]. Ψj,k,0( ) = 2 Ψ1(4 ωy)Ψ2 2 − k ωy Yi et al. [30] outline a different implementation for edge detection and analysis. In their implementation, there A frequency domain implementation for an image of size is an explicit distinction between horizontal and vertical (1) ω (2) ω N × M follows by sampling Ψj,k,l( ) and Ψj,k,l( ) in shearlets. However, the authors do not take further steps ω 2πn 2πm T to reduce the redundancy, as they choose to stay faithful the points = N M , for m = 0, ..., M − 1 and n = 0, ..., N − 1. Let B (m,n) denote the Dis- to the CST in terms of edge analysis. Further, it is not crete Fourier Transform (DFT) of the input image, then clear which cascade algorithm would be the best to do the the DFTs of the shearlet coefficient subbands can be com- inverse transform of this scheme, as no reconstruction al- puted as: gorithm is proposed. These implementations are designed for specific appli- (d) (d) 2πn 2πm cations in mind and have a redundancy factor that is lower W (m,n)= B(m,n)Ψ 0 , , j,k j,k, N M (or equal) than the redundancy factor of the undecimated DST. The following questions arise: with j =1, ..., J, d =1, 2 and k = −2j, ..., 2j − 1. 1. Can this DST be designed to use in a wide range of The DFT of the scaling coefficients is given by: applications? 2πn 2πm W s(m,n)= B(m,n)Φ 4−J , 4−J , 2. Is it possible to devise a DST that is maximally dec- N M imated, i.e. with redundancy factor as low as possi- where J denotes the number of scales. The DFT of the ble? And what can be said about the shift-variance reconstructed image B˜(m,n) is obtained by applying the of such schemes? appropriate reconstruction filters to the shearlet subbands In the next subsection, we will explain how do design such and by summing the results: a transform. 2πn 2πm B˜(m,n) = W s(m,n)Φ 4−J , 4−J + 4. NEW DESIGN OF THE DISCRETE SHEARLET N M TRANSFORM 2πn 2πm W (d)(m,n)Ψ(d) , As we explainedin Section 2.2, for the CST there is an ex- j,k j,k,0 N M j,k,dX plicit separation of the horizontal cone C1 and the vertical

¢ £

cone . An obvious discrete realization would be to use C

2 ¤ hourglass-shaped ﬁlters. We prefer not to do this, as this ¥ either increases the redundancy factor by 2, or causes angular aliasing when including decimations in the angular ﬁltering.2 The presence of angular aliasing is very cum- bersome in practical applications as it severely degrades

the directional selectivity of the basis functions. Instead,

¦ § ¨

we apply only one angular filtering stage at each scale to directly split up all orientation subbands, which also has ¡ the advantage that the corresponding filterbank is concep- tually more clean. To proceed, we will define shearlet filters in pseudo- polar frequency coordinates(FC). Every shearlet filter will

extract a wedge-shapedregion of the 2-D frequencyplane;

these wedge ﬁlters can be easily described in pseudo-polar FC. Figure 4. Pseudo-polar coordinate system. 4.1. Pseudo-polar coordinate system

We use FC (ωr, ϑ) in a pseudo-polar grid [34], that is con- sense that we use a frequency partitioning into Kj ≥ 2 sistent to a polar grid, in the sense that the pseudo-angleis orientation subbands and a lowpass subband at each scale. in the range ϑ ∈ [−π, π]. The corresponding conversion In our scheme, the specific way of decimating the horizon- from Cartesian coordinates (ωx,ωy) to pseudo-polar FC tal and vertical orientation subbands, is different, as well (ωr, ϑ) is given by: as the pseudo-polar grid for defining the filters and the filters being used. This will allow us to further subsample 2 2 the orientation subbands. 1+max |ωx| , |ωy| v At each scale of our DST, we use a (Kj + 1)-band re- ωr(ωx,ωy)= u (15) u 1+ π−2 cursive decomposition into (bandpass or highpass) ori- t Kj π ωy entation bands and a lowpass band. Let us denote the scal- |ωx|>|ωy| and ωx ≥0 4 ωx ω  ing analysis filters as H( ) and the shearlet analysis filters π ωx  2- |ωx|<|ωy| and ωy ≥0 as ω , with the index of the orientation  4 ωy Gk( ) k =1, ..., Kj   π ωy band. The scaling synthesis filters and shearlet synthesis ϑ(ωx,ωy)=  4+ |ωx|>ωy ≥0 and ωx <0  4 ωx ˜ ω ˜ ω  filters are H( ) and Gk( ), respectively. The analysis π ωy 4 -4+ ω |ωx|>-ωy >0 and ωx <0 and synthesis filter bank is shown in Figure 5. In our filter  x  π ωx bank, the image is first filtered using the oriented shearlet  -2- |ωx|<|ωy| and ωy <0  4 ωy ω  filters Gk( ), subsequently the result is decimated with  (16)  a scale-dependent factor qj , in the direction orthogonal to the main filter orientation (i.e. horizontal or vertical). where we replace the fractions ωx/ωy and ωy/ωx by 0 Next, the filter bank is iterated on the decimated output of whenever the denominator becomes 0. The denomina- the scaling filters, where the decimation factor for the scal- tor in (15) has been chosen such that ωr(±π, ±π) = π. ing step j is denoted by pj . The synthesis filter bank is en- Important to note is that adding a constant to the pseudo- tirely analogous, hence when designing the filters appro- angle ϑ corresponds to a vertical shear transform if both priately, the filter bank can be made to be self-inverting. (ωx,ωy) ∈ C1 and the transformed point also belong to To design the filters, we express the perfect recon- C1. Equivalently, if (ωx,ωy) ∈ C2, adding a constant to struction equations for Figure 5, and try to find filters that ϑ corresponds to a horizontal shear transform if the trans- satisfy these equations. The first perfect reconstruction formed point also belongs to C2. Transiting from C1 to (PR) condition for this filter bank is given by: C2 (or vice versa) can be done using a cascade of a horizontal and vertical shear transform. The pseudopolar grid Kj defined above is illustrated in Figure 4. It can be noted H(ω)H˜ (ω)+ Gk(ω)G˜k(ω)=1, ω ∈ Ω (17) that contours of equal pseudo-radial frequencies ωr define kX=1 concentric squares around the origin, instead of circles as is the case with a polar grid. with Ω = [−π, π] × [−π, π]. For a decimated transform, other PR conditions are needed, to state that the aliasing 4.2. Filter bank caused by the downsampling operations should cancel it- self. We will call these conditions the aliasing canceling The filter bank design that we propose is mostly related to conditions. Note that for some combinations of (p, q) PR the design of the Steerable Pyramid transform [2], in the is not even possible (e.g. (p, q)=(4, 2)), in that case, the 2Such an angular filterbank with decimation is used in e.g. the con- PR conditions are conflicting. tourlet transform [13]. We investigate p = 4 and q = 1 with anisotropic di-

6 7 6 8 9 : ; : : 9 7 < = > : ; :

G ω

? @ A

@ A

( ) ?

BC BD 7 > ω

G˜ 3 ( )

'* + ) ( ,

'* + ) ( , 1 &

X(ω) … 4 X(ω)

Y Z [ \ ] ^

H IJKL

: G :

'* + ) ( ,

'* + ) ( , 5 &

… 4

_ ` a b c d e f

M N OPQRS T

BC BE

7 >

' ( )* + ,

& ' ( )* + ,

… 4

U V WX

V WX

: U :

' ( )* ,

' ( )* , … 4

ω ω

G ( ) G˜ 3 ( )

'* )(,

/'* )(, 0

# # 0

5!" $" %

!" # $"#% … 4 H ω : : H ω

( ) 1 ˜

'* )(,

- 0 '* 0 ) ( , ( )

… 4

/ '()* 0 , ' ( )* 0 , … 4

: :

- ' ( )* 0 ,

'()* 0 ,

… 4

!" . $".%

" . $".% ! …

H(ω) H˜(ω)

Figure 5. Shearlet analysis and synthesis filterbank. lation matrices as in Section 2. The aliasing canceling PR Section 4.1): conditions are: H(ωr, ϑ)= H0(ωr), +∞ (ϑ + iπ)Kj ˜ mπ nπ Gk(ωr, ϑ)= G0(ωr) R − k +1 , H ωx,ωy H ωx+ ,ωy+ =0, (18) π 2 2 i=X−∞ H˜ (ωr, ϑ)= H˜0(ωr), ∞ (ϑ + iπ)K with m = 0, ..., 3, n = 0, ..., 3 and (m,n) 6= 0. No- G˜ (ω , ϑ)= G˜ (ω ) R˜ j − k +1 tably, equation (18) only affects the scaling filter and not k r 0 r π i=X−∞ ω the shearlet filters: the scaling filter H( ) must have fre- (19) π π π π ω quency support [− 4 , 4 ] × [− 4 , 4 ]. Consequently, H( ) cannot have compact support in spatial domain (see [31]). with H0(ωr) the frequency response of a 1-D scaling fil- Nevertheless, because of the lack of aliasing, the trans- ter, G0(ωr) the frequency response of a 1-D wavelet filter form can be made to be shift-invariant, in a similar way and R(ϑ) a real-valued compactly supported bump func- as done for the steerable pyramid transform [1,35]. To do tion. In equation (19), the bump function is periodized so, we define the filters in separable pseudo-polar FC (see in ϑ with period π to construct filters with real-valued impulse responses. In practice we can assume that ϑ ∈ 1 [−π, π], by the construction of the pseudo-polar grid (see 0.5 Section 4.1). Consequently, the summation in (19) only v(x) needs to iterate over a finite number of values for i. 0 Using the above filters, the PR conditions come down to: −0.2 0 0.2 0.4 0.6 0.8 1 1.2 x H0(ωr)H˜0(ωr)+ G0(ωr)G˜0(ωr)=1, ωr ∈ [−π, π] K +∞ Figure 6. The interpolation function v(x). (ϑ + iπ)K R j − k +1 · 2π Xk=1 i=X−∞ scale j pj qj (ϑ + iπ)K 1 2 1 R˜ j − k +1 =1, ϑ ∈ [−π, π] 2π 2 4 1 (20) 3 4 2 π 4 4 4 H (ω )= H˜ (ω )=0, |ω | > 0 r 0 r r 4 5 4 4 (21) Table 1. Proposed decimation factors for the DST with In Figure 7a, an example of radial filters satisfying these anisotropic dilation (also see Figure 5). equationsis shown. It can be seen thatthe scaling filter has a band center frequency ∼ π/8, as a result the frequency Similarly, angular filters satisfying (21) are given by: resolution of this DST may be rather poor (for the second scale, this frequency becomes ; hence much of 1+α ∼ π/32 0 x< − 2 the frequency content of the image is contained in the first sin π v α+2x+1 x + 1 ≤ α scale). Therefore, we propose to replace (21) by a less  2 2α 2 2 R(x)= R˜(x)=  1 −α 1 | x| < 2 strong condition:  cos π v α+2x−1 x − 1 ≤ α π 2 2α 2 2 H0(ωr)= H˜0(ωr)=0, |ωr| > 0 else 2   1  and modify the decimation operations appropriately, such with α ∈ 0, 2 a constant parameter that determines the that there is no information loss and hence PR is still pos- bandwidth of the angular filters. In Figure 7c, R(x) is de- sible. For the first scale, we set p1 = 2, q1 = 1 and picted for different values of α. Higher values of α corre- starting from the second scale (j > 1), we use pj = 4 spond to a slower decay of the transition bandwidth. The ˜ π 1 and qj = 2. In Table 1, the decimation factors pj and corresponding filters Gk(ωr, ϑ) for ωr = 2 and α = 2 qj are listed per scale. The modified Meyer wavelet and are shown in Figure 7d. The choice of α has an influence scaling filters with adjusted frequency scaling is shown in on the redundancy factor of the DST. We will go deeper Figure 7b. The use of the Meyer wavelet here is an ap- into this in Section 4.4. pealing choice due to its excellent localization properties 4.3. Folding and angular decimation in both time and frequency and also because the filters are defined directly in frequency domain [31]: By the compact support of R(x), the filters Gk(ωr, ϑ) are supported on trapezoidal wedges in the frequency plane. π 1 |ωr| < 4 Outside these wedges, the filtered DFT coefficients are 0.  π 4|ω| π π In case of more than two orientations ( ), we can H0(ωr)= cos 2 v π − 1 4 ≤ |ωr|≤ 2 , K > 2  partially get rid of the extra redundancy in two different 0 else ways (see Figure 8):  0 |ω | < π r 4 •(Folding) Shear the filtered subbands such that the  π 4|ω| π π G0(ωr)= sin 2 v π − 1 4 ≤ |ωr|≤ 2 , frequency support is fully contained in the central  1 else rectangles as shown in Figure 8a. Subsequently, a  vertical decimation can be applied to the subbands H˜0(ωr)= H0(ωr), in cone C1 and a horizontal decimation to the sub- G˜0(ωr)= G0(ωr) bands in cone C2. Note that a suitable (possibly non-integer) decimation factor needs to be chosen, where we use the following interpolation function (Fig- we will explain this further on. For the shear trans- ure 6): form, we rely on bandlimitedinterpolation (most ef- 2 3 3x − 2x 0 ≤ x ≤ 1 ficiently implemented in the DFT domain). For the v(x)= 0 x< 0 (22) exact details of the shear transform implementation,  1 1 < x we refer to [36]. for x ∈ [0, 1]. This function is chosen such that it satisfies •(Non-folding) Perfect reconstruction is possible even v(x)=1−v(1−x) (see [31]), while being a C3 function. without folding. Therefore we need to make sure

x u | } ~ y z y

t u v wx y zy {

i m n o j p i q r r s

g h i j k l g j o i g h i j k

(a)

¡ ¢ £ ¤

(b)

Figure 8. Strategies to reduce the redundancy of the DST. (a) Perfect reconstruction by shear operations and decimating (Folding), (b) Perfect reconstruction by decimating without shearing (Non-folding). See text also.

that the aliasing caused by the decimations does not number of α 1 1 1 contaminate the content of the wedges of interest. scales J 32 8 2 This can again be done by chosing the decimation 1 2.19 2.56 4.06 factor suitably (actually the same as in the fold- 2 2.66 3.13 5.00 ing strategy). This approach is illustrated in Fig- 3 2.67 3.14 5.03 ure 8b. Even though many aliasing copies are pro- 4 2.67 3.15 5.03 duced during decimation, the original wedges can 5 2.67 3.15 5.03 be perfectly reconstructed after applying the reconstruction filter G˜k(ωr, ϑ). Table 2. Redundancy factors for J scales, computed using equation (24). Both schemes have the same redundancy factor. The dif- ferenceis that in the folding strategy, the translation lattice ˜ is sheared, while without folding, the translation lattice FFTs. Because the filters H0(ωr), G0(ωr), H0(ωr) and ˜ remains Cartesian, which can be an advantage in certain G0(ωr) are bandlimited, the filters do not have compact applications. Additionally, the non-folding strategy heav- support in spatial domain. Nevertheless, it is possible to ily relies on aliasing and it is easy to show that the non- approximate the impulse response by truncation, as pro- folding strategy is not shift-invariant, whereas the folding posed e.g. in [37] for the steerable pyramid filters. strategy is shift-invariant. For a squared subband at scale j of size Nj, we com- 4.4. Computation of the redundancy factor pute the decimation factor from Figure 8 as follows: The redundancy factor for our scheme is given by the re- Nj cursive formula: dj = max 1, (23) d(1+2α)Nj / (Kj/2)e 2 1 2 1 2 1 2 In our implementation, the folding is performed in the R ≈ + 2 + 2 + 2 + · · · q1 p1 q2 p2 q3 p3 q4 DFT domain; Nj /dj then determines the integer number of DFT coefficients to keep per row or column. For this (24) −2 ÈJ p reason a ceiling function is present in equation (23). · (1+2α)+2 j=1 j . More importantly, because we have Kj orientation subbands per scale, we see that the redundancy for scale j of with pj and qj as listed in Table 1 and where the approx- the transform, that is proportional to Kj /dj ≈ 2(1+2α), imation sign is due to neglecting the ceiling operation in 3 becomes independent of Kj! equation (23). Each fraction 2/qj corresponds to shearlet Because the filters are defined in frequency domain, 2 subbands for scale j, while the fractions 1/pj are related our current implementation of this filterbank makes use of to the decimations during the scaling steps. In Table 2, re- 3Up to small deviations caused by the ceiling operation, but this is dundancy factors of the transform are given with respect usually neglectible. to the number of scales J and the parameter α. 1 redundancy to spatial locatization. H (ω ) 0 r In Figure 12, the DST subband decomposition of the G (ω ) 0.5 0 r “blackman” test image is shown. Here, we used the folding strategy to reduce the redundancy of the transform. In the magnified part of one of the subbands, it can be noted 0 Magnitude response 0 0.25 0.5 0.75 1 that the DST coefficients are not subject to spurious os- Normalized frequency cillations (aliasing) as for example shown in Figure 12d. (a) These spurious oscillations are often present in subbands 1 of shift-variant transforms and are very disturbing as the H (ω ) 0 r local energy signature of edges depends on the exact edge G (ω ) position. 0.5 0 r Another interesting experimentis the trade-off between spatial localization and the numberof orientation subbands 0 Magnitude response 0 0.25 0.5 0.75 1 while keeping α = 1/2. For our DST implementation, Normalized frequency this also means that the redundancyfactor of the transform (b) remains constant. In Figure 13, we reconstruct the zone 1 plate image from the DST coefficients from one orientation subband from the first (finest) scale of the transform, α=1 while increasing the number of orientations K for the 0.5 α=1/2 1 R(x) α=1/4 first scale. We observe that for K1 ≥ 16, edges with ori- α=1/16 entation around 135º are still well detected, however, their 0 −1 −0.5 0 0.5 1 position becomes less certain, as the responses are more x spread over the entire image. This shows that the spatial (c) localization properties of the shearlet basis elements are 1 heavily reduced in this case. As a final example, we investigate the approximation quality of different multiresolution transforms. We start 0.5|G (π/2,θ)||G (π/2,θ)| |G (π/2,θ)| |G (π/2,θ)| 1 2 3 4 from a test image, apply a given wavelet or shearlet transform to this image and we reconstruct the image from the 0 Magnitude response 0 0.5 1 1.5 2 2.5 3 2.5% largest wavelet or shearlet coefficients (in magni- θ tude). In Figure 14, the results are given for the zone plate (d) image and the barbara image, for the decimated DWT, the undecimated DWT, the dual-tree complex wavelet trans- Figure 7. (a) Shearlet radial magnitude responses for di- form (DT-CWT) and the DST. We also list the redundancy lation factor 4 (using the Meyer wavelet), (b) Shearlet fil- factor of each transform in the figure, because this factor ter radial magnitude responses with proposed adjustment plays a big role here. The DWT is a shift variant trans- to increase the lowpass center band frequency (using the form, and the aliasing creates disturbing artifacts in the Meyer wavelet), (c) Angular response R(x), (d) Shear- end result. The undecimated DWT has the largest number let filter magnitude responses for the constant radial fre- of coefficients retained in absolute terms, however the ba- quency ωr = π/2. sis functions of this transform corresponding to the HHj subbands have a poor directional selectivity, which causes here the blurring of some of the edges. The DT-CWT 5. RESULTS AND DISCUSSION basis functions have an excellent spatial localization, but are only able to distinguish 6 orientations, also causing a First, we will look closer at some of the properties of our fair amount of blurring here (see Figure 14c/g). The DST DST implementation. In Figure 9, shearlet basis func- gives here the best visual result, mainly because of its ex- tions are shown for different scales and orientations. Even cellent directional selectivity and shift-invariance. though the size of the support of these basis functions is not finite, these functions have a fast decay and are 6. CONCLUSION well localized in space, frequency and orientation. In Fig- ure 10, the frequency response of one shearlet function In this paper, we described a novel design of the Discrete is depicted. This function has a very compact support in Shearlet Transform with a redundancy factor that is very frequency domain and a clear orientation. In Figure 11, low and independent of the number of orientation sub- the influence of the parameter α on the spatial localiza- bands, while offering shift-invariance. The filters of this tion of the shearlet is illustrated: the smaller α, the larger transform are designed in pseudo-polar frequency coor- the side-lobes of the basis function, hence the worse the dinates, based on the Meyer wavelet. A special decima- spatial localization. As mentioned before, the redundancy tion scheme is applied to the filtered subbands, to further of our DST is directly related to α (values are given in Ta- reduce the redundancy. This results in a multiresolution ble 2), so we can conclude that α allows us to trade off transform, that allows to easily make a trade-off in spatial

¥ ¦ § ¨ © ª

¥ ¦ § ¨ © «

π Figure 9. Shearlets basis elements for α = 2 . For illustration purposes, we used K1 = 16 (instead of 32) orientations for the ﬁrst scale.

(a)

(c)

(d) (b)

Figure 12. (a) The “blackman” image. (b) DST decomposition (with folding) of the “blackman” image into three scales. Eight orientations are used for the ﬁrst scale. Note that the subbands for the ﬁnest scale of cone C2 have been rotated 90º. (c) Crop out of one subband, to illustrate the lack of aliasing. (d) Aliasing in a subband of the DST (implemented without folding) of the image in (a). (a) (b) K1 =2 (c) K1 =4

(d) K1 =8 (e) K1 = 16 (f) K1 = 32

Figure 13. Illustration of the directional selectivity of the DST (a) the zone plate image (b)-(f) reconstruction from one orientation subband of a DST with one scale (J =1), and for an increasing number of orientations (K1).

(a)DWT(1) (b)UDWT(7) (c)DT-CWT(4) (d)DST(4.06)

(e)DWT(1) (f)UDWT(7) (g)DT-CWT(4) (h)DST(4.06)

Figure 14. Reconstruction from 2.5% of the x-let coefﬁcients, for (a),(e) DWT with 2 scales, (b),(f) Undecimated DWT with 2 scales, (c),(g) dual-tree complex wavelet transform [6] with 2 scales, (d),(h) DST with 1 scale (because of anisotropic dilation with factor 4). Top row: crop out of the Barbara image, bottom row: crop out of the zone plate image (see Figure 13a). Between parentheses is the redundancy factor of each transform in this experiment. [3] N. G. Kingsbury, “Image processing with complex wavelets,” Philos. Trans. R. Soc. London A., Math 1 Phys. Sci., vol. 357, no. 1760, pp. 2543–2560, 1999. [4] N. G. Kingsbury, “Complex Wavelets for Shift In- 0.5 variant Analysis and Filtering of Signals,” Journal of Applied and Computational Harmonic Analysis, 0 vol. 10, pp. 234–253, May 2001. −1 [5] F. C. A. Fernandes, R. L. C. Van Spaendonck, and C. S. Burrus, “A new framework for complex 0 1 wavelet transforms,” IEEE Trans. Signal Processing, 0 vol. 51, no. 7, pp. 1825–1837, 2003.

ω /π 1 x −1 ω /π y [6] I. W. Selesnick, R. G. Baraniuk, and N. G. Kings- bury, “The Dual-Tree Complex Wavelet Transform,” Figure 10. Magnitude response of a shearlet basis ele- IEEE Signal Processing Magazine, vol. 22, pp. 123– ment. 151, Nov. 2005.

α=1/2 α=1/4 α=1/8 α=1/12 [7] C. Chaux, L. Duval, and J.-C. Pesquet, “Image analysis using a dual-tree m-band wavelet transform,” IEEE Trans. Image Processing, vol. 15, pp. 2397–2412, Aug. 2006.

[8] A. A. Barath and J. Ng, “A steerable complex wavelet construction and its application to image denoising,” IEEE Trans. Image Processing, vol. 14, no. 7, pp. 948–959, 2005.

Figure 11. Shearlets for different values of the parameter [9] D. Van De Ville and M. Unser, “Complex Wavelet α, showing the impact of the angular bandwidth on the Bases, Steerability, and the Marr-like pyramid,” spatial support of the shearlets. IEEE Trans. Image Processing, vol. 17, pp. 2063– 2080, Nov. 2008. localization, frequency localization and directional selec- [10] T. Lee, “Image Representation Using 2D Gabor tivity, while keeping the redundancy factor more or less Wavelets,” IEEE Trans. Pattern Analysis and Ma- constant. There are a number of interesting open issues chine Intelligence, vol. 18, p. 1, Oct. 1996. remaining. First is how the idea of scale and orientation shiftability from [1] can be applied to this transform, with- [11] S. Fischer, F. Šroubek, L. Perrinet, R. Redondo, out sacrificing redundancy. A second issue is that the re- and G. Cristobal, “Self-Invertible 2D Log-Gabor dundancy of the transform can possibly be decreased fur- Wavelets,” International Journal of Computer Vi- ther, while allowing certain forms of aliasing (hence being sion, vol. 75, pp. 231–246, Nov. 2007. not completely shift-invariant). Finally, another open is- [12] M. N. Do and M. Vetterli, Beyond Wavelets, chapter sue is the design of non-separable or separable shearlet Contourlets. Academic Press, 2003. filters with compact support in spatial domain, while stay- ing faithful to the CST. [13] M. N. Do and M. Vetterli, “The contourlettransform: An efficient directional multiresolution image rep- Acknowledgements: The authors would like to thank resentation,” IEEE Trans. Image Process., vol. 14, Prof. Ingrid Daubechies for the fruitful and inspiring dis- no. 12, pp. 2091–2106, 2005. cussions that initiated this work. [14] A. da Cunha, J. Zhou, and M. Do, “The nonsubsam- 7. REFERENCES pled contourlet transform: theory, design, and applications.,” IEEE Trans. Image Processing, vol. 15, [1] W. Freeman and E. Adelson, “Design and use pp. 3089–3101, Oct 2006. of steerable filters,” IEEE Transactions on Pattern analysis and Machine Intelligence, vol. 13, no. 9, [15] E. Candès, Ridgelets: Theory and Applications. PhD pp. 891–906, 1991. thesis, Departement of Statistics, Stanford Univer- sity, 1998. [2] E. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger, “Shiftable Multi-scale Transforms,” [16] M. N. Do and M. Vetterli, “The finite ridgelet trans- IEEE Trans. Information Theory, vol. 38, no. 2, form for image representation,” IEEE Trans. Image pp. 587–607, 1992. Processing, vol. 12, pp. 16–28, Jan. 2003. [17] D. L. Donoho, “Wedgelets: Nearly minimax estima- [30] S. Yi, D. Labate, G. Easley, and H. Krim, “A Shear- tion of edges,” Annals of Statistics, vol. 27, no. 3, let Approach to Edge Analysis and Detection,” IEEE pp. 859–897, 1999. Trans. Image Processing, vol. 18, pp. 929–941, May 2009. [18] J. Romberg, M. Wakin, and R. Baraniuk, “Multi- scale wedgelet image analysis: fast decompositions [31] I. Daubechies, Ten Lectures on Wavelets. Philadel- and modeling,” in Proc. International Conference on phia: Society for Industrial and Applied Mathemat- Image Processing 2002, vol. 3, pp. 585–588, 24–28 ics, 1992. June 2002. [32] P. J. Burt and E. H. Adelson, “The Laplacian pyra- [19] E. LePennec and S. Mallat, “Sparse geometricimage mid as a compact image code,” IEEE Trans. Com- representation with bandelets,” IEEE Trans. Image mun., vol. COM-31, no. 4, pp. 532–540, 1983. Processing, vol. 14, no. 4, pp. 423–438, 2005. [33] M. N. Do and M. Vetterli, “Frame reconstruction of the Laplacian pyramid,” in IEEE Conf. on Acoustics, [20] R. R. Coifman and F. G. Meyer, “Brushlets: a tool Speech and Signal Processing, vol. 6, pp. 3641– for directional image analysis and image compres- 3644, 2001. sion,” Appl. Comp. Harmonic Anal., vol. 5, pp. 147– 187, 1997. [34] A. Averbuch, R. Coifman, D. Donoho, M. Elad, and M. Israeli, “Fast and Acurrate Polar Fourier Trans- [21] J. L. Starck, E. J. Candès, and D. L. Donoho, form,” Journal on Appl. and Comp. Harm. Analysis, “The curvelet transform for image denoising,” IEEE vol. 21, pp. 145–167, 2006. Trans. on Image Process., vol. 11, pp. 670–684, 2000. [35] E. P. Simoncelli and W. T. Freeman, “The Steer- able Pyramid: A flexible architecture for Multi-scale [22] E. Candès, L. Demanet, D. Donoho, and L. Ying, Derivative Computation,” in Proc IEEE Int.Conf. “Fast Discrete Curvelet Transforms,” Multiscale Image Processing, (Washington, DC.), Oct. 1995. modeling and simulation, vol. 5, no. 3, pp. 861–899, 2006. [36] L. Condat, D. Van De Ville, and B. Forster-Heinlein, “Reversible, Fast, and High-Quality Grid Conver- [23] R. A. Gopinath, “The Phaselet Transform - An In- sions,” IEEE Trans. Image Processing, vol. 17, tegral Redundancy Nearly Shift-Invariant Wavelet pp. 679–693, May 2008. Transform,” IEEE Trans. Signal Processing, vol. 51, no. 7, pp. 1792–1805, 2003. [37] K. Castleman, M. Schulze, and Q. Wu, “Simplified design of steerable pyramid filters,” in Proceedings [24] V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, of the 1998 IEEE Int. Symposium on Circuits and and P. Dragotti, “Directionlets: anisotropic multi- Systems (ISCAS), vol. 5, pp. 329–332, May 1998. directional representation with separable filtering,” IEEE Trans. Image Processing, vol. 15, no. 7, pp. 1916–1933, 2006.

[25] Y. M. Lu and M. N. Do, “Multidimensional Direc- tional Filter Banks and Surfacelets,” IEEE Trans. Im- age Processing, vol. 16, no. 4, pp. 918–931, 2007.

[26] K. Guo and D. Labate, “Optimally Sparse Multidi- mensional Representation using Shearlets,” SIAM J Math. Anal., vol. 39, pp. 298–318, 2007.

[27] G. Easley, D. Labate, and W. Lim, “Sparse Di- rectional Image Representations using the Discrete Shearlet Transform,” Applied and Computational Harmonic Analysis, vol. 25, pp. 25–46, 2008.

[28] G. Easley, D. Labate, and F. Colonna, “Shearlet Based Total Variation for Denoising,” IEEE Trans. Image Process., vol. 18, no. 2, pp. 260–268, 2009.

[29] K. Guo, D. Labate, and W. Lim, “Edge Analysis and Identiﬁcation using the Continuous Shearlet Trans- form,” Applied and Computational Harmonic Analy- sis, vol. 27, pp. 24–46, July 2009.