Wavelet transforms for vector fields using omnidirectionally balanced multiwavelets

3018 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 12, DECEMBER 2002

Wavelet Transforms for Vector Fields UsingOmnidirectionally Balanced Multiwavelets

James E. Fowler and Li Hua

Abstract—Vector wavelet transforms for vector-valued fieldscan be implemented directly from multiwavelets; however,existing multiwavelets offer surprisingly poor performance fortransforms in vector-valued signal-processing applications. Inthis paper, the reason for this performance failure is identified,and a remedy is proposed. A multiwavelet design criterion knownas omnidirectional balancing is introduced to extend to vectortransforms the balancing philosophy previously proposed formultiwavelet-based scalar-signal expansion. It is shown that thestraightforward implementation of a vector wavelet transform,namely, the application of a scalar transform to each vectorcomponent independently, is a special case of an omnidirectionallybalanced vector wavelet transform in which filter-coefficientmatrices are constrained to be diagonal. Additionally, a familyof symmetric-antisymmetric multiwavelets is designed accordingto the omnidirectional-balancing criterion. In empirical resultsfor a vector-field compression system, it is observed that theperformance of vector wavelet transforms derived from these om-nidirectionally-balanced symmetric-antisymmetric multiwaveletsis far superior to that of transforms implemented via othermultiwavelets and can exceed that of diagonal transforms derivedfrom popular scalar wavelets.

Index Terms—Balanced multiwavelets, vector wavelet trans-forms, vector-valued signal processing.

I. INTRODUCTION

WAVELET transforms have been some of the most usefulsignal-processing tools to arise during recent years;

however, the overwhelming majority of wavelet literaturefocuses on the expansion of scalar-valued signals using scalarwavelet systems, i.e., multiresolution analyses consisting ofwavelet and scaling functions that are scalar valued. Yet, inmany applications, there is a need to process data that is inher-ently of vector form. For example, fluid flows in oceanographyand aerodynamics are usually represented as two-dimensional(2-D) or three-dimensional (3-D) vector fields in 2-D or 3-Dspace, while images with multiple spectral components can beconsidered to be 2-D fields of multidimensional vectors. Theseare just two applications out of many for which there is need ofa vector wavelet transform (VWT).

The concept of a vector transform has existed for sometime [1], and a comprehensive multiresolution-analysis theoryfor VWTs, which closely parallels theory for scalar-wavelet

Manuscript received June 22, 2001; revised July 18, 2002. This work wassupported in part by the National Science Foundation under the Large Data andScientific Software Visualization Program Grant ACI-9982344. The associateeditor coordinating the review of this paper and approving it for publication wasDr. Ta-Hsin Li.

The authors are with the Department of Electrical Computer Engineering, En-gineering Research Center, Mississippi State University, Starkville, MS 39762USA (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TSP.2002.805488

expansion, was outlined by Xia and Suter [2]. Although Xiaand Suter focused on the theoretical infrastructure for VWTsrather than the design of coefficient matrices for vector filterbanks, they recognized that multiwavelets present a naturalconstruction for VWTs. Since their introduction [3], multi-wavelets have garnered an extraordinary amount of attentionfrom both theorists and engineers, but mostly for the expansionof scalar-valued signals. By expanding a scalar function usingseveral scaling functions and wavelet functions instead of asingle pair, multiwavelet-transform systems circumvent certainlimitations posed by traditional scalar wavelets, such as thefact that scalar wavelets cannot possess simultaneously orthog-onality and linear phase. However, multiwavelets can easilyprovide expansions for vector-valued signals; in fact, usingmultiwavelets for vector-valued processing initially appearssimpler in that there is no need for scalar-to-vector conversion(which usually involves a polyphase decomposition of thescalar signal and some form of prefiltering [4]).

Given the discussion above, it appears obvious that the largebody of existing multiwavelet transforms and filters existing inthe literature could be brought to bear directly on the VWT-de-sign problem. However, we demonstrate below that contraryto expectations, existing multiwavelets perform exceedinglypoorly for vector-valued signal-processing applications. Asthe primary contribution of this paper, we analyze extensivelythis performance failure and develop a remedy for it. Inspiredby the work of Lebrun and Vetterli [5], we find that existingmultiwavelets are not suitably “balanced” for vector-valuedsources and adopt a solution we callomnidirectional balancing(OB) that greatly improves performance. Employing ourOB design criterion, we solve for a family of biorthogonal,symmetric-antisymmetric (SA) multiscaling and multiwaveletfunctions and corresponding filterbanks. Using these OBSAmultiwavelets in a VWT, we obtain performance in a simplevector-field compression system far superior to that of existingmultiwavelets.

Of course, the most straightforward way to transformvector-valued data is to apply scalar transforms directly bytransforming each vector component individually. We showbelow that such a “product” of multiple scalar transformsis tantamount to an OB VWT with a constrained structure;specifically, the filter-coefficient matrices are constrained to bediagonal. Empirical results from our vector-field compressionsystem indicate that our family of biorthogonal OBSA VWTsoffers performance that can exceed that of diagonal VWTsbased on the best-performing scalar wavelets.

In the following, we provide a brief overview of VWTtheory and its relation to multiwavelets in Section II and

1053-587X/02$17.00 © 2002 IEEE

FOWLER AND HUA: WAVELET TRANSFORMS FOR VECTOR FIELDS 3019

examine the balancing issue for multiwavelets and VWTs inSection III. We present the details of the construction of ourOBSA multiwavelets in Section IV and explore their VWTperformance in Section V. Finally, we make some concludingremarks in Section VI.

II. V ECTORWAVELET TRANSFORMS

A. Biorthogonal VWT Theory

We now present a brief overview of vector-valued wavelettheory as presented in [2], suitably generalized to the biorthog-onal case as employed in [6]. Let be the set of real numbersand be the set of integers. The realmatrix-valued signal spaceof dimension , which is , is defined as theset of allmatrix-valued signals , which are matricesof scalar signals, i.e.,

......

......

(1)

where the are scalar-valued functionsand . The integration of matrix-valued function isdefined as , i.e., the matrix ofthe integrals of the scalar functions. The inner product of twomatrix-valued functions and is defined as

. Note that this is not an inner product in thecommon sense in which it must be scalar-valued ( is an

matrix); however, it can be shown [2] to satisfy propertiesthat are necessary to be considered to be an inner product formatrix-valued signal spaces. A set of matrix-valued functions

, is orthogonalto set if

(2)

where is the Kronecker delta sequence, andis theidentity matrix. The dual sets and

are called abiorthogonal basisif (2) holds and, for every, there existscoefficient matrices

and such that . Notethat the coefficients in these expansions are matrices;however, as usual, they can be obtained via inner products

, .A biorthogonal multiwavelet multiresolution analysis is

driven by two matrix-valuedscaling functions andand two matrix-valuedwavelet functions and , whichsatisfy matrix-valueddilation equations

(3)

(4)

(5)

(6)

and biorthogonality conditions

(7)

(8)

(9)

where the coefficient sequences, , , and arematrices, and the scales and translates are 2 2

, . Translates and scales of scaling functionsand and wavelet functions and form a

nested sequence of closed subspaces (scaling spaces) and theirorthogonal-complement spaces (wavelet spaces) decomposing

. Therefore, function canbe expanded as

(10)

where is an arbitrary “starting scale,” whereas scaling coef-ficients and wavelet coefficients are matrices.Matrix filterbank equations in the style of Mallat’s algorithm [7](the ubiquitous implementation of scalar discrete wavelet trans-forms) can be derived easily [2].

In most cases, we are actually interested in transforming dataconsisting of 1 vectors rather than matrices. How-ever, the theory outlined above can still apply. To see this, definesignal using identical copies of a given vector source asits rows . Then, each rowin (10) becomes

(11)

where each row of the coefficient matrices and areidentical and equal to and , respectively. In this case,the above theory simplifies since the matrix rows are identical.Since we do not need to calculate identical rows multiple times,the matrix-valued form of Mallat’s algorithm (see [2])

Analysis: (12)

(13)

Synthesis:

(14)

simplifies to

Analysis: (15)

(16)

Synthesis:

(17)

where and are scaling and wavelet coefficient vectors ofdimension 1. We note that an orthonormal VWT is just aspecial case of a biorthogonal VWT with , ,

, and .


As a consequence of the above definitions, the matrices,, , and satisfy a matrix version of the perfect-recon-

struction (PR) conditions [6]

(18)

(19)

(20)

for . Finally, we definetwo-scale matrix symbolsas

(21)

B. Diagonal VWTs From Scalar Wavelets

The most straightforward approach to constructing abiorthogonal VWT is to directly apply biorthogonal scalarwavelets, independently, to the individual components of thevector signal. It is fairly straightforward to see that this trivialextension of scalar transformation to the vector case is a specialcase of a biorthogonal VWT constrained to have scaling andwavelet functions that are diagonal matrix-valued functions.That is, if is the biorthogonal primary scalar scalingfunction used in the expansion of each vector component(assume we use the same scalar transform for each component),then the corresponding primary vector scaling function isdiagonal diag . , , and

are likewise diagonal. The matrix biorthogonality in(7)–(9) holds due to the biorthogonality of scalar functions

, , , and . From the dilation (3)–(6), we seethat the , , , and matrices are also diagonaland are diag , diag ,

diag , and diag , where, , , and are the scalar filters for the biorthogonal

transform.

C. Multiwavelet Construction of VWTs

Scalar-valued multiwavelet-based multiresolution anal-ysis consists of scales and translates of a finite numberof primary scaling functions , dualscaling functions , primary waveletfunctions , and dual wavelet functions

. In either the scalar case (a special caseof multiwavelets wherein ) or in the general multiwaveletcase (wherein ), we expand a singlescalar-valuedfunc-tion using linear combinations of scales andtranslates of these scaling and wavelet functions. In contrast, toconstruct a VWT, we want to expandvector-valuedfunctions.However, multiwavelets and VWTs are closely related, as wasestablished in [2]. Specifically, in implementing a VWT withmultiwavelets, each column of matrix-valued scaling functions

and contains a set of multiscaling functions, whereaseach column of the matrix-valued wavelet functions and

contains a set of multiwavelet functions. The VWT isimplemented as in (15)–(17) directly with , , , and

being the filter coefficient matrices for the multiwaveletsystem.

III. M ULTIWAVELET BALANCING

A. Scalar Balancing

Using biorthogonal multiwavelet filter-coefficient matrices, , , and in (15)–(17) yields an analysis-synthesis

filter-bank pair with perfect reconstruction for use with vectorsources. Additionally, if we couple this filterbank with a proce-dure for “vectorizing” a scalar source, these equations also pro-vide the mechanisms for implementing the forward and inversediscrete multiwavelet transform (DMWT) of a scalar signal;in this case, the multi-input, multi-output filterbank is called a“multifilter” after [8]. The most straightforward way to imple-ment the required vectorization is to separate the scalar signalinto its polyphase components.

It is well known that the mere satisfaction of (18)–(20) is notsufficient to give a DMWT reasonable signal-processing per-formance for scalar signals. That is, even though (18)–(20) pro-vide perfect reconstruction, the distortions normally introducedby signal-processing operations between analysis and synthesissteps of the multifilter can have dramatically detrimental results.

For example, consider a simple compression system for scalarsignals that consists of merely the application of one scale ofDMWT analysis, the zeroing of the highpass or “detail” coef-ficients [ in (16)], and one scale of DMWT synthesis. Sup-pose this compression system uses the Geronimo–Hardin–Mas-sopust (GHM) multiwavelets [3], which are orthogonal and have

. Further, suppose we employ a polyphase vectorizationthat assembles a vector source by placing even samples fromthe scalar source into the first vector component and odd sam-ples into the second component. If the scalar signal input to thesystem is the constant signal [ ], we wouldexpect that the system would reproduce this signal exactly sinceour intuition holds that a constant signal would pass perfectlythrough the “lowpass” branch of the multifilter, and since we arediscarding only highpass coefficients, no change should result.This is, however, not the case. In reality, the output of the systemis [ ], as was observed in [5]. That is,an oscillatory distortion of the scalar constant signal occurs dueto the suppression of the detail coefficients from the multifilter.For many other multiwavelets, a similar effect occurs, althoughthe exact values of the oscillations depend on the multiwaveletused. The problem represented by this example is serious as it islikely to lead to significant distortions in any system that modi-fies coefficients between analysis and synthesis transform steps.This issue is particularly problematic for those processes, no-tably compression systems, that tend to preserve scaling coeffi-cients at the expense of wavelet coefficients under the assump-tion that the scaling coefficients provide a “low-resolution” ap-proximation to the original data.

The traditional approach to handling the oscillatory distor-tions described above is to compensate for them before the for-ward DMWT is applied, that is, one applies a so-called “pre-filter” to the input data before it enters the analysis DMWT fil-terbank. The net effect, however, is that the transform itself ischanged, usually losing orthogonality or linear phase [5]. An al-ternative approach was proposed recently [5]. In this technique,it was realized that the root of the problem lies in that the vector

is not generally an eigenvector of the two-scale matrix


symbols and of (21) when (corresponding toa zero-frequency, or constant source). To rectify this situation,a similarity transformation was proposed in order to “redesign”the and matrices such that is an eigenvector. Thisapproach was called multiwaveletbalancing[5] due to the factthat it tends to “balance” out the treatment of the vector compo-nents by the filterbank.

Specifically, in our formulation of analysis and synthesis,(15)–(17), the conditions imposed by scalar balancing are thatwe want a constant scalar signal to pass through the lowpassanalysis filter unchanged (up to a constant gain). That is,regarding (15), we want . Fromthe definitions of (21), we have, directly, that is a righteigenvector of for . That is

(22)

Additionally, we want a constant signal to pass throughthe synthesis filter (17) unchanged up to a gain, i.e.,

. Thus, we have that, or

equivalently, . Consequently, wehave that [1,1] is a left eigenvector of when , i.e.,

(23)

We note that in the original development of [5], only or-thonormal multiwavelets were considered, in which case,

, and (22) and (23) become conditions on the leftand right eigenvectors of .

Initially, it may appear that the balancing issue describedabove applies only to the situation in which a scalar source isprocessed by first vectorizing it and then applying a DMWTsince it is in the polyphase nature of the vectorization that theoscillations arise. However, as we will see in the next section,purely vector transforms, in which the input data is originally invector form so that no vectorization is needed, are not immuneto this effect. In fact, we will see that a new kind of balancingis needed to rectify the problem, and, in its absence, usingmultiwavelets for VWTs produces surprisingly poor results forcompression.

B. Omnidirectional Balancing

The vector analog of a constant scalar source is a vector fieldin which each vector is the same, i.e., .Define the orientation of the constant vector field as

. From the previous discussion, we wouldexpect some vector other than to be output from the“lowpass” branch of the multifilter unless andhappen to be a left eigenvector of and a right eigenvectorof , respectively, but we could apply the scalar balancingto ensure that this is the case. However, doing so would meanthat the resulting multiwavelet is “balanced” only for thespecific constant source at hand. That is, if we had anotherconstant vector source at a different orientation, , themultiwavelet would no longer be balanced for this source. Thedifference between balancing for vector data and balancing forscalar data is that there is only one constant scalar source towithin a gain factor but infinitely many constant vector sources,

Fig. 1. Original data. A 2-D field of 2-D vectors representing ocean currentsas measured on the surface of the Pacific ocean.

Fig. 2. Reconstruction from the baseband of a VWT implemented with theChui-Lian length-3 multiwavelet [13],MSE = 0:0334.

all with different orientation angles. This vector balancingproblem is exacerbated when the data source is not constant.

Figs. 2 and 3 illustrate the vector balancing problem for thereal vector-valued data given in Fig. 1. For Figs. 2 and 3, weperform a three-scale VWT implemented via a multiwavelet,discard all the wavelet coefficients, and reconstruct using theinverse VWT on just the baseband subband. We implement the2-D VWT in the usual separable fashion—a 1-D VWT is takenalong each row of vectors and then along each column, yieldingthree subbands of wavelet coefficients and one baseband ofscaling coefficients, repeating then on the baseband. We seethat regardless of whether we use a nonbalanced (see Fig. 2)or balanced (see Fig. 3) multiwavelet for our VWT, we getextremely poor results for our compressor. Specifically, thebaseband doesnotconsist of a low-resolution approximation ofthe original data as we are led to expect from scalar multiresolu-tion analysis. We have observed similarly poor results for every


Fig. 3. Reconstruction from the baseband of a VWT implemented with theLebrun–Vetterli balanced length-3 multiwavelet [5],MSE = 0:0394.

multiwavelet we have found in the literature.1 Since real-worldsignal-processing algorithms often discard or otherwise modifywavelet coefficients, the practical implications of the vectorbalancing problem are clear.

To solve the vector balancing problem to construct VWTs thatare resilient to data loss amongst wavelet coefficients, we pro-pose the following constraint to the VWT-design process. Re-alizing that the multiwavelet used for the VWT needs to be bal-anced forall vectors lying on the unit circle, we propose a newtype of “balancing” that is insensitive to the orientation angleof the vector data so that constant vector sources, regardless oforientation, are reproduced by the “lowpass” branch of the mul-tifilter. Specifically, in terms of (22) and (23), what is requiredis that all unit vectors are right eigenvectors of and lefteigenvectors of when . Such is the case when

(24)

where is the identity matrix. We call the impositionof (24) OB, as it “balances” the multiwavelet for all orientationsin a manner similar to the balancing proposed in [5] for a singledirection.

It is well known that if and are the scaling filters for ascalar biorthogonal wavelet system, then

(25)

The diagonal VWT discussed earlier in Section II-B havingdiag and diag thus

clearly satisfies (24). Diagonal VWTs constructed from any setof scalar biorthogonal wavelets are thereby OB.

IV. CONSTRUCTION OFOBSA MULTIWAVELET FILTERS

To construct the primary and dual multiscaling functions forour OBSA VWT, we impose the following set of equations: the

1Specifically, we have investigated the multiwavelets appearing in [3], [5],and [9]–[12], and only the multiwavelet based on the complex Daubechies fil-ters of [5] does not suffer from poor performance due to the balancing issue.However, the cascade algorithm for this latter multiwavelet does not converge;therefore, its performance is not competitive either.

PR condition (18), the OB condition (24), and the SA condition[6], [13]

(26)

where [ ] and [ ] are the intervals of sup-port of the FIR filters and , respectively, and

diag . We occupy as many re-maining degrees of freedom as possible by placing zeros at

, corresponding to the technique of vanishing momentswidely employed in scalar wavelet design. Specifically, toobtain zeros at , we have

(27)

for .After application of the above design steps, there are usu-

ally several remaining degrees of freedom, and additional de-sign criteria are necessary to fully determine a solution. In [6],a filter-optimization technique is proposed to occupy degreesof freedom in biorthogonal multiwavelet design. This approachcalls for the minimization of the deviation of the magnitude re-sponse of the equivalent scalar filter from the ideal or “brick-wall,” scalar lowpass filter. However, the motivation in [6] is todesign “good” multifilters for scalar signals; hence, this crite-rion is not entirely suitable for vector-valued signals. Since it isunclear how to define an ideal vector lowpass filter, we adaptthe scalar approach of [6] to our vector design needs as follows.

The technique of [6] is to make the equivalent scalar magni-tude response approach that of the ideal lowpass filter by mini-mizing the objective function

(28)

where is a weighting parameter, and denotes the pass-band of the ideal lowpass filter. In [6], and arethe magnitude responses of the equivalent scalar filters. For ourvector design problem, we use

(29)

while a variety of definitions exist for the matrix norm. It isunclear which matrix norm is best from a theoretical perspec-tive; however, our empirical observations indicate that all per-form equally well. We thus define the matrix norm of

as

(30)

with the division by included to normalize the dc gain to1 since our OB condition gives . Definitionssimilar to (29) and (30) are used for .

For the multiwavelets, we impose biorthogonality betweenthe multiwavelet and multiscaling functions (19), biorthogo-nality of the dual multiwavelets (20), and a SA condition similar


Fig. 4. Primary scaling and wavelet functions of our OBSA7–5 multiwavelets.

to (26). Similar to (28), we occupy any remaining the degreesof freedom by minimizing a highpass objective function

(31)

with denoting the stopband of the ideal highpass filter.To date, we have attempted solutions for only the case of

, as the overwhelming majority of multiwavelet litera-ture focuses on this multiplicity-2 case, and results are easilyvisualized for 2-D vectors. Additionally, we fix and

. We present a detailed outline of the solu-tion procedure in the Appendix. We have constructed biorthog-onal OBSA multiwavelets with this procedure for lengths 7–5and 5–3; we show the scaling and wavelet functions for theOBSA7–5 filters in Figs. 4 and 5. Filter coefficients for bothmultiwavelets can be found in the Appendix.

V. EXPERIMENTAL RESULTS

Figs. 6 and 7 repeat the experiment of Figs. 2 and 3 fortwo examples of omnidirectionally balanced VWTs, namely,a diagonal VWT derived from the popular 9–7 biorthogonalscalar wavelet (CDF9–7) [14] and a nondiagonal VWT usingour OBSA7–5 multiwavelets. As can be seen, in contrast toFigs. 2 and 3, the basebands of OB VWTs do indeed provide alow-resolution approximation to the original data. Additionally,the mean squared error (MSE) between the reconstructed andoriginal vector fields for Figs. 6 and 7 are significantly smallerthan those for Figs. 2 and 3.

To investigate the performance of our OBSA VWTs ina real signal-processing application, we built the followingvector-field compression system. Three scales of a 2-D VWTfor 2-D vectors is followed by vector quantization (VQ) ofscaling and wavelet coefficient vectors. We use the successiveapproximation VQ (SAVQ) of [15], which is a hybrid ofgain-shape VQ [16] and multistage VQ [16]. Finally, we finish

Fig. 5. Dual scaling and wavelet functions of our OBSA7–5 multiwavelets.

Fig. 6. Reconstruction from the baseband of a diagonal VWT implementedwith scalar CDF9-7 biorthogonal wavelets [14],MSE = 0:0076.

with runlength coding of all insignificant vectors (labeled as“zero” during each approximation pass of the SAVQ coder)and arithmetic coding with multiple contexts. As a whole, thesystem, which produces an embedded bitstream, is roughly anextension to vector data of a coder we developed recently forscalar-valued oceanographic imagery [17], [18].

We have compared the performance of VWTs derived froma number of orthogonal and biorthogonal multiwavelets of boththe nonbalanced [3], [12], [13] and scalar-balanced [5], [10]variety to that of VWTs created from our OBSA multiwavelets.Rate-distortion performance results for the above compressionsystem using the ocean-current data of Fig. 1 are shown inFig. 8. We see that our OBSA multiwavelets provide VWTswith performance that is far superior to VWTs derived fromother known multiwavelets. Additionally, in Fig. 9, we havecompared the compression performance of diagonal VWTsderived from popular orthonormal [19] and biorthogonal [14]


Fig. 7. Reconstruction from the baseband of a VWT implemented with ourOBSA7-5 biorthogonal multiwavelets,MSE = 0:0082.

Fig. 8. Compression performance for the ocean-current data of Fig. 1 forVWTs derived from nonbalanced multiwavelets (CL3 [13], GHM [3], Strela2[12]), scalar-balanced multiwavelets (BAT3 [5], BAT7 [5], Sel8 [10]), and ourOBSA multiwavelets.

scalar wavelets to that of VWTs created from our OBSA multi-wavelets in the above compression system. From Fig. 9, we seethat the compression performance of both of our OBSA VWTsis comparable with that of a diagonal VWT derived from theCDF9–7 biorthogonal wavelet [14], which is long popular forcompression applications. We have repeated these experimentsfor a number of other vector-valued datasets from a variety ofother applications. For example, Fig. 11 repeats the experi-ments for sea-wind data, as illustrated in Fig. 10, and Figs. 13and 15 compare VWT performance on the two optical-flow[21] fields shown in Figs. 12 and 14, respectively. We see thatboth of our OBSA VWTs offer performance comparable withthe diagonal transforms for the sea-winds data, whereas ourOBSA5–3 VWT outperforms the diagonal transforms for bothoptical-flow datasets. Thus, we conclude that although thecompression performance of our VWTs is somewhat mixedand dependent on the application producing the vector fields,

Fig. 9. Compression performance for the ocean-current data of Fig. 1for VWTs from our OBSA multiwavelets, a diagonal VWT from a scalarorthonormal wavelet (D6 [19]), and a diagonal VWT from a scalar biorthogonalwavelet (CDF9-7 [14]).

Fig. 10. Two-dimensional field of 2-D vectors representing ocean-surfacewinds derived from scatterometer observations from space.

our nondiagonal OBSA VWTs have the potential to outperformdiagonal VWTs based on popular scalar wavelets.

VI. CONCLUSIONS

In this paper, we have revealed that nearly all existingmultiwavelets perform poorly when used to implement VWTs.To remedy this situation, we have proposed the incorpora-tion of an additional criterion in the multiwavelet-designprocedure that results in multiwavelets that are balanced inan omnidirectional sense. Using this criterion, we design afamily of multiplicity-2 OBSA multiwavelets that substantiallyoutperforms existing multiwavelets when used for VWTs in asimple compression system for 2-D vectors. Additionally, weshow that the straightforward approach to transforming vectordata, namely, the independent application of scalar transformsin a component-wise fashion, is equivalent to an OB VWT


Fig. 11. Compression performance for the sea-wind data of Fig. 10 for VWTsfrom our OBSA multiwavelets, a diagonal VWT from a scalar orthonormalwavelet (D6 [19]), a diagonal VWT from a scalar biorthogonal wavelet (CDF9-7[14]), a VWT from a nonbalanced multiwavelet (CL3 [13]]), and a VWT froma scalar-balanced multiwavelet (BAT3 [5]).

Fig. 12. Two-dimensional field of 2-D vectors representing optical flow [22]in a computer-generated rendering of a rotating sphere.

Fig. 13. Compression performance for the optical-flow data of Fig. 12 for theVWTs used in Fig. 11.

Fig. 14. Two-dimensional field of 2-D vectors representing optical flow in acomputer-generated rendering of camera zoom in an office.

Fig. 15. Compression performance for the optical-flow data of Fig. 14 for theVWTs used in Fig. 11.

with coefficient matrices that are constrained to be diagonal.In empirical observations, we find that not only does ourfamily of OBSA multiwavelets provide the best compressionperformance for VWTs derived from known multiwavelets, butit also offers performance comparable with or exceeding thatof diagonal OB VWTs based on popular scalar wavelets.

In the future, we hope to incorporate into our OBSAdesign procedure higher order balancing constraints akinto those developed in [20] for scalar expansion. Specifi-cally, we propose considering samplings of signals of theform , where is some(scalar-valued) polynomial of degree, and is some arbitraryorientation angle . In the order-1 omnidirectionalbalancing case , is equal to some scalar constant,and we have the constant vector field we considered in Sec-tion III-B. For more general order-omnidirectional balancing,we would want vector-valued polynomial signals of degree

and less to be preserved to within a gain factor by boththe lowpass analysis and synthesis operations. We anticipate


that factorizations of the and matrices may provideadditional design constraints in a manner similar to those usedin [20], resulting in even greater vector-field compressionperformance than what we have demonstrated here.

APPENDIX

To find OBSA filter coefficients, we use a combination ofsymbolic solution via Matlab’s symbolic toolbox and numer-ical solution via a gradient descent. Full details of the solutionprocedure can be found in [22].

Step 1)For given filter lengths, we define coefficient matriceswith matrix elements as symbolic variables; the SA condition(26) greatly reduces the number of variables.

Step 2)We impose the PR condition (18), the OB condition(24), and the vanishing-moments conditions (27) and useMatlab’s symbolic solve function to obtain symbolic solutions.Multiple solutions involving undetermined variables andcomplex numbers are possible. For example, in solving forour biorthogonal OBSA5–3 filters, we eliminate from furtherconsideration all complex-valued solutions as well as solutionsthat lead to a purely diagonal structure. We are left with severalnondiagonal solutions that differ only in a change of sign.Arbitrarily choosing one, we have

(32)

Step 3)We use a gradient descent to numerically find valuesof undetermined variables to locally minimize the ideal-lowpassobjective function (28). For our OBSA5–3 solution, we obtain

.Step 4)To solve for and filters, we again use Matlab’s

symbolic tools. SA conditions similar to (26) reduce the numberof variables, and the biorthogonality (19) and (20) are solved forsymbolic solutions. In the case of our OBSA5–3 filter, we arriveat (33), shown at the bottom of the page.

Step 5: We use a gradient descent to numerically findvalues of undetermined variables so as to locally mini-mize the ideal-highpass objective function (31). For ourOBSA5–3 solution, we find and

.Using the above procedure, we also obtained OBSA filters

of length 7/5, with vanishing moments of order (3,1). In thesymbolic solution of Step 2, we arrive at

(34)

whereas in the numerical optimization of Step 3, we obtainand . For

the highpass filters, the symbolic solution of step 4 yields (35),shown at the top of the next page, and the numeric optimiza-tion of Step 5 gives us and

.

ACKNOWLEDGMENT

The ocean-current data was obtained from D. N. Fox atNRL-SSC, the sea-wind data from the NASA/NOAA spon-sored data system Seaflux at JPL through the courtesy of W.T. Liu and W. Tang, and the optical-low data from the Com-puter Vision Research Group at the Department of ComputerScience, University of Otago, New Zealand.

(33)


(35)

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James E. Fowlerreceived the B.S. degree in computer and information scienceengineering and the M.S. and Ph.D. degrees in electrical engineering in 1990,1992, and 1996, respectively, all from the Ohio State University, Columbus.

In 1995, he was an intern researcher at AT&T Labs, Holmdel, NJ, and, fromJanuary to July of 1997, he held an NSF-sponsored postdoctoral assignment atthe Université de Nice, Sophia Antipolis, France. He is currently an associateprofessor with the Department of Electrical and Computer Engineering, as wellas a researcher in the Engineering Research Center, at Mississippi State Uni-versity, Starkville. His research interests include image and video coding, datacompression, and wavelets.

Li Hua received the B.S. and the M.S. degrees in electrical engineering in 1997and 1999, respectively, from Nanjing University, Nanjing, China. She is cur-rently pursuing the Ph.D. degree in electrical engineering at Mississippi StateUniversity (MSU), Starkville.

Since 2000, she has been a research assistant with the Engineering ResearchCenter at MSU. Her research interests include image and signal processing,video coding, and wavelets.

Wavelet transforms for vector fields using omnidirectionally balanced multiwavelets

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