IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 7...

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 7, JULY 2008 1069

Coherent Multiscale Image Processing UsingDual-Tree Quaternion Wavelets

Wai Lam Chan, Student Member, IEEE, Hyeokho Choi, and Richard G. Baraniuk, Fellow, IEEE

Abstract—The dual-tree quaternion wavelet transform (QWT)is a new multiscale analysis tool for geometric image features. TheQWT is a near shift-invariant tight frame representation whose co-efficients sport a magnitude and three phases: two phases encodelocal image shifts while the third contains image texture informa-tion. The QWT is based on an alternative theory for the 2-D Hilberttransform and can be computed using a dual-tree filter bank withlinear computational complexity. To demonstrate the properties ofthe QWT’s coherent magnitude/phase representation, we developan efficient and accurate procedure for estimating the local geomet-rical structure of an image. We also develop a new multiscale algo-rithm for estimating the disparity between a pair of images that ispromising for image registration and flow estimation applications.The algorithm features multiscale phase unwrapping, linear com-plexity, and sub-pixel estimation accuracy.

Index Terms—Coherent processing, dual-tree, multiscale dis-parity estimation, phase, quaternion, wavelets.

I. INTRODUCTION

THE encoding and estimation of the relative locations ofimage features play an important role in many image

processing applications, ranging from feature detection andtarget recognition to image compression. In edge detection, forexample, the goal is to locate object boundaries in an image.In image denoising or compression, state-of-the-art techniquesachieve significant performance improvements by exploitinginformation on the relative locations of large transform coeffi-cients [1]–[4].

An efficient way to compute and represent relative locationinformation in signals is through the phase of the Fourier trans-form. The Fourier shift theorem provides a simple linear rela-tionship between the signal shift and the Fourier phase. Whenonly a local region of the signal is of interest, the short-timeFourier transform (STFT) provides a local Fourier phase foreach windowed portion of the signal. The use of the Fourierphase to decipher the relative locations of image features is well

Manuscript received April 4, 2008. This work was supported in part by theNational Science Foundation under Grant CCF-0431150, in part by the Officeof Naval Research under Grant N00014-02-1-0353, in part by the AFOSR underGrant FA9550-04-1-0148, in part by the AFRL under Grant FA8650-05-1850,and in part by the Texas Instruments Leadership University Program. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Dr. LJubisa Stankovic.

W. L. Chan and R. G. Baraniuk are with the Department of Electrical andComputer Engineering, Rice University, Houston, TX 77005 USA (e-mail:[email protected]; [email protected]).

H. Choi, deceased, was with the Department of Electrical and Computer En-gineering, North Carolina State University, Raleigh, NC 27695 USA.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIP.2008.924282

Fig. 1. Three real wavelets (from the horizontal, vertical, and diagonal sub-bands, respectively) from the 2-D DWT basis generated using the length-14Daubechies filter.

established in the image processing and computer vision com-munities for applications such as stereo matching and imageregistration [5]–[7]. Indeed, the classic experiment of Lim andOppenheim [8] demonstrated that for natural images the Fourierphase contains a wealth of information beyond the magnitude.By the Fourier shift theorem, estimating location informationusing local phase provides more robust estimates with sub-pixelaccuracy and requires less computational effort compared topurely amplitude-based approaches.

For signals containing isolated singularities, such as piece-wise smooth functions, the discrete wavelet transform (DWT)has proven to be more efficient than the STFT. The localityand zooming properties of the wavelet basis functions leadto a sparse representation of such signals that compacts thesignal energy into a small number of coefficients. Waveletcoefficient sparsity is the key enabler of algorithms such aswavelet-based denoising by shrinkage [9]. Many natural imagesconsist of smooth or textured regions separated by edges andare well-suited to wavelet analysis and representation. Otheradvantages of wavelet analysis include its multiscale structure,invertibility, and linear complexity filter-bank implementation.2-D DWT basis functions are easily formed as the tensorproducts of 1-D DWT basis functions along the vertical andhorizontal directions; see Fig. 1.

The conventional, real-valued DWT, however, suffers fromtwo drawbacks. The first drawback is shift variance: a smallshift of the signal causes significant fluctuations in wavelet co-efficient energy, making it difficult to extract or model signalinformation from the coefficient values. The second drawbackis the lack of a notion of phase to encode signal location infor-mation as in the Fourier case.

Complex wavelet transforms (CWTs) provide an avenue toremedy these two drawbacks of the DWT. It is interesting tonote that the earliest modern wavelets, those of Grossmann andMorlet [10], were in fact complex, and Grossman continuallyemphasized the power of the CWT phase for signal analysis andrepresentation. Subsequent researchers have developed orthog-onal or biorthogonal CWTs; see, for example, [11]–[16].

A productive line of research has developed over the pastdecade on the dual-tree CWT, which in 1-D combines two or-thogonal or biorthogonal wavelet bases using complex algebra

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1070 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 7, JULY 2008

Fig. 2. Six complex wavelets from the 2-D dual-tree CWT frame generatedfrom orthogonal near-symmetric filters [23] in the first stage and Q-filters [24]in subsequent stages. (a) Real parts, with approximate even symmetry; (b) imag-inary parts, with approximate odd symmetry.

into a single system, with one basis corresponding to the “realpart” of the complex wavelet and the other to the “imaginarypart” [17]. Ideally, the real and imaginary wavelets are a Hilberttransform pair (90 out of phase) and form an analytic waveletsupported on only the positive frequencies in the Fourier do-main, just like the cosine and sine components of a complex si-nusoid. The 1-D dual-tree CWT is a slightly (2 ) redundanttight frame, and the magnitudes of its coefficients are nearlyshift invariant [17]. There also exists an approximately linear re-lationship between the dual-tree CWT phase and the locationsof 1-D signal singularities [18] as in the Fourier shift theorem.

The 2-D dual-tree CWT for images is based on the theory ofthe 2-D Hilbert transform (HT) and 2-D analytic signal as firstsuggested by Hahn [19]. In particular, a 2-D dual-tree complexwavelet is formed using the 1-D HT of the usual 2-D real DWTwavelets in the horizontal and/or vertical directions. The resultis a 4 redundant tight frame with six directional subbandsoriented at multiples of 15 ; see Fig. 2 [17], [20]. The 2-D CWTis near shift-invariant, and its magnitude-phase representationhas a complex phase component that encodes shifts of local 1-Dstructures in images such as edges and ridges [21]. As a result,the 2-D dual-tree CWT has proved useful for a variety of tasksin image processing [3], [4], [21], [22].

Each 2-D dual-tree CWT basis coefficient has a single phaseangle, which encodes the 1-D shift of image features perpen-dicular to its orientation. This may be sufficient for analyzinglocal 1-D structures such as edges. However, when the featureunder analysis is intrinsically 2-D [25]—for example, an imageT-junction [26]—then its relative location is defined in both thehorizontal and vertical directions. This causes ambiguity in theCWT phase shift, whereby we cannot resolve the image shiftsin both the horizontal and vertical directions from the change ofonly one CWT coefficient phase. To overcome this ambiguity,we must conduct a joint analysis with two CWT phases from dif-ferently oriented subbands, which can complicate image anal-ysis and modeling considerably.

In this paper, we explore an alternative theory for the 2-D HTand analytic signal due to Bülow [25], [27] and show that it leadsto an alternative to the 2-D dual-tree CWT. In Bülow’s HT, the2-D analytic signal is defined by limiting the 2-D Fourier spec-trum to a single quadrant. Applying this theory within the dualtree framework, we develop and study a new dual-tree quater-nion wavelet transform (QWT), where each quaternion waveletconsists of a real part (a usual real DWT wavelet) and threeimaginary parts that are organized according to quaternion al-gebra; see Fig. 3. Our QWT, first proposed in [28] and [29], is a4 redundant tight frame with three subbands (horizontal, ver-tical, and diagonal). It is also near shift-invariant.

Fig. 3. Three quaternion wavelets from the 2-D dual-tree QWT frame.Each quaternion wavelet comprises four components that are 90 phaseshifts of each other in the vertical, horizontal, and both directions. (a) Hor-izontal subband, from left to right: � (x) (y) (a usual, real DWT tensorwavelet), � (x) (y), � (x) (y), � (x) (y), j (x; y)j. (b) Verticalsubband, from left to right: (x)� (y) (a usual, real DWT tensor wavelet), (x)� (y), (x)� (y), (x)� (y), j (x; y)j. (c) Diagonal sub-band, from left to right: (x) (y) (a usual, real DWT tensor wavelet), (x) (y), (x) (y), (x) (y), j (x; y)j. The image on the farright is the quaternion wavelet magnitude for each subband, a nonoscillatingfunction. The same dual-tree wavelet filters are used as the 2-D dual-tree CWTin Fig. 2.

The QWT inherits a quaternion magnitude-phase representa-tion from the quaternion Fourier transform (QFT). The first twoQWT phases encode the shifts of image features in theabsolute horizontal/vertical coordinate system, while the thirdphase encodes edge orientation mixtures and texture informa-tion. One major focus of this paper is to demonstrate coherent,multiscale processing using the QWT, or in other words, the useof its magnitude and phase for multiscale image analysis.

To illustrate the power of coherent processing, we considertwo image processing applications. In the first application,we develop a new magnitude-and-phase-based algorithm foredge orientation and offset estimation in local image blocks.Our algorithm is entirely based on the QWT shift theorem andthe interpretation of the QWT as a local QFT analysis. In thesecond application, we design a new multiscale image disparityestimation algorithm. The QWT provides a natural multiscaleframework for measuring and adjusting local disparities andperforming phase unwrapping from coarse to fine scales withlinear computational efficiency. The convenient QWT encodingof location information in the absolute horizontal/vertical co-ordinate system facilitates averaging across subband estimatesfor more robust performance. Our algorithm offers sub-pixelestimation accuracy and runs faster than existing disparity es-timation algorithms like block matching and phase correlation[30]. When many sharp edges and features are present and theunderlying image disparity field is smooth, our method alsoexhibits superior performance over these existing techniques.

Previous work in quaternions and the theory of the 2-D HTand analytic signal for image processing includes Bülow’sextension of the Fourier transform and complex Gabor filtersto the quaternion Fourier transform (QFT) [25]. Our QWTcan be interpreted as a local QFT and, thus, inherits manyof its interesting and useful theoretical properties such as thequaternion phase representation, symmetry properties, and shifttheorem. In addition, the dual-tree QWT sports a linear-time

CHAN et al.: COHERENT MULTISCALE IMAGE PROCESSING USING DUAL-TREE QUATERNION WAVELETS 1071

and invertible computational algorithm. A different extensionof the QFT yields the quaternion wavelet pyramid introducedby Bayro–Corrochano [31]; however, the use of Gabor filterslimits its performance and renders it noninvertible. There arealso interesting connections between the dual-tree QWT andthe (nonredundant) quaternion wavelet representations of Atesand Orchard [32] and Hua and Orchard [33].

Finally, we note that there exists a third alternative 2-D HTcalled the Riesz transform and its associated analytic signalcalled the monogenic signal [34]. The monogenic signal,generated by spherical quadrature filters, has a vector-valuedphase that encodes both the orientations of intrinsically 1-D(edge-like) image features and their shift normal to the edgeorientation. Felsberg’s extension of the monogenic signalcan analyze intrinsically 2-D signals with two phase anglesbut requires complicated processing such as local orientationestimation and steering of basis filters [35].

This paper is organized as follows. We start by briefly re-viewing the DWT and dual-tree CWT in Section II. Section IIIdevelops the dual-tree QWT, and Section IV discusses some ofits important properties, in particular its phase response to sin-gularities. We develop and demonstrate the QWT-based edgegeometry and disparity estimation algorithms in Section V. Sec-tion VI concludes the paper with a discussion of the QWT’s po-tential for future applications. The Appendix contains detailedderivations and proofs of some of the QWT properties and the-orems from Sections IV and V.

II. REAL AND COMPLEX WAVELET TRANSFORMS

This section overviews the real DWT and the dual-tree CWT.We also develop a new formulation for the 2-D dual-tree CWTusing the theory of 2-D HTs.

A. Real DWT

The real DWT represents a 1-D real-valued signal interms of shifted versions of a scaling function and shiftedand scaled versions of a wavelet function [36]. The func-tions and ,

, form an orthonormal basis, and we can representany as

(1)

where and arethe scaling and wavelet coefficients, respectively. The param-eter sets the coarsest scale space that is spanned by .Behind each wavelet transform is a filterbank based on lowpassand highpass filters.

The standard real 2-D DWT is obtained using tensor prod-ucts of 1-D DWTs over the horizontal and vertical dimensions.The result is the scaling function and three subbandwavelets , , and that are orientedin the horizontal, vertical, and diagonal directions, respectively[36] (see Fig. 1).

The real wavelet transform suffers from shift variance; thatis, a small shift in the signal can greatly perturb the magnitude

Fig. 4. One-dimensional dual-tree CWT is implemented using a pair of filterbanks operating on the same data simultaneously. Outputs of the filter banks arethe dual-tree scaling coefficients, c and c , and the wavelet coefficients,d and d , at scale ` and shift p. The CWT coefficients are then obtainedas d + jd .

of wavelet coefficients around singularities. It also lacks a no-tion of phase to encode signal location information and suffersfrom aliasing [37]. These issues complicate modeling and infor-mation extraction in the wavelet domain.

B. Dual-Tree CWT

The 1-D dual-tree CWT expands a real-valued signal in termsof two sets of wavelet and scaling functions obtained from twoindependent filterbanks [17], as shown in Fig. 4. We will usethe notation and to denote the scaling and waveletfunctions and and to denote their corresponding co-efficients, where specifies a particular set of wavelet filters.The wavelet functions and from the two trees playthe role of the real and imaginary parts of a complex analyticwavelet . The imaginary waveletis the 1-D HT of the real wavelet . The combined system isa 2 redundant tight frame that, by virtue of the fact thatis nonoscillating, is near shift-invariant.1

It is useful to recall that the Fourier transform of the imagi-nary wavelet equals when andwhen . Thus, the Fourier transform of the complexwavelet function has no energy (orlittle in practice) in the negative frequency region,2 making itan analytic signal [17].

C. Hilbert Transforms and 2-D CWT

Extending the 1-D CWT to 2-D requires an extension of theHT and analytic signal. There exist not one but several differentdefinitions of the 2-D analytic signal that each zero out a dif-ferent portion of the 2-D frequency plane [27]. We will considertwo definitions. The first, proposed by Hahn in [19], employscomplex algebra and zeros out frequencies on all but a singlequadrant ( , , for example, where ( , ) indexes the2-D frequency plane). In this formulation, the complete 2-D an-alytic signal consists of two parts: one having spectrum on theupper right quadrant ( , ) and the other on the upperleft quadrant ( , ) [27].

1A finitely supported function can never be exactly analytic [37]. In practice,we can only design finite-length complex wavelets that are approximately ana-lytic, and, thus, the CWT is only approximately shift-invariant [17], [20].

2Note that the Fourier transform of the complex scaling function, � (!) +j� (!) = � (!), is only approximately analytic in practice, and so its supportwill leak into the negative frequency region.


Definition 1 [19]: Let be a real-valued, 2-D function. Thecomplete 2-D complex analytic signal is defined in the spacedomain, , as the pair of complex signals

(2)

(3)

where

(4)

(5)

(6)

The function is the total HT; the functions andare the partial HTs; and are impulse sheets along the

axis and axis, respectively; and denotes 2-D convolution.The 2-D complex analytic signal in (2)–(3) is the notion

behind the 2-D dual-tree CWT [17], [20]. Each 2-D CWT basisfunction is a 2-D complex analytic signal consisting of a stan-dard DWT tensor wavelet plus three additional real waveletsobtained by 1-D HTs along either or both coordinates. Forexample, starting from real DWT’s diagonal-subband tensorproduct wavelet from above, we obtainfrom (4)–(6) its partial and total HTs

From Definition 1, we then obtain the two complex wavelets

(7)

(8)

having orientations, 45 and , respectively. Similar expres-sions can be obtained for the other two subbands and

) based on and .Each 2-D CWT coefficient has only a single phase angle,

which encodes the 1-D shift of image features perpendicular toits subband direction. Fig. 5(a) illustrates this phase-shift prop-erty. This encoding may be sufficient for local 1-D structuressuch as edges, since we can define edge shifts uniquely by asingle value, say , in the direction perpendicular to the edge.However, even in this case, the analysis is not so straightforwardwhen the edge does not align with the six orientations of theCWT subbands. Moreover, shifts of intrinsically 2-D (nonedge)image features such as in Fig. 5(a) require two valuesin the and directions, respectively. This creates ambiguityin the CWT phase shift. We can resolve this ambiguity by usingthe coefficients from two CWT subbands, but this complicatesthe use of the CWT for image analysis, modeling, and otherimage processing applications. In contrast, Fig. 5(b) illustrates

Fig. 5. (a) CWT coefficient’s single phase angle responds linearly to imageshift r in a direction orthogonal to the wavelet’s orientation. (b) Two of theQWT coefficient’s three phase angles respond linearly to image shifts (r ; r )in an absolute horizontal/vertical coordinate system.

a more convenient encoding of image shifts in absolute -co-ordinates (with two phase angles) using the quaternion phasesof our new QWT, to which we now turn our attention.

III. QUATERNION WAVELET TRANSFORM (QWT)

A. Quaternion Hilbert Transform

There are several alternatives to the 2-D analytic signal ofDefinition 1; we focus here on one due to Bülow [27]. It com-bines the partial and total HTs from (4)–(6) to form an analyticsignal comprising a real part and three imaginary componentsthat are manipulated using quaternion algebra [25].

The set of quaternionshas multiplication rules and, as well as component-wise addition and multiplication by

real numbers [38]. Additional multiplication rules include:, , and . Note that

quaternionic multiplication is not commutative. The conjugateof a quaternion is defined by

while the magnitude is defined as.

An alternative representation for a quaternion is through itsmagnitude and three phase angles: [25],where are the quaternion phase angles, computedusing the following formulae (for normalized, i.e., )

(9)

and, in the regular case (i.e., when )

(10)

(11)

In the singular case, i.e., when , , and are notuniquely defined. Only the sum (if ) or the differ-ence (if ) of and is unique [25]. Ifcalculated from (9)–(11) satisfy , sub-tract by if ; add to if . As a result,


each quaternion phase angle is uniquely defined within the range.

The operation of conjugation in the usual set of complex num-bers, , where and , is a so-calledalgebra involution that fulfills the two following properties forany : and . In , there arethree nontrivial algebra involutions

(12)

(13)

(14)

Using these involutions we can extend the definition of Her-mitian symmetry. A function is called quaternionicHermitian if, for each

and

(15)

Bülow introduces an alternative definition of 2-D analyticsignal based on the quaternion Fourier transform (QFT) [25].The QFT of a 2-D signal is given by

(16)

where denotes the QFT operator, indexes theQFT domain, and the quaternion exponential

(17)

is the QFT basis function. The real part of (17) is, while the other three quaternionic

components are its partial and total HTs as defined in (4)–(6).Note that the QFT of a real-valued signal is quaternionicHermitian, and each QFT basis function satisfies the definitionof a 2-D quaternion analytic signal.

Definition 2 [27]: Let be a real-valued 2-D signal. The 2-Dquaternion analytic signal is defined as

(18)

where the functions , , and are defined as in(4)–(6).

B. QWT Construction

Our new 2-D dual-tree QWT rests on the quaternion defi-nition of 2-D analytic signal. By organizing the four quadra-ture components of a 2-D wavelet (the real wavelet and its 2-DHTs) as a quaternion, we obtain a 2-D analytic wavelet andits associated quaternion wavelet transform (QWT). For ex-ample, for the diagonal subband, with

, we ob-tain the quaternion wavelet

(19)

Fig. 6. Quaternion Fourier domain relationships among the four quadraturecomponents of a quaternion wavelet (x; y) in the diagonal subband. TheQFT spectra of the real wavelet (x) (y) in the first to fourth quadrantsare denoted by (F ; F ; F ; F ), respectively. The partial and totalHilbert transform operations are equivalent to multiplying the quadrants ofF f (x) (y)g in (a) by �j , or �j , or both. (a) F f (x) (y)g,(b) F f (x) (y)g, (c) F f (x) (y)g, (d) F f (x) (y)g.

To compute the QWT coefficients, we can use a separable 2-Dimplementation [20] of the dual-tree filter bank in Fig. 4. Duringeach stage of filtering, we independently apply the two sets ofand wavelet filters to each dimension ( and ) of a 2-D image;for instance, applying the set of filters to both dimensionsyields the scaling coefficients and the diagonal, vertical,and horizontal wavelet coefficients, , , and ,respectively. Therefore, the resulting 2-D dual-tree implemen-tation comprises four independent filter banks [corresponding toall possible combinations of wavelet filters applied to each di-mension ( , , , and )] operating on the same 2-D image.We combine the wavelet coefficients of the same subband fromthe output of each filter bank using quaternion algebra to obtainthe QWT coefficients; for example, for the diagonal subband:

.Fig. 3(c) illustrates the four components of a quaternion

wavelet and its quaternion magnitude for the diagonal subband.The partial and total HT components resemble butare phase-shifted by 90 in the horizontal, vertical, and bothdirections. The magnitude of each quaternion wavelet (squareroot of the sum-of-squares of all four components) is a smoothbell-shaped function. We can also interpret the four componentsof in the QFT domain as multiplying the quadrantsof the QFT of by or , or both, as shownin Fig. 6. Note that the order of multiplication is importantbecause quaternion multiplication is noncommutative. Thisquaternion wavelet, , has support in only a singlequadrant of the QFT domain (see Appendix A).

The construction and properties are similar for the othertwo subband quaternion wavelets based on and

[see the horizontal and vertical subband wavelets,and in Fig. 3(a) and (b), respectively].


In summary, in contrast with the six complex pairs of CWTwavelets (12 functions in total), the QWT sports three quater-nion sets of four QWT wavelets (12 functions in total).

Finally, note that the quaternion wavelet transform isapproximately a windowed quaternion Fourier transform(QFT) [25]. In contrast to the QFT in (16), the basis func-tions for the QWT are scaled and shifted versions of thequaternion wavelets ,

, and.

IV. QWT PROPERTIES

Since the dual-tree QWT is based on combining 1-D CWTfunctions, it preserves many of the attractive properties of theCWT. Furthermore, the quaternion organization and manipula-tion provide new features not present in either the 2-D DWT orCWT. In this section, we discuss some of the key properties ofthe QWT with special emphasis on its phase.

A. Tight Frame

The QWT comprises four orthonormal basis sets and, thus,forms a 4 redundant tight frame. The components of the QWTwavelets at each scale can be organized in matrix form as

(20)

The frame contains shifted and scaled versions of the functionsin plus the scaling functions. Each column of the matrix

contains the four components of the quaternion waveletcorresponding to a subband of the QWT. For example, thefirst column contains the quaternion wavelet components inFig. 3(c), that is, the tensor product wavelet andits 2-D partial and total HTs in (19). Each row of containsthe wavelet functions necessary to form one orthonormal basisset. Since has four rows, the total system is a 4 redundanttight frame. An important consequence is that the QWT isstably invertible. The wavelet coefficients corresponding to theprojections onto the functions in can be computed using a2-D dual-tree filter bank with linear computational complexity.

B. Relationship to the 2-D CWT

A unitary transformation links the QWT frame elements andcoefficients and the 2-D CWT frame elements and coefficients.The components of the CWT wavelets at each scale can be or-ganized in matrix form, as shown in (21) at the bottom of the

page. The columns of contain the complex wavelets orientedat , , and , respectively.

We obtain the CWT wavelets by multiplying the matrix in(20) by the unitary matrix

(22)

Since , the CWT also satisfies the tight-frame propertywith the same 4 redundancy factor. As we will see in the Sec-tion IV-D, both the CWT phase and the QWT phases encode2-D image feature shifts; however, there exists no straightfor-ward relationship between the phase angles of the QWT andCWT coefficients.

C. QWT and QFT

To make concrete the interpretation of the QWT as a localQFT, we derive a QFT Plancharel theorem and an inner productformula in the QFT domain.QFT Plancharel Theorem. Let be a real-valued 2-D signal,let be a quaternion-valued 2-D signal, and let and

be their respective QFTs. Then the inner product ofand in the space domain equals the following inner

product in the QFT domain:

(23)

where and are the algebra involutions defined in(12)–(14). The functions and are, respectively, the evenand odd components of with respect to the spatial coordinate

, as defined in the QFT convolution theorem [25]

(24)

(25)

We call the right side of (23) the QFT inner product betweenand . The proof appears in Appendix B.

The QFT Plancharel Theorem enables us to interpret theQWT as a local or windowed QFT. Let be a quaternionwavelet at a particular scale and subband and be a realimage under analysis. Their QFT inner product in (23) givesthe corresponding QWT coefficient. Since quaternion waveletshave a single-quadrant QFT spectrum as shown in Appendix A,

(21)


the integration limit of the QFT inner product reduces toin the QWT case.

D. QWT Phase Properties

Recall from Section III-A that each QWT coefficientcan be expressed in terms of its magnitude and phase as

. We seek a shift theorem for the QWTphase that is analogous to that for the CWT. Since the QWTperforms a local QFT analysis, the shift theorem for the QFT[25] holds approximately for the QWT. When we shift animage from to , the QFT phase undergoes thefollowing transformation:

(26)

where denotes the shift in the horizontal/verticalspatial coordinate system.

To transfer the shift theorem from the QFT to the QWT, weexploit the fact that the QWT is approximately a windowedQFT. That is, each quaternion wavelet is approximately a win-dowed quaternion exponential from (17), and each QWT coef-ficient is the inner product [as in (23)] between this windowedquaternion exponential and the image. The scale of analysiscontrols the center frequency of the windowed exponen-tial in the QFT plane.

The magnitude and phase of the resulting coefficient are de-termined by two factors: the spectral content of the imageand the center frequency of the wavelet. These two fac-tors determine the frequency parameters we should use inthe shift theorem for the QFT (26). We term the effectivecenter frequency for the corresponding wavelet coefficient. For-tunately, for images having a smooth spectrum over the supportof the quaternion wavelet in the QFT domain, .Note that should always lie in the first quadrant of theQFT domain (that is, ).

Thus, to apply the shift theorem in QWT image analysis, wefirst estimate the effective center frequency for each QWT coef-ficient (or assume that ). Then, using (26), wecan estimate the shift of one image relative to a secondimage from the phase change . Conversely, we canestimate the phase shift once we know the image shift.

Finally, a quick word on the quirky third QWT phase angle. We can interpret as the relative amplitude of image energy

along two orthogonal directions as in [25], which is useful foranalyzing texture information. For example, Fig. 7 depicts thefunction for the diagonal subband as we adjust of thewavelet coefficient . We see a gradual change in appearancefrom an oriented function to texture-like and back. This propertycould prove useful for the analysis of images with rich textures[25]. As we describe below in Section V-A, the third phase alsorelates to the orientation of a single edge.

V. QWT APPLICATIONS

In this section, we demonstrate the utility of the QWT withtwo applications in geometrical edge structure and image dis-parity estimation.

Fig. 7. Effect of varying � on the structure of the corresponding weightedquaternion wavelet for the diagonal subband (from left to right): � =�(�=4),�(�=8), 0,�=8, (�=4). The corresponding weighted wavelet changesfrom textured (� = 0) to oriented (� = �(�=4)).

A. Edge Geometry Estimation

Edges are the fundamental building blocks of manyreal-world images. Roughly speaking, a typical natural imagecan be considered as a piecewise smooth signal in 2-D thatcontains blocks of smooth regions separated by edge dis-continuities due to object occlusions. Here we use the QWTmagnitude and phase to extend the multiscale edge geometryanalysis of [18] and [21].

1) Theory: Consider an image containing a dyadicimage block that features a single step edge, as parameterizedin Fig. 8(a). Note that for an edge oriented at angle , any shift

in the directions satisfying the constraint

(27)

is identical to a shift from the center of the block by in thedirection perpendicular to .

Our goal in this section is to analyze the phaseof the quaternion wavelet coefficient (e.g., for the verticalsubband) corresponding to the quaternion waveletwhose support aligns with the dyadic image block containingthe edge (the other subbands behave similarly). We will showthat and provide an accurate means with which toestimate the edge offset and edge orientation , respectively.

First, we establish a linear relationship betweenand the edge offset using the shift theorem in (26). Recallfrom Section IV-D that the effective center frequencydepends on both the image QFT spectral content andthe center frequency of the quaternion wavelet, and it alwayslies in the first quadrant. Since the spectral energy of the edgeQFT, , concentrates along two impulse ridges throughthe origin having orientations and in theQFT domain [see Fig. 8(b) and Appendix C], we can write

, where is a positive constantthat depends on , the subband, and the scale of analysis. When the edge passing through the image block center

displaces perpendicularly by , the changes in phase anglessatisfy and .

Plugging and using (27), we obtainthe concise formula

(28)

where we choose when , andwhen . We have verified this relationship via

experimental analysis of straight edges in detail in an earlierpaper [28].

Based on the interpretation of the QWT as a local QFT, weuse the inner product formula in (23) to analyze the behavior of


Fig. 8. (a) Parameterization of a single edge in a dyadic image block (awedgelet [39]). (b) QFT spectrum of the edge; shaded squares represent thequaternion wavelets in the horizonal, vertical, and diagonal subbands. Theenergy of the edge is concentrated along the two dark lines crossing at theorigin and is captured by the vertical subband with effective center frequencyat quaternion frequency (u; v). The region bounded by the dashed line demon-strates the spectral support of the QWT basis “leaking” into the neighboringquadrant. (a) Single-edge model (wedgelet); (b) edge QFT spectrum.

for the same edge block. The QWT coefficient can becomputed from the QFT inner product between and theQFT of the edge image (similarly for the other subbands).Our analysis in Appendix D states that if the quaternion waveletis perfectly analytic, then regardless of , when

and when . Note that thiscorresponds to the singular case in the quaternion phase calcu-lation in Section III-A.

However, practical quaternion wavelets will not be perfectlyanalytic, and so their QFT support will leak into other quadrantsof the QFT domain as in Fig. 8(b). This necessitates the morein-depth analysis of Appendix E, which shows that in this case

(29)

where is a measure of the ratio of local signal energy in thepositive quadrant to the energy in the leakage quadrant. For thevertical subband as shown in Fig. 8(b), when the edge orien-tation changes from 0 to 45 , this ratio changes from 1to 0 and, thus, changes from 0 to . We model this be-havior of in the horizontal and vertical subbands to design anedge orientation estimation in Section V-A-II. Since the diag-onal subband wavelet has QFT support distant from the leakagequadrants, the QWT subband coefficients are almost unaf-fected by leakage (i.e., ). Their corresponding approx-imately equal and do not vary with .

2) Practice: Based on the above analysis, we propose a hy-brid algorithm to estimate the edge geometry based onthe QWT phase and the magnitude ratios betweenthe three subbands. We generate a set of wedgelets with known

and [see Fig. 8(a)] [39] for analysis and testing. Our algo-rithm is reminiscent of the edge estimation scheme in [21].

To estimate the edge orientation , we use both the magni-tude ratios among the three subbands and of the subband withthe largest magnitude. The subband with the largest magnitudegives the approximate orientation of the edge ( for diag-onal, for horizontal, and for vertical); the sign of

tells whether the direction is positive or negative. We exper-imentally analyze the QWT magnitude ratios and of the setof generated wedgelets corresponding to changing edge orien-tations by multiples of 5 . Using standard curve-fitting tech-niques, we develop a simple relationship between these param-eters and for our orientation estimation scheme. The resultingorientation estimation algorithm achieves a maximum error ofonly in practice for ideal edges.

To estimate the offset of the edge, we apply the relationshipbetween and in (28). We use from either thehorizontal or vertical subband (whichever has a larger magni-tude). Depending on , we compute the sum (or difference)of the change in phase angles for the edge underanalysis, using as the reference edge. Upon analysis ofthe simulated wedgelets with known , we estimate ,to be used in (28). Our final edge offset estimation algorithmachieves a maximum error of approximately relative tothe normalized unit edge length of the dyadic block (that is,sub-pixel accuracy). More details of the experimental analysisfor the wedgelet model can be found in [28]. According to (28),within a -range of , the range of is limited to aninterval of length , which ensures that the edge stayswithin the image block under analysis. Therefore, in our offsetestimation, we need only consider one -range ofand do not need to perform any “phase-unwrapping”.

Finally, we estimate the polarity of the edge. By first ob-taining an offset estimate for each polarity of the edge with ori-entation estimated above, namely and , we use the innerproduct between the image block and two wedgelets with the es-timated edge parameters and to determine thecorrect polarity. Although our calculation of estimation accu-racy is based on the wedgelet model, our algorithm also workswell for real-world images such as the popular “cameraman”image in Fig. 9.

Our results demonstrate the close relationship between edgegeometry and QWT magnitude and phases, in particular, theencoding of edge location in the QWT phases and theencoding of edge orientation in the QWT magnitude and thethird phase .

B. Image Disparity Estimation

In this section, as another example of QWT-based data pro-cessing, we present an algorithm to estimate the local dispari-ties between the target image and the reference image

. Disparity estimation is the process of determining thelocal translations needed to align different regions in two im-ages, that is, the amount of 2-D translation required to move alocal region of a target image centered at pixel to alignwith the region in a reference image centered at the same lo-cation . This problem figures prominently in a range ofimage processing and computer vision tasks, such as video pro-cessing to estimate motion between successive frames, time-lapse seismic imaging to study changes in a reservoir over time,medical imaging to monitor a patient’s body, super-resolution,etc.

Recall that the QWT phase property states that a shiftin an image changes the QWT phase from to

. Thus, for each QWT coefficient, if we


Fig. 9. Local edge geometry estimation using the QWT. (a) Several edgy re-gions from the “cameraman” image; (b)–(e) edge estimates from the corre-sponding QWT coefficients. The upper row shows the original image region,the lower row shows a wedgelet [see Fig. 8(a)] having the edge parameter esti-mates (�; r) (no attempt is made to capture the texture within the block).

know , the effective center frequency of the local image re-gion analyzed by the corresponding QWT basis functions, thenwe can estimate the local image shifts from the phasedifferences.

However, the center frequency is image dependent, ingeneral. To be able to estimate image shifts from QWT phasedifferences, we first need to estimate for each QWT co-efficient. For this estimate, we can again use the QWT phaseproperties. If we know the image shifts and measure the phasedifference, then we can compute .

By manually translating the reference image byknown small amounts both horizontally and vertically, weobtain two images and . Aftercomputing the QWTs of and , wecan use the phase differences between the QWTcoefficients to obtain estimates for the effective spectral center

for each dyadic block across all scales asand . The range of QWT phase angles limits ourestimates to andfor horizontal and vertical shifts, respectively, where is thelength of one side of the dyadic block corresponding to eachcoefficient.

Once we know the center frequency for each QWTcoefficient, we can estimate the local image shifts by measuringthe difference between the QWT phase corresponding to thesame local blocks in image and .

A key challenge in phase-based disparity estimation is re-solving the phase wrap-around effect due to the limited rangeof phase angles. Due to phase wrapping, each observed phasedifference can be mapped to more than one disparity estimate.Specifically, for QWT phase differences between

the reference and target images, we can express the possibleimage shifts of each dyadic block as

(30)

where and . Depending on , is chosensuch that it equals 0 when is even and equals 1 when is odd.The special wrap-around effect in (30) is due to the limited rangein and (to and , respectively).

In our multiscale disparity estimation technique, we usecoarse scale shift estimates to help unwrap the phase in thefiner scales. If we assume that the true image shift is smallcompared to the size of dyadic squares at the coarsest scale

, then we can set in (30) at this scale (nophase wrap-around) and obtain correct estimates for and

. Effectively, this assumption of no phase wrap-around atthe coarsest scale limits the maximum image shift that we canestimate correctly. Once we have shift estimates at scale , foreach block at scale , we estimate the shifts as follows.

1) Interpolate the estimates from the previous scale(s) to ob-tain predicted estimates .

2) Substitute into (30) and determine thesuch that is closest to .

3) Remove any unreliable .4) Repeat Steps 1)–3) for the finer scalesStep 1) uses either nearest-neighbor interpolation (which

gives higher estimation accuracy) or bilinear interpolation(which results in a smoother disparity field for better visualquality). We choose the latter for our simulations in this paper.In Step 3), we use a similar reliability measure as in the con-fidence mask [25] to threshold unreliable phase and offsetestimates. We also threshold based on the magnitude of theQWT coefficients. We iterate the above process until a fineenough scale (e.g., ), since estimates typically becomeunreliable at this scale and below. The QWT coefficients forthe small dyadic blocks have small magnitudes, and so theirphase angles are very sensitive to noise.

We can improve upon the basic iterative algorithm by fusingestimates across subbands and scales. First, with proper interpo-lation, we can average over estimates from all scales containingthe same image block. Second, we can average estimates fromthe three QWT subbands for the same block to yield more ac-curate estimates, but we need to discard some unreliable sub-band estimates (for example, horizontal disparity in the hori-zontal subband and in the vertical subband). We incorporatethese subband/scale averaging steps into each iteration of Steps1)–4).3

Fig. 10 illustrates the result of our QWT phase-based dis-parity estimation scheme for two frames from the rotatingRubik’s cube video sequence [25]. This is an interestingsequence, because a rotation cannot be captured by a singleglobal translation but can be closely approximated by localtranslations. The arrows indicate both the directions and mag-nitudes of the local shifts, with the magnitudes stretched forbetter visibility. We can clearly see the rotation of the Rubik’scube on the circular stage, with larger translations closer to the

3Matlab code available at http://www.dsp.rice.edu/software/qwt.shtml.


Fig. 10. Multiscale QWT phase-based disparity estimation results. (a), (b) Ref-erence and target images from the “Rubik’s cube” image sequence [25]. (c) Dis-parity estimates between two images in the sequence, shown as arrows overlaidon top of the reference image (zoomed in for better visualization, arrow lengthsproportional to magnitude of disparity).

viewer (bottom of the image) and smaller translations furtherfrom the viewer (top of the image). In our experiments, weobtained the most robust estimations by averaging over bothscales and subbands.

Fig. 11 demonstrates the multiscale nature of our disparityestimation approach on an image sequence of a living heart [40].The presence of sharp edges plus the smooth cardiac motionin these images is well-matched by our wavelet-based analysis.Our algorithm averages over several scales to obtain motionfields at various levels of detail. Fig. 11(c) and (e) clearly showthe coarse-scale contraction and relaxation motions of the heart,while Fig. 11(d) and (f) displays more detailed movements ofthe heart muscles, arteries, and blood flow (in particular, see thearrows toward left region of the heart near the artery).

In addition to visualizing the changes from one image toanother, we can use our algorithm as the feature matching stepof an image registration algorithm, using the disparity infor-mation to build a warping function to align the images. Oneimportant note is that our QWT-based method is region-basedin that it does not require any detection of feature pointsor other image landmarks. Traditional region-based featurematching methods, which estimate the spatial correlation (orcorrespondence) between two images, include block-matchingand Fourier methods. For comparison, we compare our ap-proach to the exhaustive search (ES) block matching technique,which is very computationally demanding but offers the bestperformance among all general block-matching techniques. Wealso compare to a Fourier sub-pixel motion estimation method

Fig. 11. Multiscale QWT phase-based disparity estimation results for the“heart” image sequence. (a), (b) Reference and target “heart ”images fromtwo frames during heart contraction (systole). Estimation results showcoarse-scale and fine-scale detailed motion of the heart and blood flow duringthe (c)–(d) contraction and (e)–(f) relaxation (systole and diastole) phases of thecardiac cycle, illustrating the multiscale nature of our algorithm. (a) Referenceimage (heart) during contraction; (b) target image (heart) during contraction;(c) coarse-scale estimates (during contraction); (d) fine-scale estimates (duringcontraction); (e) coarse-scale estimates (during relaxation); (f) fine-scaleestimates (during relaxation).

known as gradient correlation (GC), which has been shownto have better PSNR performance than other recent Fouriermethods [30].

As a performance measure, we use the peak signal-to-noiseratio (PSNR) between the motion compensated imageand the target image , which is given by

(31)

where is the number of image pixels. The motion compen-sated image is obtained by shifting each image blockin the reference image according to the estimated mo-tion vectors. Fig. 12 compares the results for three image se-quences: the “Rubik” and “Taxi” sequences commonly usedin the optical flow literature, and the “Heart” sequence from


Fig. 12. Comparison of multiscale QWT phase-based disparity estimationwith two motion estimation algorithms, gradient correlation (GC) [30] andexhaustive search (ES). The performance measure is PSNR (in decibels)between the motion-compensated image and the target image of three testimage sequences (“Rubik,” “Heart,” and “Taxi”). (a) Frame-by-frame PSNRperformance comparison in the “Rubik” sequence. (b) Table of average PSNRperformance (over all frames) for each test sequence. The multiscale QWTphase-based method demonstrates the best performance among the three testalgorithms for the “Rubik” sequence and shows comparable performance tothe other algorithms for the “Heart” and “Taxi” sequences. Last row of tableshows the computational complexity of each algorithm.

Fig. 11. Fig. 12(a) demonstrates the superior performance of ourQWT phase-based algorithm over the other algorithms for the“Rubik” sequence, which has piecewise-smooth image framesand a smooth underlying disparity flow. While its PSNR per-formance is relatively far from ES for the “Heart” sequence,we note that the QWT phase-based approach provides a motionfield that is more useful for patient monitoring and diagnosis.For the “Taxi” sequence, which contain discontinuities in theirunderlying flows, the QWT phase-based algorithm sports com-parable performance [see the table in Fig. 12(b)]. Since the mul-tiscale averaging step in our algorithm tends to smooth out theestimated flow, it should not be expected to perform as well fordiscontinuous motions fields of rigid objects moving past eachother.

Additional advantages of our QWT-based algorithm includeits speed (linear computational complexity) and sub-pixel esti-mation accuracy. For an -pixel image, our algorithmis more efficient than the FFT-based GC and sig-nificantly faster than ES, which can take up to com-putations with the search parameter on the order of . Gen-eral block-matching techniques such as ES can only decipherdisparities in an integer number of pixels. On the other hand,our QWT-based algorithm can achieve sub-pixel estimation anddemonstrates greater accuracy for the “Rubik” sequence thanexisting phase-based sub-pixel estimation methods such as GC.Besides gradient correlation, there exist other phase-based al-gorithms for disparity estimation and image registration [25],

[31], [41]–[44]. These approaches use phase as a feature map, where the phase function maps 2-D , -coordinates

to phase angles. They assume the phase function to stay constantupon a shift from the reference image to the target image; thatis, where is the phase function for thereference image and for the target image. Then, the disparityestimation problem is simplified to calculating the optical flowfor these phase functions [41], [45]. In contrast, our algorithmis entirely based on the multiscale dual-tree QWT and its shifttheorem.

Our approach is similar to Bayro-Corrochano’s QWT dis-parity estimation algorithm in [31] in its use of quaternion phaseangles. However, the latter approach requires the design of aspecial filter to compute the phase derivative function in ad-vance, while our approach need only estimate the local frequen-cies . Our implementation also uses a dual-tree filterbank,as compared to the quaternion wavelet pyramid of Gabor fil-ters in [31]. Provided a continuous underlying disparity flow,our algorithm yields a denser and more accurate disparity map,even for smooth regions within an image. Incorporating an op-timization procedure such as in [44] or a statistical model intoour current algorithm can further improve estimation accuracy,particularly for blocks with phase singularity, but requires extracomputation time.

Kingsbury et al. have developed a multiscale displacementestimation algorithm based on the 2-D CWT [41], [42]. Theirapproach combines information from all six CWT subbands inan optimization framework based on the optical flow assump-tions. In addition to disparity estimation, they simultaneouslyregister the target image to the reference image. Both their CWTmethod and our QWT method are multiscale and wavelet-basedand, thus, in general, best for smooth underlying disparity flows.However, our QWT algorithm is much simpler and easier to use,because it does not involve the tuning of several parameters forthe iterative optimization procedures as in the CWT algorithm.While our method estimates local disparities without warpingthe image, we can apply any standard warping procedure toeasily register the two images from the estimated disparities.Thanks to the ability of the QWT to encode location informationin absolute horizontal/vertical coordinates, we can easily com-bine the QWT subband estimates to yield more accurate flowestimation results. Combining subband location information inthe 2-D CWT is more complicated, since each subband encodesthe disparities using complex phase angles in a reference framerotated from other subbands.

VI. CONCLUSION

We have introduced a new 2-D multiscale wavelet represen-tation, the dual-tree QWT, that is particularly efficient for co-herent processing of relative location information in images.This tight-frame image expansion generalizes complex waveletsto higher dimensions and inspires new processing and analysismethods for wavelet phase.

Our development of the dual-tree QWT is based on analternative definition of the 2-D HT and 2-D analytic signaland on quaternion algebra. The resulting quaternion waveletshave three phase angles; two of them encode phase shifts in anabsolute horizontal/vertical coordinate system while the third


encodes textural information. The QWT’s approximate shifttheorem enables efficient and easy-to-use analysis of the phasebehavior around edge regions. We have developed a novelmultiscale phase-based disparity estimation scheme. Throughefficient combination of disparity estimates across scale andwavelet subbands, our algorithm clearly demonstrates theadvantages of coherent processing in this new QWT domain.Inherited from its complex counterpart, the QWT also featuresnear shift-invariance and linear computational complexitythrough its dual-tree implementation.

Beyond 2-D, the generalization of the Hilbert transform to-D signals using hypercomplex numbers can be used to de-

velop higher dimensional wavelet transforms suitable for sig-nals containing low-dimensional manifold structures [46]. TheQWT developed here could play an interesting role in the anal-ysis of ( )-D manifold singularities in -D space. This effi-cient hypercomplex wavelet representation could bring us newways to solve high-dimensional signal compression and pro-cessing problems.

APPENDIX

A) Single Quadrant QFT of a QWT Basis Function: Fora real-valued signal with QFT , we can derive thefollowing relationships from (16):

(32)

(33)

(34)

We begin by taking the QFT of (19). Equations (32)–(34) applyto the second to fourth components on the right hand side of (19)respectively becauseare real-valued signals. Given the QFT relationship of the four

quadrature components of in Fig. 6,has support only on a single quadrant . The same also holdsfor the quaternion wavelets in other subbands and scales.

B) Proof of QFT Plancharel Theorem: Starting from thespace domain inner product in (23), we have (35), shown at thebottom of the page. One can also uses the QFT convolutiontheorem (Theorem 2.6) in [25] for this proof. The integrand onthe right hand side of (23) is the QFT of the function

where denotes 2-D convolution.C) QFT of a Step Edge: First, we express the step edge

in Fig. 8(a) as a 2-D separable function (a constant functionalong the -direction multiplied by a 1-D step function along the

-direction). The QFT of such a function is . Then,applying the QFT affine theorem ([25, Theorem 2.12]) with thetransformation matrix involving rotation and using the QFTshift theorem with offset satisfying (27) yield the QFTof the step edge

(36)

D) QWT Phase Angles for a Step Edge: This calculationcombines the results from (23) and (36). Let be the in-tegrand of the inner product formula in (23) in the QFT do-main. The integration limit only involves because of thesingle-quadrant support of the QWT basis. Consider the spe-cial case when the edge signal has zero offset .When , its QFT component in iswhere is the component involving the -ridge in (36).Let the QFT of a QWT basis, , be

. Substituting andinto (23), the QWT coefficient can be expressed

(35)


as

(37)

where and are the integrals involving , , etc.After normalizing (37) by its magnitude , we com-pute the third phase angle as

(38)

When , in , which gives.

Moreover, this special quaternion in (37) with isin the singular case; as described in [25], its other phase angles

are nonunique but the difference is unique withthe following expression:

(39)

whose value largely depends on the QFT spectrum of the basisand on the edge orientation and offset .

E) Theoretical Analysis of Leakage Effect: Consider theinner product of an edge signal and a QWT basisin both the main quadrant and the leakage quadrantwhen .4 We can express this inner product in in asimilar fashion as in (37) and yield

(40)

Again, and are integrals involving , , etc. Com-bining (37) and (40) gives the QWT coefficient, i.e., the innerproduct between the nonideal basis and the edge signal

(41)

whose magnitude is . Its third phaseangle can be expressed as

(42)

4The leakage quadrant can be either S or S depending on the spectral sup-port of the basis element w(x).

In spite of leakage, the relationship between QWT phase an-gles and edge offset still holds as in the case withoutleakage, i.e., varies linearly with edge offset . Sinceis not necessarily for the QFT of the edge signal, i.e., thesingular case no longer holds, there exists unique expressionsfor both and . However, our derivations show that the dif-ference of and is the same as in the ideal case (withoutleakage) as in (39).

ACKNOWLEDGMENT

While this paper was in final preparation, H. Choi passedaway. We will forever remember his broad vision, his keen in-sights, and our lively discussions. His legacy will live on throughhis many contributions to the signal processing community. Wethank I. Selesnick for many discussions and his Matlab dual-treeCWT code, N. Kingsbury for discussions on CWT-based imageregistration, T. Gautama et al. for providing us with their imagesequences, J. Barron et al. for making their image sequencespublicly available, and V. Argyriou et al. who generously sharedtheir code for gradient correlation motion estimation for videosequences. We would also like to the reviewers for their in-sightful comments and for providing a more concise derivationfor the single-quadrant QFT of a QWT basis in Appendix A anda more general Plancharel theorem in Appendix B.

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[46] W. L. Chan, H. Choi, and R. Baraniuk, “Directional hypercomplexwavelets for multidimensional signal analysis and processing,” in Proc.IEEE Int. Conf. Acoustics, Speech, Signal Processing, May 2004, pp.996–999.

Wai Lam Chan (S’03) received the B.S. degree inelectrical and computer engineering and computerscience from Duke University, Durham, NC, in 2002(summa cum laude), and the M.S. degree in electricaland computer engineering from Rice University,Houston, TX, in 2006.

He is currently a graduate student in the Depart-ment of Electrical and Computer Engineering, RiceUniversity. He held a visiting research position inseismic imaging at ConocoPhillips in 2005. Hisresearch interests include THz imaging, compressed

sensing, and hyper-complex wavelets with applications to image processing.Mr. Chan received the Presidential Fellowship and the Texas Instruments Fel-

lowship from Rice University and is a member of Tau Beta Pi and Eta KappaNu.

Hyeokho Choi received the B.S. degree in controland instrumentation engineering from Seoul NationalUniversity Seoul, Korea, in 1991, and the M.S. andPh.D. degrees in electrical engineering from the Uni-versity of Illinois at Urbana-Champaign, Urbana, in1993 and 1998, respectively.

In 1998, he joined Rice University, Houston, TX,as a Research Associate. From 2000 to 2005, he wasa Research Professor at Rice University. In Fall 2005,he joined North Carolina State University, Raleigh, asan Assistant Professor in the Electrical and Computer

Engineering Department. His research interests were in the areas of signal andimage processing and computer vision.

Dr. Choi received the Presidential Honor at Seoul National University at grad-uation in 1991 and the Texas Instruments Postdoctoral Fellowship in 1998. Hewas an Associate Editor for the IEEE TRANSACTIONS ON IMAGE PROCESSING.

Richard G. Baraniuk (S’85–M’93–SM’98–F’01)received the B.Sc. degree from the University ofManitoba, Manitoba, Canada, in 1987, the M.Sc.degree from the University of Wisconsin, Madison,in 1988, and the Ph.D. degree from the University ofIllinois at Urbana-Champaign, Urbana, in 1992, allin electrical engineering.

After spending 1992 and 1993 with the EcoleNormale Supérieure, Lyon, France, he joined RiceUniversity, Houston, TX, where he is currently theVictor E. Cameron Professor of Electrical and Com-

puter Engineerng and Founder of the Connexions Project. He spent sabbaticalsat the Ecole Nationale Supérieure de Télécommunications, Paris, France, in2001, and the Ecole Fédérale Polytechnique de Lausanne, Switzerland, in2002. His research interests in signal and image processing include waveletsand multiscale analysis, statistical modeling, and sensor networks.

Dr. Baraniuk received a NATO postdoctoral fellowship from NSERC in1992, the National Young Investigator award from National Science Foundationin 1994, a Young Investigator Award from the Office of Naval Research in1995, the Rosenbaum Fellowship from the Isaac Newton Institute of Cam-bridge University in 1998, the C. Holmes MacDonald National OutstandingTeaching Award from Eta Kappa Nu in 1999, the Charles Duncan JuniorFaculty Achievement Award from Rice University in 2000, the ECE YoungAlumni Achievement Award from the University of Illinois in 2000, and theGeorge R. Brown Award for Superior Teaching at Rice University in 2001 and2003. He was elected a Plus Member of AAA in 1986.

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