Wavelet Steerability and the Higher-Order Riesz Transform Michael Unser, Fellow, IEEE, and Dimitri Van De Ville, Member, IEEE

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 1 Wavelet steerability and the higher-order Riesz transform Michael Unser, Fellow, IEEE, and Dimitri Van De Ville, Member, IEEE

Abstract— Our main goal in this paper is to set the founda- particular, Simoncelli and co-workers were able to design a tions of a general continuous-domain framework for designing complete family of directional multi-scale transforms which is steerable, reversible signal transformations (a.k.a. frames) in indexed by the number of orientation bands k [9]–[11]. Their multiple dimensions (d ≥ 2). To that end, we introduce a self-reversible, Nth-order extension of the Riesz transform. We steerable pyramid is over-complete by a factor 4k/3 + 1 and prove that this generalized transform has the following remark- has the important property of being self-inverting. Steerable able properties: shift-invariance, scale-invariance, inner-product wavelets have also been proposed in the redundant frame- preservation, and steerability. The pleasing consequence is that work of the continuous wavelet transform, including some the transform maps any primary wavelet frame (or basis) of d constructions on the sphere [12]. While steerability is attractive L2(R ) into another “steerable” wavelet frame, while preserving the frame bounds. The concept provides a functional counterpart conceptually, it is obviously not a strict requirement; other to Simoncelli’s steerable pyramid whose construction was primar- popular solutions to the problem of directional multiresolu- ily based on filterbank design. The proposed mechanism allows tion image representations include the 2-D Gabor transform for the specification of wavelets with any order of steerability in [13], curvelets [14], the dual-tree complex wavelet transform any number of dimensions; it also yields a perfect reconstruction [15], [16], directional wavelet frames [17], contourlets [18], filterbank algorithm. We illustrate the method with the design of a novel family of multi-dimensional Riesz-Laplace wavelets that bandelets [19], and directionlets [20]. essentially behave like the Nth-order partial derivatives of an The steerable pyramid has been used quite extensively isotropic Gaussian kernel. in image processing. Typical examples of applications include local orientation analysis, adaptive angular filtering, deformable templates [4], contour detection, shape from shading I.INTRODUCTION [2], [9], texture analysis and retrieval [21]–[23], directional Scale and directionality are key considerations for visual pattern detection [24], feature extraction for motion tracking processing and perception. Researchers in image processing [25] and, of course, image denoising [26], [27]. Mathemati- and applied mathematics have worked hard on formalizing 2 cally, the steerable pyramid is a tight frame of `2(Z ), due to these notions and on developing some corresponding feature the self-reversibility of the underlying discrete filterbank. The extraction algorithms and/or signal representation schemes. link with wavelet theory would be complete if there was a The proper handling of the notion of scale in image process- continuous-domain formulation of the transform, which has ing owes much to the work of Mallat who set the foundation not been worked out explicitly so far. The other point is of the multiresolution theory of the wavelet transform which d that Simoncelli’s construction is quite specific and difficult involves continuously-defined functions in L2(R ) [1]. The to generalize to higher dimensions. remarkable aspect of this theory is that it yields a reversible The purpose of this paper is to revisit the construction of one-to-one decomposition of a signal across scale; unfor- steerable wavelets and pyramids using a continuous-domain tunately, the underlying wavelet bases (which are typically operator-based formalism. The cornerstone of our approach separable) have uneven angular responses; they are strongly is the Riesz transform which constitutes the natural multi- biased towards the vertical and horizontal directions. dimensional generalization of the Hilbert transform. The Riesz The steerable filters of Freeman and Adelson constitute transform has a long tradition in mathematics [28] and has another elegant framework that is more specifically tuned to been studied extensively in the context of Calderon-Sygmund’s´ orientation [2]. The primary focus there is on the image theory of singular integral operators [29]. Its introduction to analysis aspect of the problem and the search for an efficient signal processing is more recent. Felsberg used the transform computational scheme for the filtering of an image with a to define the “monogenic signal” as a 2-D generalization of reference template that can be oriented arbitrarily. Several the analytical signal representation [30], [31]. This pioneering solutions have been suggested for the design of optimized work inspired Metikas and Olhede to specify a monogenic filtering templates [3]–[5]. Researchers have also proposed a version of the continuous wavelet transform which is a fully unifiying Lie-group formulation of steerability that allows for redundant signal representation [32]. At about the same time the extension of the concept to others classes of transforma- as Felsberg, Larkin independently introduced a complexified tions (e.g. translation and dilation) [6]–[8]. version of the 2-D Riesz transform in optics—calling it It is possible—and highly desirable conceptually—to rec- the spiral quadrature phase transform—and applied it to the oncile the two types of approaches by allowing for wavelet demodulation of interferograms and the analysis of fringe decompositions with a certain level of redundancy [9]. In patterns [33], [34]. In recent work, we took advantage of the The authors are with the Biomedical Imaging Group (BIG), Ecole´ Poly- unitary property of this latter transform to specify a proper, technique Fed´ erale´ de Lausanne (EPFL), CH-1015 Lausanne, Switzerland. reversible wavelet-domain monogenic signal analysis with a IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 2 minimal amount of redundancy [35]. Riesz-Laplace wavelets. The distinguishing features of these Our present focus on the Riesz transform is motivated by its wavelets is that they correspond to the Nth-order derivatives invariance with respect to the primary coordinate transforma- of a multi-dimensional B-spline kernel. The Gaussian shape tions: translation, dilation and rotation (cf. Subsection II.D). of the underlying smoothing kernel, which is common to all The first two invariance properties are ideally matched to the wavelets, makes the transform attractive for multi-dimensional framework of wavelets, while the latter provides a connection feature extraction. We conclude the paper by summarizing the with steerable filters and gradient-like signal analyses [35], key features of our approach and by discussing the similarities [36]. The formal extension of this link calls for a higher-order and differences with existing directional transforms. version of the transform, which we introduce in Subsection II.E. Thanks to this operator, we are able to take any multires- II.THE RIESZTRANSFORMANDITSHIGHER-ORDER d olution decomposition of L2(R ) and to map it into a series EXTENSIONS directional counterparts. The direct benefits of the proposed methodology are: The purpose of this section is to lay the mathematical A general framework for constructing families of steer- foundations for our approach in a fashion that is self-contained d and geared towards a signal processing audience. We start able wavelet frames and pyramids of L2(R ) with arbitrary orders1 in any number of dimensions d ≥ 2. by reviewing the primary features of the Riesz transform A clear decoupling between the multiresolution and the with a special emphasis on its invariance properties. We then steerability aspects of the representation. In the present introduce a novel higher-order extension of the transform that work, we adopt a less constraining definition of a steerable is specifically designed to preserve inner-products (subsection transform which does not require the basis functions to E). The important aspect that we want to bring forth is the be rotated versions of one another—the basic requirement link between these generalized Riesz operators and steerable is that the component subspaces be rotation-invariant [7]. filterbanks, as introduced by Freeman and Adelson. This offers design flexibility and also carries over naturally to dimensions greater than two. A. Notations A continuous-domain formulation that formalizes and extends the technique proposed by Simoncelli et al.; in We consider a generic d-dimensional signal f indexed by d essence, it represents the multiresolution theory of L ( d) the continuous-domain space variable x = (x1, . . . , xd) ∈ R . 2 R d counterpart of their original filterbank design. The perfect The multidimensional Fourier transform of f ∈ L1(R ) is d specified as reconstruction property and closedness in L2(R ) is automatically guaranteed. Z ˆ −jhω,xi f(ω) = f(x)e dx1 ··· dxd, A full control over the regularity (Sobolev smoothness) d and approximation theoretic properties (vanishing mo- R d ments, order of approximation, etc.) of the basis functions. with ω = (ω1, . . . , ωd) ∈ R . This definition is extended The perspective of a purely analytical design of novel appropriately for Lebesgue’s space of finite energy functions d families of bona-fide steerable wavelets—in particular, L2(R ), as well as for Schwartz’s class of tempered distri- 0 d higher-order Riesz-Laplace wavelets which admit explicit butions S (R ). We recall that the squared L2-norm of f is closed-form formulas in any number of dimensions. given by Z The paper is organized as follows. In Section II, we review 2 2 kfk d = |f(x)| dx ··· dx L2(R ) 1 d the mathematical properties of the Riesz transform; we then d proceed with the specification of a higher-order transform that R Z 1 ˆ 2 has the fundamental property of conserving energy. Another = d |f(ω)| dω1 ··· dωd (2π) d crucial property is that the components of our higher-order R Riesz transform define a subspace that is rotation-invariant, where the right-hand side follows from Parseval’s identity. d d which is the key to steerability. In Section III, we prove The adjoint of a linear operator L: L2(R ) → L2(R ) is ∗ ∗ that the Riesz transform and its higher-order extension have denoted by L ; it is such that hf, LgiL2 = hL f, giL2 , ∀f, g ∈ d L ( d). the property of mapping a wavelet frame of L2(R ) onto 2 R another one, while preserving the frame bounds. We thereby The fractional Laplacian (−∆)s/2 with s ∈ R+ is the establish a natural correspondence between a (quasi-isotropic) isotropic differential operator of order s whose Fourier-domain wavelet frame and its directional Riesz counterparts. In Sec- definition in the sense of distributions is tion IV, we explicitly make the connection between the 2-D s F s (−∆) 2 f(x) ←→ kωk fˆ(ω). (1) version of our generalized Riesz transform and Simoncelli’s construction. In doing so, we also introduce an extended For s/2 = n ∈ N, it is a purely local operator that is angular parametrization of a rotating filterbank that ensures proportional to the classical n-fold Laplacian. Otherwise, its self-reversibility. In Section V, we illustrate our concept with effect is non-local and described by the following distribu- the specification of an extended family of high-order steerable tional impulse response [37]

1 C In this work, we do equate the notions of order of steerability and the s,d ←→F kωks, (2) order of the Riesz transform. kxks+d IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 3

s d+s 2 Γ( 2 ) d where Cs,d = d/2 is an appropriate normalization where the filters (hn)n=1 are characterized by their frequency π Γ(−s/2) ˆ constant and Γ(r) = (r − 1)! is Euler’s gamma function. The responses hn(ω) = −jωn/kωk. To obtain explicit space- definition (1) of the fractional Laplacian can be extended to domain expressions, it is helpful to view these filters as partial −1 −1 negative orders as well, which leads to a fractional integral- derivatives of the function g(x) = F {kωk }(x) (the − 1 like behavior. The so-defined family of fractional operators impulse response of the isotropic integral operator (−∆) 2 ). satisfies the composition rule (−∆)α1 (−∆)α2 = (−∆)α1+α2 Specifically, by making use of the Fourier-transform relation which follows directly from their Fourier-domain definition, (2) (which is also valid for negative s provided that s + d∈ / with the convention that (−∆)0 = Identity. Note, however, −2N [37]), we get that the fractional integral (−∆)−s/2f with s > 0 is well- C 1 Γ d−1 1 defined only for functions (or distributions) f whose moments −1,d 2 2 g(x) = d−1 = d+1 · d−1 , up to order dse are zero, due to the singularity of the frequency kxk π 2 kxk response at the origin. which we then differentiate to obtain We will use the multi-index notation to state some of our results concisely: specifically, n = (n , . . . , n ) is a (d−1)C−1,d 1 d z }| { d-dimensional multi-index vector with non-negative integers d+1 ∂ Γ 2 xn Pd n n1 nd hn(x) = − g(x) = · . entries; |n| = i=1 ni; z = z1 ··· zd for any z = ∂x d+1 kxkd+1 d n π 2 (z1, . . . , zd) ∈ C ; and n! = n1!n2! ··· nd!. With these conventions, the multinomial theorem reads The case d = 1 is excluded from the above argument; in that particular instance, we have g(x) = log |x| whose derivative N X N! n π (z1 + ··· + zd) = z , dg(x) n! yields h(x) = − dx = 1/(πx). The explicit form of these |n|=N various operators for d = 1, 2, 3, 4 is recapitulated in Table 1 where the summation is over all multi-indices n satisfying the for further reference. condition |n| = N. Note that the impulse responses of the Riesz transform are all anti-symmetric: Rn{δ}(x) = hn(x) = −hn(−x) and B. The 1-D Hilbert transform essentially decaying like 1/kxkd where d is the number of dimensions. While the Riesz transform remains a non-local The Hilbert transform plays a central role in the analytical operator, it is interesting to observe that the delocalization signal formalism which is a powerful tool for AM/FM analysis becomes less pronounced in higher dimensions. It is also [38]. It is a 1-D linear, shift-invariant operator whose defining known from functional analysis that the Riesz transform is property is to maps all cosine functions into their corre- a bounded operator for the primary Lebesgue spaces; i.e., sponding sine functions, without affecting their amplitude. The Hilbert operator, denoted by H, therefore acts as an allpass d ∀f ∈ Lp(R ), kRfkLd ≤ Ad,p · kfkLp . 1 < p < ∞ (5) filter; it is characterized by the transfer function hˆ(ω) = p −jsign(ω) = −jω/|ω|. Its impulse response (in the sense where the Ad,p are some suitable constants [29]. The singular of distributions) is h(x) = 1/(πx), which is slowly decaying, aspect of the transform is that the extreme cases p = 1, +∞ expressing the fact that H is a non-local operator. A potential are excluded; this also reflects the fact that the Riesz transform mathematical difficulty is that h(x) ∈/ L1(R) which means is not stable in the ordinary BIBO sense. Fortunately for us, that the corresponding filter is not BIBO-stable (bounded in- the situation is particularly favorable for p = 2, as will be put/bounded output), even though it is well-behaved in the L2- seen next (cf. Property 3). sense (e.g., bounded and perfectly reversible). A remarkable property in this latter respect is that H−1 = −H = H∗, indicating that H is a unitary operator; this follows directly D. Properties of the multidimensional Riesz transform from the Fourier-domain definition. We now present and discuss the key properties of the operator R. C. The Riesz transform Property 1 (Invariances): The Riesz transform is The Riesz transform is the natural multidimensional exten- translation- and scale-invariant: sion of the Hilbert transform [29]. It is the scalar-to-vector d signal transformation R whose frequency-domain definition ∀x0 ∈ R , R{f(· − x0)}(x) = R{f(·)}(x − x0) is ∀a ∈ +, R{f(·/a)}(x) = R{f(·)}(x/a) ω R Rdf(ω) = −j fˆ(ω) (3) The translation invariance directly follows from the definition, kωk while the scale invariance is easily verified in the Fourier where fˆ(ω) is the Fourier transform of the d-dimensional domain. input signal f(x). This corresponds to a multidimensional d- Remarkably, the Riesz transform is also rotation-invariant. channel filterbank whose space-domain description is To formalize this property, we consider the group of rotation d     matrices in R . Specifically, let Ru denote a d × d rotation R1f(x) (h1 ∗ f)(x) matrix whose first row is the unit vector u = (u1, . . . , ud); Rf(x) =  .  =  .  (4)  .   .  i.e., [Ru]1,n = [u]n = un (Ru is such that it rotates the first Rdf(x) (hd ∗ f)(x) coordinate vector e1 into u). IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 4

TABLE I RIESZ TRANSFORM COMPONENTS AND RELATED DIFFERENTIAL OPERATORS FOR d = 1, 2, 3, 4

− 1 ∂ − 1 1 Operator (−∆) 2 Rn = − (−∆) 2 (−∆) 2 ∂xn

1 −jωn Frequency response kωk kωk kωk Impulse responses log |x| 1 −1 d = 1 π πx π|x|2

1 x −1 d = 2 n 2πkxk 2πkxk3 2πkxk3

1 x −1 d = 3 n 2π2kxk2 π2kxk4 π2kxk4

1 3x −3 d = 4 n 4π2kxk3 4π2kxk5 4π2kxk5

Property 2 (Steerability): The Riesz transform filterbank It is instructive to determine the Riesz transform of a pure defined by (4) is steerable in the sense that cosine wave with frequency (or wave number) ω0. A simple d Fourier calculation yields X h1(Rux) = hu, Rδ(x)i = unhn(x), T ω0 T R{cos(ω0 x)}(x) = sin(ω0 x). n=1 kω0k ◦ 2 while the component filters are 90 rotated versions of each We can then apply (6) and recover the corresponding sine wave other; i.e., hn(x) = h1(Ren x) where [en]k = δk. by steering the directional Hilbert transform in the appropriate This is shown by using the rotation property of the Fourier ω0 direction u0 = kω k : transform 0 T T d Hu0 {cos(ω0 x)}(x) = sin(ω0 x). F [Ruω]1 X unωn h (R x) ←→ hˆ (R ω) = −j = −j 1 u 1 u kR ωk kωk This desirable behavior follows from the fact that the central u n=1 d cut of the frequency response of Hu along the direction u X ˆ perfectly replicates the behavior the 1-D Hilbert transform: = unhn(ω). n=1 Hu(ωu) = −jsign(ω). The only limitation of the technique is that the ideal Hilbert-transform-like behavior falls off like The above result implies that the Riesz transform commutes hu, u i (the cosine of the angle), which calls for a precise with rotations, as expressed by the following property of its 0 adjustment of the analysis direction u . impulse response 0 Property 3 (Inner-product preservation): The Riesz trans- R{δ}(Rux) = RuR{δ}(x) form satisfies the following Parseval-like identity

Qd d where δ(x) = δ(xn) is the multidimensional Dirac n=1 d X ∀f, φ ∈ L2( ), hRf, Rφi d = hRnf, RnφiL distribution. R L2 2 Property 2 suggests linking the Riesz transform to the n=1 following directional version of the Hilbert transform = hf, φiL2 . This is established by using Parseval’s relation for the Fourier d X transform Huf(x) = unRnf(x) = hu, Rf(x)i (6) d n=1 X hRnf, RnφiL2 whose impulse response hu(x) = h1(Rux) simply corre- n=1 sponds to the rotated version of h1(x). Note that the associa- tion between the Riesz and the directional Hilbert transforms d Z is essentially the same as the link between the gradient and 1 X ωn ˆ ωn ˆ∗ = d −j f(ω) j φ (ω) dω1 ··· dωd (2π) d kωk kωk the directional derivative (more on this later). n=1 R d 2 2 1 Z P ω In general, a single vector (or direction) is not sufﬁcient to specify a = n=1 n fˆ(ω) φˆ∗(ω) dω ··· dω rotation matrix in d dimensions. In the present case, however, the functions d 2 1 d (2π) d kωk h (x) are isotropic within the hyperplane perpendicular to the corresponding R i 1 coordinate vector ei so that our “geometrical” statements implicitly refer to = hf,ˆ φˆi = hf, φi a whole equivalence class of rotation matrices. (2π)d L2 L2 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 5

A similar manipulation leads to the identification of the where (−∆)s/2 is the isotropic derivative operator of order s adjoint operator, which is specified by [cf. (1)]. The Sobolev smoothness equivalence then directly ωT follows from the energy preservation property (8) and the ∗ ∗ ∗ F 2 R r(x) = R1r1(x) + ··· + Rdrd(x) ←→ j rˆ(ω), translation invariance of R, which implies that kfk = kωk L2 Pd kR fk2 > kR fk2 , and k(−∆)s/2fk2 = (7) n=1 n L2 n L2 L2 where r(x) = (r (x), . . . , r (x)) is a generic finite-energy Pd kR (−∆)s/2fk2 > k(−∆)s/2R fk2 . 1 d n=1 n L2 n L2 ∗ d-vector signal and Rn = −Rn. It also corresponds to the In this work, we will use the Riesz transform to map a set −1 ∗ left inverse of the transform R = R , owing to the fact of primary wavelets into another augmented one. Property 1 ωT ω that j kωk × −j kωk = 1. A direct consequence is the and 3 are essential for that purpose because they ensure that energy conservation property the mapping preserves the scale- and shift-invariant structure d as well as the L2-stability of the representation. Note that the 2 X 2 2 latter goes hand-in-hand with the reversibility of the transform kRfk d = kRnfkL = kfkL , (8) L2 2 2 n=1 and its ability to perfectly represent all finite-energy signals. which emphasizes the unitary character of the transform. Thanks to Property 5, we also have the guarantee that the We conclude this section by listing a few additional func- Riesz wavelets are functionally well-behaved in that they retain tional properties of the components of the Riesz transform for the same regularity (degree of differentiability) as the primary specific classes of signals or basis functions. templates from which they are derived. Property 4 is relevant d as well because it enforces a certain level of coordinate-wise Property 4 (Orthogonality): Let ψ(x) ∈ L2(R ) be a real-valued function whose Fourier transform ψˆ satisfies orthogonality which is potentially favorable for local feature ˆ 2 extraction and signal analysis. Finally, we will be able to take the coordinate-wise symmetry relations: |ψ(ω1, . . . , ωd)| = ˆ 2 advantage of Property 2 to design wavelet transforms whose |ψ(±ω1,..., ±ωd)| . Then, basis functions are steerable.

hRnψ, ψiL2 = 0, for n = 1, . . . , d

hRnψ, RmψiL2 = 0, when m 6= n. Proof: The direct application of Parseval’s relation yields E. Higher-order Riesz transform hR ψ, R ψi n m L2 It is also possible to define higher-order Riesz transforms 1 Z ω ω by considering individual signal components of the form = −j n ψˆ(ω) j m ψˆ∗(ω) dω ··· dω d 1 d Ri1 Ri2 ···RiN f with i1, i2, ··· iN ∈ {1, ··· , d}. While there (2π) d kωk kωk R are dN possible ways of forming such Nth order terms, there Z ˆ 2 1 |ψ(ω)| are actually much less distinct Riesz components due to the = d ωnωm 2 dω1 ··· dωd (2π) d kωk R commutativity and factorization properties of the underlying 1 Z Z |ψˆ(ω)|2 convolution operators. Our definition of higher-order transform = ω ω dω ··· dω dω = 0 d n m 2 1 d n removes this intrinsic redundancy and is based on the follow- (2π) d−1 kωk R R | {z } ing operator identity. φ(ωn) Theorem 1: The Nth-order Riesz transform achieves the The partial integration with respect to all frequency variables following decomposition of the identity except ωn yields the auxiliary function φ(ωn) which is symmetric, as a consequence of our hypotheses. The product func- X N! n1 nd ∗ n1 nd (R1 ···R ) (R1 ···R ) = Id tion ωnφ(ωn) is therefore anti-symmetric, which necessarily n! d d leads to a vanishing outer integral. |n|=N While the assumptions for Property 4 do not hold for arbi- using the multi-index vector n = (n , ··· , n ). trary signals, they are met by most basis functions commonly 1 d k used for image processing. Indeed, the required symmetry Here, Rn denotes the k-fold iterate of Rn, while the weighting relations are satisfied by all functions that are either separable factors in the expansion are the multinomial coefficients of (because of the Hermitian symmetry of the 1-D Fourier trans- order |n| = N form), isotropic, or coordinate-wise symmetric/anti-symmetric |n|! N N! in the space domain (up to some arbitrary translation). = = . The smoothness of a function is often specified by its n! n1, . . . , nd n1! n2! ··· nd! inclusion in the Sobolev space W s (the class of functions 2 Proof: By applying the Theorem’s left-hand side decomposi- that are “s-times” differentiable in the L2-sense) [39]. A nice feature of the Riesz transform is that it fully preserves Sobolev tion to an input signal f, we obtain a series of components of regularity. the form Property 5 (Sobolev smoothness): f ∈ W s( d) ⇔ 2 R n1 nd ∗ n1 nd s d gn ,...,n (x) = (R ···R ) (R ···R ) f(x). Rnf ∈ W2 (R ) for n = 1, . . . , d. 1 d 1 d 1 d Proof: The explicit definition of Sobolev’s space of order s is: n o This corresponds to a convolution whose Fourier-domain W s( d) = f(x), x ∈ d : kfk2 + k(−∆)s/2fk2 < +∞ 2 R R L2 L2 equivalent is IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 6

Property 7: The Nth-order Riesz transform satisﬁes the n n following Parseval-like identity jω −jω ˆ gˆn1,...,nd (ω) = f(ω) d (N) (N) kωk kωk ∀f, φ ∈ L2(R ), hR f, R φi p(N,d) = L2 2 n1 2 nd |ω1| |ωd| X n n ˆ hR f, R φiL = hf, φiL . = 2 ··· 2 f(ω) 2 2 kωk kωk |n|=N |ω |2n1 · · · |ω |2nd In particular, by setting f = φ in the above property, = 1 d fˆ(ω). (9) kωk2N we observe that the higher-order Riesz transform perfectly preserves the L2-norm of the signal. The generalized version Now, we obviously have the identity of this result is the boundedness of the higher-order Riesz transform for the classical Lebesgue spaces (with the notable N kωk2N |ω |2 + ··· + |ω |2 exception of p = 1, +∞). fˆ(ω) = fˆ(ω) = 1 d fˆ(ω). kωk2N kωk2N Property 8: The higher-order Riesz transform satisﬁes the following norm inequalities: Next, we apply the multinomial theorem to expand the central d (N) ∀f ∈ Lp( ), kR fk p(N,d) ≤ Ad,p,N ·kfkLp 1 < p < ∞ factor R Lp

2n1 2n X N! |ω1| · · · |ωd| d where the Ad,p,N are appropriate constants. fˆ(ω) = fˆ(ω) n! kωk2N Since the higher-order Riesz transform can be decomposed |n|=N into a succession of first-order transforms, the result is a con- X N! = gˆ (ω) sequence of the Lp boundedness of all elementary component n! n1,··· ,nd |n|=N transformations (cf. (5)). The final important feature is rotation invariance; it is inti- where we have identified the relevant higher-order reconstruc- mately linked to the concept of steerability which is discussed tion components using (9). in the next section. From the above, it is clear that an Nth order Riesz transform decomposes a signal into p(N, d) distinct components, where N+d−1 F. Higher-order steerability p(N, d) = d−1 is the cardinality of the indexing set {n ∈ Nd s.t. |n| = N}. We also want to ensure that the transform The most interesting aspect of the higher-order Riesz trans- preserves energy which calls for an appropriate normalization. form is its steerability which also comes hand-in-hand with We therefore specify the Nth order Riesz transform of the improved angular selectivity. To best explain this behavior, we (N) signal f as the p(N, d)-vector signal transformation whose relate R to the N-fold version of the directional Hilbert components are given by transform Hu along the direction specified by the unit vector u. To that end, we write the Fourier-domain equivalent of (6) r N! f n = Rnf = Rn1 ···Rnd f. (10) d n ! ··· n ! 1 d X −jωi ˆ 1 d Hduf(ω) = ui f(ω). (11) kωk In some instances, we will use the more concise vector i=1 notation R(N)f, keeping in mind that each component is We then iterate the operator N times and expand the right- associated with its own normalized, Nth-order Riesz operator hand side using the multinomial theorem R(n1,...,nd), as defined above. hu, ωiN N ˆ Note that the number of higher-order components roughly H[u f(ω) = −j f(ω) (12) increases like the (d − 1)th power of the dimension. For kωk instance, p(N, 1) = 1, p(N, 2) = N + 1, and p(N, 3) = (N + d !N X −jωi ˆ 2)(N + 1)/2. The good news is that the increase is moderate = ui f(ω) kωk in dimension 2 (the context of image processing) and still i=1 n manageable in dimension 3 (volumetric imaging). The case X N! −jω = un fˆ(ω). d = 1 is uninteresting since the higher-order transform cycles n! kωk |n|=N back and forth between the identity and the conventional Hilbert transform. Finally, we go back to the space domain, which yields Not too surprisingly, the higher-order Riesz transform inher- X N! its all the fundamental properties of the operator R, as these HN f(x) = un1 Rn1 ··· und Rnd f(x) u n! 1 1 d d are preserved through the iteration mechanism. |n|=N (N) r Property 6: The higher-order Riesz transforms R are X N! = unRnf(x) translation and scale-invariant. n! What is more remarkable is the higher-order inner-product |n|=N X (n1,...,nd) preservation property—it comes as a corollary of the decom- = cn1,...,nd (u) f (x) (13) position of the identity given by Theorem 1. |n|=N Higher-orderIEEE TRANSACTIONS ON IMAGEdirectional PROCESSING, VOL. Hilbert X, NO. XX, 2010 transform 7

N hu,ωi N cN,0, ,0(u) frequency response HdN (ω) = −j = (−j cos θ) (N,0, ,0) ··· u kωk (N,0, ,0) f ··· ··· does not depend upon the radial frequency ω = kωk, but R θ ω . . only upon the angle between and the direction vector . . u. It is maximum along the direction θ = 0; along that cn , ,n (u) 1 ··· d N N (n1, ,nd) N ray, Hd(uω) = (−jsgn(ω)) which coincides (up to a f ··· u f(x) (n1, ,nd) u f(x) ··· R H sign change) with the 1-D identity when N is even (sym- . . . metric operator) and the 1-D Hilbert transform otherwise . c0, ,0,N (u) (0, ,0,N) ··· (anti-symmetric operator). As θ increases from 0 to π/2, the (0, ,0,N) f ··· N ··· response falls off like cos(θ) —a clear indication that angular R selectivity improves with N.

Fig. 1. Steerable ﬁlterbank implementation of the Nth order directional Hilbert transform in multiple dimensions. The output signal is formed by G. Connection with the gradient and partial-derivatives taking an appropriate linear combination of the components of the Nth order Riesz transform of the signal. The scheme is adaptive in the sense the direction There is a direct connection between the Riesz transform of analysis u may vary spatially. and the gradient, which is easily seen in the frequency domain. Speciﬁcally, we have that

1 where the “steering” coefficients are given by 1 ∇f(x) = (−1)(−∆) 2 Rf(x) (16) 1 r where (−∆) 2 is the square-root-Laplace operator whose fre- N! n1 nd cn1,...,nd (u) = (u1 ··· ud ). (14) quency is kωk [cf. Table 1]. The converse relation is n1! ··· nd! − 1 Rf(x) = (−1)(−∆) 2 ∇f(x), (17) The signal processing transcription of (13) leads to the block diagram in Fig. 1. The corresponding system falls within which should be interpreted as a smoothed version of the gra- − 1 the general framework of steerable filterbanks of Freeman dient (up to a sign change); the integral operator (−∆) 2 acts and Adelson [2], although these authors did not pursue their on all derivative components and has an isotropic smoothing concept for dimensions larger than 3. The main point is that we effect. can apply a p(N, d)-channel filterbank with impulse responses The directional counterparts of this last equation is: n {hn = R δ}(|n|=N) to the signal f and then compute − 1 N Huf(x) = (−1)(−∆) 2 Duf(x) (18) Hu f(x) along any direction u by forming an appropriate linear combination with the weights given by (14). While some where Hu is defined by (6) and where Duf = h∇f, ui is the authors define the order of steerability as the dimension of the derivative of f along the direction u. invariant subspace k = p(N, d), By iteration, we obtain the link with the partial derivatives The argument also extends for generalized Riesz operators of order N = n1 + ··· + nd: T (N) of the form a R where a is a given p(N, d)-vector N n n N − N ∂ f(x) of weights. This implies that the function space VN,d = 1 d 2 R1 ···Rd f(x) = (−1) (−∆) (19) ∂n1 x ··· ∂nd x span{hn(x)}(|n|=N) is rotation-invariant and that there is a 1 d simple weighting mechanism for steering generalized Riesz and the higher-order directional derivatives:

operators in arbitrary directions. Following this line of thought, N N N − 2 N it is also possible to define generalized higher-order Riesz Hu f(x) = (−1) (−∆) Du f(x) (20) (N) operators of the form AR where A is a p(N, d) × p(N, d) The crucial element here is the fractional integral operator of non-singular complex matrix. This generalized operator will − N F N order N: (−∆) 2 ←→ 1/kωk . It is an isotropic lowpass share all the fundamental properties of R(N) (shift-invariance, filter whose smoothing strength increases with N. The global scale-invariance and steerability) and will preserve inner- interpretation of these relations is that there is a one-to-one products provided that A = U is a unitary matrix. Indeed, connection between Riesz components and spatial derivatives such a representations leads to the following decomposition with the former being smoothed versions of the latter. The of the identity which comes as a corollary of Theorem 1: Riesz transform therefore captures the same directional in- ∗ ∗ UR(N) UR(N) = R(N) UH U R(N) formation as derivatives but it has the advantage of being | {z } much better conditioned since there is no amplification of high I ∗ frequencies. We will now see how we can exploit its invariance (N) (N) = R R = Id (15) and reversibility properties to construct new steerable wavelet- type decompositions. This simple mechanism offers additional degrees of freedom for shaping higher-order impulse responses and designing steerable, reversible filterbanks in arbitrary dimensions. III.GENERAL CONSTRUCTION OF STEERABLE WAVELET It is also instructive to investigate the directional behavior FRAMES N of the operator Hu which corresponds to the u-directional The pleasing consequence of Property 3 is that the Riesz N d version of the operator R1 . The important point is that its transform will automatically map any frame of L2(R ) into IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 8 another one. This fundamental observation was made indepen- properties. Specifically, let us consider the following wavelet dently by Held et al. [36] in full generality, and by us in the decomposition of a finite energy signal: more constrained 2-D setting, considering a complex version X X ∀f ∈ L ( d), f(x) = hf, ψ i ψ˜ (x), (22) of the Riesz transform that defines a unitary mapping from 2 R i,k L2 i,k d 2 i∈Z k∈Z L2(R ) into itself [35]. Here, we will present a proper Nth- order extension of these ideas. Before establishing the results, where the wavelets at scale i are dilated versions of the ones −id/2 i we start by recalling a few basic facts about frames [40]–[42]. at the reference scale i = 0: ψi,k(x) = 2 ψ0,k(x/2 ).

Definition 1: {φ } d General wavelet schemes also have an inherent translation- A family of functions k k∈Z is called a d invariant structure in that the ψ ’s are constructed from the frame of L2( ) iff. there exists two strictly positive constants 0,k R d A and B < ∞ such that integer shifts (index k) of up to 2 distinct generators (mother wavelets). X 2 ∀f ∈ L ( d),A kfk2 ≤ |hφ , fi | ≤ B kfk2 . 2 R L2 k L2 L2 For convenience, we now use a multi-index representation d k∈Z of the corresponding Nth-order Riesz wavelets, which are (21) defined as follows The frame is called tight iff. the frame bounds can be chosen r N! such that A = B. If A = B = 1, we have a Parseval frame that n n n1 nd ψi,k(x) = R ψi,k = R1 ···Rd ψi,k, (23) satisfies the remarkable decomposition/reconstruction formula n1! ··· nd! Pd d X with n = (n1, . . . , nd) and |n| = i=1 ni = N. ∀f ∈ L2(R ), f = hφk, fiL2 φk. Proposition 2: Let {ψi,k} be a primal wavelet frame of d k∈Z d L2(R ) associated with the reconstruction formula (22). n n While the form of this expansion is the same as that associated {ψ } d {ψ˜ } d Then, i,k (|n|=N,k∈Z ) and i,k (|n|=N,k∈Z ) form a with an orthonormal basis, the fundamental aspect of the frame dual set of wavelet frames satisfying the general decompo- generalization is that the family (φk) may be redundant. More sition/reconstruction formula generally, we have that d X X X n ñ ∀f ∈ L2(R ), f(x) = hf, ψi,kiL2 ψi,k(x). d X ˜ i∈ k∈ d |n|=N ∀f ∈ L2(R ), f = hφk, fiL2 φk Z Z d k∈Z Proof: Theorem 1 is equivalent to the following decompo- ˜ where (φk) is a so-called dual frame. While the dual frame is sitions of the identity operator: X X not unique in the redundant case, there is one—the so-called Rn∗Rn = RnRn∗ = Id, (24) canonical one—that is used preferentially since it maps into |n|=N |n|=N the minimum-norm inverse [42]. n n∗ We are now ready to state the fundamental frame preser- which allows us to interchange the role R and R vation property of the Nth order Riesz transform, which when analyzing (resp. synthesizing) a signal. First, we apply generalizes the first-order result of Held et al. the wavelet decomposition (22) to the transformed signals n∗ d Proposition 1: The Nth order Riesz transform maps R f ∈ L2(R ) for all n such that |n| = N: d the frame {φ } d of L ( ) into another frame n∗ X X n∗ ˜ k k∈Z 2 R R f(x) = hR f, ψi,kiL ψi,k(x) n 2 {R φk}(|n|=N,k∈ d) that has the same frame bounds and a d Z i∈Z k∈Z redundancy increased by p(N, d) = N+d−1. X X d−1 = hf, Rnψ i ψ˜ (x) (by duality). Proof: First we observe that kRnfk = kRn∗fk ≤ i,k L2 i,k L2 L2 d n∗ N n i∈Z k∈Z kfkL2 where R = (−1) R is the adjoint of the n- component Riesz operator Rn defined by (10). We can We then make use of (24) to resynthesize f: X therefore apply the frame inequality (21) to each (adjoint) f(x) = RnRn∗f(x), component of the Nth-order Riesz transform: |n|=N X 2 A kRn∗fk2 ≤ |hφ , Rn∗fi | ≤ B kRn∗fk2 which yields the desired reconstruction formula by linear- L2 k L2 L2 d ity. The final ingredient for this construction is Property 6, k∈Z which ensures that the wavelet structure is conserved; i.e., Next, we sum up these inequalities over the multi-index Rnψ (x) = 2−id/2Rnψ (x/2i), as well as the basic shift- components |n| = N, which yields i,k 0,k invariant structure that involves a finite number of generators X X 2 A kfk2 ≤ |hφ , Rn∗fi | ≤ B kfk2 which are the higher-order Riesz transforms of the primary L2 k L2 L2 d mother wavelets. k∈Z |n|=N If, in addition, the mother wavelets in (22) are (quasi- where we have made use of the energy preservation feature )isotropic, then we end up with a decomposition that is of the Riesz transform which is implied by Property 7. The steerable (cf. Section II.F). For instance, we can use the proposition then simply follows from the adjoint-defining steering equation (13) to rotate the first component wavelet n∗ n relation hφk, R fiL = hR φk, fiL . (N,0,...,0) (N,0,...,0) 2 2 ψ = R ψi,k to any desired orientation: What is of even greater significance to us is that the Nth- i,k N X n (N,0,...,0) order Riesz transform will transform any wavelet frame into Hu ψi,k(x) = cn(u)ψi,k = ψi,k (Rux), (25) another one, thanks to its translation- and scale-invariance |n|=N IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 9 where the right-hand side of the equality is valid provided that the primal wavelet ψ(x) = ψ(kxk) is isotropic. Similar steering equations can be derived for all other Riesz wavelet components. Concretely, this means that the global wavelet analysis is rotation-covariant. Finally, if the wavelet decomposition (22) is build around (1,0) (0,1) a Mallat-type multiresolution analysis and admits a fast filterbank algorithm, then there is a corresponding perfect reconstruction filterbank implementation for its Nth-order Riesz counterpart in Proposition 2.

IV. THEGENERALIZED RIESZTRANSFORMAND 2-D (2,0) (1,1) (0,2) STEERABLE FILTERBANKS In 2-D, there is an interesting connection between the generalized Riesz transform (as defined in §II.E), steerable filterbanks that are polar-separable, and the steerable pyramid of Simoncelli et al. [11]. (3,0) (2,1) (1,2) (0,3) The Nth-order Riesz transform for d = 2 has N + 1 distinct components. The corresponding frequency responses Fig. 2. Frequency responses (real or imaginary part) of the components filters N ˆ nˆ o of the first, second and third order Riesz transform. The origin of the 2-D span the subspace VN,2 = span hn1,N−n1 (ω) , which frequency domain is in the center and the intensity scale is stretched linearly n1=0 is rotation-invariant—this is simply the frequency-domain for maximum contrast. The plots are labeled by (n1, n2) with n2 = N −n1. transposition of what has already been stated for the space domain in §II.F. The specific form of these filters is s n1 N−n1 ˆ N −jω1 −jω2 hn1,N−n1 (ω) = n1 kωk kωk s N (a) θ =0 θ = π/3 θ =2π/3 = (−j)N (cos θ)n1 (sin θ)N−n1 0 0 0 n1 jθ = Hn1,N−n1 (e ) where we have made the polar-coordinate substitution cos θ = √ ω1 √ ω2 2 2 and sin θ = 2 2 . The main point is that these ω1 +ω2 ω1 +ω2 basis functions are purely polar and that they are completely (b) θ0 =0 θ0 = π/4 θ0 = π/2 θ0 =3π/4 characterized by the 2π-periodic radial profile functions Fig. 3. Frequency responses of the angular filters of Simoncelli’s third (a) and v fourth (b) order steerable pyramids. The origin of the 2-D frequency domain u ! „ −1 «m „ −1 «n m+nu m + n z + z z − z is in the center and the intensity scale is stretched linearly for maximum Hm,n(z) = (−j) t . m 2 2j contrast. (26) with z = ejθ. The complete set of 2-D basis functions for Fourier series H (ejθ) = PN c [k]ejkθ with N = 2, 3, 4 are displayed in Fig. 2. n1,n2 k=−N n1,n2 c [±N] 6= 0, as is readily checked by expanding (26). For any given order, we can associate the filters in pairs n1,n2 The idea of the generalized Riesz transform is to construct (hˆ (ω), hˆ (ω)) that are rotated with respect to each n1,n2 n2,n1 new basis functions by taking suitable linear combinations of other by 90◦. As the order increases, the frequency patterns the primary ones. Each of these basis functions will be polar become more intricate and distinct from those observed in as well and characterized by a generic profile function conventional steerable filterbanks where all components are rotated versions of a single template. The case N = 2 calls N jθ X jkθ for a comment: even though the central pattern looks visually H(e ) = c[k]e . (27) similar to the two others, the (1, 1) filter is not a rotated version k=−N of the two others because one also has to take the range of A truly remarkable property is that we can use any profile the display into consideration; e.g., (black=-1, white=0) for function to define a self-inverting generalized Riesz transform (n1, n2) ∈ {(2, 0), (0, 2)} and (black=-1, white=1) otherwise. that has the structure of a steerable filterbank corresponding As far as steerability is concerned, the important property to the block diagram in Fig. 4. jθ PN jkθ is that the profile functions are angularly “bandlimited” [2]; Theorem 2: Let H(e ) = k=−N c[k]e where the that is, that their Fourier series have a limited number of c[k]’s are arbitrary real-valued (or purely imaginary) coeffi- terms. In fact, they all have the form of a N-term truncated cients. Then, the (N + 1)-channel filterbank whose filters for Generalized self-inverting Riesz transform

N General radial proﬁle function: H(ejθ) = c[k]ejkθ k= N IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX,! 2010− 10

f0 h (x) h ( x) 0 0 − . . f(x) . . f(x)

fN hN (x) hN ( x) − Fig. 5. Original 256 × 256 cameraman with subimage.

Fig. 4. 2-D self-invertible steerable filterbank. The filters are rotated versions of each other with an equi-angular spacing. Their1 frequency responses are j(Hθ+ (πenjθ) = (cos θ)N to construct their steerable pyramid, hˆn(ω cos θ, ω sin θ) = H(e NN+1 ) purely polar and angularly bandlimited.(N They+1) spanN the samec[k space] 2 · as the having noted that this function would satisfy the required self- components of the Nth order Riesz transform. k= N | | − N q P reversibility condition (up to some unspecified normalization constant).hˆ (ω The) 2 autocorrelation= 1 filter in that particular case is ⇒ | n | n = 0,...,N are given by n=0 −1 2N N −1 z + z 1 X 2N −2k πn ! j(θ+ ) HN (z)HN (z ) = = 2N z , ˆ H(e N+1 ) 2 2 k + N hn(ω cos θ, ω sin θ) = q k=−N PN 2 (N + 1) k=−N |c[k]| 9 1 2N allowing for the identification of aN [0] = 22N N . The is steerable and self-inverting. application of Theorem 2 then yields the normalized version of Simoncelli’s original reproduction formula, Proof: We make the substitution z = ejθ and consider the N Laurent polynomial H(z) = PN c[k]zk. The crucial ob- (2N N!)2 X πn k=−N · cos2N θ + = 1, servation is that the coefficients a[k] of the product polynomial (2N)! (N + 1) N + 1 n=0 2N −1 X k which is given here for reference. This particular choice of H(z)H(z ) = a[k]z polar profile corresponds to a steerable implementation of k=−2N N the directional Hilbert transform operator H(cos θ,sin θ), which correspond to the autocorrelation of the sequence c[k] (of clearly falls within the framework of the generalized Riesz length 2N + 1); in particular, this implies that the terms transform. The corresponding frequency responses for N +1 = with index |k| > 2N are necessarily zero. Therefore, if 3 and N + 1 = 4 are given in Fig. 3a and 3b, respectively, we down-sample the sequence a[k] by a factor (2N + 2), and should be compared to the second and third row in Fig. we are left with a single non-zero coefficient at the origin: 2. The filterbanks are identical for N = 1, but not otherwise. PN 2 a[0] = k=−N |c[k]| . In the frequency domain, this down- Yet, they span the same spaces and define a self-invertible sampling operation corresponds to a periodization, leading to transform in either formulation. the identity We would like to emphasize that the results in this section

2N+1 are also directly relevant for the design of general 2-D polar- 1 X j(θ+ 2πn ) 2 |H(e 2N+2 )| = a[0], separable steerable filters whose polar frequency response 2N + 2 ˆ jθ n=0 is of the form h(ω, θ) =w ˆ(ω)H(e ) where wˆ(ω) is a given radial frequency window. This factorization allows for where the right-hand side is the Fourier transform of the a complete decoupling of the radial and angular parts of remaining impulse. Next we note that |H(ejθ)|2 is π-periodic the design. Since the self-reversibility of the angular part since the coefficients are real-valued (or purely imaginary). is automatically guaranteed (thanks to Theorems 1 and 2), We can therefore divide the sum in two equal parts, which the designer can concentrate on the problem of specifying a leads to the partition of unity formula radial (or quasi-isotropic) multi-resolution decomposition that N ensures a perfect (or near perfect) reconstruction, which is 1 X j(θ+ πn ) 2 |H(e N+1 )| = 1. a[0] (N + 1) essentially the approach that was taken by Simoncelli et al. in n=0 [11]. An important point, though, is that the present framework This is equivalent to gives access to a larger variety of angular profiles and steerable basis functions than what has been considered so far. N X ˆ 2 |hn(ω)| = 1, n=0 V. HIGH-ORDER STEERABLE RIESZ-LAPLACE WAVELETS which ensures that the filterbank in Fig. 4 has the perfect While the construction method outlined in §III is applicable reconstruction property. to any primary wavelet transform, it will give the most While the proof relies on rather basic signal processing con- satisfactory results when the underlying primary wavelets are cepts, this result does not appear to have been reported before; nearly isotropic. We will therefore illustrate the concept using at least not in such general terms. It allows for generalizations a primary Laplacian-like (or Mexican hat) decomposition of d of the design of Simoncelli et al. who used the profile function L2(R ) [43]. IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 11

γ =5 (3,0) (2,1) (1,2) (3,0)

(a) (b)

(c) Fig. 8. Reversible wavelet decompositions of Cameraman image (zoom on a 64 × 64 subimage). (a) Primary Laplace wavelet pyramid with γ = 5. (b) 3rd order Riesz-Laplace wavelet transform. (c) Rotated ﬁlterbank spanning the same steerable wavelet subspaces as (b).

(3,0) (2,1) (1,2) (0,3) (a) (b) (c)

Fig. 9. Frequency responses (real or imaginary part) of the wavelet analysis ﬁlters associated with Fig. 8. (a) Primary Laplace wavelets ψbiso,i(ω) with (n1,n2) γ = 5. (b) 3rd-order Riesz-Laplace wavelets ψbi (ω) with n1 + n2 = 3. (c) Corresponding “a` la Simoncelli” directional ﬁlterbank. IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 12

Laplace Riesz-Laplace Laplace Riesz-Laplace

γ =3 (1,0) (0,1)

(a) (b)

(1,0) (0,1) (a) (b)

γ =4 (2,0) (1,1) (0,2)

Fig. 6. Reversible wavelet decompositions of Cameraman image (zoom on a 64 × 64 subimage). (a)-(b): Primary Laplace wavelet pyramid with γ = 3 and its corresponding 1st order Riesz-Laplace (or Marr) wavelet transform. (c)-(d): Primary Laplace wavelet pyramid with γ = 4 and its corresponding 2nd order Riesz-Laplace wavelet transform. (2,0) (1,1) (0,2) A. Primary Laplacian-like wavelets (c) (d) Polyharmonic splines are the natural spline functions associated with the Laplace operator; they can be used to specify Fig. 7. Frequency responses (real or imaginary part) of the wavelet analysis d ﬁlters associated with Fig. 6. (a)-(b) Primary Laplace wavelets ψbiso,i(ω) a family of multiresolution analyses L2( ) which is indexed (1,0) (0,1) R with γ = 3 and 1st-order Riesz-Laplace wavelets ψb (ω) and ψb (ω). by the order of the Laplacian [44]. In prior work, we have i i (c)-(d) Primary Laplace wavelets ψbiso,i(ω) with γ = 4 and 2nd-order introduced a pyramid-like representation of the underlying (2,0) (1,1) (0,2) Riesz-Laplace wavelets ψbi (ω), ψbi (ω), and ψbi (ω). The scale i wavelet subspaces that involves a single quasi-isotropic anal- is increasing from i = 0 upto 2 as one moves down. ysis wavelet (cf [43, Section IV.B]):

γ/2 ψiso(x) = (−∆) β2γ (2x) (28) and their good isotropy properties, especially over the lower frequency range. (3,0) (2,1) (1,2) (0,3) d where γ > 2 is the order (possibly fractional) of the wavelet. The function β2γ (x) is the “most-isotropic” polyharmonic B- spline of order 2γ; it is a smoothing kernel that converges to B. Steerable Riesz-Laplace wavelets a Gaussian as γ increases [44, Proposition 2]. The associated Nth-order steerable Riesz-Laplace wavelet The corresponding “Laplacian” analysis wavelets in (22) transform is obtained through the direct application of Proposi- −id/2 i are given by ψi,k(x) = 2 ψiso(x/2 − k) and the wavelet tion 1. The explicit form of the underlying directional analysis transform has a fast reversible filterbank implementation, as wavelets is r described in [43]. The resulting wavelet decomposition has a N! γ (n1,...,nd) n1 nd 2 pyramid-like structure that is similar to the Laplacian pyramid ψγ (x) = R1 ···Rd (−∆) β2γ (2x) n1! ··· nd! of Burt and Adelson [45]. The fundamental difference, how- (29) ever, is that the proposed transform involves a single analysis for all multi-indices n = (n1, . . . , nd) with |n| = N. The wavelet (28) that is specifically tied to the operator (−∆)γ/2. enlightening aspect in this formula is that the directional be- Two examples of wavelet decompositions with γ = 3, 4 of havior of these wavelets is entirely encoded in the differential Cameraman (cf. Fig. 5) are displayed in Fig. 6a and 6c; operators that are acting on the B-spline smoothing kernel: the we have zoomed on a fragment of the image to facilitate Riesz transform factor is steerable while the Laplacian part, the visualization. The decomposition is a frame with a small which is responsible for the vanishing moments, is perfectly d redundancy factor (R = 4/3 in 2-D and R = 6/5 for d = 3). isotropic. The order parameter γ > 2 is a degree of freedom of The corresponding 2-D wavelet frequency responses are shown the transform that can be tuned to the spectral characteristics in Fig. 7a and 7c. Note the bandpass behavior of these filters of the signal; this is especially handy if we are dealing with IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 13 fractal processes because of the whitening properties of the ∂2 ∂2 fractional Laplacian [46]. In the sequel, we will set it to ∂ ∆ 2 + 2 ∆ ∂x1 ∂x2 γ = N + 2m with integer m ≥ 0 to establish a link with ∂x1 ! " conventional (non-fractional) derivative operators. The proposed wavelets appear to be well suited for feature extraction and analysis because they provide multiscale estimates of the Nth order derivatives of the image. Indeed, by using Eq. (19) that links the Riesz transform and partial derivatives, we get r N! ∂N ξ(x) (n1,...,nd) N (a) surface detector (b) line detector ψN+2m (x) = (−1) n n (30) n1! ··· nd! ∂ 1 x1 ··· ∂ d xd m Fig. 10. Isosurface representation of the first and second-order 3-D Riesz- where the smoothing kernel ξ(x) = −∆ β2N+4m(2x) is (n,0,0) Laplace wavelets ψn+2 (x) (white: positive threshold, black : negative isotropic-bandpass (mth Laplacian of a Gaussian-like func- threshold). (a) : surface or contour detector with n = 1. (b) : line or ridge tion). Note that for N > 1, we can select m = 0 to impose detector with n = 2. a Gaussian-like smoothing kernel instead of a Mexican hat. If we now consider the analysis of the signal f(x) at scale i with in Fig. 8. The first involves the Riesz-Laplace wavelets which the rescaled wavelet ψ(n1,...,nd)(x) = 2−id/2ψ(n1,...,nd)(x/2i) N+2m,i N+2m produces a local description of image features in terms of at location x , we obtain that 0 third-order derivatives [cf. (31)]. The second corresponds to N (n1,...,nd) ∂ (f ∗ ξi)(x) an equivalent directional filterbank with a basic detector (first hf, ψN+2m,i (· − x0)iL2 ∝ (31) ∂n1 x ··· ∂nd x Riesz wavelet) that is rotated in the four principal directions 1 d x=x0 θ = π/2, π/4, 0, −π/4 using the steering equations (13) and where we have made use of the adjoint and scale-covariance 0 (14) with u = (cos θ , sin θ ). This decomposition represents properties of the partial derivative operators. The interpre- 0 0 the “a` la Simoncelli” counterpart of the basic Riesz-Laplace tation of this relation is that the Riesz wavelet coefficients (n1,...,nd) (n ,...,n ) transform; it is obtained through the application of Theorem 2 w [k] = hf, R 1 d ψ i correspond to the partial N i i,k with H(ejθ) = (cos θ) with N = 3. These filters also have derivatives of order N (sampled at location x = 2ik) 0 a differential interpretation: they all correspond to third-order of the bandpass-filtered version (f ∗ ξ ) of the signal with i directional derivatives, in accordance with (20) and (25). The ξ (x) = 2−id/2ξ(x/2i). The non-trivial aspect of these i latter decomposition obviously achieves a better decomposi- partial-derivative wavelets is that the corresponding signal tion of the image in terms of directional components, while decomposition is fully reversible (frame property) and that the the former has a greater pattern diversity since it involves two transform admits an efficient perfect-reconstruction filterbank distinct templates (3, 0) and (2, 1) which can, in principle, also implementation. be steered in arbitrary directions. Interestingly, the simplest case (d, N, m) = (2, 1, 1) yields The comparison of the frequency responses of these two two gradient-like wavelets that happen to be mathematically systems is given in Fig 9. The directional character of the equivalent to the Marr wavelets described in [47]. An example “rotated-filterbank” decomposition is quite apparent in Fig of such a wavelet decomposition is shown in Fig. 6(b). 9(c). The steering is not perfect in the higher portion of In our earlier work, we have argued that such a transform the spectrum due to the “square-shaped” outer frequency could be interpreted as a multiscale edge detector and that profile of the primary wavelet decomposition, which is only it could yield a compact description of an image in terms approximately isotropic [cf. Fig 9(a)]. While this constitutes a of its primal wavelet sketch [47]. The interest of the present limitation of the current Laplacian-based design, the distorting formulation is that it gives the 3-D generalization of this effect is far less noticeable in the space domain because the complex representation without resorting to any sophisticated underlying polyharmonic B-spline smoothing kernel β in algebra (e.g., quaternions). 2γ (29) is guaranteed to converge to an isotropic Gaussian as γ The steerable Riesz-Laplace transforms for N > 1 are all increases. new. For instance, for (d, N, m) = (2, 2, 1), we obtain three parallel wavelet channels that provide a multiscale Hessian analysis. The corresponding three-channel directional wavelet VI.DISCUSSION transform is shown in Fig. 6(b). The frequency responses of The proposed operator-based framework is very general: it the 1st and 2nd order wavelet filters are displayed in Fig. 7. can yield a large variety of multidimensional wavelets with It is also interesting to consider the 3-D versions of these arbitrary orders of steerability. The goal that we are pursuing wavelets (i.e., (d, N, m) = (3, n, 1) with n = 1, 2), which are here is quite similar to that achieved by Simoncelli with his represented in Fig. 10. The first-order wavelet may serve as steerable pyramid [11]. However, there are two fundamental a contour/surface detector, while the second order wavelet is differences. First, Simoncelli’s approach is primarily based on more suitable for detecting lines or ridges. digital filterbank design. Ours, on the other hand, is analytical The scheme can be pushed further to obtain multiscale and rooted in functional analysis; the distinction is similar to image representations with higher orders of steerability. Two that between Mallat’s multiresolution theory, which provides a examples of 3rd order decompositions with γ = 5 are shown continuous-domain interpretation of wavelets, and the design ∂ ∆ β8 ∂x1

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 14 of critically-sampled perfect reconstruction ﬁlterbanks that actually predate wavelets but contain the algorithmic essence of this type of decomposition. The second difference is that, for N > 1, the prototypical Riesz wavelets are no longer rotated version of a single template, as is the case with traditional N=2

steerable filters. This is quite apparent in Figures 8 and 9. gamma=4 The “a` la Simoncelli” counterpart of the 3rd-order Riesz- ∂2 ∂2 1 ∂2 Laplace transform involves four rotated versions of the first ∆ 2 ∆ 2 2 ◦ ◦ ◦ ∂x ∂x − 3 ∂x filter uniformly spread over the circle (e.g., at 0 , 45 , 90 and 1 ! 1 2 " ◦ 135 ). In fact, the Riesz-Laplace wavelets that we introduced Fig. 11. Two examples of steerable 2nd-order wavelets. (a): Hessian-like (2,0) (2,0) (1,1) in Section V to illustrate our concept are much closer to wavelet ψ . (b) improved ridge detector: ψ − √1 ψ . 4 4 3 2 4 derivatives than to rotated filterbanks; we believe that this could be an advantage for analysis purposes. Our approach is obviously not limited to polyharmonic to be investigated. splines. It works best—i.e., provides exact steerability—when 3) Simplicity of implementation: The Riesz compo- the primary wavelets are isotropic; it also automatically yields nents/wavelet coefficients are straightforward to compute in a tight frame whenever the primary decomposition is tight to the Fourier domain using a FFT-based version of the fast start with. Other relevant considerations—including topics for wavelet transform algorithm. They can then be recombined lin- future research—are as follows: early as desired to generate a large variety of steerable wavelet 1) Decoupling between multiresolution and multiorienta- transforms, including rotated filterbanks. This computational tion properties: The proposed framework decouples the direc- approach simplifies the implementation and frees us from the tional and multi-scale aspects of wavelet design. While this is requirement of using finitely-supported filters. For the present valuable conceptually, it also suggests some new design chal- work, we did implement the 2-D Riesz-Laplace transforms lenges; i.e., the specification of wavelet frames with improved for orders up to 11, even though we did only show results for isotropy properties. The Riesz-Laplace wavelets that we have N = 1, 2, 3 (Figs. 6b-c, 8b). The “a` la Simoncelli” version of presented are very nearly isotropic, but they are not tight. these transforms (Fig. 8(c)) were obtained by performing the Alternatively, we could have considered a primary orthogonal adequate linear combinations of the Riesz wavelet coefficients; wavelet transformation, such a quincunx decomposition with we found this to be a convenient, cost-effective solution. good isotropy properties [48], which would have automatically 4) Exact reconstruction property: The exact reconstruction yielded a tight Riesz frame. The practical difficulty is to property comes as a direct consequence of Proposition 3 which optimally integrate the requirements of self-reversibility and ensures that the generalized Riesz transform preserves the isotropy (or radial symmetry) into a given computational frame property. In other words, we have a general mechanism structure; this is a research subject in its own right which for mapping any reversible multi-resolution decomposition we plan to investigate in future work. into its directional counterpart which is automatically guaran- 2) Steerable wavelets diversity: Our extended subspace- teed to be reversible as well. This is not necessarily so in the based definition of steerability opens up the perspective of a case of Simoncelli’s steerable pyramid. Indeed the structure of much larger variety of steerable wavelets than what has been the underlying filterbank is such that it will only yield perfect used so far. We are proposing two possible design strategies, reconstruction when the aliasing is completely suppressed; that both of which preserve self-reversibility. The first is to consider is, when the radial lowpass/highpass filters are bandlimited. a generalized Riesz transform of the form UR(N) where U The implication is that the reconstruction with finite length is any user-specified unitary matrix; for a justification, see filters will only be approximate. There is no such constraint (15). An attractive possibility is to derive U using some kind in the present scheme because the computational structure of principal component analysis. The second option, which is of the primary “isotropic” multiresolution decomposition is limited to 2-D, is to take advantage of Theorem 2 to specify essentially irrelevant, as long as it satisfies the frame property. a rotating filterbank profile that is optimized for a specific 5) Preservation of approximation theoretic properties: task. Since there is no constraint on the selection of the Because the Riesz transform and its higher order extension d coefficients of the filter profile (27), one can optimize these are well-defined convolution operators in L2(R ), they will for better angular selectivity or tune them to the characteristics preserve all important approximation theoretic properties of of a given type of pattern. For instance, we can optimize the primary wavelet transform that is used for the construction. the 2nd-order (or Hessian-like) wavelets for the task of ridge These include: detection using a Canny-like criterion [5]. The fact that such the order of approximation γ, which means that the a weight adjustment can improve the central elongation of the approximation error of a scale-truncated expansion decays wavelet is illustrated in Fig. 11; the first detector corresponds like the γth power of that scale. to the conventional choice (cos θ)2, while the second uses the initial number of vanishing moments; this implies that the optimal weights given in [5, Table 2]. The remarkable the Riesz wavelet coefficients are essentially zero in smooth feature is that the two proposed design methodologies are image areas where the image is well represented by its essentially constraint-free. The extent to which this increased lower-order Taylor series. flexibility can improve performance is an aspect that remains the Sobolev degree of smoothness of the basis functions. IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. XX, 2010 15

Another important aspect is that, even though the Riesz [5] M. Jacob and M. Unser, “Design of steerable filters for feature detection transform is a non-local operator, it will minimally impact the using Canny-like criteria,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 8, pp. 1007–1019, August 2004. decay of the wavelet functions. This is because its Fourier- [6] M. Michaelis and G. Sommer, “A Lie group approach to steerable domain singularity at the origin is tempered by the zeros filters,” Pattern Recognition Letters, vol. 16, no. 11, pp. 1165–1174, (vanishing moments) of the primary wavelets. 1995. [7] Y. Hel-Or and P. C. Teo, “Canonical decomposition of steerable func- 6) Generalization to higher dimensions: To the best of our tions,” Journal of Mathematical Imaging and Vision, vol. 9, no. 1, pp. knowledge, this is the first scheme that can provide steerable 83–95, 1998. wavelets in dimensions greater than 2. Such transforms could [8] P. C. 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