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

636 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 3, MARCH 2010 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 foundations one-to-one decomposition of a signal across scale; unfor- of a general continuous-domain framework for designing steer- tunately, the underlying wavelet bases (which are typically able, reversible signal transformations (a.k.a. frames) in multiple separable) have uneven angular responses; they are strongly dimensions ( ). To that end, we introduce a self-reversible, th-order extension of the Riesz transform. We prove that this biased towards the vertical and horizontal directions. generalized transform has the following remarkable properties: The steerable filters of Freeman and Adelson constitute an- shift-invariance, scale-invariance, inner-product preservation, and other elegant framework that is more specifically tuned to ori- steerability. The pleasing consequence is that the transform maps entation [2]. The primary focus there is on the image analysis any primary wavelet frame (or basis) of into another aspect of the problem and the search for an efficient computa- “steerable” wavelet frame, while preserving the frame bounds. The concept provides a functional counterpart to Simoncelli’s tional scheme for the filtering of an image with a reference tem- steerable pyramid whose construction was primarily based on plate that can be oriented arbitrarily. Several solutions have been filterbank design. The proposed mechanism allows for the speci- suggested for the design of optimized filtering templates [3]–[5]. fication of wavelets with any order of steerability in any number Researchers have also proposed a unifiying Lie-group formula- of dimensions; it also yields a perfect reconstruction filterbank tion of steerability that allows for the extension of the concept to algorithm. We illustrate the method with the design of a novel family of multidimensional Riesz-Laplace wavelets that essentially others classes of transformations (e.g., translation and dilation) behave like the th-order partial derivatives of an isotropic [6]–[8]. Gaussian kernel. It is possible—and highly desirable conceptually—to recon- Index Terms—Directional derivatives, frames, Hilbert trans- cile the two types of approaches by allowing for wavelet decom- form, multiresolution decomposition, Riesz transform, steerable positions with a certain level of redundancy [9]. In particular, Si- filters, wavelet transform. moncelli and co-workers were able to design a complete family of directional multiscale transforms which is indexed by the number of orientation bands [9]–[11]. Their steerable pyramid I. INTRODUCTION is over-complete by a factor and has the important property of being self-inverting. Steerable wavelets have also CALE and directionality are key considerations for vi- been proposed in the redundant framework of the continuous S sual processing and perception. Researchers in image wavelet transform, including some constructions on the sphere processing and applied mathematics have worked hard on for- [12]. While steerability is attractive conceptually, it is obviously malizing these notions and on developing some corresponding not a strict requirement; other popular solutions to the problem feature extraction algorithms and/or signal representation of directional multiresolution image representations include the schemes. 2-D Gabor transform [13], curvelets [14], the dual-tree complex The proper handling of the notion of scale in image pro- wavelet transform [15], [16], directional wavelet frames [17], cessing owes much to the work of Mallat who set the foundation contourlets [18], bandelets [19], and directionlets [20]. of the multiresolution theory of the wavelet transform which The steerable pyramid has been used quite extensively in involves continuously-defined functions in [1]. The image processing. Typical examples of applications include remarkable aspect of this theory is that it yields a reversible local orientation analysis, adaptive angular filtering, deformable templates [4], contour detection, shape from shading [2], [9], Manuscript received April 03, 2009; revised September 22, 2009. First texture analysis and retrieval [21]–[23], directional pattern published December 22, 2009; current version published February 18, 2010. detection [24], feature extraction for motion tracking [25] This work was supported in part by the Swiss National Science Foundation and, of course, image denoising [26], [27]. Mathematically, under Grants 200020-121763 and PP00P2-123438 and in part by the Center for Biomedical Imaging of the Geneva-Lausanne Universities and the EPFL, the steerable pyramid is a tight frame of , due to the as well as the foundations Leenaards and Louis-Jeantet. The associate editor self-reversibility of the underlying discrete filterbank. The coordinating the review of this manuscript and approving it for publication was link with wavelet theory would be complete if there was a Prof. Brian L. Evans. M. Unser is with the Biomedical Imaging Group (BIG), École Polytech- continuous-domain formulation of the transform, which has nique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (e-mail: not been worked out explicitly so far. The other point is that michael.unser@epfl.ch). Simoncelli’s construction is quite specific and difficult to V. Van De Ville is with the Institute of Bioengineering, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland , and also with generalize to higher dimensions. the Department of Radiology and Medical Informatics, University of Geneva, The purpose of this paper is to revisit the construction of steer- Switzerland(e-mail: dimitri.vandeville@epfl.ch). able wavelets and pyramids using a continuous-domain oper- Color versions of one or more of the fiures in this paper are available online at http://ieeexplore.ieee.org. ator-based formalism. The cornerstone of our approach is the Digital Object Identifier 10.1109/TIP.2009.2038832 Riesz transform which constitutes the natural multidimensional

Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 24,2010 at 10:23:30 EST from IEEE Xplore. Restrictions apply. UNSER AND VAN DE VILLE: WAVELET STEERABILITY AND THE HIGHER-ORDER RIESZ TRANSFORM 637 generalization of the Hilbert transform. The Riesz transform has has the fundamental property of conserving energy. Another a long tradition in mathematics [28] and has been studied exten- crucial property is that the components of our higher-order sively in the context of Calderón-Sygmund’s theory of singular Riesz transform define a subspace that is rotation-invariant, integral operators [29]. Its introduction to signal processing is which is the key to steerability. In Section III, we prove that more recent. Felsberg used the transform to define the “mono- the Riesz transform and its higher-order extension have the genic signal” as a 2-D generalization of the analytic signal rep- property of mapping a wavelet frame of onto another resentation [30], [31]. This pioneering work inspired Metikas one, while preserving the frame bounds. We thereby establish and Olhede to specify a monogenic version of the continuous a natural correspondence between a (quasi-isotropic) wavelet wavelet transform which is a fully redundant signal represen- frame and its directional Riesz counterparts. In Section IV, we tation [32]. At about the same time as Felsberg, Larkin inde- explicitly make the connection between the 2-D version of our pendently introduced a complexified version of the 2-D Riesz generalized Riesz transform and Simoncelli’s construction. In transform in optics—calling it the spiral quadrature phase trans- doing so, we also introduce an extended angular parametriza- form—and applied it to the demodulation of interferograms and tion of a rotating filterbank that ensures self-reversibility. In the analysis of fringe patterns [33], [34]. In recent work, we Section V, we illustrate our concept with the specification took advantage of the unitary property of this latter transform to of an extended family of high-order steerable Riesz-Laplace specify a proper, reversible wavelet-domain monogenic signal wavelets. The distinguishing features of these wavelets is that analysis with a minimal amount of redundancy [35]. they correspond to the th-order derivatives of a multidimen- Our present focus on the Riesz transform is motivated by its sional B-spline kernel. The Gaussian shape of the underlying invariance with respect to the primary coordinate transforma- smoothing kernel, which is common to all wavelets, makes the tions: translation, dilation and rotation (cf. Section II-D). The transform attractive for multidimensional feature extraction. first two invariance properties are ideally matched to the frame- We conclude the paper by summarizing the key features of work of wavelets, while the latter provides a connection with our approach and by discussing the similarities and differences steerable filters and gradient-like signal analyses [35], [36]. The with existing directional transforms. formal extension of this link calls for a higher-order version of the transform, which we introduce in Section II-E. Thanks to II. RIESZ TRANSFORM AND ITS HIGHER-ORDER EXTENSIONS this operator, we are able to take any multiresolution decomposition of and to map it into a series of directional coun- The purpose of this section is to lay the mathematical foun- terparts. The direct benefits of the proposed methodology are as dations for our approach in a fashion that is self-contained and follows. geared towards a signal processing audience. We start by re- • A general framework for constructing families of steerable viewing the primary features of the Riesz transform with a spe- wavelet frames and pyramids of with arbitrary or- cial emphasis on its invariance properties. We then introduce a ders1 in any number of dimensions . novel higher-order extension of the transform that is specifically • A clear decoupling between the multiresolution and the designed to preserve inner-products (Section II-E). The impor- steerability aspects of the representation. In the present tant aspect that we want to bring forth is the link between these work, we adopt a less constraining definition of a steer- generalized Riesz operators and steerable filterbanks, as intro- able transform which does not require the basis functions duced by Freeman and Adelson. to be rotated versions of one another—the basic requirement is that the component subspaces be rotation-invariant Notations [7]. This offers design flexibility and also carries over nat- We consider a generic -dimensional signal indexed by the urally to dimensions greater than two. continuous-domain space variable . The • A continuous-domain formulation that formalizes and multidimensional Fourier transform of is specified extends the technique proposed by Simoncelli et al.; in as essence, it represents the multiresolution theory of counterpart of their original filterbank design. The perfect reconstruction property and closedness in is automatically guaranteed. with . This definition is extended appro- • A full control over the regularity (Sobolev smoothness) and priately for Lebesgue’s space of finite energy functions , approximation theoretic properties (vanishing moments, as well as for Schwartz’s class of tempered distributions . order of approximation, etc.) of the basis functions. We recall that the squared -norm of is given by • The perspective of a purely analytical design of novel families of bona-fide steerable wavelets—in particular, higher- order Riesz-Laplace wavelets which admit explicit closed- form formulas in any number of dimensions. The paper is organized as follows. In Section II, we review the mathematical properties of the Riesz transform; we then proceed with the specification of a higher-order transform that where the right-hand side follows from Parseval’s identity. The adjoint of a linear operator is denoted 1In this paper, we do equate the notions of order of steerability and the order of the Riesz transform. by ; it is such that .

Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 24,2010 at 10:23:30 EST from IEEE Xplore. Restrictions apply. 638 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 3, MARCH 2010

The fractional Laplacian with is the B. Riesz Transform isotropic differential operator of order whose Fourier-domain The Riesz transform is the natural multidimensional exten- definition in the sense of distributions is sion of the Hilbert transform [29]. It is the scalar-to-vector signal transformation whose frequency-domain definition is (1) (3) For , it is a purely local operator that is propor- tional to the classical -fold Laplacian. Otherwise, its effect is where is the Fourier transform of the -dimensional input nonlocal and described by the following distributional impulse signal . This corresponds to a multidimensional -channel response [37]: filterbank whose space-domain description is

(2) . . . . (4) where is an appropriate normalization constant and is where the filters are characterized by their frequency Euler’s gamma function. The definition (1) of the fractional responses . To obtain explicit space-do- Laplacian can be extended to negative orders as well, which main expressions, it is helpful to view these filters as partial leads to a fractional integral-like behavior. The so-defined derivatives of the function (the im- family of fractional operators satisfies the composition rule pulse response of the isotropic integral operator ). which follows directly from Specifically, by making use of the Fourier-transform relation (2) their Fourier-domain definition, with the convention that (which is also valid for negative provided that . Note, however, that the fractional integral [37]), we get with is well-defined only for functions (or distributions) whose moments up to order are zero, due to the singularity of the frequency response at the origin. We will use the multi-index notation to state some of our results concisely: specifically, is a which we then differentiate to obtain -dimensional multi-index vector with non-negative inte- gers entries; ; for any ; and . With these conventions, the multinomial theorem reads The case is excluded from the above argument; in that particular instance, we have whose derivative yields . The explicit form of these various operators for , 2, 3, 4 is recapitulated in where the summation is over all multi-indices satisfying the Table I for further reference. condition . Note that the impulse responses of the Riesz transform are all anti-symmetric: and essen- A. One-Dimensional Hilbert Transform tially decaying like where is the number of dimensions. While the Riesz transform remains a nonlocal operator, The Hilbert transform plays a central role in the analytic it is interesting to observe that the delocalization becomes less signal formalism which is a powerful tool for AM/FM anal- pronounced in higher dimensions. It is also known from func- ysis [38]. It is a 1-D linear, shift-invariant operator whose tional analysis that the Riesz transform is a bounded operator defining property is to maps all cosine functions into their for the primary Lebesgue spaces; i.e., corresponding sine functions, without affecting their ampli- tude. The Hilbert operator, denoted by , therefore, acts as (5) an allpass filter; it is characterized by the transfer function . Its impulse response (in the where the are some suitable constants [29]. The singular sense of distributions) is , which is slowly aspect of the transform is that the extreme cases are decaying, expressing the fact that is a nonlocal operator. A excluded; this also reflects the fact that the Riesz transform is potential mathematical difficulty is that which not stable in the ordinary BIBO sense. Fortunately for us, the means that the corresponding filter is not BIBO-stable (bounded situation is particularly favorable for , as will be seen next input/bounded output), even though it is well-behaved in the (cf. Property 3). -sense (e.g., bounded and perfectly reversible). A remarkable property in this latter respect is that , C. Properties of the Multidimensional Riesz Transform indicating that is a unitary operator; this follows directly We now present and discuss the key properties of the operator from the Fourier-domain definition. .

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

1) Property 1 (Invariances): The Riesz transform is transla- The above result implies that the Riesz transform commutes tion- and scale-invariant with rotations, as expressed by the following property of its impulse response

where is the multidimensional Dirac The translation invariance directly follows from the defini- distribution. tion, while the scale invariance is easily verified in the Fourier Property 2 suggests linking the Riesz transform to the fol- domain. lowing directional version of the Hilbert transform Remarkably, the Riesz transform is also rotation-invariant. To formalize this property, we consider the group of rotation (6) matrices in . Specifically, let denote a rotation matrix whose first row is the unit vector ; whose impulse response simply corre- i.e., ( is such that it rotates the first sponds to the rotated version of . Note that the association coordinate vector into ). between the Riesz and the directional Hilbert transforms is 2) Property 2 (Steerability): The Riesz transform filterbank essentially the same as the link between the gradient and the defined by (4) is steerable in the sense that directional derivative (more on this later). It is instructive to determine the Riesz transform of a pure cosine wave with frequency (or wave number) . A simple Fourier calculation yields while the component filters are 90 rotated versions2 of each other; i.e., where . This is shown by using the rotation property of the Fourier We can then apply (6) and recover the corresponding sine wave transform by steering the directional Hilbert transform in the appropriate direction

This desirable behavior follows from the fact that the central cut of the frequency response of along the direction perfectly replicates the behavior of the 1-D Hilbert transform: 2In general, a single vector (or direction) is not sufﬁcient to specify a rotation . The only limitation of the technique is that the ideal matrix in dimensions. In the present case, however, the functions are Hilbert-transform-like behavior falls off like (the cosine isotropic within the hyperplane perpendicular to the corresponding coordinate vector so that our “geometrical” statements implicitly refer to a whole equiv- of the angle), which calls for a precise adjustment of the analysis alence class of rotation matrices. direction .

Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 24,2010 at 10:23:30 EST from IEEE Xplore. Restrictions apply. 640 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 3, MARCH 2010

3) Property 3 (Inner-Product Preservation): The Riesz The partial integration with respect to all frequency variables transform satisfies the following Parseval-like identity except yields the auxiliary function which is symmetric, as a consequence of our hypotheses. The product function is, therefore, anti-symmetric, which necessarily leads to a vanishing outer integral. While the assumptions for Property 4 do not hold for arbitrary signals, they are met by most basis functions commonly used for image processing. Indeed, the required symmetry relations are This is established by using Parseval’s relation for the Fourier satisfied by all functions that are either separable (because of the transform Hermitian symmetry of the 1-D Fourier transform), isotropic, or coordinate-wise symmetric/anti-symmetric in the space domain (up to some arbitrary translation). The smoothness of a function is often specified by its inclu- sion in the Sobolev space (the class of functions that are “ -times” differentiable in the -sense) [39]. A nice feature of the Riesz transform is that it fully preserves Sobolev regularity. 5) Property 5 (Sobolev Smoothness): for . Proof: The explicit definition of Sobolev’s space of order is

A similar manipulation leads to the identiﬁcation of the adjoint operator, which is speciﬁed by

(7) where is the isotropic derivative operator of order [cf. (1)]. The Sobolev smoothness equivalence then directly follows from the energy preservation property (8) and the translation in- where is a generic finite-energy variance of , which implies that -vector signal and . It also corresponds to the left inverse of the transform , owing to the fact that , and . A direct consequence is the . energy conservation property In this work, we will use the Riesz transform to map a set of primary wavelets into another augmented one. Property 1 and 3 are essential for that purpose because they ensure that (8) the mapping preserves the scale- and shift-invariant structure as well as the -stability of the representation. Note that the which emphasizes the unitary character of the transform. latter goes hand-in-hand with the reversibility of the transform We conclude this section by listing a few additional functional and its ability to perfectly represent all finite-energy signals. properties of the components of the Riesz transform for specific Thanks to Property 5, we also have the guarantee that the Riesz classes of signals or basis functions. wavelets are functionally well-behaved in that they retain the 4) Property 4 (Orthogonality): Let be same regularity (degree of differentiability) as the primary tem- a real-valued function whose Fourier transform satisfies plates from which they are derived. Property 4 is relevant as well the coordinate-wise symmetry relations: because it enforces a certain level of coordinate-wise orthogo- . Then nality which is potentially favorable for local feature extraction and signal analysis. Finally, we take advantage of Property 2 to design wavelet transforms whose basis functions are steerable.

Proof: The direct application of Parseval’s relation yields D. Higher-Order Riesz Transform

It is also possible to deﬁne higher-order Riesz transforms by considering individual signal components of the form with . While there are possible ways of forming such th-order terms, there are actually much less distinct Riesz components due to the commutativity and factorization properties of the underlying convolution operators. Our deﬁnition of higher-order transform removes this intrinsic redundancy and is based on the following operator identity.

Theorem 1: The th-order Riesz transform achieves the fol- In some instances, we will use the more concise vector notation lowing decomposition of the identity , keeping in mind that each component is associated with its own normalized, th-order Riesz operator , as defined above. Note that the number of higher-order components roughly increases like the th power of the dimension. For instance, , , and using the multi-index vector . . The good news is that the increase is moderate in dimen- Here, denotes the -fold iterate of , while the sion 2 (the context of image processing) and still manageable weighting factors in the expansion are the multinomial coeffi- in dimension 3 (volumetric imaging). The case is unin- cients of order teresting since the higher-order transform cycles back and forth between the identity and the conventional Hilbert transform. Not too surprisingly, the higher-order Riesz transform in- herits all the fundamental properties of the operator , as these are preserved through the iteration mechanism. Proof: By applying the Theorem’s left-hand side decom- 1) Property 6: The higher-order Riesz transforms are position to an input signal , we obtain a series of components translation and scale-invariant. of the form What is more remarkable is the higher-order inner-product preservation property—it comes as a corollary of the decomposition of the identity given by Theorem 1. 2) Property 7: The th-order Riesz transform satisfies the This corresponds to a convolution whose Fourier-domain equiv- following Parseval-like identity alent is

In particular, by setting in the above property, we observe that the higher-order Riesz transform perfectly preserves (9) the -norm of the signal. The generalized version of this result is the boundedness of the higher-order Riesz transform for Now, we obviously have the identity the classical Lebesgue spaces (with the notable exception of ). 3) Property 8: The higher-order Riesz transform satisﬁes the following norm inequalities:

Next, we apply the multinomial theorem to expand the central factor where the are appropriate constants. Since the higher-order Riesz transform can be decomposed into a succession of first-order transforms, the result is a consequence of the boundedness of all elementary component transformations [cf. (5)]. The final important feature is rotation invariance; it is inti- mately linked to the concept of steerability which is discussed where we have identified the relevant higher-order reconstruc- in the next section. tion components using (9). From the above, it is clear that an th-order Riesz trans- E. Higher-Order Steerability form decomposes a signal into distinct components, The most interesting aspect of the higher-order Riesz trans- where is the cardinality of the indexing form is its steerability which also comes hand-in-hand with set . We also want to ensure that the improved angular selectivity. To best explain this behavior, we transform preserves energy which calls for an appropriate nor- relate to the -fold version of the directional Hilbert malization. We, therefore, specify the th-order Riesz trans- transform along the direction specified by the unit vector form of the signal as the -vector signal transformation . To that end, we write the Fourier-domain equivalent of (6) whose components are given by

(10) (11)

Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 24,2010 at 10:23:30 EST from IEEE Xplore. Restrictions apply. 642 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 3, MARCH 2010

Fig. 1. Steerable ﬁlterbank implementation of the th-order directional Hilbert transform in multiple dimensions. The output signal is formed by taking an appropriate linear combination of the components of the th-order Riesz transform of the signal. The scheme is adaptive in the sense the direction of analysis may vary spatially.

We then iterate the operator times and expand the right-hand invariant subspace , the argument also extends for side using the multinomial theorem generalized Riesz operators of the form where is a given -vector of weights. This implies that the function space is rotation-invariant and that there is a simple weighting mechanism for steering generalized Riesz operators in arbitrary directions. Following this line of thought, it is also possible to deﬁne generalized higher-order Riesz operators of the form where A is a nonsingular complex matrix. This generalized operator will share all the fundamental properties of (shift-invariance, scale-invariance and steerability) and (12) will preserve inner-products provided that is a unitary matrix. Indeed, such a representations leads to the following Finally, we go back to the space domain, which yields decomposition of the identity which comes as a corollary of Theorem 1

(15) (13)

This simple mechanism offers additional degrees of freedom where the “steering” coefficients are given by for shaping higher-order impulse responses and designing steerable, reversible filterbanks in arbitrary dimensions. (14) It is also instructive to investigate the directional behavior of the operator which corresponds to the -directional version The signal processing transcription of (13) leads to the block of the operator . The important point is that its frequency diagram in Fig. 1. The corresponding system falls within response does the general framework of steerable filterbanks of Freeman not depend upon the radial frequency , but only upon and Adelson [2], although these authors did not pursue their the angle between and the direction vector . It is max- concept for dimensions larger than 3. The main point is that imum along the direction ; along that ray, we can apply a -channel filterbank with impulse re- which coincides (up to a sign change) with the sponses to the signal and then compute 1-D identity when is even (symmetric operator) and the 1-D along any direction by forming an appropriate Hilbert transform otherwise (anti-symmetric operator). As in- linear combination with the weights given by (14). While some creases from 0 to , the response falls off like —a authors define the order of steerability as the dimension of the clear indication that angular selectivity improves with .

F. Connection With the Gradient and Partial-Derivatives Deﬁnition 1: A family of functions is called a There is a direct connection between the Riesz transform frame of iff. there exists two strictly positive constants and the gradient, which is easily seen in the frequency domain. and such that Speciﬁcally, we have that

(21) (16) The frame is called tight iff. the frame bounds can be chosen such that . If , we have a Parseval frame that where is the square-root-Laplace operator whose fre- satisﬁes the remarkable decomposition/reconstruction formula quency is [cf. Table I]. The converse relation is

(17) which should be interpreted as a smoothed version of the gra- While the form of this expansion is the same as that associated dient (up to a sign change); the integral operator acts with an orthonormal basis, the fundamental aspect of the frame on all derivative components and has an isotropic smoothing generalization is that the family may be redundant. More effect. generally, we have that The directional counterparts of this last equation is

(18) where is defined by (6) and where is the where is a so-called dual frame. While the dual frame is derivative of along the direction . not unique in the redundant case, there is one—the so-called By iteration, we obtain the link with the partial derivatives of canonical one—that is used preferentially since it maps into the order minimum-norm inverse [42]. We are now ready to state the fundamental frame preservation property of the th-order Riesz transform, which generalizes the first-order result of Held et al. (19) Proposition 1: The th-order Riesz transform maps the frame of into another frame and the higher-order directional derivatives that has the same frame bounds and a (20) redundancy increased by . Proof: First we observe that The crucial element here is the fractional integral operator of where is the adjoint of the -com- order : . It is an isotropic ponent Riesz operator defined by (10). We can, therefore, lowpass filter whose smoothing strength increases with . apply the frame inequality (21) to each (adjoint) component of The global interpretation of these relations is that there is a the th-order Riesz transform one-to-one connection between Riesz components and spatial derivatives with the former being smoothed versions of the latter. The Riesz transform, therefore, captures the same directional information as derivatives but it has the advantage of Next, we sum up these inequalities over the multi-index com- being much better conditioned since there is no amplification ponents , which yields of high frequencies. We will now see how we can exploit its invariance and reversibility properties to construct new steerable wavelet-type decompositions.

where we have made use of the energy preservation feature of III. GENERAL CONSTRUCTION OF STEERABLE the Riesz transform which is implied by Property 7. The propo- WAVELET FRAMES sition then simply follows from the adjoint-defining relation The pleasing consequence of Property 3 is that the Riesz . transform will automatically map any frame of into an- What is of even greater significance to us is that the th-order other one. This fundamental observation was made indepen- Riesz transform will transform any wavelet frame into another dently by Held et al. [36] in full generality, and by us in the one, thanks to its translation- and scale-invariance properties. more constrained 2-D setting, considering a complex version of Specifically, let us consider the following wavelet decomposi- the Riesz transform that defines a unitary mapping from tion of a finite energy signal: into itself [35]. Here, we will present a proper th-order extension of these ideas. Before establishing the results, we start by (22) recalling a few basic facts about frames [40]–[42].

Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 24,2010 at 10:23:30 EST from IEEE Xplore. Restrictions apply. 644 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 3, MARCH 2010 where the wavelets at scale are dilated versions of the ones where the right-hand side of the equality is valid provided that at the reference scale : . the primal wavelet is isotropic. Similar steering General wavelet schemes also have an inherent translation-in- equations can be derived for all other Riesz wavelet compo- variant structure in that the ’s are constructed from the in- nents. Concretely, this means that the global wavelet analysis teger shifts (index ) of up to distinct generators (mother is rotation-covariant. wavelets). Finally, if the wavelet decomposition (22) is build around a For convenience, we now use a multi-index representation of Mallat-type multiresolution analysis and admits a fast filterbank the corresponding th-order Riesz wavelets, which are defined algorithm, then there is a corresponding perfect reconstruction as follows: filterbank implementation for its th-order Riesz counterpart in Proposition 2. (23) IV. GENERALIZED RIESZ TRANSFORM AND 2-D STEERABLE FILTERBANKS with and . In 2-D, there is an interesting connection between the gen- Proposition 2: Let be a primal wavelet frame of eralized Riesz transform (as defined in Section II-E), steerable associated with the reconstruction formula (22). Then, filterbanks that are polar-separable, and the steerable pyramid and form a dual set of Simoncelli et al. [11]. of wavelet frames satisfying the general decomposition/recon- The th-order Riesz transform for has dis- struction formula tinct components. The corresponding frequency responses span the subspace , which is rotation-invariant—this is simply the frequency-domain transpo- sition of what has already been stated for the space domain in Proof: Theorem 1 is equivalent to the following decompo- Section II-F. The specific form of these filters is sitions of the identity operator:

(24) which allows us to interchange the role and when ana- lyzing (resp. synthesizing) a signal. First, we apply the wavelet decomposition (22) to the transformed signals for all such that where we have made the polar-coordinate substitution and . The main point is that these basis functions are purely polar and that they are completely characterized by the -periodic radial proﬁle functions

We then make use of (24) to resynthesize (26) with . The complete set of 2-D basis functions for 2, 3, 4 are displayed in Fig. 2. For any given order, we can associate the filters in pairs that are rotated with respect to each which yields the desired reconstruction formula by linearity. other by 90 . As the order increases, the frequency patterns The final ingredient for this construction is Property 6, which en- become more intricate and distinct from those observed in sures that the wavelet structure is conserved; i.e., conventional steerable filterbanks where all components are rotated versions of a single template. The case calls , as well as the basic shift-invariant struc- for a comment: even though the central pattern looks visually ture that involves a finite number of generators which are the similar to the two others, the filter is not a rotated version higher-order Riesz transforms of the primary mother wavelets. of the two others because one also has to take the range of the display into consideration; e.g., ( , ) If, in addition, the mother wavelets in (22) are for and ( , ) (quasi-)isotropic, then we end up with a decomposition otherwise. that is steerable (cf. Section II-F). For instance, we can As far as steerability is concerned, the important property is use the steering (13) to rotate the first component wavelet that the profile functions are angularly “bandlimited” [2]; that to any desired orientation is, their Fourier series have a limited number of terms. In fact, they all have the form of a -term truncated Fourier series (25) with , as is readily checked by expanding (26).

Fig. 2. Frequency responses (real or imaginary part) of the components ﬁlters of the ﬁrst, second and third-order Riesz transform. The origin of the 2-D frequency domain is in the center and the intensity scale is stretched linearly for maximum contrast. The plots are labeled by with .

The idea of the generalized Riesz transform is to construct correspond to the autocorrelation of the sequence (of length new basis functions by taking suitable linear combinations of ); in particular, this implies that the terms with index the primary ones. Each of these basis functions will be polar as are necessarily zero. Therefore, if we down-sample well and characterized by a generic proﬁle function the sequence by a factor , we are left with a single nonzero coefﬁcient at the origin: . In the frequency domain, this down-sampling operation corresponds (27) to a periodization, leading to the identity

A truly remarkable property is that we can use any profile function to define a self-inverting generalized Riesz transform that has the structure of a steerable filterbank corresponding to the where the right-hand side is the Fourier transform of the re- block diagram in Fig. 4. maining impulse. Next, we note that is -periodic Theorem 2: Let where the since the coefficients are real-valued (or purely imaginary). We ’s are arbitrary real-valued (or purely imaginary) coeffi- can, therefore, divide the sum in two equal parts, which leads to cients. Then, the -channel filterbank whose filters for the partition of unity formula are given by

This is equivalent to is steerable and self-inverting. Proof: We make the substitution and consider the which ensures that the filterbank in Fig. 4 has the perfect recon- Laurent polynomial . The crucial ob- struction property. servation is that the coefficients of the product polynomial While the proof relies on rather basic signal processing con- cepts, this result does not appear to have been reported before; at least not in such general terms. It allows for generalizations of the design of Simoncelli et al. who used the profile function to construct their steerable pyramid,

Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 24,2010 at 10:23:30 EST from IEEE Xplore. Restrictions apply. 646 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 3, MARCH 2010

Fig. 3. Frequency responses of the angular ﬁlters of Simoncelli’s third (a) and fourth (b) order steerable pyramids. The origin of the 2-D frequency domain is in the center and the intensity scale is stretched linearly for maximum contrast.

Fig. 4. Two-dimensional self-invertible steerable filterbank. The filters are rotated versions of each other with an equi-angular spacing. Their frequency responses are purely polar and angularly bandlimited. They span the same space as the components of the th-order Riesz transform. having noted that this function would satisfy the required self- reversibility condition (up to some unspecified normalization constant). The autocorrelation filter in that particular case is

allowing for the identiﬁcation of . The application of Theorem 2 then yields the normalized version of Simoncelli’s original reproduction formula

Fig. 5. Original 256 256 cameraman with subimage.

We would like to emphasize that the results in this section are which is given here for reference. This particular choice of polar also directly relevant for the design of general 2-D polar-sepa- profile corresponds to a steerable implementation of the direc- rable steerable filters whose polar frequency response is of the tional Hilbert transform operator , which clearly form where is a given radial fre- falls within the framework of the generalized Riesz transform. quency window. This factorization allows for a complete de- The corresponding frequency responses for and coupling of the radial and angular parts of the design. Since the are given in Fig. 3(a) and (b), respectively, and self-reversibility of the angular part is automatically guaranteed should be compared to the second and third row in Fig. 2. The (thanks to Theorems 1 and 2), the designer can concentrate on filterbanks are identical for , but not otherwise. Yet, they the problem of specifying a radial (or quasi-isotropic) multi-res- span the same spaces and define a self-invertible transform in olution decomposition that ensures a perfect (or near perfect) either formulation. reconstruction, which is essentially the approach that was taken

Fig. 6. Reversible wavelet decompositions of Cameraman image (zoom on a 64 64 subimage). (a)-(b) Primary Laplace wavelet pyramid with and its corresponding first-order Riesz-Laplace (or Marr) wavelet transform. (c)-(d) Primary Laplace wavelet pyramid with and its corresponding second-order Riesz-Laplace wavelet transform. by Simoncelli et al. in [11]. An important point, though, is that subspaces that involves a single quasi-isotropic analysis wavelet the present framework gives access to a larger variety of angular (cf [43, Section IV-B]) profiles and steerable basis functions than what has been considered so far. (28) where is the order (possibly fractional) of the wavelet. V. H IGH-ORDER STEERABLE RIESZ-LAPLACE WAVELETS The function is the “most-isotropic” polyharmonic B-spline of order ; it is a smoothing kernel that converges to While the construction method outlined in Section III is ap- a Gaussian as increases [44, Proposition 2]. plicable to any primary wavelet transform, it will give the most The corresponding “Laplacian” analysis wavelets in (22) are satisfactory results when the underlying primary wavelets are given by and the wavelet nearly isotropic. We will, therefore, illustrate the concept using transform has a fast reversible filterbank implementation, as a primary Laplacian-like (or Mexican hat) decomposition of described in [43]. The resulting wavelet decomposition has a [43]. pyramid-like structure that is similar to the Laplacian pyramid of Burt and Adelson [45]. The fundamental difference, however, is that the proposed transform involves a single analysis wavelet A. Primary Laplacian-Like Wavelets (28) that is specifically tied to the operator . Two examples of wavelet decompositions with , 4 of Cameraman Polyharmonic splines are the natural spline functions associ- (cf. Fig. 5) are displayed in Fig. 6(a) and (c); we have zoomed on ated with the Laplace operator; they can be used to specify a a fragment of the image to facilitate the visualization. The de- family of multiresolution analyses which is indexed by composition is a frame with a small redundancy factor ( the order of the Laplacian [44]. In prior work, we have intro- in 2-D and for ). The corresponding 2-D wavelet duced a pyramid-like representation of the underlying wavelet frequency responses are shown in Fig. 7(a) and (c). Note the

Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 24,2010 at 10:23:30 EST from IEEE Xplore. Restrictions apply. 648 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 3, MARCH 2010

Laplacian [46]. In the sequel, we will set it to with integer to establish a link with conventional (nonfrac- tional) derivative operators. The proposed wavelets appear to be well suited for feature extraction and analysis because they provide multiscale estimates of the th-order derivatives of the image. Indeed, by using (19) that links the Riesz transform and partial derivatives, we get

(30)

where the smoothing kernel is isotropic-bandpass ( th Laplacian of a Gaussian-like function). Note that for , we can select to impose a Gaussian- like smoothing kernel instead of a Mexican hat. If we now consider the analysis of the signal at scale with the rescaled wavelet at location , we obtain that

(31)

where we have made use of the adjoint and scale-covariance properties of the partial derivative operators. The interpretation of this relation is that the Riesz wavelet coefficients correspond to the partial derivatives of order (sampled at location ) of the bandpass-filtered version of the signal with . The nontrivial aspect of these partial-derivative wavelets is that the corresponding signal decomposition is fully reversible (frame property) and that the Fig. 7. Frequency responses (real or imaginary part) of the wavelet analysis transform admits an efficient perfect-reconstruction filterbank filters associated with Fig. 6. (a)-(b) Primary Laplace wavelets implementation. with and first-order Riesz-Laplace wavelets and . Interestingly, the simplest case yields (c)-(d) Primary Laplace wavelets with and second-order Riesz-Laplace wavelets , and . The scale is two gradient-like wavelets that happen to be mathematically increasing from upto 2 as one moves down. equivalent to the Marr wavelets described in [47]. An example of such a wavelet decomposition is shown in Fig. 6(b). In our earlier work, we have argued that such a transform could bandpass behavior of these filters and their good isotropy prop- be interpreted as a multiscale edge detector and that it could erties, especially over the lower frequency range. yield a compact description of an image in terms of its primal wavelet sketch [47]. The interest of the present formulation B. Steerable Riesz-Laplace Wavelets is that it gives the 3-D generalization of this complex repre- The associated th-order steerable Riesz-Laplace wavelet sentation without resorting to any sophisticated algebra (e.g., transform is obtained through the direct application of Proposi- quaternions). tion 2. The explicit form of the underlying directional analysis The steerable Riesz-Laplace transforms for are all wavelets is new. For instance, for , we obtain three parallel wavelet channels that provide a multiscale Hessian analysis. The corresponding three-channel directional wavelet trans- (29) form is shown in Fig. 6(c). The frequency responses of the first for all multi-indices with . The en- and second-order wavelet filters are displayed in Fig. 7. It is also lightening aspect in this formula is that the directional behavior interesting to consider the 3-D versions of these wavelets (i.e., of these wavelets is entirely encoded in the differential opera- with 1, 2), which are represented in tors that are acting on the B-spline smoothing kernel: the Riesz Fig. 10. The first-order wavelet may serve as a contour/surface transform factor is steerable while the Laplacian part, which is detector, while the second-order wavelet is more suitable for de- responsible for the vanishing moments, is perfectly isotropic. tecting lines or ridges. The order parameter is a degree of freedom of the The scheme can be pushed further to obtain multiscale transform that can be tuned to the spectral characteristics of the image representations with higher orders of steerability. Two signal; this is especially handy if we are dealing with fractal examples of third-order decompositions with are shown processes because of the whitening properties of the fractional in Fig. 8. The first involves the Riesz-Laplace wavelets which

Fig. 8. Reversible wavelet decompositions of Cameraman image (zoom on a 64 64 subimage). (a) Primary Laplace wavelet pyramid with . (b) Third-order Riesz-Laplace wavelet transform. (c) Rotated filterbank spanning the same steerable wavelet subspaces as (b). produces a local description of image features in terms of VI. DISCUSSION third-order derivatives [cf. (31)]. The second corresponds to an equivalent directional filterbank with a basic detector (first The proposed operator-based framework is very general: it Riesz wavelet) that is rotated in the four principal directions can yield a large variety of multidimensional wavelets with ar- using the steering relations (13) and bitrary orders of steerability. The goal that we are pursuing here (14) with . This decomposition represents is quite similar to that achieved by Simoncelli with his steer- the “à la Simoncelli” counterpart of the basic Riesz-Laplace able pyramid [11]. However, there are two fundamental differ- transform; it is obtained through the application of Theorem 2 ences. First, Simoncelli’s approach is primarily based on dig- with with . These filters also have ital filterbank design. Ours, on the other hand, is analytical and a differential interpretation: they all correspond to third-order rooted in functional analysis; the distinction is similar to that be- directional derivatives, in accordance with (20) and (25). The tween Mallat’s multiresolution theory, which provides a contin- latter decomposition obviously achieves a better decomposition uous-domain interpretation of wavelets, and the design of crit- of the image in terms of directional components, while the ically-sampled perfect reconstruction filterbanks that actually former has a greater pattern diversity since it involves two predate wavelets but contain the algorithmic essence of this type distinct templates and which can, in principle, also of decomposition. The second difference is that, for , the be steered in arbitrary directions. prototypical Riesz wavelets are no longer rotated version of a The comparison of the frequency responses of these two sys- single template, as is the case with traditional steerable filters. tems is given in Fig. 9. The directional character of the “ro- This is quite apparent in Figs. 8 and 9. The “à la Simoncelli” tated-filterbank” decomposition is quite apparent in Fig. 9(c). counterpart of the third-order Riesz-Laplace transform involves The steering is not perfect in the higher portion of the spec- four rotated versions of the first filter uniformly spread over the trum due to the “square-shaped” outer frequency profile of the circle (e.g., at 0 , 45 , 90 and 135 ). In fact, the Riesz-Laplace primary wavelet decomposition, which is only approximately wavelets that we introduced in Section V to illustrate our con- isotropic [cf. Fig. 9(a)]. While this constitutes a limitation of the cept are much closer to derivatives than to rotated filterbanks; we current Laplacian-based design, the distorting effect is far less believe that this could be an advantage for analysis purposes. noticeable in the space domain because the underlying polyhar- Our approach is obviously not limited to polyharmonic monic B-spline smoothing kernel in (29) is guaranteed to splines. It works best—i.e., provides exact steerability—when converge to an isotropic Gaussian as increases. the primary wavelets are isotropic; it also automatically yields

Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 24,2010 at 10:23:30 EST from IEEE Xplore. Restrictions apply. 650 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 3, MARCH 2010

Fig. 9. Frequency responses (real or imaginary part) of the wavelet analysis ﬁlters associated with Fig. 8. (a) Primary Laplace wavelets with . (b) Third-order Riesz-Laplace wavelets with . (c) Corresponding “à la Simoncelli” directional ﬁlterbank.

Fig. 11. Two examples of steerable second-order wavelets. (a): Hessian-like wavelet . (b) improved ridge detector: .

than what has been used so far. We are proposing two possible Fig. 10. Isosurface representation of the first and second-order 3-D design strategies, both of which preserve self-reversibility. The Riesz-Laplace wavelets (white: positive threshold, black: first is to consider a generalized Riesz transform of the form negative threshold). (a) Surface or contour detector with . (b) Line or ridge detector with . where is any user-specified unitary matrix; for a justification, see (15). An attractive possibility is to derive a tight frame whenever the primary decomposition is tight to using some kind of principal component analysis. The second start with. Other relevant considerations—including topics for option, which is limited to 2-D, is to take advantage of The- future research—are as follows. orem 2 to specify a rotating filterbank profile that is optimized A. Decoupling Between Multiresolution and Multiorientation for a specific task. Since there is no constraint on the selection Properties of the coefficients of the filter profile (27), one can optimize these for better angular selectivity or tune them to the charac- The proposed framework decouples the directional and teristics of a given type of pattern. For instance, we can opti- multiscale aspects of wavelet design. While this is valuable mize the second-order (or Hessian-like) wavelets for the task of conceptually, it also suggests some new design challenges; ridge detection using a Canny-like criterion [5]. The fact that i.e., the specification of wavelet frames with improved isotropy such a weight adjustment can improve the central elongation properties. The Riesz-Laplace wavelets that we have presented of the wavelet is illustrated in Fig. 11; the first detector corre- are very nearly isotropic, but they are not tight. Alternatively, sponds to the conventional choice , while the second we could have considered a primary orthogonal wavelet trans- uses the optimal weights given in [5, Table 2]. The remarkable formation, such a quincunx decomposition with good isotropy feature is that the two proposed design methodologies are essen- properties [48], which would have automatically yielded a tight tially constraint-free. The extent to which this increased flexi- Riesz frame. The practical difficulty is to optimally integrate bility can improve performance is an aspect that remains to be the requirements of self-reversibility and isotropy (or radial investigated. symmetry) into a given computational structure; this is a research subject in its own right which we plan to investigate in C. Simplicity of Implementation future work. The Riesz components/wavelet coefficients are straightforward to compute in the Fourier domain using a FFT-based ver- B. Steerable Wavelets Diversity sion of the fast wavelet transform algorithm. They can then Our extended subspace-based definition of steerability opens be recombined linearly as desired to generate a large variety up the perspective of a much larger variety of steerable wavelets of steerable wavelet transforms, including rotated filterbanks.

This computational approach simplifies the implementation and VII. CONCLUSION frees us from the requirement of using finitely-supported filters. For the present work, we did implement the 2-D Riesz-Laplace We introduced a general operator framework that provides transforms for orders up to 11, even though we did only show re- a mathematical bridge between the continuous-domain theory sults for 1, 2, 3 [Figs. 6(b)-(c), 8(b)]. The “à la Simoncelli” of the wavelet transform and steerable filterbanks. In effect, we version of these transforms [Fig. 8(c)] were obtained by per- are taking advantage of the invariance properties (with respect forming the adequate linear combinations of the Riesz wavelet to translation, scaling and rotation) of the generalized Riesz coefficients; we found this to be a convenient, cost-effective so- transform, together with its self-reversibility, to decouple the lution. directional and multiscale aspects of wavelet design. The primary theoretical contribution is a rigorous and straightforward D. Exact Reconstruction Property formalism for the specification of steerable wavelet transforms The exact reconstruction property comes as a direct con- in the continuous-domain—i.e., as frames of —while sequence of Proposition 2 which ensures that the generalized ensuring that they are endowed with all important approxima- Riesz transform preserves the frame property. In other words, tion-theoretic wavelet properties: order of approximation, van- we have a general mechanism for mapping any reversible ishing moments, regularity, etc. multiresolution decomposition into its directional counterpart The present work is also practically relevant because of the which is automatically guaranteed to be reversible as well. new perspective that it offers for image processing. Interesting This is not necessarily so in the case of Simoncelli’s steerable topics for future research include the design of novel families of pyramid. Indeed the structure of the underlying filterbank is steerable wavelet transforms (in particular, tight frames), the de- such that it will only yield perfect reconstruction when the velopment of steerable wavelet algorithms for the processing of aliasing is completely suppressed; that is, when the radial volumetric data, and the investigation of the concept of steerable lowpass/highpass filters are bandlimited. The implication is wavelet diversity—in particular, signal-adapted design—which that the reconstruction with finite length filters will only be holds good promises for data processing (e.g., denoising) and approximate. There is no such constraint in the present scheme feature extraction. because the computational structure of the primary “isotropic” multiresolution decomposition is essentially irrelevant, as long REFERENCES as it satisfies the frame property. [1] S. G. Mallat, “A theory of multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell., vol. E. Preservation of Approximation Theoretic Properties 11, no. 7, pp. 674–693, Jul. 1989. [2] W. T. Freeman and E. H. Adelson, “The design and use of steerable Because the Riesz transform and its higher order extension filters,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 13, no. 9, pp. are well-defined convolution operators in , they will pre- 891–906, Sep. 1991. serve all important approximation theoretic properties of the pri- [3] E. Simoncelli and H. Farid, “Steerable wedge filters for local orientation analysis,” IEEE Trans. Image Process., vol. 5, no. 9, pp. mary wavelet transform that is used for the construction. These 1377–1382, Sep. 1996. include: [4] P. Perona, “Deformable kernels for early vision,” IEEE Trans. Pattern • the order of approximation , which means that the approx- Anal. Mach. Intell., vol. 17, no. 5, pp. 488–499, May 1995. [5] M. Jacob and M. Unser, “Design of steerable filters for feature detec- imation error of a scale-truncated expansion decays like the tion using Canny-like criteria,” IEEE Trans. Pattern Anal. Mach. Intell., th power of that scale; vol. 26, no. 8, pp. 1007–1019, Aug. 2004. • the initial number of vanishing moments; this implies [6] M. Michaelis and G. Sommer, “A Lie group approach to steerable filters,” Pattern Recognit. Lett., vol. 16, no. 11, pp. 1165–1174, 1995. that the Riesz wavelet coefficients are essentially zero in [7] Y. Hel-Or and P. C. Teo, “Canonical decomposition of steerable func- smooth image areas where the image is well represented tions,” J. Math. Imag. Vis., vol. 9, no. 1, pp. 83–95, 1998. by its lower-order Taylor series; [8] P. C. Teo and Y. Hel-Or, “Lie generators for computing steerable functions,” Pattern Recognit. Lett., vol. 19, no. 1, pp. 7–17, 1998. • the Sobolev degree of smoothness of the basis functions. [9] E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger, Another important aspect is that, even though the Riesz trans- “Shiftable multiscale transforms,” IEEE Trans. Inf. Theory, vol. 38, no. form is a nonlocal operator, it will minimally impact the decay 2, pp. 587–607, 1992. [10] A. Karasaridis and E. Simoncelli, “A filter design technique for of the wavelet functions. This is because its Fourier-domain sin- steerable pyramid image transforms,” in Proc. IEEE Int. Conf. Acous- gularity at the origin is tempered by the zeros (vanishing mo- tics, Speech, and Signal Processing, May 7–10, 1996, vol. 4, pp. ments) of the primary wavelets. 2387–2390. [11] E. Simoncelli and W. Freeman, “The steerable pyramid: a flexible ar- F. Generalization to Higher Dimensions chitecture for multi-scale derivative computation,” in Proc. Int. Conf. Image Processing, Oct. 23–26, 1995, vol. 3, pp. 444–447. To the best of our knowledge, this is the first scheme that can [12] Y. Wiaux, L. Jacques, and P. Vandergheynst, “Correspondence principle between spherical and Euclidean wavelets,” Astrophys. J., vol. provide steerable wavelets in dimensions greater than 2. Such 632, p. 15, 2005. transforms could turn out to be valuable for the analysis and [13] J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural processing of 3-D biomedical data sets (e.g., X-ray tomograms, networks for image-analysis and compression,” IEEE Trans. Acoust. Speech Signal Process., vol. 36, no. 7, pp. 1169–1179, Jul. 1988. MRI, and confocal micrographs). By contrast, the extension of [14] E. J. Candès and D. L. Donoho, “Ridgelets: A key to higher-dimen- Simoncelli’s steerable pyramid, which uses rotated filters that sional intermittency?,” Phil. Trans. Roy. Soc. Lond. A., pp. 2495–2509, are angularly uniformily distributed, to 3-D does not seem fea- 1999. [15] N. Kingsbury, “Image processing with complex wavelets,” Philos. sible because of the lack of a systematic mechanism for uni- Trans. Roy. Soc. Lond. A, Math. Phys. Eng. Sci., vol. 357, no. 1760, formly tiling the sphere. pp. 2543–2560, 1999.

Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 24,2010 at 10:23:30 EST from IEEE Xplore. Restrictions apply. 652 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 3, MARCH 2010

[16] I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, “The dual-tree [43] M. Unser and D. Van De Ville, “The pairing of a wavelet basis with a complex wavelet transform,” IEEE Signal Process. Mag., vol. 22, no. mildly redundant analysis via subband regression,” IEEE Trans. Image 6, pp. 123–151, 2005. Process., vol. 17, no. 11, pp. 2040–2052, Nov. 2008. [17] P. Vandergheynst and J.-F. Gobbers, “Directional dyadic wavelet trans- [44] D. Van De Ville, T. Blu, and M. Unser, “Isotropic polyharmonic forms: Design and algorithms,” IEEE Trans. Image Process., vol. 11, B-splines: Scaling functions and wavelets,” IEEE Trans. Image no. 4, pp. 363–372, Apr. 2002. Process., vol. 14, no. 11, pp. 1798–1813, Nov. 2005. [18] M. N. Do and M. Vetterli, “The contourlet transform: An efficient di- [45] P. Burt and E. Adelson, “The Laplacian pyramid as a compact image rectional multiresolution image representation,” IEEE Trans. Image code,” IEEE Trans. Commun., vol. 31, no. 11, pp. 532–540, Apr. 1983. Process., vol. 14, pp. 2091–2106, 2005. [46] P. Tafti, D. Van De Ville, and M. Unser, “Invariances, Laplacian-like [19] E. Le Pennec and S. Mallat, “Sparse geometric image representations wavelet bases, and the whitening of fractal processes,” IEEE Trans. with bandelets,” IEEE Trans. Image Process., vol. 14, pp. 423–438, Image Process., vol. 18, no. 4, pp. 689–702, Apr. 2009. 2005. [47] D. Van De Ville and M. Unser, “Complex wavelet bases, steerability, [20] V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. Dragotti, “Di- and the Marr-like pyramid,” IEEE Trans. Image Process., vol. 17, no. rectionlets: Anisotropic multidirectional representation with separable 11, pp. 2063–2080, Nov. 2008. filtering,” IEEE Trans. Image Process., vol. 15, pp. 1916–1933, 2006. [48] M. Feilner, D. Van De Ville, and M. Unser, “An orthogonal family of [21] H. Greenspan, S. Belongie, R. Goodman, P. Perona, S. Rakshit, and C. quincunx wavelets with continuously adjustable order,” IEEE Trans. Anderson, “Overcomplete steerable pyramid filters and rotation invari- Image Process., vol. 14, no. 4, pp. 499–510, Apr. 2005. ance,” in Proc. CVPR, Jun. 21–23, 1994, pp. 222–228. [22] M. Do and M. Vetterli, “Rotation invariant texture characterization and Michael Unser (M’89–SM’94–F’99) received the retrieval using steerable wavelet-domain hidden markov models,” IEEE M.S. (summa cum laude) and Ph.D. degrees in elec- Trans. Multimedia, vol. 4, pp. 517–527, 2002. trical engineering in 1981 and 1984, respectively, [23] G. Tzagkarakis, B. Beferull-Lozano, and P. Tsakalides, “Rotation-in- from the École Polytechnique Fédérale de Lausanne variant texture retrieval with gaussianized steerable pyramids,” IEEE (EPFL), Switzerland. Trans. Image Process., vol. 15, pp. 2702–2718, 2006. From 1985 to 1997, he was a Scientist with the [24] F. Denis and A. Baskurt, “Multidirectional curvilinear structures de- National Institutes of Health, Bethesda, MD. He is tection using steerable pyramid,” J. Electron. Imag., vol. 13, no. 4, pp. now full Professor and Director of the Biomedical 756–765, 2004. Imaging Group, EPFL. His main research area [25] A. Jepson, D. Fleet, and T. El-Maraghi, “Robust online appearance is biomedical image processing. He has a strong models for visual tracking,” in Proc. IEEE CVPR, 2001, vol. 1, pp. interest in sampling theories, multiresolution algo- I–415–I–422. rithms, wavelets, and the use of splines for image processing. He has published [26] J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image over 150 journal papers on those topics and is one of ISI’s Highly Cited authors denoising using scale mixtures of Gaussians in the wavelet domain,” in Engineering (http://isihighlycited.com). IEEE Trans. Image Process., vol. 12, no. 11, pp. 1338–1351, Nov. Dr. Unser has held the position of associate Editor-in-Chief (2003–2005) 2003. for the IEEE TRANSACTIONS ON MEDICAL IMAGING and has served as [27] A. A. Bharath and J. Ng, “A steerable complex wavelet construction Associate Editor for the same journal (1999–2002; 2006–2007), the IEEE and its application to image denoising,” IEEE Trans. Image Process., TRANSACTIONS ON IMAGE PROCESSING (1992–1995), and the IEEE SIGNAL vol. 14, no. 7, pp. 948–959, Jul. 2005. PROCESSING LETTERS (1994–1998). He is currently member of the editorial [28] M. Riesz, “Sur les fonctions conjuguées,” Math. Z., vol. 27, pp. boards of Foundations and Trends in Signal Processing, the SIAM Journal of 218–244, 1927. Imaging Sciences, and Sampling Theory in Signal and Image Processing. He [29] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean co-organized the first IEEE International Symposium on Biomedical Imaging Spaces. Princeton, NJ: Princeton Univ. Press, 1971. (ISBI2002). He was the founding chair of the technical committee of the [30] M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. IEEE-SP Society on Bio Imaging and Signal Processing (BISP). He received Signal Process., vol. 49, pp. 3136–3144, 2001. the 1995 and 2003 Best Paper Awards, the 2000 Magazine Award, and the 2008 [31] M. Felsberg and G. Sommer, “The monogenic scale-space: A unifying Technical Achievement Award from the IEEE Signal Processing Society. He approach to phase-based image processing in scale-space,” J. Math. is an EURASIP Fellow and a member of the Swiss Academy of Engineering Imag. Vis., vol. 21, no. 1, pp. 5–26, 2004. Sciences. [32] G. Metikas and S. C. Olhede, “Multiple multidimensional Morse wavelets,” IEEE Trans. Signal Process., vol. 55, pp. 921–936, 2007. [33] K. G. Larkin, “Natural demodulation of two-dimensional fringe pat- Dimitri Van De Ville (M’02) received the M.S. terns. II. Stationary phase analysis of the spiral phase quadrature trans- degree in engineering and computer sciences from form,” J. Opt. Soc. Amer. A, vol. 18, no. 8, pp. 1871–1881, 2001. Ghent University, Ghent, Belgium, in 1998, and the [34] K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation Ph.D. degree in 2002. of two-dimensional fringe patterns. I. General background of the spiral He received a grant as Research Assistant with phase quadrature transform,” J. Opt. Soc. Amer. A, vol. 18, no. 8, pp. the Fund for Scientific Research Flanders Bel- 1862–1870, 2001. gium (FWO). In 2002, he joined Prof. M. Unser’s [35] M. Unser, D. Sage, and D. Van De Ville, “Multiresolution monogenic Biomedical Imaging Group at the École Polytech- signal analysis using the Riesz-Laplace wavelet transform,” IEEE nique Fédérale de Lausanne (EPFL), Lausanne, Trans. Image Process., vol. 18, no. 11, pp. 2402–2418, Nov. 2009. Switzerland. Since December 2005, he has been [36] S. Held, M. Storah, P. Massopust, and B. Forster, “Steerable wavelet heading the Signal Processing Unit at the University frames based on the Riesz transform,” IEEE Trans. Image Process., to Hospital of Geneva, Geneva, Switzerland, as part of the Centre d’Imagerie be published. Biomédicale (CIBM). In 2009, he was the recipient of a Swiss National Science [37] I. Gelfand and G. Shilov, Generalized Functions. New York: Aca- Foundation professorship and currently holds a joint position at the University demic, 1964, vol. 1. of Geneva and the EPFL. His research interests include wavelets, statistical [38] S. L. Hahn, Hilbert Transforms in Signal Processing. Boston, MA: analysis, multidimensional splines, and applications in biomedical imaging, Artech House, 1996. such as functional magnetic resonance imaging, spectroscopy, electro-en- [39] R. A. Adams, Sobolev Spaces. New York: Academic, 1975. cephalography, and microscopy. [40] R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier se- Dr. Van De Ville has served as an Associate Editor for the IEEE ries,” Trans. Amer. Math. Soc., vol. 72, pp. 341–366, Mar. 1952. TRANSACTIONS ON IMAGE PROCESSING (2006–2009) and the IEEE SIGNAL [41] A. Aldroubi, “Portraits of frames,” Proc. Amer. Math. Soc., vol. 123, PROCESSING LETTERS (2004–2006). Since 2003, he has also been an Editor and no. 6, pp. 1661–1668, 1995. Webmaster of The Wavelet Digest. He is co-chair of the Wavelets XII (2007) [42] O. Christensen, An Introduction to Frames and Riesz Bases. New Wavelets XIII (2009) international conferences, together with V. Goyal and M. York: Birkhauser, 2003. Papadakis.

Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 24,2010 at 10:23:30 EST from IEEE Xplore. Restrictions apply.