Symmetric Orthogonal Complex-Valued Filter Bank Design by Semidefinite Programming

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 9, SEPTEMBER 2007

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Symmetric Orthogonal Complex-Valued Filter Bank Design by Semidefinite Programming Ha Hoang Kha, Hoang Duong Tuan, Ba-Ngu Vo, and Truong Q. Nguyen

Abstract—A new design method for complex-valued twochannel finite impulse response (FIR) filter banks with both orthogonality and symmetry properties is developed. Based on a novel linear matrix inequality (LMI) characterization of trigonometric curves, the optimal design of perfect-reconstruction filter banks is cast into a semidefinite programming (SDP) problem. The dimension of the resulting SDP problem is further reduced by exploiting convex duality. Consequently, the globally optimal solution can be found for any practical filter length and desired regularity order. Index Terms—Complex-valued filter bank, semidefinite programming (SDP), trigonometric polynomial.

I. INTRODUCTION RTHOGONAL filter banks with symmetric FIR filters are of great interest in certain applications of image and video processing. The symmetry property of the filters is important for handling boundary distortions of finite length signals effectively [21]. From an implementation point of view, the symmetry allows us to halve the total number of multiplications [17]. On the other hand, the orthogonality property in filter banks preserves the energy of the input signal in the subbands, which guarantees that errors arising from quantization or transmission will not be amplified. This property is especially convenient for coding system design since the quantizer design can be carried out completely in the transform domain. In addition, orthogonality usually leads to high energy compaction [19]. Thus, it is desirable to design filter banks that are both symmetric and orthogonal. However, real-valued two-channel filter banks with simultaneous orthogonality and symmetry do not exist except for the trivial Haar filters [15], [19]. In contrast, nontrivial orthogonal and symmetric complex-valued filter banks do exist and are capable of providing many potentially beneficial properties. They produce orthogonal and symmetric complex wavelets, which can offer both nearly shift invariance and good directional

O

Manuscript received October 2, 2005; revised September 29, 2006. This work was supported by the Australian Research Council by Grant ARC Discovery Project 0556174. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Yuan-Pei Lin. H. H. Kha and H. D. Tuan are with the School of Electrical Engineering and Telecommunication, University of New South Wales, UNSW Sydney, NSW 2052, Australia (e-mail: [email protected]; [email protected]). B.-N. Vo is with the Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Vic 3052, Australia (e-mail: [email protected]). T. Q. Nguyen is with the Department of Electrical and Computer Engineering, University of California in San Diego, La Jolla, CA 92093 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2007.896288

selectivity, compared to shift variance and poor directional selectivity of real-valued wavelets [9], [10]. Furthermore, they can be applied to complex systems such as radars and discrete multi-tone modulation (DMT) systems [6]. This paper proposes a new design method for complex-valued two-channel FIR filter banks, which achieves both orthogonality and symmetry. A previously known method for designing complex-valued filter banks is based on the lattice structure, where the filter bank is factorized into a cascade of elementary building blocks. These blocks are characterized by a finite set of rotation angles [19]. Though this method provides an efficient and robust structure suitable for hardware implementation, it has a drawback in that it requires solving a system of highly nonlinear complex equations [6]. Therefore, this technique is not practical for large length filters due to numerical intractability in solving nonlinear complex equations in high dimensions. Another method reported in [21] is based on the product filter and spectral factorization. However, this approach does not guarantee that the symmetric prototype filter exists, and hence a symmetric filter bank cannot always be obtained. Our main result in this paper is a set of necessary and sufficient conditions on the product filter so that symmetric prototype filters can be obtained via spectral factorization (Theorem 1). Furthermore, we show that the conditions for symmetry, orthogonality and regularity are completely characterized by standard linear constraints and linear matrix inequality (LMI) constraints on the product filter. Consequently, the design problem is shown to be a semidefinite programming (SDP) problem. In addition, convex duality allows us to reduce it to yet another SDP problem with much smaller dimension. Subsequently, the globally optimal solution can be efficiently computed for any filter length and desired regularity order. The organization of this paper is as follows. Section II presents the formulation of the orthogonal and symmetric filter bank design problem. In Section III, this design problem is then converted to an equivalent SDP problem that can be readily solved by standard software packages. Design examples that serve to verify the effectiveness and efficiency of the proposed approach are presented in Section IV. Some concluding remarks are given in Section V. Relevant mathematical proofs are provided in the appendices. The notations used in the paper are rather standard. In particular, the notation denotes a (symmetric) positive semidefinite matrix. The inner product beand is defined as , i.e., tween the matrices . The symbols and denote the vector space of real and complex -vectors, respectively. For a given set its convex hull (conic hull), denoted by ( ), is the smallest convex set (cone) in

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Fig. 1. Maximally decimated two-channel filter bank.

that contains . For a given function , we use to denote , where the asterisk subscript denotes the conjugation of coefficients without conjugating . stands for . With a slight abuse of the unit vector to refer to . notation, we use Our preliminary results have been reported in a conference paper [8]. This paper presents a more complete version of the work. II. MATHEMATICAL MODEL OF ORTHOGONAL AND SYMMETRIC FILTER BANKS Two-channel filter banks have been of primary interest in the theory and applications of multirate filter banks and wavelets. They are used as the basic building blocks in the construction of dyadic multi-resolution transform and hence implemented in resolution-scalable compression schemes. Furthermore, recursive cascade of a two-channel filter bank along one or both of the low-pass and high-pass subbands provides a conceptually simple way to create a wide variety of different structures [17]. A two-channel maximally decimated uniform filter bank is illustrated in Fig. 1. An analysis filter bank with low-pass filter and high-pass filter decomposes the input signal into the subband signals and . This is followed by a synthesis filter bank with the low-pass filter and high-pass filter , which reconstructs the output signal from the subband signals. It can be easily shown that the output of the two-channel filter bank is given by

[7]. In this case, the output for some odd signal is a delayed and scaled version of the input . Note that the conditions (1), (2) hold in general cases of orthogonal filter banks where filters can be either FIR or infinite impulse response (IIR). It is well-known that filter banks with IIR filters provide good frequency selectivity and low computational complexity [20]. However, the relations in (1) imply that the synthesis filters are time-reversed versions of the analysis filters. Therefore, if the analysis filters are causal, then the synthesis filters are anticausal. The anticausal IIR filters cannot in general be implemented precisely except for a case where the signal to be filtered is of finite length [20]. Due to this drawback, orthogonal IIR filter banks are less popular in practical applications. In this paper, we restrict our consideration on designing orthogonal filter banks with FIR filters. , are It follows from (1) that the low-pass filters are antisymsymmetric and the high-pass filters is chosen to be symmetric, metric, if the prototype filter i.e., (3) is the vector of (complex-valued) where . coefficients of Hence, the problem of designing an orthogonal and symmetric filter bank boils down to designing a prototype filter satisfying the orthogonality condition (2) and symmetry condition (3). While the symmetry condition is linear in the filter coefficients , the orthogonality condition is highly nonlinear. To characterize these constraints by LMIs, we first introduce the product filter (4) If the prototype filter coefficients satisfy the symmetry condition (3), the product filter is a positive real filter with symmetric coefficients . For convenience, let

The frequency response of the product filter is In an orthogonal two-channel filter bank, all filters are obtained as follows: from a single low-pass prototype filter

(1) where is the filter order. These relations imply that the magnitude responses of and are exactly the same as and , respectively. those of In addition to the conditions in (1), the filter bank is said to be orthogonal if and only if the prototype filter satisfies the following condition:

(5) In terms of the product filter (2) is given by

which is equivalent to the following linear constraints in the coefficients : (6) where

(2)

, the orthogonality condition

if

is the Kronecker delta function ( and ). However, not all positive real filters can

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KHA et al.: SYMMETRIC ORTHOGONAL COMPLEX-VALUED FILTER BANK DESIGN

be factorized into the form (4) with satisfying the symmust metry condition (3). In fact, in view of (3) and (4), have the following specific form:

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III. CONVERSION TO SDP The LMI characterizations of the nonlinear constraint (7) and semi-infinite constraints (10) are given in Section III-A, while the reduction of the problem dimension is given in Section III-B. A. SDP Formulation

(7) . for some In addition, certain applications require the low-pass filters to is said to be -regular if it has mulbe regular. A filter or, equivalently, has multiple tiple zeros at . This translates into the following linear conzeros at straints in the filter coefficients :

The constraint (7) is a highly nonlinear relationship between the product filter coefficients and the symmetric complex prototype filter coefficients . It can be seen that if in (7) is restricted to be real only, then the set of all satisfying (7) is nonconvex, and, thus, it is not possible to express (7) as LMIs. Interestingly, when is allowed to be complex, the set of all such is convex, and can be described by LMIs. Let

(8) Finally, to ensure that the filters have good responses and stopband , where in the passband , and is a positive constant depending on the required transition width, we require that the square error

then, the th-order trigonometric moment matrix is depositive semidefinite fined as the following matrix, shown at the bottom of the page. The matrix with is defined as the matrix obtained by applying the change of variables

(9)

(12) to

is minimized, and that the standard peak-error constraints

, that is,

(10) are satisfied [1]. In summary, the design of a two-channel filter bank with orthogonality, symmetry and regularity properties is now formulated as an optimization problem which minimizes the convex quadratic objective function (9) subject to the linear constraints (6), (8), the nonlinear constraint (7), and the semi-infinite linear constraints (10) in the variable s.t. (6), (8), (7), (10)

(11)

The next section is devoted to converting the nonlinear constraint (7) and semi-infinite constraints (10) into LMIs, so that our filter bank design problem can be formulated as an SDP problem. We also exploit convex duality to reduce the dimension of this SDP problem for efficient computation.

Furthermore, we also define

Similarly, is defined as the matrix obtained by applying . the change of variables (12) to The following theorem shows that the nonlinear constraint (7) can indeed be cast as LMIs. which contains all possible Theorem 1: Define the set product filter coefficients such that there exist the symmetric prototype filters, i.e.,

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(13)

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Then, the set

can be alternatively described by LMIs

if and only if there exist positive semidefinite matrices and such that

(14) (19) Proof: See Appendix I. We remark that by comparing the ’s with the coefficients of in , linear the same “power” of relations between and in (14) can be established. These constitute the LMIs linear relations and the constraint that describe in (14). It is clear from the LMI representation in Theorem 1 that is convex. Next, we observe that the constraints (10) are linear in the variable , thus the feasible region of the constraints will be convex. However, the constraints have to be satisfied at each frequency , so there is an infinite number of inequality constraints. Since there are a finite number of variables and an infinite number of constraints, the constraints are called semi-infinite trigonometric constraints. These semi-infinite constraints can be approximated by discretizing the continuous frequency axis. The discretizing method has a problem in that the inequality constraints are not guaranteed to be satisfied between the discrete frequencies. To alleviate this problem, recently, it has been shown that the semi-infinite trigonometric constraints (10) can be cast as a finite number of linear equations and LMIs [18], [5]. To achieve this, we appeal to a result in [18], which is briefly outlined as follows. A trigonometric is defined as curve

Proof: The proof of this theorem is derived from Markov-Lucacs theorem for nonnegative algebraic polyno. For more mial and Chebysev algebraic polynomial in details, we refer the reader to [18]. From Theorem 2, it is clear that each of constraints in (17) [or in (18)] can be expressed by linear equations of coefficients and , and two LMIs . elements of matrices Summarizing, we have the following. is the following convex • The objective function quadratic function in :

where

• The constraint (6) is the linear equation (20)

(15) where

and its polar cone is given by (16) It is obvious that the constraints (10) can be written as for (17)

Similarly, the regularity constraints (8) are the linear constraints (21)

where with

• The filter bank design problem (11) is reformulated as the following problem:

The constraints (17) can be compactly rewritten as (18)

(22)

To show that constraints (17) involve a finite number of linear equations and LMIs, we summarize the trigonometric MarkovLucacs theorem for the case of odd order trigonometric polynomial. Theorem 2: A trigonometric polynomial , where , is nonnegative for

This is an SDP problem because (20), (21) are linear constraints, and (14), (18) are LMIs. It should be noted that (18) consists of four LMIs involving eight additional variables each of which is positive semidefinite matrix. Furthermore, a symmetric (14) is also an LMI involving one positive semidefinite matrix variable . Thus, the total number

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of scalar variables for the SDP problem in (22) is , which is very high for large .

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, and, hence,

can be given explicitly as

follows:

B. Dimension Reduction Here, we apply convex duality to convert the SDP problem (22) into another SDP problem which has much less number of (sometimes called variables. We first define the polar cone as the dual cone) of a given set (23) It is straightforward to see that . Theorem 3: The polar cone of the set given by

We then determine the dual function, which by definition is the minimum of the Lagrangian over described by (14) is

(24) Proof: Applying the change of variables (12) to (14) yields where

From the definition of the polar cone As a result, the dual problem can be expressed as

subject to This completes the proof. Theorem 4: [18] The convex hull of the trigonometric curve defined by (15) is fully characterized by LMI constraints

(27) Finally, using Schur’s complement [3], the dual problem (27) can be rewritten as the following SDP problem:

(25) Consequently, the conic hull of

is given by subject to (26)

Proof: See Appendix II. Following [18] we can also reduce the dimension of the SDP problem (22) by exploiting convex duality. We first form the Lagrangian associated with the primal problem (22) as follows:

(28) Once the optimal solutions and of the dual problem have been found, the optimal solution of the primal problem (22) is retrieved by the following equation:

where = and are the Lagrange multipliers associated with the equality constraints. The dual variable is an element of the set

and the dual variable . Note that

(the dual cone of

),

Note that in contrast to the primal optimization problem (22), the , SDP problem (28) involves five variables each of dimension , one variable of dimension and other scalar variables (the s and ), so the total number . Hence, from a numerical of scalar variables is

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Fig. 2. (a), (b) Magnitude responses of the analysis filters with of the scaling functions and wavelets.

N = 11. (c), (d) The solid line shows the real part, and the dashed line shows the imaginary part

viewpoint, the dimension of the problem is not an issue even for large . satisfying (7) is recovered The optimal prototype filter from the optimal product filter by spectral factorization [4], [21]. It should be emphasized that the spectral factorization method will not achieve the symmetric prototype filter if the are not even multiplicity real zeros of the product filter (see [21] for example). However, our optimization method only searches over the feasible region where there exist symmetric prototype filters. In particular, the possible coefficients , which guarantee the existence of the symmetric prototype filters, have been restricted by the constraint (7). As a result, our methods can always obtain the symmetric prototype from spectral factorization. IV. DESIGN EXAMPLES In this section, we provide two design examples of symmetric orthogonal complex-valued two-channel filter banks to verify

the effectiveness of the proposed method. In these examples we use the SDP library SeDuMi [16] to solve the SDP problem (28). Example 1: A symmetric orthogonal complex-valued twochannel filter bank is designed using the above method. The has the following specifications: low-pass prototype filter , passband edge frequency , filter order , and stopband attenuation stopband edge frequency dB. We consider regularity orders of 1 and 3. The magnitude responses of the analysis filters are shown in Fig. 2(a) and (b). It can be observed that the stopband attenuation becomes worse as the regularity order increases. This is because a portion of the free coefficients of the filter have been used to accommodate the regularity constraints. Like real-valued wavelet transform, the discrete dyadic complex-valued wavelet transform can be implemented by iterating a two-channel filter bank on its low-pass output. As a result, the scaling and wavelet functions associated with the optimal design are shown in Fig. 2(c) and (d). The properties of the filters in the optimal design are intimately related to those of the

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KHA et al.: SYMMETRIC ORTHOGONAL COMPLEX-VALUED FILTER BANK DESIGN

Fig. 3. (a), (b) Magnitude responses of the analysis filters with of the scaling functions and wavelets.

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N = 63. (c), (d) The solid line shows the real part, and the dashed line shows the imaginary part

scaling and wavelet functions. First, the largest possible support of the wavelets and scaling functions is the interval as the FIR filters are causal with order . Wavelets with compact support are often desired in applications in which time location properties are important [2]. Second, the scaling functions and the low-pass filter have the same symmetric property, while the wavelet functions and the high-pass filter are antisymmetric. Thirdy, the regularity order of the filters is closely related to the smoothness of the wavelets and scaling functions [12], [14]. It can be seen from Fig. 2(c) and (d) that the scaling and wavelet are smoother than functions associated with the filters for those for . Example 2: Although the prefect reconstruction filter banks are designed to completely cancel aliasing, in practice the aliasing components can still exist in the reconstructed signal due to different levels of quantization noise in the subbands. The leakage of quantization noise can cause undesirable artifacts in encoded images [2], [15]. These artifacts can be mitigated by filters with small transition band and high stopband atten-

uation. Moreover, filter banks with high stopband attenuation are essential in audio signal processing and signal detection [15]. Small transition band and high stopband attenuation can only be achieved with longer filters, which cannot be handled adequately by existing approaches. In this example, a filter bank with long filter is designed. The low-pass prototype filter has the , following specifications: stopband attenuation dB, and . Fig. 3 shows the magnitude responses of the analysis filters and the corresponding scaling functions and wavelets. It can be seen that the magnitude responses meet the required specifications. The filters have sharp transition bands and high stopband attenuations. The scaling functions are symmetric and the wavelets are antisymmetric. V. CONCLUDING REMARKS In this paper, a novel method for designing symmetric orthogonal complex-valued filter banks has been presented. The key contribution is the formulation of the design problem as an

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SDP problem that can be efficiently solved by available software. Compared to previous complex-valued filter bank design methods [6], [21], our proposed method has several advantages. First, globally optimal filters and wavelets of any length and regularity order can be obtained. Second, (LMI) constraints on the product filter have been incorporated in our formulation to ensure that the symmetric prototype filter can always be obtained via spectral factorization. Our design examples have verified the effectiveness and efficiency of the proposed method.

APPENDIX I THE PROOF OF THEOREM 1

Therefore, for in (7) is for

, an alternative expression

Now, consider the following alternative expression for the set defined by (13)

For the proof of Theorem 1, we need the following version of the Markov–Lucacs Theorem [11]. Theorem 5: Any th-order nonnegative real polynomial

can be expressed by a sum of squares of two polynomials with order or less

(32) , it is clear that the cone defined by Since (32) is a subset of the cone defined by the right hand side of (14). on the right-hand On the other hand, for any side (RHS) of (14), we have

(33)

(29) where [18]

or, equivalently

is a nonsingular matrix such that

(30) An immediate consequence of Theorem 5 is that for a given th-order nonnegative real polynomial, there exists a positive semidefinite matrix such that (31) Now, we detail the proof of Theorem 1. Under (3), there is an alternative expression of in the unit disc [13]

(the existence of follows from the fact that is the Chebyshev polynomial of order in ). Considering the , it folright hand side of (33) as an algebraic polynomial in and lows from Theorem 5 that there exist such that

Consequently

for

for

. Hence

Thus, the set defined by the right hand side of (14) is also a subset of the cone defined by (32). This completes the proof of Theorem 1. Authorized licensed use limited to: Univ of Calif San Diego. Downloaded on February 24, 2009 at 15:26 from IEEE Xplore. Restrictions apply.

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APPENDIX II THE PROOF OF THEOREM 4 The proof exploits the results reported in [18]. In order to make this paper self-contained, we briefly recall the results that are relevant for the proof of Theorem 4. is convex as it is described by LMIs. For any The set , it is evident that

(34) Hence, and so by definition of the convex hull. . For the set It remains to prove that in , we define its support function of every (35) It follows that

(36) and according to Theorem 2 there exist that the following representation holds

and

such

(37) Using the change of variables (12) on both sides of (37) yields

(38) , and also for

Since

, it follows that

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[2] A. N. Akansu and R. A. Haddad, Multiresolution Signal Decomposition: Transforms, Subbands, Wavelets. New York: Academic, 2001. [3] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2003. [4] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992. [5] T. N. Davidson, Z. Q. Luo, and J. F. Sturm, “Linear matrix inequality formulation of spectral mask constraints with applications to FIR filter design,” IEEE Trans. Signal Process., vol. 50, no. 11, pp. 2702–2715, Nov. 2002. [6] X. Q. Gao, T. Q. Nguyen, and G. Strang, “A study of two-channel complex-valued filterbanks and wavelets with orthogonal and symmetry properties,” IEEE Trans. Signal Process., vol. 50, no. 4, pp. 824–833, Apr. 2002. [7] C. Herley and M. Vetterli, “Wavelets and recursive filter banks,” IEEE Trans. Signal Process., vol. 41, no. 8, pp. 2536–2556, Aug. 1993. [8] H. H. Kha, H. D. Tuan, B. Vo, and T. Q. Nguyen, “Symmetric orthogonal complex-valued filter bank design by semidefinite programming,” in Proc. 2006 IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP), Toulouse, France, May 2006. [9] N. Kingsbury, “Shift invariant properties of the dual-tree complex wavelet transform,” in Proc. IEEE Conf. Acoust., Speech Signal Process., Phoenix, AZ, Mar. 1999, paper SPTM 3.6. [10] N. Kingsbury, “Complex wavelets for shift invariant analysis and filtering of signals,” J. Appl. Computat. Harmon. Analysis, vol. 10, no. 3, pp. 234–253, May 2001. [11] M. Krein and A. Nudel’man, “The markov moment problem and extremal problems,” Transl. Math. Monographs, vol. 50, 1977. [12] S. Oraintara, T. D. Tran, P. N. Heller, and T. Q. Nguyen, “Lattice structure for regular paraunitary linear-phase filterbanks and m-band orthogonal symmetric wavelets,” IEEE Trans. Signal Process., vol. 49, no. 11, pp. 2659–2672, Nov. 2001. [13] J. Proakis and D. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. ed. Englewood Cliffs, NJ: Prentice-Hall. [14] P. Steffen, P. N. Heller, A. Gopinath, and C. S. Burrus, “Theory of regular m-band wavelet bases,” IEEE Trans. Circuits Syst., vol. 41, no. 12, pp. 3497–3511, Dec. 1993. [15] G. Strang and T. Q. Nguyen, Wavelets and Filter Banks. Wellesley, MA: Wellesley-Cambridge, 1997. [16] J. F. Sturm, “Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones,” Optimiz. Methods Software, vol. 11–12, pp. 625–653, 1999. [17] D. S. Taubman and M. W. Marcellin, JPEG2000: Image Compression Fundamental, Standards and Practice. New York: Springer Science+Business Media, Inc., 2002. [18] H. D. Tuan, T. T. Son, B. Vo, and T. Q. Nguyen, “LMI characterization for the convex hull of trigonometric curves and applications,” in Proc. 2005 IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP), Philadelphia, PA, May 2005. [19] P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice-Hall, 1993. [20] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding. Upper Saddle River, NJ: Prentice-Hall PRT, 1995. [21] X. Zhang, M. D. Desai, and Y. Peng, “Orthogonal complex filter banks and wavelets: Some properties and design,” IEEE Trans. Signal Process., vol. 47, no. 4, pp. 1039–1048, Apr. 1999.

(39) Hence, it concludes that plete.

. The proof is com-

ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments which helped to improve the quality of the paper. REFERENCES [1] J. Adams, “FIR digital filters with least-square stopbands subject to peak-gain constraints,” IEEE Trans. Circuits Syst., vol. 39, no. 4, pp. 376–388, Apr. 1991.

Ha Hoang Kha received the B.E. and M.E. degrees in electrical engineering and telecommunications from HoChiMinh City University of Technology, in 2000 and 2003, respectively. He is presently pursuing the Ph.D. degree with the School of Electrical Engineering and Telecommunications, University of New South Wales, since 2004. His research interests are in digital signal processing, filter banks and wavelets, and their applications in image processing and digital communications.

Hoang Duong Tuan was born in Hanoi, Vietnam. He received the diploma and the Ph.D. degrees, both in applied mathematics, from Odessa State University, Ukraine, in 1987 and 1991, respectively. From 1991 to 1994, he was a Researcher with Optimization and Systems division, Vietnam National Center for Science and Technologies. He spent nine

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academic years in Japan as an Assistant Professor with the Department of Electronic-Mechanical Engineering, Nagoya University, from 1994 to 1999, and then as an Associate Professor with the Department of Electrical and Computer Engineering, Toyota Technological Institute, Nagoya, from 1999 to 2003. Presently, he is an Associate Professor with the School of Electrical Engineering and Telecommunications, the University of New South Wales, Sydney, Australia. His research interests include theoretical developments and applications of optimization-based methods in many areas of control, signal processing, and communication.

Ba-Ngu Vo was born in Saigon, Vietnam. He received the B.Sc./B.E. degree (with first class honors) and the Ph.D. degree in 1994 and 1997, respectively. He is currently an Associate Professor with the Department of Electrical and Electronic Engineering, University of Melbourne, Melbourne, Australia. His research interests are stochastic geometry, random sets, multitarget tracking, optimization, and signal processing.

Truong Q. Nguyen received the B.S., M.S., and Ph.D. degrees in electrical engineering from the California Institute of Technology, Pasadena, in 1985, 1986, and 1989, respectively. He was with the Massachusetts Institute of Technology Lincoln Laboratory, Cambridge, from June 1989 to July 1994, as a member of Technical Staff. Since August 1994 to July 1998, he was with the Electrical and Computer Engineering Department, University of Wisconsin, Madison. He was with Boston University, Boston, MA, from 1996 to 2001, and is currently with the Electrical and Computer Engineering Department, University of California at San Diego. His research interests are in the theory of wavelets and filter banks and applications in image and video compression, telecommunications, bioinformatics, medical imaging and enhancement, and analog/digital conversion. He is the coauthor (with Prof. Gilbert Strang) of a popular textbook, Wavelets and Filter Banks (Wellesley, MA.: Wellesley-Cambridge Press, 1997) and the author of several Matlab-based toolboxes on image compression, electrocardiogram compression, and filter bank design. He also holds a patent on an efficient design method for wavelets and filter banks and several patents on wavelet applications including compression and signal analysis. Prof. Nguyen received the IEEE TRANSACTIONS IN SIGNAL PROCESSING Paper Award (Image and Multidimensional Processing area) for the paper he cowrote with Prof. P. P. Vaidyanathan on linear-phase perfect-reconstruction filter banks in 1992. He received the NSF Career Award in 1995 and is currently the Series Editor (Digital Signal Processing) for Academic Press. He served as Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1994 to 1996 and for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS from 1996 to 1997.

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