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Multiresolution monogenic signal analysisusing the Riesz-Laplace wavelet transform

Michael Unser, Fellow, IEEE,, Daniel Sage, and Dimitri Van De Ville, Member, IEEE

Abstract— The monogenic signal is the natural 2-D counterpartof the 1-D analytical signal. We propose to transpose the conceptto the wavelet domain by considering a complexified versionof the Riesz transform which has the remarkable property ofmapping a real-valued (primary) wavelet basis of L2(R2) into acomplex one. The Riesz operator is also steerable in the sensethat it give access to the Hilbert transform of the signal along anyorientation. Having set those foundations, we specify a primarypolyharmonic spline wavelet basis of L2(R2) that involves asingle Mexican-hat-like mother wavelet (Laplacian of a B-spline).The important point is that our primary wavelets are quasi-isotropic: they behave like multi-scale versions of the fractionalLaplace operator from which they are derived, which ensuressteerability. We propose to pair these real-valued basis functionswith their complex Riesz counterparts to specify a multiresolutionmonogenic signal analysis. This yields a representation whereeach wavelet index is associated with a local orientation, anamplitude and a phase. We give a corresponding wavelet-domainmethod for estimating the underlying instantaneous frequency.We also provide a mechanism for improving the shift androtation-invariance of the wavelet decomposition and show how toimplement the transform efficiently using perfect-reconstructionfilterbanks. We illustrate the specific feature-extraction capa-bilities of the representation and present novel examples ofwavelet-domain processing; in particular, a robust, tensor-basedanalysis of directional image patterns, the demodulation ofinterferograms, and the reconstruction of digital holograms.

I. INTRODUCTION

The analytical signal is a complex extension of a 1-D signalthat is based upon the Hilbert transform; it was introduced tosignal theory by Denis Gabor in 1946 [1]. This representationgives access to the instantaneous amplitude and phase ofa signal and is widely used in applications involving somekind of amplitude or frequency modulations [2], [3]. Thistype of AM/FM analysis can also be performed in a mul-tiresolution framework using Kingsbury’s dual-tree wavelettransform, which consists of two wavelet transforms whichare (approximately) Hilbert transforms of one another [4]–[7]. An exact implementation is actually possible within theframework of fractional splines [8], which is closed underfractional differentiation1.

Several attempts to generalize the analytical signal to twodimensions have been reported in the literature [9]–[11]. Thefirst class of representations relies on a separable applicationof the Hilbert transform along the x and y coordinates. In the

The authors are with the Biomedical Imaging Group (BIG), Ecole Poly-technique Federale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland.

1The Hilbert transform can be interpreted as the anti-symmetric (fractional)derivative of order 0, while the identity is the symmetric derivative of order0. Likewise, we may view the Riesz transform, which is central to this work,as the (fractional) gradient of order 0.

approach of Hahn [9], the analytical signal is represented bytwo complex components whose Fourier spectrum is restrictedto the upper-right and lower-right quadrant, respectively. Thetransposition of this approach to the wavelet domain resultsin a 2-D dual-tree complex wavelet transform that has aredundancy of four. The transform allows for a separationinto six distinct orientation channels via a proper combinationof positive and negative frequency bands [4], [12]. The clearadvantage of this type of representation over the traditionalreal-valued separable wavelet transforms is improved shift-invariance (thanks to the notions of wavelet-domain amplitudeand phase) and better orientation selectivity [4], [5], [12].There is also an alternative of the 2-D analytic signal—due to Bulow and Sommer [10]—that starts with the sameseparable Hilbert-transform constituents, but combines theminto four “quadrature” components that are manipulated usinga more sophisticated quaternion algebra. The transfer of thisconcept to the wavelet domain leads the dual-tree quaternionwavelet transform [13], which is well suited for the coherentprocessing of relative positional information.

The third generalization of the analytic signal—called themonogenic signal—was introduced by Felsberg and Sommerin 2001 [11]. Its distinguishing property is that the underlyingfeature extraction process (phase, amplitude and orientationestimation) is truly rotation-invariant which explains its suc-cess for image analysis. The method is build around the Riesztransform which is the vector-valued extension of the Hilberttransform favored by mathematicans [14]. To make an analogy,the Riesz transform is to the Hilbert transform what thegradient is to the derivative operator. The image-processing ap-plications of the monogenic signal are numerous: they includecontour detection and local structure analysis [15], [16]; stereo,motion estimation, and image registration [17], [18]; as wellas image segmentation and phase-contrast imaging [19]. Themonogenic signal is also closely linked to the spiral quadraturephase transform in optics which constitutes a powerful toolfor the processing of fringe patterns and the demodulationof interferograms [20]–[22]. Recently, Metikas and Olhedehave transposed the approach to the wavelet-domain, albeitwithin the framework of the continuous wavelet transformwhich is an overly-redundant representation [23]. In practice,the monogenic analysis is often performed on some bandpass-filtered versions of the input signal which also leads to theidea of multiresolution [15], [24]. Our goal in this paper isto fill the gap by presenting the minimally-redundant waveletcounterpart of Felsbergs monogenic signal representation. Thenon-trivial aspect here is that the multiscale versions of thesignal are critically sampled and that the scheme is fully

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reversible (exact reconstruction property).The main features of the proposed “Riesz-Laplace” wavelet

decomposition are:• The wavelet transform has three monogenic components

for any scale/location index.• The primary mother analysis wavelet is Laplacian-like

and essentially isotropic, while the two others are thecomponents of the Riesz transform of the former. In ourformulation, the two Riesz wavelets are combined into asingle complex transform.

• The three wavelet components give access to the localorientation—that is, the dominant orientation of imagefeatures in a local neighborhood—as well as to the keyAM/FM parameters in the preferred orientation: ampli-tude, phase, and instantaneous frequency. These latterparameters are specific to the monogenic formalism andare not accessible in a conventional wavelet transform.

• The monogenic wavelet analysis is essentially rotation-invariant because of: (1) the isotropy of the motherwavelet, and (2) the fact that the Riesz transform issteerable.

• The wavelet transform has a fast filterbank algorithm.In fact, the decomposition involves the concatenation oftwo wavelet bases of L2(R2), the second of which iscomplex-valued.

The paper is organized as follows. In Section II, we re-view the key properties of the complex Riesz transform, andreinterpret Felsberg’s monogenic signal using the directionalHilbert transform; we also present a more robust, tensor-basedestimation of the local orientation. In Section III, we usethe invariance properties of the complex Riesz operator toconstruct a new complex wavelet basis of L2(R2) through theunitary mapping of a primary real-valued wavelet transform.Our primary wavelet basis involves a single Mexican-hat-likemother wavelet. It is specified within a polyharmonic splineframework that forces the wavelet to replicate the isotropicbehavior of the (fractional) Laplacian. This ensures that thesecondary complex wavelet, which is the Riesz transformof the former, is steerable. In Section IV, we propose touse the so-defined pair of Riesz-Laplace wavelet transformsto specify a multiresolution monogenic signal analysis. Wealso show how to extract wavelet-domain information thatcharacterizes the local structure of the image (orientation,coherency), as well as its modulation/localization properties(amplitude, phase and spatial frequency). In Section V, wepropose two simple solutions for improving the shift androtation-invariance of the decomposition. We illustrate ourmethod with concrete examples of wavelet-domain monogenicsignal analyses, including a novel application of wavelets tocoherent optical imaging.

II. THE RIESZ TRANSFORM AND THE MONOGENIC SIGNAL

We start with a brief review of the Hilbert transform andthe analytical signal formalism and then proceed with theextension of those concepts to two dimensions. In particular,we present an alternative formulation of Feldberg’s monogenicsignal representation that rests upon a “complexified” version

of the Riesz transform which is due to Larkin [21]. Thisoperator has a number of remarkable of properties that willbe exploited in our construction.

A. The Hilbert transformThe 1-D Hilbert transform is the linear, shift-invariant

operator H that maps all 1-D cosine functions into theircorresponding sine functions. It is an allpass filter that ischaracterized by the transfer function H(!) = !jsign(!) =!j!/|!|. Its impulse response (in the sense of distributions)is h(x) = 1/("x). Note that h(x) is neither in L1(R) nor inL2(R) due to the singularity at the origin. The other importantmathematical feature is the slow decay of h(x) which impliesthat the Hilbert transform is a non-local operator; in particular,it will generally map compactly supported functions into non-compactly supported ones.

The Hilbert transform plays a central role in the analyticalsignal formalism which is a powerful tool for AM/FM anal-ysis. Given a real valued 1-D signal f(x), one defines thecomplex analytical signal

fanal(x) = f(x) + jHf(x) = A(x)ej!(x). (1)

The advantage of this representation is that it allows one to re-trieve the time-varying amplitude and instantaneous frequencyof an AM/FM signal of the form f(x) = A(x) cos(2"#(x)x+$0), where A(x) and #(x) are slowly-varying. Specifically,A(x) = |fanal(x)| and #(x) = d!(x)

dx .

B. The Riesz transformThe Riesz transform is the natural multidimensional exten-

sion of the Hilbert transform [14]. It is the scalar-to-vectorsignal transformation whose frequency response is !j!/"!".In 2-D, which is the case of interest here, it can be expressedas

fR(x) =!

f1(x)f2(x)

"=

!hx # f(x)hy # f(x)

"(2)

where f(x) with x = (x, y) is the input signal, and wherethe filters hx and hy are characterized by the 2-D frequencyresponses Hx(!) = !j!x/"!" and Hy(!) = !j!y/"!"with ! = (!x,!y). By using the well-known property thatF{ 1

!x!}(!) = 2"!!! and performing a partial differentiation

with respect to x or y, we readily derive the correspondingimpulse responses hx(x) = x

(2")|x|3 and hy(x) = y(2")|x|3 .

For the present work, we have chosen to work with a“complexified” version of the Riesz transform that combinesthe two Riesz components into a single complex signal; it wasintroduced by Larkin in optics under the name of the spiralphase quadrature transform [20], [21]. The correspondingdefinition of the 2-D complex Riesz operator is

Rf(x) F$% (!j!x + !y)"!" f(!), (3)

where f(!) =#

R2 f(x)e"j#!,x$dxdy is the 2-D Fouriertransform of f . The equivalent space-domain formulation is

Rf(x) = ((hx + jhy) # f) (x)

=12"

$

R2

x% + jy%

|x%|3 f(x! x%)dx%dy%, (4)

3

or, equivalently, Rf(x) = f1(x)+jf2(x), in accordance with(2). Note that this operator defines an allpass filter and thatits impulse response is anti-symmetric: R{%}(x) = hR(x) =!hR(!x).

C. Properties of the complex Riesz operatorWe now present and discuss the key properties of R [14],

[20].Property 1 (Invariances): The complex Riesz operator is

both translation- and scale-invariant:

&x0 ' R2, R{f(·! x0)}(x) = R{f(·)}(x! x0)&a ' R+, R{f(·/a)}(x) = R{f(·)}(x/a).

The translation invariance directly follows from the defini-tion while the scale-invariance is easily verified in the Fourierdomain.

Property 2 (Steerability): The impulse response of theRiesz transform satisfies the rotation-covariance relation

hR(R#x) = e"j#hR(x)

where R# =%

cos & sin &! sin & cos &

&is the matrix that implements

a spatial rotation by the angle &.This is established in the Fourier domain by showing thathR(R#!) = e"j#hR(!).

The above result means that the Riesz transform can berotated by simple multiplication with a complex number. Anequivalent interpretation is that the component filters of theRiesz operator form a steerable2 filterbank in the sense definedby Freeman and Adelson [25]. This becomes most apparentwhen we link the operator to the following directional versionof the Hilbert transform

H#f(x) = cos & f1(x)+sin & f2(x) = Re'e"j#Rf(x)

((5)

whose impulse response corresponds to the rotated versionof hx(x) = Re (hR(x)). Note that the association betweenthe Riesz and the directional Hilbert transforms is essentiallythe same as the link between the gradient and the directionalderivative (more on this later).

It is also instructive to determine the Riesz transform of apure cosine function of frequency !0 = (!0 cos &0,!0 sin &0).A simple Fourier-domain calculation yields

R{cos((!0, ·))}(x) = ej#0 sin((!0,x)).

We can then apply (5) to recover a pure sine wave by steeringthe directional Hilbert transform in the appropriate direction:

H#0{cos((!0, ·))}(x) = sin((!0,x)).

This desirable “quadrature” behavior—as Larkin calls it [20]—follows from the fact that the central cut of the frequencyresponse of H# along the direction & perfectly replicates thebehavior of the 1-D Hilbert transform: H#(! cos &, ! sin &) =!jsign(!). The only limitation of the technique is that theideal Hilbert-transform-like behavior falls off like cos(&!&0),

2Steerability designates the property of being able to synthesize theresponse of any rotated version of a filter through an appropriate linearcombination of basis functions.

which calls for a precise adjustment of the analysis direction&0.

Note that the above properties can also be stated using themore conventional vector formulation of the Riesz transformand that they are transposable to higher dimensions [14]. Thisis not so for the final one which is truly specific to the present2-D complex formulation and central to our argumentation (cf.Subsection III.A).

Property 3 (Unitary transform): The complex Riesz trans-form is a unitary mapping from L2(R2) into L2(R2)(Lebesgue’s space of 2-D complex-valued finite energy func-tions).This is equivalent to the property |hR(!)|2 = 1 which isobviously satisfied. Thus, the adjoint of the Riesz operator isspecified by

R&f(x) F$% (+j!x + !y)"!" f(!), (6)

with the property that R"1 = R&.

D. Monogenic signal and directional Hilbert analysis

Given a 2-D signal f(x),x ' R2, Felsberg and Sommerdefine the three-component monogenic signal

fm(x) = (f(x), Re(Rf(x)), Im(Rf(x))) = (f, f1, f2).(7)

The local amplitude of the signal is given by A(x) ="fm(x)" =

)f2 + f2

1 + f22 , while its local orientation & and

instantaneous phase $ are specified by the following relations:

f = A cos $, f1 = A sin $ cos &, f2 = A sin $ sin &. (8)

As an alternative to the elaborate quaternion formalismproposed by Felsberg and Sommer, we prefer to view themonogenic signal analysis as a directional transposition of the1-D analytical signal analysis (1). Specifically, we define r =)

f21 + f2

2 = |Rf | and interpret this quantity as the maximumresponse of the directional Hilbert transform operator; i.e.,r(x) = max#{H#f(x)} = max#{Re(e"j#Rf(x))}. Thismaximum response occurs in the direction specified by u =(cos &, sin &) = (f1/r, f2/r), as a direct consequence of (5).Next, we consider the complex variable z = f + jr = Aej! =A cos $ + jA sin $, which is then treated as if it was a 1-Dentity. In particular, we recover the instantaneous amplitudeA =

)f2 + r2 and phase $ = arctan

*rf

+of the signal f

along the direction u. Likewise, we estimate the instantaneousfrequency along the direction u by # = Du$ = (*$, u)where Du and * denote the directional derivative and gradientoperators, respectively.

The conceptual advantage of viewing the monogenic sig-nal analysis this way is that one can easily extend it toany other direction u% = (cos &%, sin &%) by taking r(x) ={Re(e"j#!Rf(x)) = ((f1, f2),u%). In particular, we mayselect a direction of analysis that is associated with an area ofinterest, rather than a point location. This is justifiable whenthe signal is locally coherent and may be a good strategy foradding robustness to the orientation determination.

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E. Improved tensor-based estimation of the local orientation

The practical scheme that we propose is inspired by thestructure tensor formalism [26]. In accordance with Felsberg’smonogenic signal analysis, we select the orientation thatmaximizes H#f(x), but instead of doing it pointwise, weoptimize the response over a local neighborhood specified bya symmetric, positive weighting function v(x) = v(!x) + 0centered on the current position. Specifically, we determine thelocal orientation at x0 by solving the optimization problem

&v(x0) = arg max#'["","]

$

R2v(x! x0) |H#f(x)|2 dxdy.

To simplify the derivation, we introduce the weighted inner-product

(f, g)v =$

R2v(x)f(x)g(x) dxdy

and rewrite the criterion to maximize as$

R2v(x! x0) |H#f(x)|2 dxdy =

= "f1(·+ x0) cos & + f2(·+ x0) sin &"2v

='

cos & sin &(

J(x0)!

cos &sin &

"

where J(x0) is the 2, 2 symmetric matrix whose entries aregiven by

[J(x0)]mn = (fm(·+ x0), fn(·+ x0))v

=$

R2v(x! x0)fm(x)fn(x) dxdy

with m,n ' {1, 2}. Because of the underlying inner-productstructure, J(x0) is guaranteed to be positive-definite; in fact,it is the Riesz-transform counterpart of the local structuretensor which is build around the gradient. The determination ofthe orientation therefore amounts to maximizing the quadraticform uT J(x0)u under the constraint "u" = 1, which is a clas-sical eigenvalue problem. The optimal direction is specifiedby the eigenvector u1 corresponding to the largest eigenvalue'max of J(x0). This eigenvalue gives the magnitude ofthe maximum response. By combining it with the smallereigenvalue, we obtain a coherency index

0 - ( ='max ! 'min

'max + 'min- 1,

which provides us with a precious indication of the degreeof directionality of the local neighborhood. The directionalHilbert transform analysis will obviously perform best whenthe local neighborhood has a structure that is predominantlyone-dimensional (i.e., ( . 1), which suggests using thecoherency as a general reliability index.

In this paper, we will go one step further by transfering thesenotions to the wavelet domain, which essentially amounts toperforming a directional Hilbert analysis on bandpass-filtered(and critically-sampled) versions of the signal. The remarkableaspect is that this can be done in a stable, reversible way usingbona fide wavelet bases.

III. COMPLEX RIESZ-LAPLACE WAVELETS

Our method of construction is simple conceptually. Westart with a wavelet basis that is essentially isotropic andthen use the unitary Riesz transform mapping to construct acorresponding complex steerable wavelet basis. The technicaldifficulty is to specify a good primary decomposition thatreplicates the isotropic behavior of the Laplace operator; weshow that this can be done using polyharmonic splines.

A. A general method for constructing complex wavelet basesThe interesting consequence of Property 3 is that the

complex Riesz operator will automatically map any basis ofL2(R2) into another one. Indeed, we have that

&)k,)l ' L2(R2), ()k,)l)L2 = (R&R)k,)l)L2

= (R)k,R)l)L2 . (9)

What is of even greater interest to us is that it will transformany wavelet basis into another complex one, thanks to thetranslation- and scale-invariance properties. Specifically, letus consider a dual pair of real-valued wavelets * and * thatgenerate a non-separable biorthogonal basis of L2(R2) suchthat

&f ' L2(R2), f(x) =,

i'Z

,

k'Z2\2Z2

(f, *i,k)L2 *i,k(x) (10)

with the short-hand notation *i,k(x) = 2i*(2ix! 12k). Then,

the complex wavelets *% = R* and *% = R* also form abiorthogonal pair with the property that

&g ' L2(R2), g(x) =,

i'Z

,

k'\2Z2

(g, *%i,k)L2 *%

i,k(x).

The latter expansion is obtained by considering the complexsignal g = R{g1 + jg2} where the real-valued functions g1

and g2 are both decomposed according to (10) and by makinguse of the unitary property (9). A crucial ingredient for thisconstruction is Property 1, which ensures that R*i,k = *%

i,kwhere *% = R*.

If, in addition, the real wavelets * or * are isotropic, thenwe end up with a decomposition that is steerable and thereforerotation-invariant, thanks to Property 2. Indeed, we can use (5)to rotate the wavelet corresponding to the real part of *% = R*to any desired orientation:

Re'e"j#R*(x)

(= H#*(x) = Re (*%(R#x)) , (11)

where the right-hand side of the equality is valid providedthat the initial wavelet *(x) = *(|x|) is isotropic. The sameargument obviously also holds if we substitute R by its adjointR& = R"1.

While this complex-wavelet construction technique looksstraightforward, at least formally, there are some technical dif-ficulties associated with the specification of the correspondingcomplex multiresolution analysis of L2(R2) because the Rieszoperator will not map a scaling function into another one;in particular, it will destroy the partition of unity and inducesome delocalization due to the Fourier-domain singularity atthe origin (suppression of the DC component). Fortunately

5

for us, its effect is much less damaging on the wavelet sidebecause of their vanishing moment property which tempersthe singularity up to the order of the transform.

B. Fractional differential and integral operatorsThe fractional Laplacian (!!)$ with + ' R+ is the

isotropic differential operator of order 2+ whose Fourier-domain definition in the sense of distributions is

(!!)$f(x) F$% "!"2$f(!). (12)

For + = n ' N, it is a purely local operator that is proportionalto the classical n-fold Laplacian. Otherwise, its effect is non-local and described by the following distributional impulseresponse (cf. [27])

C2$,d

"x"2$+d

F$% "!"2$ (13)

where d = 2 is the number of dimensions and Cs,d =2s!( d+s

2 )"d/2!("s/2)

is an appropriate normalization constant; "(r) =(r ! 1)! is Euler’s gamma function. Interestingly, the Fouriertransform pair (13) is also valid for negative + provided that!(++d/2) is non-integer; otherwise, the left-hand side needsto be replaced by a term that is proportional to "x"2n log("x")with n = !(+ + d/2). This suggests extending the definitionof the fractional Laplacian to negative orders as well, whichleads to a fractional integral-like behavior. The so-definedfamily of fractional operators satisfies the composition rule(!!)$1(!!)$2 = (!!)$1+$2 which follows from theirFourier-domain definition, with the convention that (!!)0 =Identity. Note that the fractional integral (!!)"$f with + > 0is well-defined only for functions (or distributions) f whosemoments up to order /+0 are zero, due to the singularity ofthe frequency response at the origin.

The operators (!!) 12 and (!!)" 1

2 are of special relevancebecause they allow us to make a direct link between the Riesztransform and the complex-gradient (or Wirtinger) operator.Specifically, we have that

!,

,x+ j

,

,y

"f(x) = !R(!!)

12 f(x) (14)

= R!

12""x"3 # f

"(x),

where the first equality is trivially checked in the Fourierdomain. The converse relation is

Rf(x) = !!

,

,x+ j

,

,y

"(!!)"

12 f(x) (15)

= !!

,

,x+ j

,

,y

" !1

2""x" # f

"(x),

where we are assuming that f is zero-mean to avoid any ill-posedness problem. The latter formula allows us to interpretthe Riesz transform as a smoothed version of the imagegradient. Interestingly, the underlying smoothing kernel, whichcorresponds to a radial filtering with "!""1, is the same asthe one encountered in tomography when the reconstructionis performed by simple backprojection [28], [29]. This pointstowards a mathematical connection between the Radon and

Riesz transforms, which is discussed in [11]; recently, Wietzkeand Sommer have exploited this link to derive higher-ordersignal descriptors by iterating the transform [30].

The directional versions of relations (14) and (15) are:

Duf(x) = !H#(!!)12 f(x) (16)

H#f(x) = !Du(!!)"12 f(x), (17)

where Duf = (*f,u) is the derivative of f along thedirection u = (cos &, sin &). This is consistent with the 1-Dinterpretation of the Hilbert transform being a lowpass-filteredversion of the derivative operator.

C. Polyharmonic splines

The polyharmonic splines are the spline functions associatedwith the fractional Laplacian operators. To introduce them, weconsider the cardinal setting where the spline knots are on thecartesian grid Z2.

Definition 1: The continuously-defined function s(x) 'L2(R2) is a cardinal polyharmonic spline of order - iff. (cf.[31])

(!!)%/2f(x) =,

k'Z2

d[k]%(x! k)

where %(x) is the 2-D Dirac distribution.The interpretation is that these splines have singularities

of order - at the integers and that they have one degreeof freedom per grid point. We can formally integrate theabove equation by applying the inverse operator (!!)"%/2

whose impulse response is denoted by .%(x). This leadsto the following characterization of the space of cardinalpolyharmonic splines:

V0 = span{.%(x! k)}k'Z2 1 L2(R2)

which involves the integer shifts of the Green function ofthe defining operator; i.e., (!!)%/2.% = %. A more practicalShannon-like representation of these splines is

V0 =

-s(x) =

,

k'Z2

s[x]/%(x! k)

.,

where s[x] = s(x)|x=k are the integer samples of s(x)and where /%(x) is the unique cardinal polyharmonic splineinterpolator such that /%(k) = %[k] (Kronecker delta). Thisbasis function is best characterized by its Fourier transform

/%(x) F$% /%(!) ="!""%

/k'Z2 "! + 2"k""%

=1

1 +/

k'Z2\{0}

*!!!

!!+2"k!

+% . (18)

It is endowed with the following remarkable properties whichcan all be inferred from the above definition [31]:

• /% is a polyharmonic spline of order -. This simply fol-lows from the fact that "!"% /%(!) [the Fourier transformof (!!)%/2/%] is 2"-periodic.

• /% is interpolating, which is equivalent to/

k'Z2 /%(!+2"k) = 1 (through Poisson’s summation formula).

6 IEEE TRANSACTIONS ON IMAGE PROCESSING, TO APPEAR

• Its Fourier transform is bounded (0 - /%(!) - 1)and decaying away from the origin like O("!"%). Inparticular, this implies that /% has up to -!1 derivativesin the L2-sense.

• /% generates a Riesz3 basis for the space of cardinalpolyharmonic splines V0 provided that the order - >1 [32]–[34].

• /% has approximation order - and has the ability tointerpolate all polynomials of total degree less than - [35,Chapter 4]. This follows from the property that /%(!) haszeros of order - at ! = 2"k,k ' Z2\{0} (Strang-Fixconditions of order -).

• /%(x) decays like O("x""(%+2)) when - is non-even[35, Theorem 4.3], and exponentially-fast otherwise (- '2N+) [31], [36].

• /%(!) = 1 +O("!"%) as ! % 0, meaning that it has a-th order of flatness at the origin (similar to a classicalButterworth filter of order -).

• /%(x) is symmetric and converges to the separable sincinterpolator sinc(x) as the order - goes to infinity [32].

D. Laplacian-like waveletsFollowing Bacchelli et al. [34], we define the symmetric,

Laplacian-like spline wavelet of (fractional) order - > d2

*(x) = (!!)%/2/2%(Dx) (19)

where /2%(x) is the polyharmonic spline interpolator of order2- [cf. (18)] and where D is a given admissible 2-D dilation(or subsampling) matrix (e.g., cartesian or quincunx). We thenspecify the basis functions

*i,k(x) = |det(D)|i/2 *(Dix!D"1k), (20)

which are dilated (scale index i) and translated versions (spaceindex k) of one another. For a given i, these functions arecentered on the rescaled lattice D"(i+1)Z2 which also cor-responds to the location of the spline knots. Observe that thecorresponding lattice is an upsampled version of the canonicalgrid D"iZ2 which is included as a subset.

Using such (normalized) wavelets, we specify the waveletsubspace at resolution i:

Wi = span {*i,k(x)}k'Z2\DZ2 , (21)

which involves all wavelets at scale i except the ones thatare located on the canonical lattice (D"iZ2). Note that this isa special instance of a non-separable wavelet construction inwhich the usual M = det(D) ! 1 wavelet channels at scalei involve basis functions that are all identical up to a shift.Likewise, we introduce the scaling functions at resolution i

)i,k(x) = |det(D)|i/2 /%(Dix! k),

and specify the corresponding approximation space

Vi = span {)i,k(x)}k'Z2 , (22)

3A Riesz basis is a fundamental concept in functional analysis that wasintroduced by the Hungarian mathematician Frigyes Riesz; it has not muchto do with the Riesz transform that is due to Marcel Riesz (Frigyes’ youngerbrother), except perhaps that the complex version of the Riesz transform mapsa Riesz basis into another one.

which is the subspace of polyharmonic splines with knots onthe grid at resolution i: D"iZ2.

The remarkable property of this construction is that thesubspaces Vi and Wi are orthogonal and that their direct sumyields the finer resolution space: Vi+1 = Wi 2 Vi; that is, thesubspace of polyharmonic splines of order - with knots onthe lattice D"(i+1)Z2. This is best understood by focusing onthe wavelets at resolution i = !1.

Proposition 1: The wavelets *"1,k(x) =|det(D)|"1/2 (!!)%/2/2%(x! k) are cardinal polyharmonicsplines of order - (i.e., with knots on the lattice Z2).Moreover, they are orthogonal to V"1 whenever k /' DZ2.

Proof: To establish the first part, we simply evaluate theFourier transform of (!!)%/2*"1,k(x):

"!"% "!"% /2%(!)e"j#!,k$0 12 3

&"1,k(!)

=e"j#!,k$

/k'Zd "! + 2"k""2%

,

which is clearly 2"-periodic. This proves that *"1,k(x) is acardinal polyharmonic spline of order -. For the second part,we use the property that V"1 = span{.%(x !Dk%)}k!'Z2 1L2(R2) where .% is the Green function of (!!)%/2. We thenperform the inner product manipulation

|det(D)|12 (*"1,k, .%(·!Dk%))

= ((!!)12 /2%(·! k), .%(·!Dk%))

= (/2%(x! k), (!!)12 .%(x!Dk%)) (by duality)

= (/2%(·! k), %(·!Dk%)) (Green function property)= (/2%(·), %(·+ k !Dk%)) = %[k !Dk%]

Since /2% is interpolating and k!Dk% is necessarily integer,the expression vanishes except when k = Dk%.

This combined with the fact that the polyharmonic splinesfor - > 1

2 provide a valid multiresolution analysis of L2(R2)yields the theorem below, which is a slight extension ofpreviously published results. Specifically, Bacchelli et al. [34],[37] investigated the dyadic, non-fractional case (i.e., D = 2Iand - ' N+), while Van De Ville et al. [33] fully characterizedthe underlying fractional wavelet bases, but in the quincunxcase only (i.e., D = 2I).

Theorem 1: The polyharmonic spline wavelet *(x) givenby (19) generates a Riesz basis for W0. Moreover,{*i,k(x)}i'Z, k'Z2\DZ2 yields a semi-orthogonal basis ofL2(R2) for all (fractional) orders - > 1

2 .

The direct implication is that any function f ' L2(R2) hasthe following stable wavelet decompositions

f(x) =,

i'Z

,

k'Z2\DZ2

(f, *i,k) *i,k(x)

=,

i'Z

,

k'Z2\DZ2

(f,*i,k) *i,k(x), (23)

which are termed primal or dual depending on the type ofreconstruction wavelets. The complementary wavelet functions*i,k 'Wi are the unique duals of *i,k in the sense that theysatisfy the biorthogonality property: (*i,k,*i!,k!) = %i"i!,k"k!

with k ' Z2\DZ2. While the primal wavelets are generated

7

from a single template, there are M ! 1 distinct generatorsfor the dual ones, as is usually the case for any generalmultidimensional wavelet construction with M = det(D).Structurally, the wavelet decomposition/reconstruction algo-rithm corresponds to an M -channel perfect reconstructionfilterbank. The dual wavelets are defined implicitly throughthe perfect reconstruction conditions in Mallat’s fast filterbankalgorithm [33], [34].

E. Complex, Riesz-Laplace wavelet basis of L2(R2)Having set the mathematical foundations, we can now

apply the (adjoint) Riesz transform to the above polyharmonicwavelet basis to obtain a bona fide complex transform. Thecorresponding complex, Riesz-Laplace wavelet of order - > 1

2is given by

*%(x) = R&*(x) = R&(!!)%/2/2%(Dx). (24)

The reason for considering R& instead of R will becomeclearer in Section IV; it does not fundamentally change any-thing, except for the sign of the real part of the transform.By referring back to the argument in Subsection III.A, weimmediately obtain the following corollary of Theorem 1.

Corollary 1: The complex Riesz-Laplace wavelet*%(x) = R&* given by (24) generates a Riesz basis forW %

0 = R&W0 = span{*%(x !D"1k)}k'Z2\DZ2 . Moreover,4*%

i,k(x) = |det(D)|i/2 *%(Dix!D"1k)5

i'Z, k'Z2\DZ2

yields a complex Riesz basis of L2(R2) provided that - > 12 .

Interestingly, these wavelets are special instances (up tosome proportionality factor) of the complex rotation-covariantwavelets that we had identified and characterized mathemati-cally in previous work [38], [39]. Our earlier formulation didnot involve the Riesz transform but rather a combination ofiterated Laplace and complex gradient (or Wirtinger) operatorsof the form

L%,N =!

,

,x! j

,

,y

"N

(!!)!"N

2 (25)

The link with (24) is obtained by making use of (15) whichyields the equivalence with the case N = 1

*%(x) =!

,

,x! j

,

,y

"(!!)%/2"1/2%(Dx) = L%,1/2%(Dx).

The advantage of the present formulation is that we haveidentified the unitary mapping that transforms the complexwavelets into the polyharmonic ones (and vice versa) whichsimplifies the theory considerably. In particular, this allows usto step through the mathematical properties of the polyhar-monic spline wavelets and to transpose them almost literallyto the complex case. The argument rests upon the invarianceproperties of the complex Riesz transform which preserve thewavelet character of the transform, as well as the structureof the underlying Gram matrix (inner products between basisfunctions).

Property 4: (Semi-orthogonality). The polyharmonic splineand complex Riesz-Laplace wavelets are orthogonal acrossscales:&k, l ' Z2\DZ2, i 3= j,

(*i,k,*j,l) = 0 4 (R&*i,k,R&*j,l) = 0.

This follows from Proposition 1.

Property 5: (Smoothness). The critical Sobolev exponent ofthe polyharmonic spline and complex Riesz-Laplace waveletsof order - is smax = - ! 1.

The smoothness (or degree of differentiability) of a waveletis often characterized by its inclusion in the Sobolev spacesW s

2 = {f ' L2(R2) | (!!)s/2f ' L2(R2)} which representthe class of functions with s derivatives in the L2-sense. Weknow that * ' W s

2 for s < smax = - ! 1 by construction,since it is a polyharmonic spline of order -. R&* is includedin the same Sobolev spaces as a direct consequence of thenorm-preservation property of the complex Riesz transform;in particular, "*"L2 = "R&*"L2 and "(!!)s/2*"L2 ="(!!)s/2R&*"L2 .

Property 6: (Operator like-behavior). The polyharmonicspline wavelet transform implements a multiscale version ofthe Laplace operator:

(f,*(Di ·!x)) = (!!)%/2(f # )(Di·))(x)

where the smoothing kernel is )(x) = /2%(Dx), while theRiesz-Laplace wavelet transform does the same for the Riesztransform of the input signal:

(f,R&*(Di ·!x)) = (Rf,*(Di ·!x))= (!!)%/2(Rf # )(Di·))(x).

Property 7: The polyharmonic spline and complex Riesz-Laplace wavelet transforms with parameter - have approxima-tion order -. Specifically, this means that the error of the scale-truncated approximation of a function f ' W %

2 at resolutioni0 decays like the -th power of that scale666666

i0"1,

i="(

,

k'Z2\DZ2

(f, *i,k) *i,k

666666L2

- C%(2"i0)% "(!!)!2 f"L2

666666

i0"1,

i="(

,

k'Z2\DZ2

(f,R&*i,k) R&*i,k

666666L2

-

C%(2"i0)% "(!!)!2 f"L2

where C% is a constant. As i0 % +5, the approximation errortends to zero which proves that the both wavelet representa-tions are dense in L2(R2).

This is a standard approximation-theoretic result for thepolyharmonic splines [31], [33], [34]. It carries over to thecomplex Riesz wavelets by considering the expansion of Rfinstead of f and by noting that all L2-norms in the aboveinequality remain unchanged if we replace f by Rf or R&f(unitary property).

Property 8: (Vanishing moments). The polyharmonic splineand Riesz-Laplace wavelets of order - have /-0 vanishingmoments.

The vanishing-moments property for the polyharmonicspline wavelets follows from the fact that the interpolator /%

satisfies the Strang-Fix conditions of order - (cf. subsectionIII.C). In the frequency domain, this translates into the van-ishing of *(!) and of all its partial derivatives up to order6-7 at the origin [40]. Applying the Riesz transform does

8 IEEE TRANSACTIONS ON IMAGE PROCESSING, TO APPEAR

not fundamentally affect this behavior because the frequencyresponse is also vanishing at the origin. Another way tosee this is to observe that 7*(!) 8 "!"% and 8R&*(!) 8(j!1 +!2)"!"%"1,! % 0, as a consequence of the operator-like behavior in Property 6.

Property 9: (Fast implementation). The polyharmonicspline and complex Riesz-Laplace wavelet transformsare perfectly reversible. They both have a fast filterbankimplementation.

This latter aspect is further developed in subsection V.B.

IV. MONOGENIC WAVELET ANALYSIS

Referring back to Section II-D, we now show how to takeadvantage of the paired transforms to perform a wavelet-domain monogenic signal analysis. In essence, this amountsto performing the monogenic analysis of the bandpass-filteredsignals (f #*i)(x) where *i(x) = |det(D)|i/2 *(Dix) is thenormalized and rescaled polyharmonic spline wavelet at scalei. Indeed, since the primary (real-valued) analysis wavelet*i(x) is symmetric, the relevant set of wavelet coefficientsof the signal f can be expressed as

wi[k] = (f,*i,k)= (*i # f)(x)|x=D"(i+1)k (26)

r1,i[k] + jr2,i[k] = (f,R&*i,k)= (Rf,*i,k)= R(*i # f)(x)|x=D"(i+1)k, (27)

where r1,i[k] and r2,i[k] are the real and imaginary partsof the complex-valued Riesz wavelet coefficients; the sam-pling lattice at resolution i is D"(i+1)Z2 in accordance withour wavelet definition (20). The above equations provide anexact correspondence between the wavelet coefficient triples(wi[k], r1,i[k], r2,i[k]) and the sample values of (*i # fm)(x)where fm(x) is Felsberg’s monogenic signal, as defined by (7).Concretely, this means that we can perform a full multireso-lution monogenic signal analysis based on the above wavelet-domain description of the signal.

There are two complementary aspects in this analysis thatwe propose to handle separately: 1) the determination of thelocal orientation, and 2) the Hilbert analysis along the prefer-ential orientation, which gives access to the local amplitude,phase and frequency information. The main point is that wecan determine these quantities for each scale and locationindex (i,k).

A. Wavelet-domain orientation analysisIn accordance with the scheme outlined in Section II.E, our

estimation of the orientation is based on the evaluation of thestructure matrix Ji(k) associated with the wavelet cell (i,k)and a given weighting sequence v[l] + 0. The entries of this2, 2 symmetric matrix are given by

[Ji(k)]mn = Jmn =,

l'Z2

v[l! k] rm,i[l]rn,i[l]

with m,n ' 1, 2. The estimation neighborhood is specifiedby the weighting sequence v[l], which is center-symmetric,

(a) (b)Fig. 1. “Psychedelic Lena”: (a) “Lena” with two superimposed concentricspherical waves originating from the eyes; the propeller effect is due to theinterference of the two waves. (b) Its complex Riesz wavelet transform with! = 4.

normalized (/

k'Z2 v[k] = 1) and decreasing away fromthe origin. It is chosen to be quite localized (e.g., Gaussianwindow with 0 ' [1, 2]). The eigen-decomposition of thismatrix yields the three primary directional wavelet features:

• Orientation:

& =12

arctan!

2J12

J22 ! J11

"(28)

• Hilbert-transform energy:

'max =12

!J11 + J22 +

9(J22 ! J11)2 + 4J2

12

"(29)

• Coherency:

( ='max ! 'min

'max + 'min=

)(J22 ! J11)2 + 4J2

12

J22 + J11(30)

The angle & gives the orientation along which the directionalHilbert transform of the bandpass-filtered signal is maximumon average within the local neighborhood specified by v. Theextent of this response is measured by the Hilbert-transformenergy. The coherency 0 - ( - 1 quantifies the degree ofdirectionality of the signal over the neighborhood of interest. Avalue close to one indicates that the structure is predominantly1-D, which reinforces the validity of the subsequent AM/FManalysis.

The local coherence hypothesis that is implicit in theanalytical signal formalism is more questionable when theeigenvalues 'max and 'min are both large. This typicallycorresponds to the signature of a localized isotropic imagestructure (peak or blob), or a corner, which may be identifiedusing a Harris-like detector [26], [41]. A possible wavelet-domain corner index is

hcorner = 'max'min ! 1('max + 'min)2

where 1 is an empirical constant in the range 0.04! 0.5.Note that a pointwise analysis can be performed by selecting

the weighting sequence v to be a unit impulse. In that case,the Hilbert-transform energy and orientation are given by thesquare modulus and the phase of (f,R&*i,k), respectively;the coherency and corner index are then meaningless becausethe rank of Ji[k] is necessarily one.

9Example: Psychedelic Lena

6

Orientation

Amplitude

(c)(a) (b)

Fig. 2. Multiresolution directional analysis derived from the Riesz wavelettransform in Fig. 1b [the orientation is encoded in the hue channel]. (a)Pointwise orientation. (b) Tensor-based orientation with Gaussian observationwindow. (c) Combined visualization of tensor-based features: orientation inhue, coherency in saturation, and local Hilbert-transform energy in brightness.

An example of directional wavelet analysis is presentedin Figures 1-2, using color coding. The scale progressionis dyadic with a slightly redundant wavelet sampling (i.e.,k ' Z2 instead of k ' Z2\2Z2); the primary analysis wavelet* is a Mexican-hat-like polyharmonic spline of order - = 4, asspecified in Subsection V.A. The intent here is to demonstratethe difference between a pointwise orientation estimation withv[k] = %[k] (cf. Fig. 2a) and a tensor-based analysis where v isa fixed Gaussian window with 0 = 1.5 (cf. Fig. 2b). The latterorientation map (wrapped within the interval [0,"]) is lessnoisy; it is more robust and in better agreement with our visualperception of local directionality. Finally, in Fig. 2c, we havecombined the three primary directional wavelet features intoa single image using the HSB channels: hue for orientation,saturation for squared coherency, and brightness for energy.Interestingly, the wave components, which have most of theirenergy in the second subband (scale i = !2) are clearly visibleat that scale.

B. Wavelet-domain AM/FM analysis

The idea is now to perform an analytical signal analysisalong the direction provided by the orientation analysis andencoded by the unit vector u = (cos &, sin &). The correspond-ing directional Hilbert component is given by

qi[k] = (H#f,*i,k) = Re'(e"j#Rf,*i,k)

(

= Re'e"j#(f,R&*i,k)

(

= r1,i[k] cos & + r2,i[k] sin &. (31)

This leads to the specification of the analytical wavelet trans-form: wi[k] + jqi[k] = Ai[k] · ej!, which is locally orientedin the preferential direction &. The wavelet-domain amplitudeand local phase are computed as

Ai[k] =9

w2i [k] + qi[k]2, (32)

$ = arctan!

qi[k]wi[k]

", (33)

where qi[k] is given by (31) with & specified by (28).

Recalling that the wavelet analysis at scale i is equivalent toa bandpass filtering of the input signal, we may therefore usethe wavelet amplitude to extract a putative AM signal compo-nent in the corresponding wavelet subband. The other usefulinformation (FM) is carried by the instantaneous frequency(or wave number) that corresponds to the spatial derivative of$ in the direction u. After some algebraic manipulations (cf.Appendix I), we obtain an explicit form for the instantaneousfrequency:

#i[k] =!! qi[k] (wx,i[k] cos & + wy,i[k] sin &) +

wi[k](r1x,i[k] + r2y,i[k])":

A2i [k],

=!qi[k]Re

*ej#(f,R&*%

i,k)+! wi[k](f,*%

i,k)

A2i [k]

(34)

which has the advantage of avoiding the use of the arctanfunction as well as the problems associated with phase-wrapping. The additional required quantities are the spatialderivatives of w, r1 and r2. These are computed economicallyby considering the two auxiliary Riesz-Laplace wavelets *%

and R&*% of reduced order - ! 1, which are defined as

*%(x) =!! ,

,x! j

,

,y

"R&*(x) = (!!)1/2*(x)

F$% "!"*(!),

R&*%(x) =!

,

,x! j

,

,y

"*(x)

F$% (j!x + !y)*(!).

The first is real, symmetric and is designed to calculatethe Laplacian-like term !(r1x,i[k] + r2y,i[k]) = (f,*%

i,k),while the second is complex and is associated with thecomplex-gradient term wx,i[k] ! j wy,i[k] = (f,R&*%

i,k).The main point is that these additional “derivative” wavelettransforms can be calculated using essentially the same fil-terbank algorithm with an appropriate modification of theanalysis filters. This means that the cost for obtaining aninstantaneous frequency estimate is about twice that of theoriginal monogenic wavelet decomposition. The advantage ofthe present formulation is that the frequency formula (34)is exact and that it does not involve any finite differenceapproximation of the spatial derivatives.

V. PRACTICAL IMPLEMENTATION AND RESULTS

We conclude the presentation by discussing some importantpractical issues and by presenting concrete analysis examples.

A. Improving the invariance properties

The monogenic wavelet transform that was introduced so faris mathematically elegant because it is tied to a basis. However,there is a price to pay for this property which is some lack ofinvariance. We will now show how to bypass this limitation.

10 IEEE TRANSACTIONS ON IMAGE PROCESSING, TO APPEAR

1) Shift invariance: Eventhough it is build around a singlewavelet, the wavelet analysis of Section III is somewhatawkward to interpret because the transform is missing thewavelets at the subsampling locations corresponding to thenext coarser grid; this is expressed mathematically in Theorem1 by having the spatial wavelet index (k) restricted to the setZ2\DZ2. For image analysis, it is tempting to include the“missing” shifts as well (i.e., consider the complete cartesiansampling set Z2), and to rearrange the wavelet coefficients intoa single, image-like subband at scale i. This is equivalent tonot subsampling the wavelet subband after digital filtering. Forinstance, in the dyadic case where D = 2I, this description isassociated with the enlarged analysis spaces

W+i = span

;2i*(2ix! k/2)

<k'Z2 . (35)

where * is the generating wavelet of order -. The correspond-ing notation for the (non-subsampled) wavelet coefficients atscale i is wi[k] (as before) and all previous formulas arestill applicable with k spanning over Z2 instead of Z2\2Z2.The same arrangement is also used for the Riesz branch ofthe transform. This extension improves the shift-invariance ofthe decomposition and its cost is moderate (total redundancyfactor of 1/3). With this sampling arrangement, the overallredundancy of the monogenic Riesz-Laplace transform is R =3

'1 + 1

4 + 116 + · · ·

(= 4.

2) Rotation invariance and steerability: The other weakpoint is that the smoothing kernel /2% of our defining waveletis not isotropic. In fact, it is sinc-like since it is an interpolatingfunction. We propose to exploit the degrees of freedom pro-vided by the enlarged wavelet spaces W+

i to tune the kernel’sshape to make the wavelet more nearly isotropic. Our solutionis to select the “isotropic” polyharmonic B-spline 22% as ournew smoothing kernel [33]. This is justified by the fact thatthis B-spline spans the same spline space as /2% , with theadvantage that is is better localized and that it converges to aGaussian as - increases [33, Proposition 2]. The frequency-domain formula for our new smoothing kernel is

22%(x) F$%!

V (ej''')"!"2

"%

, (36)

where V (ej''') is the “most-isotropic” discrete Laplacian filter

V (ej''') =83

*sin2

*!1

2

++ sin2

*!2

2

++

+23

!sin2

!!1 + !2

2

"+ sin2

!!1 ! !2

2

"". (37)

The corresponding mother wavelet to be used in replacementof the preceding one is *iso(x) = (!!)%/222%(Dx), whilethe whole monogenic analysis procedure remains the sameas described earlier. This is entirely justifiable mathematicallysince the modified Mexican-hat-like wavelet also spans theanalysis spaces W+

i . Concretely, this wavelet substitutionamounts to a simple change of wavelet filters in the algorithmin a manner that is fully reversible and transparent to the user.This simple adaptation results into a dramatic improvement inwavelet isotropy, as illustrated in Fig. 3. The effect is equallyhelpful on the Riesz wavelet for it substantially improves the

Building Mexican-Hat-like wavelets

!!(x) = (!!)!/2"2!(2x) !iso(x) = (!!)!/2"2!(2x)

(a) (b)

! = 4

Fig. 3. Improving wavelet isotropy: (a) Primary polyharmonic spline waveletbasis generator "(x) = (!!)!/2#2!(2x) with ! = 4. (b) its Mexican-hat-like replacement "iso(x) = (!!)!/2$2!(2x), which spans the same spacewhile being essentially isotropic.

Monogenic splines and wavelets

Real part

! ("!)!/2 (fractional Laplacian)Isotropic wavelet: !iso(x)

Complex Riesz wavelet: !Riesz(x)

Imaginary part

Gaussian-like kernel: !2!(2x)

! R! (adjoint Riesz transform)

Fig. 4. Summary of relevant basis functions for the monogenic wavelettransform with ! = 4. The real and imaginary parts of the complex Rieszwavelet are 90!-rotated versions of each other.

steerability of the transform; this is essential for the featureextraction process to be rotation-invariant.

A global summary and visualization of the relevant ba-sis functions for monogenic wavelet analysis is provided inFig. 4. The important point is that all basis functions arerelated to the smoothing kernel 22%(2x) via some operatortransformations (Laplacian and Riesz transform) that are trulyscale and rotation-invariant. This means that we have a fullcontrol of isotropy (and steerability) through the shaping ofthe smoothing kernel. In the proposed B-spline scenario, thiskernel is nearly isotropic and closely resembles a Gaussianwith standard deviation

)2-/12.

The frequency responses of the analysis filterbank thatimplement the corresponding polyharmonic wavelet transformand its Riesz counterpart are shown in Figs. 5a and 5b,respectively. The primary wavelet filters have a clear bandpassbehavior and their response is very nearly isotropic, especiallyin the lower portion of the spectrum. The Riesz wavelet filtersare directional along the horizontal or vertical directions. Dueto the underlying symmetries, the responses associated withthe real and imaginary part of the transform are 90)-rotatedversions of each other (in both space and frequency domains).

11

Wavelet frequency responses

4

!iso,i(!) ! = 4

Real Imaginary

!R!!iso,i(!)

(a) (b)

Fig. 5. Frequency responses of the monogenic wavelet analysis filters : (a)Mexican-hat-like wavelet b"iso,i(!) with ! = 4, (b) corresponding Rieszwavelet !R""iso,i(!). The scale i is decreasing vertically from i = 0 downto !3.

B. Implementation

The monogenic wavelet transform is implemented effi-ciently by running two perfect-reconstruction filterbanks inparallel: one for the primary (Mexican-hat-like) polyharmonicwavelet transform, and one for its Riesz counterpart whichrequires the use of complex arithmetics. We will now brieflydescribe the implementation of the primary wavelet transform,which corresponds to the block diagram in Fig. 6. The keycomponents are:

• The analysis part that consist of a lowpass channel (filterH(z"1)) and a single wavelet channel (highpass filterG(z"1)).

• A reconstruction stage that implements the dual frameoperator. It includes a conventional lowpass channel (up-sampling and filtering by H(z)), a regression modulewhich produces a critically-sampled output, and, finally, astandard wavelet reconstruction module (upsampling andmultichannel filtering by G(z)) whose input is the N -component wavelet coefficient vector wi"1 with N =|det(D)|! 1.

The non-standard aspect of the scheme is that there isno downsampling in the wavelet analysis channel, whichproduces a slightly redundant transform. The redundancy isremoved during the reconstruction process by the subbandregression module which outputs a sequence of critically-sampled wavelet coefficients wi"1 that is maximally con-sistent with wi"1 in the least-squares sense. In effect, thealgorithm projects the redundant wavelet analysis onto a non-redundant system (standard wavelet basis). Here, we will onlyspecify the analysis part of the algorithm which is sufficient tocharacterize the process. For the synthesis part, we refer thereader to [42] which contains all the details for designing thecomplementary set of regression-based reconstruction filters.

4

+

! D

! D G(z)! D

H(z)

subband

regression

ci

! D

ci

ci!1

wi!1

wi!1

G(z)

H(z)

Analysis Synthesis

Fig. 6. Perfect reconstruction filterbank algorithm for the primary redundantwavelet transform. The role of the subband regression module is to convertthe redundant wavelet representation into a non-redundant one described bythe intermediate wavelet coefficient vector wi#1.

Since we want the isotropic/monogenic wavelets to beapplied on the analysis side, we choose to represent thesuccessive multiresolution approximations of the input signalin terms of the dual basis function )(x) = 2%(x):

fi = ProjVif =

,

k'Z2

ci[k])i,k.

We recall that 2%(x) ' V0 is the unique (biorthogonal)polyharmonic spline such that (2%(·!k),2%(·!k%)) = %k"k! .

The algorithm is initialized by assuming that the input signalis bandlimited; i.e., f(x) =

/k'Z2 f [k]sinc(x ! k). The

fine-scale coefficient sequence is therefore given by c0[k] =(f,2%(·!k)) = (hinit#f)[k] where hinit[k] = (sinc,2%(·!k))is the optimal projection prefilter. The coarsening part ofthe decomposition is achieved by successive filtering anddown-sampling: ci"1[k] = (h # ci)[Dk]. The (symmetric)lowpass filter h is specified by the two-scale relation for thepolyharmonic B-spline

2%(D"1x) = |det(D)|12

,

k'Z2

h[k]2%(x! k),

and its frequency response is found by inserting the formulafor 2%(!) provided by (36):

H(ej!) = |detD|12

2%(DT !)2%(!)

= |detD|12"%

=V (ejDT !)V (ej!)

>%/2

(38)

where V (ej!) is the discrete Laplacian filter defined by (37).Finally, the wavelet coefficients are obtained by digital filteringof the ci’s (without down-sampling):

wi[k] = (f,*i,k) = (ci # g)[k].

The wavelet analysis filter is determined from the followingB-spline representation of the isotropic wavelet

*iso(D"1x) = |det(D)|12

,

k'Z2

g[k]2%(x!k) = (!!)!2 22%(x).

(39)

12 IEEE TRANSACTIONS ON IMAGE PROCESSING, TO APPEAR

Its frequency response is obtained by taking the Fouriertransform of this equation, which yields

G(ej!) = |detD|"12"!"% 22%(!)

2%(!)

= |detD|"12 V (ej!)%/2 (40)

The decomposition algorithm for the Riesz counterpart of thetransform is completely analogous and the process is perfectlyreversible in both cases. Since the analysis and synthesis filtersare non-separable and specified by their frequency responses,we have chosen to implement all filtering and re-samplingoperations in the Fourier domain assuming periodic boundaryconditions. Our wavelet code uses FFTs; it is generic andquite efficient computationally. Matlab and JAVA (e.g., ImageJplugin) versions of these transforms are available on the webat http://bigwww.epfl.ch/demo/monogenic/.

C. Wavelets versus scale space

Wavelets and scale space methods are distinct approachesto multi-scale signal analysis; they both have their proponentsand specialized conferences devoted to them. Interestingly, wecan make a connection between the present approach and the+-scale space of Duits, Felsberg and collaborators becauseboth representations are tightly coupled to the fractional Lapla-cian operator [15], [43]. Specifically, these authors perform amonogenic analysis in a scale space that is governed by thefractional diffusion equation

,f(x, s),s

= !(!!)$f(x; s)

with + ' (0, 1] and the initial condition f(x; 0) = f(x).The corresponding monogenic scale-space flow is obtainedby convolving the initial signal f and its Riesz transformwith a series of +-stable smoothing kernels K$

s (x) whosewidth increases with the scale s. For + = 1

2 , the smoothingfunction is the Poisson kernel K

12s (x) = s/(2")

(s2+!x!2)d+12

, while

it is a Gaussian with a variance proportional to s for + = 1.Therefore, apart from the Laplacian factor in (39), the scale-space version of the monogenic signal is qualitatively similarto the monogenic wavelet transform developed in this paper.However, there are four fundamental differences, which are atthe very heart of wavelet theory:

a) the proposed wavelet transform uses subsampling and isa much less redundant representation;

b) it is a fully reversible representation;c) it is a Hilbert-space method that involves a series of

embedded approximation spaces;d) the smoothing kernels at different scales are dyadic

dilations of a single reference template (B-spline).As a result, the proposed wavelet scheme is much moreefficient computationally and storage-wise. The Poisson scalespace, on the other hand, offers a more progressive scale pro-gression; it is perfectly isotropic and predominantly lowpass[15].

Example: Zoneplate

6

Synthetic zoneplate

wavenumber Orientation

Synthetic zoneplate

Fig. 7. Example of wavelet-domain monogenic image analysis. Left: inputimage (synthetic zoneplate); center: wavelet-domain wavenumber estimationwith calibrated frequency scale; right: tensor-based orientation (Hue channel).Example: Modulated Cameraman

8

(a)

Orientation(b) (c)

Fig. 8. Example of wavelet-domain amplitude demodulation. (a) inputimage (modulated “cameraman”); (b) wavelet-domain amplitude estimation;(c) tensor-based orientation (Hue channel).

D. Experimental results

All results presented here were obtained by performinga three-component monogenic Riesz-Laplace transform with- = 2 using the “improved” Mexican-hat-like analysis wavelet(Laplacian of a B-spline) described in subsection VI.A. Wesystematically used a tensor-based estimation of the orientationwith a fixed Gaussian neighborhood window (0 = 2). Thedisplayed orientation is wrapped within the interval [0,"] andencoded in the hue channel; this color coding is exemplifiedwith the circular pattern in Fig. 7, which contains a full 360-degree range of orientations.

The example in Fig. 8 illustrates the ability of the mono-genic wavelet transform to recover amplitude-modulated sig-nals when the frequency/phase of the carrier is unknown and/orslowly-varying. The test signal was generated by multiply-ing the standard ”cameraman” image with a circular wavepropagating with constant frequency. We observe that theAM-encoded information is retrieved in the amplitude of thetransform and that the original image is identifiable down

13Example: Coherence analysis of Barbara

12

HSB

Hue: Orientation

Saturation: Coherency

Brightness: Modulus

Orientation

(c)

! = 2," = 2

(b)

(a)

Coherency

Fig. 9. Example of wavelet-domain coherence analysis. (a) 512 " 512Barbara; (b) wavelet-domain coherency map; (c) composite HSB display ofresults (Hue: orientation, Saturation: coherency, Brightness: amplitude).

to the third level of resolution. A remarkable aspect is therobustness of the scheme with respect to scale, especially ifone considers that the wavelet channels are bandpass filtered.Likewise, the direction of the carrier is adequately captured bythe wavelet orientation. We also notice some ghosting effectin the orientation map, which corresponds to the edges ofthe cameraman. The best separation is achieved at the secondlevel of the pyramid, which yields the maximum amplituderesponse.

The synthetic chirp-like zoneplate image (cf. Fig. 7) wasgenerated specifically to test the validity of the wavelet-domain local frequency estimation algorithm provided by(34). The orientation is properly extracted at all levels, asis obvious from Fig. 7c. More importantly, we verified thatthe wavenumber was correctly estimated at all scales andlocations: it is increasing linearly from the center outwards.Note that the finer-scale frequency estimation fails in the centerof the image which is understandable since it is the regionwhere the attenuation effect of highpass wavelet filtering isthe strongest. In general, one should expect the reliability of agiven wavenumber estimate to improve as its value gets closerto the center frequency of the corresponding wavelet band.

Fig. 9 provides another striking example of wavelet-domainAM/FM analysis. The input picture Barbara contains severaltextured objects (jeans, table cloth, etc.) which are segmentedout by the multi-scale coherency analysis. Note how theperiodic structures give rise to near-constant white patches inthe coherency map and to nicely saturated colored regions inthe composite orientation display. The crisper aspect of thelatter is due to its brightness being controlled by the waveletamplitude; this is a more local feature than the coherencywhich is defined over a local neighborhood. The primarycontours of the picture are also properly detected in themultiscale maps.

The next example illustrates the applicability of the tech-nique to the processing of real-world interferometric data. Theimage on the left of Fig. 10 was acquired using a digitalholography microscope. It is a hologram that results from the

Example: Digital holography microscopy

10

DHM image

Amplitude WavenumberOrientation

tensor-based

orientationFig. 10. Example of wavelet-domain amplitude demodulation. Left: ex-perimental digital holography micrograph; center: wavelet-domain amplitudeestimation; right: tensor-based orientation (Hue channel).

interference between the wave reflected by the object to be im-aged (a patterned nano-structure) and a reference plane wave(or carrier) [44]. The light intensity of the object can be re-constructed by using an appropriate demodulation and filteringalgorithm [45]. Fig. 10 illustrates the ability of the monogenicwavelet transform to perform such a blind multi-resolutionreconstruction via its wavelet amplitude. The orientation mapindicates that the carrier is a diagonally-oriented plane wave.We can also estimate its frequency from the wavenumber map(data not shown) which is essentially constant within the firstthree bands. Note that the transform also separates out someartifactual lower-frequency interferences that are superimposedonto the input signal and partly visible as circular patterns. Inthe present case, these components get transfered into the twolower resolution levels of the pyramid; they are probably dueto some dust particle intersecting the optical path of the laserbeam.

The final illustration is the multiresolution analysis of afingerprint image which contains oriented ridge-like structures(cf. Fig. 11). At finer scale, the wavelet modulus acts as acontour detector, while it eventually fills in the whole fingerprint area as the scale gets coarser (level 3). The multiresolu-tion orientation map nicely captures the directionality of theunderlying pattern; it also gets smoother and less noisy withincreasing scale. The third level estimate is probably the onethat looks the most satisfactory from a visual point of view.For more advanced feature extraction, we refer to the work ofLarkin et al. which describes a related phase-based frameworkfor fingerprint analysis; it includes specific detectors for branchcuts, loops and delta structures [46].

VI. CONCLUSION

We have set the mathematical foundations of a monogenicwavelet transform that gives access to the local orientation,amplitude and phase information of a 2-D signal within aproper multiresolution setting. The transform is build aroundthe monogenic extension of a polyharmonic spline wavelet

14 IEEE TRANSACTIONS ON IMAGE PROCESSING, TO APPEARExample: Fingerprint

11

WavenumberModulus

Amplitude OrientationOrientation

Fig. 11. Example of wavelet-domain monogenic image analysis. Left:original fingerprint image; center: wavelet-domain amplitude estimation; right:tensor-based orientation (Hue channel).

basis of L2(R2). The decomposition is reasonably fast, mod-erately redundant (a factor 3 or 4, depending on the samplingconfiguration), and fully reversible.

An important property is that the complex Riesz branchof the transform, which is akin to a multi-scale gradient, issteerable. We have shown that this decomposition can yield anew set of tensor-based features (orientation, Hilbert-transformenergy, and coherency) that offer interesting perspectives forlocal orientation, texture and structure analysis; this extractionprocess is robust and essentially rotation-invariant. The latterproperty is not fulfilled by the great majority of featuresobtained by conventional separable wavelet processing.

The distinguishing aspect of the present decomposition,when we compare it to the other currently-available complexor quaternion wavelet transforms, is the presence of the Lapla-cian branch of the transform, which yields the primary com-ponent in the monogenic signal formalism. It essentially actslike an isotropic filterbank which decomposes the signal into aseries non-directional wavelet subbands. The complementaryrole of the Riesz wavelets is to estimate the predominantlocal orientation within the given band and to compute thecorresponding directional Hilbert transform through an appro-priate steering mechanism. The monogenic feature extractionprocess ultimately gives access to the modulation parametersof the signal: amplitude, phase and instantaneous frequency,which are difficult to extract otherwise. Potential applicationsinclude various type of coherent image analyses—in particular,the processing of interferograms and fringe patterns—as wellas image enhancement and denoising since the transform isreversible. More generally, one may build upon the importantbody of work on the monogenic signal and take advantageof the Riesz-Laplace transform for transposing some of thesemethods to the wavelet domain, which may result in novelalgorithms.

APPENDIX IDIRECTIONAL HILBERT ANALYSIS

The monogenic wavelet components are denoted by(w, r1, r2). Our goal is to determine the amplitude (A), thephase ($) and the instantaneous frequency (#) in the directionspecified by u = (cos &, sin &) where & is our local orientationestimate—not necessarily equal to &0 = arctan

*r2r1

+. To that

end, we first define

q = r1 cos & + r2 sin &, (41)

which is the Hilbert transform of w in the direction &. Next,we introduce the complex signal z = w + jq, which isthe directional counterpart of the analytical signal in 1-D.The amplitude and phase information is encoded in the polarrepresentation z = Aej!, which yields

A =)

w2 + q2

$ = arctan* q

w

+.

The instantaneous frequency # is obtained by taking thederivative of $ along the direction of analysis: # = Du$. To geta more practical formula, we first evaluate the phase gradientusing the property that $ = Im(log z):

*$ = Im!*z

z

"=

!q*w

A2+

w*q

A2

with the differential notation *w = ((w(x , (w

(y ) = (wx, wy). Weobtain the directional derivative by taking the inner productwith u, which gives

Du$ = (*$, u) =!q(wx cos & + wy cos &)

A2+

w(uT Hu)A2

,

where H =!

r1x r1y

r2x r2y

"is a symmetric matrix (analogous

to the Hessian) due to the properties of the Riesz transform. Ifwe now assume that the underlying signal is locally coherent(in the sense that it has a one-dimensional structure along thedirection u), then H is of rank 1 and uT Hu = 'max =trace(H) = r1x + r2y . This leads to the simplified formula

# = Du$ =!q(wx cos & + wy sin &)

A2+

w(r1x + r2y)A2

. (42)

Thus, the critical ingredient for this determination is to providea numerical procedure for computing the derived quantitieswx, wy and !(r1x + r2y), which can be done in an efficientway using two reduced-order wavelet transforms, as specifiedin the text.

ACKNOWLEDGEMENTS

The authors would like to thank Katarina Balac (KB) forher implementation of the Laplace/Riesz transform as a JAVAPlugin for ImageJ; they are also grateful to Francois Aguet forproviding the original test image (Fig. 1) and to Prof. ChristianDepeursinge for giving us access to his digital holographymicroscopy data. This work was supported by the SwissNational Science Foundation (MU, KB) under Grant 200020-109415 and by the Center for Biomedical Imaging (DVDV)of the Geneva-Lausanne Universities and the EPFL, as wellas the foundations Leenaards and Louis-Jeantet.

15

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