Optimizing the Multiwavelet Shrinkage Denoising - Semantic Scholar

240

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 1, JANUARY 2005

Optimizing the Multiwavelet Shrinkage Denoising Tai-Chiu Hsung, Member, IEEE, Daniel Pak-Kong Lun, Member, IEEE, and K. C. Ho, Senior Member, IEEE

Abstract—Denoising methods based on wavelet domain thresholding or shrinkage have been found to be effective. Recent studies reveal that multivariate shrinkage on multiwavelet transform coefficients further improves the traditional wavelet methods. It is because multiwavelet transform, with appropriate initialization, provides better representation of signals so that their difference from noise can be clearly identified. In this paper, we consider the multiwavelet denoising by using multivariate shrinkage function. We first suggest a simple second-order orthogonal prefilter design method for applying multiwavelet of higher multiplicities. We then study the corresponding thresholds selection using Stein’s unbiased risk estimator (SURE) for each resolution level provided that we know the noise structure. Simulation results show that higher multiplicity wavelets usually give better denoising results and the proposed threshold estimator suggests good indication for optimal thresholds. Index Terms—Multiwavelet, parameter estimation, prefilter, smoothing methods, wavelet transforms, white noise.

I. INTRODUCTION

S

UPPOSE we are going to estimate tion

from noisy observa(1)

is independent and identically diswhere noise. The goal of denoising is to mintributed (iid) imize the mean square error (MSE) MSE

(2)

subject to the condition that is at least as smooth as , where . Wavelet thresholding methods [1]–[6] have been proven to be effective in estimating from since the wavelet transform represents a signal using basis functions that are localized in time and scale simultaneously. Signal energy tends to cluster into a few number of wavelet coefficients with large amplitudes. It is different from the wavelet coefficients of noise that tend to scatter in the time-scale space with Manuscript received July 30, 2003; revised December 17, 2003. This work was supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China, under Project B-Q706 and a grant from the Hong Kong Polytechnic University under Project A418. The associate editor coordinating the review of this paper and approving it for publication was Proc. Ziziang Xiong. T.-C. Hsung and D. P.-K. Lun are with the Centre for Multimedia Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected], [email protected]). K. C. Ho is with the Department of Electrical and Computer Engineering, University of Missouri, Columbia, MO 65201 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2004.838927

small amplitudes. The multiwavelet [7], [8] extends the idea of the wavelet by representing signal with more than one scaling function. It is found that these scaling functions can be designed to be simultaneously symmetric, orthogonal, and have short supports, which cannot be achieved at the same time for wavelet systems using only one scaling function. In [9]–[12], it is found that multiwavelet denoising with multivariate shrinkage gives consistently better results than using wavelet shrinkage. The improvement is contributed by the multiwavelet transform as well as the multivariate shrinkage operator, which effectively exploit the statistical information of the transformed coefficient vectors of noise. In applying the discrete multiwavelet transform on the scalar signal, it is necessary to perform proper initialization to obtain the finest scale scaling coefficient vectors from the sampled scalar signal. It is usually achieved by using a prefilter [13], [14]. Although an alternate way is sometimes adopted by using the “balanced” multiwavelet basis [15], filters with longer length result. Many methods have been suggested for designing prefilters [14], [16]–[20]. They enable the resulting filterbank to possess desired approximation power and property such as orthogonality. However, there is no simple method . This limits to obtain the prefilter for higher multiplicity the application of multiwavelets of higher multiplicity, which potentially give better performance compared with the traditional ones due to their better characterization of signals. In this paper, we first suggest a simple method for the design of second-order approximation preserving orthogonal prefilters for any multiplicity, which enable higher multiplicity wavelet applications, such as denoising. Experimental results show that denoising using higher multiplicity wavelets usually gives better performance. Good selection of the parameter set is critical to the success of multiwavelet shrinkage denoising. Since the multivariate shrinkage function is different from that used in wavelet shrinkage, we cannot borrow the risk estimators suggested for scalar wavelet shrinkage to the selection of the thresholding parameters. Furthermore, the components of a transform coefficient vector are mutually dependent. It makes the problem of finding the optimal parameter set more difficult. Recently, risk estimators for multiwavelet denoising are suggested [21]. However, they can only be used to derive a single threshold for all resolution levels and may not be optimal in case of nonwhite noises. In this paper, we suggest a risk estimator that allows us to optimally select the parameter set for multivariate shrinkage based on the principle of Stein’s unbiased risk estimator (SURE) [22]. The resulting parameters are level-dependent such that they can adapt to the characteristic of signal in different resolutions. The proposed risk estimator closely resembles the mean square error (MSE), as shown in (2) for

1053-587X/05$20.00 © 2005 IEEE

HSUNG et al.: OPTIMIZING THE MULTIWAVELET SHRINKAGE DENOISING

241

any multiplicity, and hence, the thresholds obtained approach optimum. The organization of this paper is as follows. In Section II, we first fix the notations and present a brief review on multiwavelet denoising. In Section III, we present a simple method for the design of second-order approximation preserving orthogonal prefilters. In Sections IV and V, we study the effect of multivariate shrinkage and derive risk estimator based on SURE. In Sections II–V, we verify our proposed methods by applying them on the denoising of several popular test signals with various settings. Their performances are then analyzed and discussed. II. MULTIWAVELET DENOISING A. Multivariate Shrinkage In [4] and [5], Donoho suggested the method of soft-thresholding (shrinkage). It is defined as follows. 1) Compute the wavelet transform of the observed signal . 2) Apply the shrinkage function coordinatewise to the empirical wavelet transform coefficients with a threshold . 3) Obtain the denoised signal by the inverse wavelet transform. For the case of multiwavelet denoising, the denoising procedure is similar to that of wavelet denoising, but the shrinkage function is modified to multivariate shrinkage [11]. The multivariate shrinkage operator for a transform coefficient vector is defined as for

(3) for

(4)

, and denotes the shrinkage parameters . denotes the threshold for the shrinkage operation performed at a particular resolution level, is the covariance matrix of noise, and is a coefficient vector. Geometrically, all vectors are shrunk toward the origin according to the shape of the noise distribution rather toward the component axes (as shown in Fig. 1). Orthogonal multiwavelet denoising by multivariate shrinkage leads to better performance over traditional shrinkage because of two reasons. First, a higher multiplicity orthogonal wavelet allows a signal to be represented by a linear combination of several mother wavelets with various translations and scales. It enables the application of multivariate statistics [11] to the transform coefficient vectors and provides us a very clear picture on the differences of noises and signals in various scales. It is noticed that noises become multivariate normal distributed in the transform domain, whereas signals generally do not and are of stronger magnitude. It is a more reliable tool for us to differentiate signal from noise as compared with using only one mother wavelet. Second,the traditional shrinkage function, which shrinks the wavelet coefficients with respect to a single basis function, cannot achieve the best in the multivariate case because it has not taken into account the mutual behavior of the signal with respect to the multiple basis functions given by where

multiwavelets. It can be seen from Fig. 1 that the component of the coefficient vector will be shrunk to zero if using traditional shrinkage, but it remains if we use multivariate shrinkage, which is more reasonable because it does not likely behave as noise. Selection of thresholds is critical to the success of wavelet denoising, Donoho suggested the universal threshold and proved that soft thresholding with this threshold has a minimax optimality property. Similarly, Downie and Silverman derived the multivariate universal threshold [11] . for multiwavelet denoising: However, the universal threshold may not be optimal in the sense of (2). It is because the thresholding method also removes the wavelet coefficients of the signal in addition to noise if the magnitudes of observation are both smaller than the threshold. This motivates much research on the choice of the thresholding parameter , which balances on killing noisy and signal components. In [4] and [6], the principle of Stein’s unbiased risk estimator (SURE) is used to find the optimal threshold for the unknown signal. However, there is no SURE estimator for each level of the multiwavelet transform. B. Covariance Matrices Let us define the vector filterbank for the levels discrete as and the cormultiwavelet transform of multiplicity responding prefilter as . Then, we can write the discrete . The multiwavelet transform for scalar signals as discrete multiwavelet transform of the noisy signal becomes , where and . The matrix is designed such that the levels output transform by 1 vector, i.e., coefficient vectors are arranged into an , where is the th transform coefficient vector of at scale ,and each vector contains elements. For each scale , we have a different covariance matrix , where , is the noise power, as shown in (1). If we are given the covariance structures of the , we can compute the distribution noise , say parameters of the transformed noise for scale : and , where is the equivalent multiwavelet by using transform matrix at level . One can also estimate robust statistics, as suggested in [11]. We know that a multivariate normal distribution is converted into another multivariate normal distribution after the linear matrix transform. If we are given the covariance structures of the , and the matrix multivariate normal distributed noise transform, we can compute the distribution parameters of the transformed noise. Since we can treat the multiwavelet transform as a linear transform and express them in matrix form, i.e., , for levels, the transform matrix is given by

for The highpass multiwavelet transform matrix from its matrix filter coefficients follows:

(5) is constructed as

242


and are the multiscaling and multiwavelet functions, respectively. Equations (8) and (9) have the following equivalent form in the Fourier transform domain:

where

Fig. 1. Multivariate shrinkage function.

and

is a band diagonal matrix

..

From [14] and [16], we consider the expansion of scalar signals by using all the multiscaling functions, i.e., . To relate and the samples , we first construct a superfunction by a linear combiof nation of multiscaling functions as follows:

. ..

.

where is the prefilter coefficients with length . For prematrix. filtering with decimation factor , is the and are obtained in a similar way so that the dimension of the matrices multiplications in (5) are matched. Then, the distribution parameters of noise coefficient vectors for level is given by

(6) (7)

(10) where is a finite-supported sequence of vectors. It is desirable that the superfunction has a lowpass property since , we can expand bandlimited signals by can be approximated from the sampled and the coefficients for large . Furthermore, it is shown scalar signal that the superfunction will satisfy the Strang–Fix conditions of such that order if there exists

In [11] and [12], it is suggested to adopt robust statistics methods to estimate the covariance structure of the transformed noise coefficient vectors. For simulation, we assume that is known such that we can use (7) to compute the covariance matrix .

III. HIGHER MULTIPLICITY PREFILTERING

(11) for

and

(12)

This implies that the function of order up to such that

has the approximation power

A. Strang–Fix Conditions For the design of higher multiplicity prefilters, we follow the approach suggested in [14] and [16]. Let us recall the conditions for a multiscaling function to provide a given order of approximation in this section. Consider the Hilbert space of all square integrable functions on . The Fourier transform is given by . of function The Fourier transform of a vector of function is given by

for polynomial up to order . To ensure the superfunction has higher approximation power, we require the pre. filters to fulfill (11) and (12) with all values of Besides, the combined filters should have both low and highpass properties, as shown in the following equations:

. The multiwavelet can be constructed by selecting matrix filters and that satisfy the multiresolution refinement equations as follows: where

(8) (9)

is the -by-one vector with all elements zero, and ; . To design a prefilter wherein the associated superfunction satisfies the Strang–Fix conditions, it should be first noted that if there is a superfunction satisfying the Strang–Fix conditions, i.e.,


(11) and (12), it is equivalent that there exists such that generates ) and

243

(which

(13) where

denotes the th derivatives of , and satisfies

w.r.t.

,

(14) (15) The resulting superfunction given by (10) with the prefilter coefficients given by (13) has an approximation power up to order . This rewrites the Strang–Fix conditions (11)–(15). Hence, to design a prefilter where the associated superfunction satisfies the Strang–Fix conditions, we can first compute the of the eigen-equations (14) and (15). Then, we look solution that satisfies (13). for a finite vector sequence

Fig. 2.

Prefiltering and postfiltering schemes.

Fig. 3.

Combined filters.

Therefore, for

B. Second-Order Approximation Preserving Orthogonal Prefilter

(17)

In [14] and [18], the orthogonal prefilter is suggested for the prefiltering scheme, as illustrated in Fig. 2, which is in the form [2] (see also the combined filters in Fig. 3) .. .

Then, the procedure of finding the prefilter is equivalent to find parameters and that satisfy (17). In order to make (17) solvsuch that able, we need to select enough number of taps is nonsingular. It is satisfied when . Let us the matrix . Consider the case , where denote is nonsingular, and it can be rewritten into the following form: (18)

(16)

(19)

such that where is an -bymust satisfy

,

, and for , and identity matrix, and are unit vectors that

is null space of parameter vector an orthogonal matrix

, and is an . Then, we can construct (20)

such that . Then, apply right-hand sides of (18) to get To make the resulting prefilter enables second-order approximation preserving, it needs to satisfy the Strang–Fix conditions , 1. That is, , and (13)–(15) for . From (16), differentiate w.r.t. , and we have

to the left- and (21)

Let us consider the one-by-one. Since . formed into another

terms on the left-hand side of (21) is an orthogonal matrix, In addition, matrices are transset but still in their original form , where . For the last term on the right-hand side of (21), recall from . Then, (20) that diag and we arrive at the following formulation: diag

(22)

244


TABLE I THRESHOLDS AND THE CORRESPONDING SQUARE ERRORS DERIVED FROM THE SQUARE ERROR FUNCTION, BSURE AND LSURE WHEN APPLYING TO THE DENOISING OF “QUADCHIRPS,” AS SHOWN IN FIG. 4, WITH SIGNAL LENGTH 2 AND RNR 5 USING DGHO5 SETTING

Suppose that we select symmetric matrix symmetric, hence, .

; then,

is

diag where we denote

and

and (23)

Since

are unit vectors, ..

.

The prefilter is then given by (16) with the computed . Let us show a simple example. Consider the wavelet with multiand is found to be plicity three given by [24, Ex. 3.1].

must satisfy diag

Since we have a sufficient number of free parameters and , we may choose . Then, we can find the value of the matrix , hence, the prefilter. Let us summarize the procedure for finding a second-order approximation-preserving orthogonal prefilter. 1) Find and that satisfy (14) and (15). 2) Select a symmetric matrix from and and the . corresponding null space 3) Compute using (23), and set all the diagonal elements to 1 to obtain . 4) Compute the eigenvalue decomposition on , i.e., . 5) Compute . 6) The prefilter parameters are given by .

, respectively. We can see that all elare nonzero, and we may simply select diag . Then, we have from (20), and we also have the first equation shown is then given by at the bottom of the page. , shown in the second equation at the bottom of , the page. Then, , and . The solution is then given by completing step 4 to 6, shown in the third equation at the bottom of the page. For the case where are zero, we need to select another form some elements of of symmetric matrix that satisfies (19). We further apply the same method to obtain second-order approximation preserving orthogonal prefilters for multiwavelets of higher multiplicities, i.e., 4 and 5, as shown in Appendixes A and B. ements in

IV. EFFECT OF MULTIVARIATE SHRINKAGE In this section, we study the effect of applying the multivariate shrinkage function on the observation . This would give us the essential relations for making estimation on the mean square


TABLE II SQUARE ERROR WITH OPTIMAL THRESHOLDS WHEN APPLYING TO THE DENOISING OF “DOPPLER.” X AND R DENOTE THE SIGNAL LENGTH 2 AND THE RNR VALUE, RESPECTIVELY

245

error funtion (2). Let us define the square norm of the shrunk observation as follows:

(24) The corresponding expectation is

Tr

(25)

where is the difference of the shrunk observations and the shrunk true values, and is the total number of vector coefficient vectors. The noise vector is an of multivariate normal constructed from random vectors . Let us study the last term in detail:

TABLE III SQUARE ERROR WITH OPTIMAL THRESHOLDS WHEN APPLYING TO THE DENOISING OF “HYPCHIRPS.” X AND R DENOTE THE SIGNAL LENGTH 2 AND THE RNR VALUE, RESPECTIVELY

(26) It is equal to sum up the expectation of the inner products of and on different levels. The noise vectors are distributed . With an abuse of with multivariate normal notation, we consider these inner expectations without the index for the multiwavelet coefficients at a particular resolution level . To obtain these terms, we need the following lemma. , Lemma 1: For a multivariate normal distributed , , and

Tr

Tr (27)

TABLE IV SQUARE ERROR WITH OPTIMAL THRESHOLDS WHEN APPLYING TO THE DENOISING OF “QUADCHIRPS”1. X AND R DENOTE THE SIGNAL LENGTH 2 AND THE RNR VALUE, RESPECTIVELY

where . Proof: Let us first recall that for and function

Now, let

,

, where

(28) . Then, we have

(29)

246


Fig. 4. Example of the risk estimators bSURE and LSURE, as compared with square error, when applying to the denoising of “QuadChirps” with signal length 2 and RNR 5 using the DGHO5 setting.

where

Finally, we have the following formulation:

. From (3), for

Tr Taking partial derivatives on each component of w.r.t.

Tr

where is the th row of

, which is an . Then

Tr

vector, and

Tr

This proves the lemma. It can be shown that if the covariance of noise is equal to , and the last two terms will reduce to . Then, becomes zero for the case of multiplicity where the effect of reducing noise by multivariate shrinkage is equivalent to that of traditional shrinkage on each components of coefficient vector independently. By substituting (27) into (25), we


247

Fig. 5. Performance of the risk estimators bSURE and LSURE, with 100 trails of multivariate shrinkage denoising on several test signals of signal length: 2 . X axes are the RNR values. Y axes show the SER = log ((SE SE )=SE ), where SE is the total square error with estimated thresholds, and SE is the total square error with optimal thresholds (level dependent). (a): SER of “Blocks.” (b) SER of “Bumps.” (c) SER of “Cusp.” (d) SER of “HypChirps.” (e) SER of “Piecewise-Polynomial.” (f) SER of “QuadChirps.”

0

obtain the risk function for a particular parameter set . For practical implementation, (25) and (26) can be approximated as in (30). For a particular level , with the knowledge of the noise structure LSURE Tr

, and is the number of coefficient where vectors at level . The operator counts the number of nonzero coefficient vectors after shrinkage, which approximates the first term of (27). The optimal threshold is then the one that minimizes LSURE in each level. V. NUMERICAL EXPERIMENTS

Tr

Tr (30)

In this section, we show the performance of higher multiplicity wavelet denoising and the proposed LSURE. To keep the simulations simple, the prefiltering is nondecimating, whereas the discrete multiwavelet filterbank is decimating. We use linear filtering if the signal does not have energy near boundaries; otherwise, we symmetrically extend the signal at the boundaries.

248


Fig. 6. Performance of the risk estimators bSURE and LSURE, with 100 trails of multivariate shrinkage denoising on several test signals of signal length: 2 . X axes are the RNR values. Y axes show the TR = log ( ((T h Th )=T h ) ), where T h is the optimal threshold in level b, and T h

0

is the estimated threshold in level b. (a) TR of “Blocks.” (b) TR of “Bumps.” (c) TR of “Cusp.” (d) TR of “HypChirps.” (e) TR of “Piecewise-Polynomial.” (f) TR of “QuadChirps.”

The experiments were carried out on 1-D test signals with samples at several noise levels measured in terms of root signal-to-noise ratio RNR var . The test signals include “Bumps,” “Blocks,” “Cusp,” “Piecewise polynomials,” “HypChirps,” and “QuadChirps,” which are used in [4]–[6]. We measure the performance of different denoising algorithms for each resolution level in terms of mean square error . The following multiwavelets and SE prefilters settings are used: 1) wavelet of multiplicity 2 [7], [14] with Xia’s orthogonal prefilter [14] (GHMXIA);

2) wavelet of multiplicity 3 ([24, app. 2]) with the proposed orthogonal prefilter (DGHO3); 3) wavelet of multiplicity 4 ([23, table 1]) with the proposed orthogonal prefilter (DGHO4); 4) wavelet of multiplicity 5 ([23, table 2]) with the proposed orthogonal prefilter (DGHO5). The multiwavelet filter coefficients can be found in the references as mentioned above. The orthogonal prefilters are derived using the proposed algorithm in Section III-B. See the Appendix for the corresponding orthogonal prefilter parameters for DGHO4 and DGHO5. The covariance matrices can be computed as suggested in (7).


In the simulations, we test the following risk estimators: 1) SURE estimator borrowed from wavelet shrinkage bSURE [22] Tr

bSURE Tr

(31)

2) proposed LSURE , in (30). Thresholds are derived by minimizing the above risk estimators. In the simulations, we also generate a set of results by using the theoretical optimal thresholds for comparison. The optimal thresholds are obtained by exhaustively searching for the thresholds that minimize the mean square error, as shown in (2). Note that in practice, the mean square error can never be obtained since we do not have the original signal. Hence, the optimal thresholds shown here are only for comparison purposes. First of all, let us have a look at the performances of different denoising methods by using optimal thresholds with different multiwavelet bases. The results are obtained by averaging 100 trails of multiwavelet denoising. Table I shows the estimated thresholds by using bSURE and LSURE for each resolution level and the corresponding square errors. The values are compared with the optimal ones. It is seen that the performance given by LSURE is obviously much better than bSURE. The signals are contaminated with additive white Gaussian noise at RNR 10, 7, 5, and 2. Tables II–IV show the total square error of the denoised results for “Doppler,” “HypChirps,” and “QuadChirps,” respectively, using optimal thresholds. For these signals, we can see that in most cases, higher multiplicity wavelets give better performance. For the performance of the proposed risk estimator, we show in Fig. 4 an example for the denoising of “QuadChirps” with and RNR 5 using the DGHO5 setting. We can sample size see that the traditional bSURE does not work for higher multiplicity at any resolution level. On the contrary, LSURE closely resembles the mean square error function. In Figs. 5 and 6, we further show the performance of the proposed risk estimator for different test signals at different noise levels and different multiplicities. The results

249

shown are the average of 100 trails of the multivariate denoising experiment. In Fig. 5, we introduce the measure SE SE SE for the evaluation of SER the denoising performances of different risk estimators and compare them with those achieved by using the optimal thresholds. SE is the total square error with estimated thresholds is the total square error with optimal thresholds and SE (level dependent). For each diagram in Fig. 5, the -axis is the RNR values, and the -axis is the SER. We can see that LSURE consistently gives better performance than bSURE. It is also interesting to note that higher multiplicity usually gives better performance for LSURE. In Fig. 6, we introduce Th Th Th to the measure TR evaluate the accuracy of the estimated thresholds comparing is the optimal threshold with the optimal ones, where Th in level , and Th is the estimated thresholds in level . For each diagram in Fig. 6, the -axis is the RNR values, and the -axis is the TR. We can see that LSURE gives much smaller TR values, as compared with bSURE, for all test signals. VI. CONCLUSION In this paper, we studied two issues for improving the multiwavelet denoising based on multivariate shrinkage. First, we suggested a simple method for designing second-order approximation preserving orthogonal prefilter for any multiplicity. This enables applications using multiwavelets of higher multiplicity. Second, we investigated the risk estimators as applied to multivariate shrinkage. We suggested the new LSURE for finding thresholds that approach optimum in multiwavelet denoising. Numerical experiments show that first, multivariate shrinkage of higher multiplicity usually gives better performance, and second, the proposed LSURE substantially outperforms the traditional bSURE in multivariate shrinkage denoising, particularly at high multiplicity. APPENDIX A The DGH wavelet of multiplicity 4 [24] with the orthogonal prefilter computed using the proposed method (DGHO4) is shown in the equation at the bottom of the page.

250


APPENDIX B The DGH wavelet of multiplicity 5 [23] with the orthogonal prefilter computed using the proposed method (DGHO5) is shown in the equation at the top of the page. REFERENCES [1] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992. [2] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice-Hall, 1993. [3] S. Mallat and W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inform. Theory, vol. 38, pp. 617–643, Mar. 1992. [4] D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika, vol. 81, pp. 425–455, 1994. [5] D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory, vol. 41, pp. 613–627, May 1995. [6] D. Donoho, “Adapting to unknown smoothness via wavelet shrinkage,” J. Amer. Statist. Assoc., vol. 90, 1995. [7] J. S. Geronimo, D. P. Hardin, and P. R. Massopust, “Fractal functions and wavelet expansions based on several scaling functions,” J. Approximation Theory, vol. 78, pp. 373–401, 1994. [8] C. K. Chui and J.-A. Lian, “A study of orthonormal multi-wavelets,” Applied Numer. Math., vol. 20, pp. 273–298, 1996. [9] G. Strang, “Short wavelets and matrix dilation equations,” IEEE Trans. Signal Process., vol. 43, pp. 108–115, Jan. 1995. [10] V. Strela, P. N. Heller, G. Strang, P. Topiwala, and C. Heil, “The application of multiwavelet filterbanks to image processing,” IEEE Trans. Image Process., vol. 8, pp. 548–563, Apr. 1999. [11] T. R. Downie and B. W. Silverman, “The discrete multiple wavelet transform and thresholding methods,” IEEE Trans. Signal Process., vol. 46, pp. 2558–2561, Sep. 1998. [12] T. D. Bui and G. Chen, “Translation-invariant denoising using multiwavelet,” IEEE Trans. Signal Process., vol. 46, pp. 3414–3420, Dec. 1998. [13] X.-G. Xia, J. S. Geronimo, D. P. Hardin, and B. W. Suter, “Design of prefilters for discrete multiwavelet transforms,” IEEE Trans. Signal Process., vol. 44, pp. 25–35, Jan. 1996. [14] X.-G. Xia, “A new prefilter design for discrete multiwavelet transforms,” IEEE Trans. Signal Process., vol. 46, pp. 1558–1570, Jun. 1998.

[15] J. Lebrun and M. Vetterli, “High-order balanced multiwavelets: Theory, factorization, and design,” IEEE Trans. Signal Process., vol. 49, pp. 1918–1930, Sep. 2001. [16] G. Plonka, “Approximation properties of multiscaling functions: A fourier approach,” Rostocker Mathematische Kolloquium, vol. 49, pp. 115–126, 1995. [17] , “Approximation order provided by refinable function vectors,” Constructive Approx., vol. 13, pp. 221–244, 1997. [18] Y. Xinxing, J. Licheng, and Z. Jiankang, “Design of orthogonal prefilter with the strang-fix condition,” Electron. Lett., vol. 35, no. 2, pp. 117–119, Jan. 1999. [19] D. P. Hardin and D. W. Roach, “Multiwavelet prefilters -I: Orthogonal prefilters preserving approximation order p 2,” IEEE Trans. Circuits Syst. II, vol. 45, no. 8, pp. 1106–1112, Aug. 1998. [20] K. Attakitmongcol, D. P. Hardin, and D. M. Wilkes, “Multiwavelet prefilters—Part II: Optimal orthogonal prefilters,” IEEE Trans. Image Process., vol. 10, pp. 1476–1487, Oct. 2001. [21] E. Bala and A. Ertuzun, “Applications of multiwavelet techniques to image denoising,” in Proc. IEEE Int. Conf. Image Process., vol. 3, New York, Sep. 22–25, 2002, pp. 581–584. [22] M. Jansen, M. Malfait, and A. Bultheel, “Generalized cross validation for wavelet thresholding,” Signal Process., vol. 56, no. 1, pp. 33–44, Jan. 1997. [23] G. C. Donovan, J. S. Geronimo, and D. P. Hardin, “Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets,” SIAM J. Mathematical Anal., vol. 30, no. 5, pp. 1029–1056, 1998. , “Intertwining multiresolution analysis and the construction of [24] piecewise polynomial wavelets,” SIAM J. Math. Anal., vol. 27, no. 6, pp. 1791–1815, 1996.

Tai-Chiu Hsung (M’93) received the B.Eng. (Hons.) and Ph.D. degrees in electronic and information engineering in 1993 and 1998, respectively, from the Hong Kong Polytechnic University, Hong Kong. In 1999, he joined the Hong Kong Polytechnic University as a Research Fellow. His research interests include wavelet theory and applications, tomography, and fast algorithms. Dr. Hsung is a member of IEE.


Daniel Pak-Kong Lun (M’91) received the B.Sc.(Hons.) degree from the University of Essex, Essex, U.K., and the Ph.D. degree from the Hong Kong Polytechnic University (formerly called Hong Kong Polytechnic), Hong Kong, in 1988 and 1991, respectively. He is now an Associate Professor and the Associate Head of the Department of Electronic and Information Engineering, Hong Kong Polytechnic University. His research interests include digital signal processing, wavelets, multimedia technology, and internet technology. Dr. Lun participates actively in professional activities. He was the Secretary, Treasurer, Vice-Chairman, and Chairman of the IEEE Hong Kong Chapter of Signal Processing in 1994, 1995–1996, 1997–1998, and 1999–2000, respectively. He was the Finance Chair of 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, which was held in Hong Kong in April 2003. He is a Chartered Engineer and a corporate member of the IEE.

251

K. C. Ho (S’89–M’91–SM ’00) was born in Hong Kong. He received the B.Sc. degree with First Class Honors in electronics and the Ph.D. degree in electronic engineering from the Chinese University of Hong Kong, in 1988 and 1991, respectively. He was a research associate with the Department of Electrical and Computer Engineering, Royal Military College of Canada, Ottawa, ON, Canada, from 1991 to 1994. He joined Bell-Northern Research, Montreal, QC, Canada, in 1995 as a member of scientific staff. He was a faculty member with the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK, Canada, from September 1996 to August 1997. Since September 1997, he has been with the University of Missouri-Columbia, where he is currently an Associate Professor with the Electrical and Computer Engineering Department. He is also an Adjunct Associate Professor at the Royal Military College of Canada. His research interests are in source localization, wavelet transform, wireless communications, subsurface object detection, statistical signal processing, and the development of efficient adaptive signal processing algorithms for various applications, including landmine detection, echo cancellation, and time delay estimation. He has been active in the development of the ITU Standard Recommendation G.168: Digital Network Echo Cancellers since 1995. He is the editor of the ITU Standard Recommendation G.168. He has three patents from the United States in the area of telecommunications. Dr. Ho is an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING. He was the recipient of the Croucher Foundation Studentship from 1988 to 1991, and he received the Junior Faculty Research Award from College of Engineering of the University of Missouri-Columbia in 2003.