Deconvolution Filtering of 2-D Digital Systems

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 9, SEPTEMBER 2002 2319 H Deconvolution Filtering of 2-D Digital Systems Lihua Xie, Senior Member, IEEE, Chunling Du, Cishen Zhang, and Yeng Chai Soh Abstract This paper deals with the problem of two-dimensional (2-D) digital system deconvolution under the performance specification. The 2-D digital system under consideration is expressed by the Fornasini Marchesini local state-space (FM LSS) model. For a given 2-D digital system, a 2-D deconvolution filter is designed to reconstruct the 2-D signal to meet a prescribed performance specification. A key analytical means for the deconvolution filter design is the 2-D bounded realness property derived in this paper. Applying this property, the 2-D deconvolution filter design is formulated into a convex optimization problem characterized by linear matrix inequalities (LMIs). A procedure for the design of an deconvolution filter for a given 2-D digital system with measurement noise is developed first. The LMI approach to the 2-D deconvolution filtering problem is further extended to the 2-D robust deconvolution filtering problem with polytopic modeling uncertainties in the 2-D system. The obtained results on the 2-D deconvolution filtering and robust deconvolution filtering problems are demonstrated with examples. Index Terms Deconvolution filter, norm, linear matrix inequality, 2-D discrete system. I. INTRODUCTION IT IS known that a signal transmitted through a transmission channel is subject to a convolution operation of the transmission channel, and to reconstruct the transmitted signal from the output of the transmission channel is the deconvolution problem, which is to inverse the convolution operation on the transmitted signal. Often, in practice, the received signal from the output of the transmission channel is embedded in noise. Deconvolution is a fundamental problem in signal processing and has been widely explored in the literature. It has many application areas, for example, data transmission, equalization, reverberation cancellation, seismic deconvolution, image restoration, speech signal processing, e.g., [1] [3], [5]. The deconvolution problem for one-dimensional (1-D) systems has attracted a lot of interest from researchers, and several methods have been proposed to deal with the problem. An optimal deconvolution filter was first designed via least squares prediction error by Silvia and Robinson in [6]. In general, the 1-D system deconvolution problem can be solved via the optimal method by Wiener filtering technique from the Manuscript received April 11, 2001; revised May 6, 2002. The associate editor coordinating the review of this paper and approving it for publication was Dr. Masaaki Ikehara. L. Xie and Y. C. Soh are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-mail: elhxie@ntu.edu.sg). C. Du is with the MMS Group, Data Storage Institute, Singapore. C. Zhang is with the Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Australia. Publisher Item Identifier 10.1109/TSP.2002.800401. frequency domain perspective, such as outer inner factorization and orthogonal principle, e.g., [1], or by Kalman filtering technique from the time domain perspective, e.g., [3], [5]. In addition, a minimax deconvolution filter design based on theory has attracted great attention for its robustness property, e.g., [2], [28]. Two-dimensional (2-D) digital systems have found many applications, especially in signal and image processing and telecommunications. The techniques of 2-D inverse filtering were widely applied to digital image restoration from the mid-1960s; see, for example, [10], [17], [23], and references cited therein. In [23], 2-D deconvolution was used to restore images from the early planetary exploration missions. Harris deconvoluted the blurring due to atmospheric turbulence in telescope images in [17]. In addition, the inverse filtering problem of shift-variant imaging systems was discussed in [24], the technique of blind deconvolution is applied in [27], and invertibility of 2-D state-space periodically shift-variant discrete systems was studied in [25]. Despite the extensive research on the optimal solutions to the deconvolution filtering problem for 1-D systems, little progress has been made for the 2-D deconvolution filtering problem with respect to some optimal system performance. The aim of this paper is to present a state-space solution to the 2-D deconvolution problem under the performance specification. The deconvolution problem in this paper is to design a 2-D deconvolution filter to reconstruct the input signal to a given 2-D digital system in the presence of measurement noise. The performance of the deconvolution filter is described by the norm of the deconvolution error system. Similar to the result for the 1-D deconvolution problem, the bounded realness property of 2-D systems plays a key role for the solution to the 2-D problem [11]. We will first derive an improved bounded realness property for 2-D systems from that in [11] in terms of a linear matrix inequality (LMI). Based on this property, solvability conditions for the deconvolution filter are derived in terms of LMIs. The solutions of the LMI s, if they exist, are then used to construct a feasible 2-D deconvolution filter. We further extend the result of the deconvolution filter to 2-D systems with polytopic modeling uncertainties, which may be caused by system modeling and quantization errors. As a result, conditions and an LMI solution for the robust 2-D filter subject to modeling uncertainties are derived. An advantage of the solutions for the 2-D deconvolution filtering problem is that the powerful LMI Tool [14] can be directly applied to compute the 2-D deconvolution filter. The rest of this paper is organized as follows. Section II presents formulation and preliminaries for the 2-D deconvolution problem. Section III derives an improved bounded real lemma for 2-D systems. In Section IV, a solution for the 2-D deconvolution problem that is characterized by a 1053-587X/02$17.00 2002 IEEE

2320 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 9, SEPTEMBER 2002 set of LMIs is presented. Following the result of Section IV, Section V gives examples to demonstrate the LMI solutions and the deconvolution filter performance. The following notations that are used are fairly standard. Set of non-negative integers. Set of real matrices. identity matrix. diag Block diagonal matrix. Matrix is symmetric and positive definite Transpose of the matrix. Euclidean vector norm. The norm of a 2-D signal is given by In addition, introduce Then, a state-space representation of the filtering error system is obtained as where, and (2.5) (2.6) (2.7) where is said to be in the space or, for simplicity, if. II. PROBLEM FORMULATION AND PRELIMINARIES Consider a discrete 2-D signal, with, which is transmitted through a 2-D digital system described by the FM LSS model [12] (2.1) (2.2) where is the 2-D system state, and is the received 2-D signal that is corrupted by the receiver noise., with,, and are real matrices with appropriate dimension. It is assumed that the boundary condition is zero, i.e., for, and no information on the statistics of the input signal and the receiver noise is known, except that they are energy bounded, i.e.,. The problem under consideration is to find a 2-D deconvolution filter to reconstruct the input signal, in some optimal sense, from the received signal. Let the 2-D deconvolution filter of the given 2-D system be represented by the FM LSS model as (2.3) (2.4) where ; ; ; ;. The boundary condition of the filter is set to be zero, i.e., for. Denote by the deconvolution filtering error, i.e., (2.8) Before giving a precise statement of the problem under investigation, we introduce the notion of asymptotic stability and performance for the 2-D error system as follows. It is known that the transfer function of the error system is given by and the norm of the system is (2.9) where denotes the maximum singular value. Definition 2.1 [19]: The 2-D system is asymptotically stable if under the zero input and any boundary condition such that, where Lemma 2.1 [21]: The error system ( ) is asymptotically stable if there exist matrices and such that where. Definition 2.2: Given a positive scalar, the 2-D system has an performance if it is asymptotically stable and. Remark 2.1: Similar to the 1-D system case, by using the 2-D Parseval s theorem [20], it is not difficult to show that under the stability of, the condition of is equivalent to (2.10) under the zero boundary condition. It is also noted that is referred to as the bounded realness of the transfer function. For the filtering error system (2.5) and (2.6), we now state the deconvolution filtering problem as follows: Find, if it exists, a deconvolution filter of the form (2.3) and (2.4) such

XIE et al.: DECONVOLUTION FILTERING OF 2-D DIGITAL SYSTEMS 2321 that the filtering error system (2.5) and (2.6) has a prespecified performance. or III. IMPROVED BOUNDED REAL LEMMA An earlier version of the bounded real lemma for 2-D systems was derived in [11] in terms of a 2-D algebraic Riccati equation. We now present a new improved version of the bounded real lemma for 2-D systems in terms of a linear matrix inequality. This is stated as follows. Theorem 3.1: Given a scalar, the 2-D error system has an performance if there exist matrices and satisfying the following LMI: Hence, it follows from Lemma 2.1 that the error system is asymptotically stable. To establish the performance, we note from [11, proof of Th. 1] that (3.14) implies that This is equivalent to (3.11) where is as given in Lemma 2.1, and (3.16) diag diag (3.12) Proof: The existence of and in the LMI (3.11) implies that. Now, let,, and. It follows that Under the zero-boundary condition,, it follows (2.6) that and. Thus, it follows from (3.16) that Substituting this into (3.11) gives (3.17) (3.13) It can be known from (3.15) that. Therefore and ap- Pre- and post-multiplying (3.13) by diag plying Schur complements yields and It follows from (3.14) that i.e., (3.14) (3.15) This completes the proof. Remark 3.1: The difference between the new bounded real lemma of Theorem 3.1 and its old version in [11] is that the new version does not involve searching for scaling parameters or scaling matrices, as required in [11]. Moreover, the use of more general and flexible scaling matrices and in Theorem 3.1 instead of a scaling parameter as used in [11] can lead to tighter bound, i.e., better quantification, on the system norm than that of [11]. This is a key advantage of the new bounded real lemma over the old version. Remark 3.2: Observe that given the deconvolution filter (2.3) and (2.4) and a scalar, (3.11) is linear in unknown variables and. Therefore, the problem of solving (3.11) is a convex optimization, and a solution of (3.11), if it exists, can be obtained by implementing the LMI using the Matlab LMI Toolbox [14]. If, for a, the solution for the inequality (3.11) does not exist, one may relax the performance level to some until it is sufficiently large

2322 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 9, SEPTEMBER 2002 for an admissible solution. On the other hand, if a solution exists for a given, then an admissible solution always exists for any. We now use an example to show that the new bounded real lemma can provide a tighter bound on the system norm over the old version. Consider the 2-D digital filter in form of the following transfer function [13]: where,,,,,,, and It can be represented in form of the FM LSS model (2.5) and (2.6) with The minimum that is derived by using Theorem 3.1 is computed to be, whereas the old version of the bounded real lemma given by [11, Th. 2] yields. Note that the actual norm of the filter is. Hence, this example demonstrates that the new bounded real lemma of Theorem 3.1 is much less conservative in evaluating the performance of 2-D systems. The improved bounded real lemma presented in Theorem 3.1 will be applied in the following sections to solve the 2-D deconvolution problem. (4.2) Proof: See the Appendix. Remark 4.1: Theorem 4.1 gives a sufficient condition for existence of the Deconvolution filter. Observe that the matrix inequality (4.1) is linear in the unknown matrix variables, and. Therefore, a solution to (4.1), if it exists, can be easily obtained by implementing the LMI using the Matlab LMI Toolbox [14]. Once a solution to the LMI (4.1) is obtained, an deconvolution filter can be constructed as follows. First, note that the LMI (4.1) implies. Thus, the matrix is invertible, and nonsingular matrices and can be computed to satisfy. It also follows from the proof of Theorem 4.1 that the matrices, and solved from the LMI (4.1) can be expressed as (4.3) (4.4) (4.5) where and. Hence, the matrices of the deconvolution filter,, and can be computed as IV. DECONVOLUTION FILTER DESIGN This section presents an LMI approach to the design of 2-D deconvolution filter. The main result is stated in the following theorem. Theorem 4.1: Consider the 2-D system (2.1) and (2.2). Then, for a given positive scalar, the deconvolution filtering problem is solvable if there exist matrices,,,,, and such that Note that the filter parameter the solution of the LMI (4.1). has been derived directly from V. ROBUST DECONVOLUTION FILTER DESIGN It is assumed in the preceding section that the 2-D system is known exactly. In practice, there may exist modeling uncertainties, e.g., modeling and quantization errors, in, and the designed deconvolution filter based on the nominal model may perform poorly when the actual system differs from the nominal model. In this section, we consider the deconvolution filtering problem under system modeling uncertainties. Our objective is to find deconvolution filters with guaranteed performance under all admissible parameter uncertainties. Consider the following 2-D system model: (4.1) (5.1) (5.2)

XIE et al.: DECONVOLUTION FILTERING OF 2-D DIGITAL SYSTEMS 2323 where,,, and are the signals of the 2-D system as specified in (2.1) and (2.2), and the matrices,,,,, and are appropriately dimensioned with partially unknown parameters. It is assumed that parameters of the matrices,,,,, and reside within the following uncertainty polytope: (5.3) where, are known vertices of the polytope and are unknown parameters. Remark 5.1: Polytopic representation of uncertainty has been adopted for 1-D systems, e.g., [4]. An advantage of this representation is that it can provide less conservative design than other types of uncertainty representation such as the norm-bounded uncertainty [29]. We aim at designing a fixed deconvolution filter of the form (2.3) and (2.4) for the uncertain 2-D model (5.1) and (5.2) such that (5.4) is satisfied under the zero boundary condition, where, and is a given positive scalar. This problem is referred to as the robust deconvolution filtering. It follows from the convexity of the uncertain system matrices that a deconvolution filter solves the robust deconvolution filtering problem if there exist, and a fixed (vertex independent) matrix such that (5.5), shown at the bottom of the page, is satisfied, where However, due to the requirement of a fixed Lyapunov matrix for all the vertices, the LMI solution based on (5.5) can be very conservative. To reduce the conservativeness of the solution, we follow [16] to modify (5.5) and present an improved LMI solution to the robust deconvolution filter. This result is given in the following theorem. Theorem 5.1: Consider the 2-D uncertain system (5.1) and (5.2). For a given positive scalar, the robust deconvolution filtering problem is solvable if there exist matrices,,,,,,,, and such that is satisfied for all the vertices,, where we have (5.8) (5.10), shown at the bottom of the next page, and denotes an entry that can be deduced from the symmetric property of the matrices and. Furthermore, if the above LMIs are solvable, then a robust deconvolution filter in the form of (2.3) and (2.4) can be obtained as (5.11) Proof: See the Appendix. Remark 5.2: Theorem 5.1 presents an LMI solution to the robust deconvolution filtering problem for 2-D digital systems with parameter uncertainty residing in a polytope. We observe again that (5.8) is linear in all unknown matrix variables. Therefore, the LMI solution can be readily solved by convex optimization and computed by Matlab LMI Toolbox [14]. The matrices of the 2-D robust deconvolution filter can be directly computed from the solution of the LMIs through (5.11). VI. EXAMPLE This section presents examples to demonstrate the LMI solutions to the 2-D deconvolution filtering and robust deconvolution filtering problems and performance of the designed 2-D deconvolution and robust filters. First, consider a 2-D channel that is modeled by a 2-D FIR lowpass filter in the transfer function form (5.6) (5.7) The magnitude response of the channel model is shown in Fig. 1. (5.5)

2324 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 9, SEPTEMBER 2002 Note that the model can be represented in the state equation form (2.1) and (2.2) with [13] Fig. 1. Magnitude response of the 2-D channel. Given an bound, it follows from Theorem 4.1 and by employing the Matlab LMI Toolbox [14] that a solution can be obtained, and an deconvolution filter in the form (2.3) and (2.4) can then (5.8) (5.9) (5.10)

XIE et al.: DECONVOLUTION FILTERING OF 2-D DIGITAL SYSTEMS 2325 be computed based on Remark 4.1. The filter parameters are given by We further consider a 2-D channel model with the following uncertainty polytope of two vertices: To evaluate the performance of the designed deconvolution filter under different levels of measurement noise, we introduce the reconstruction signal-to-noise ratio (SNR) as SNR and the measurement SNR as SNR where and are the horizontal and vertical horizons. The reconstruction SNR will be compared with the SNR of the system before deconvolution processing as defined by SNR With an bound of and by applying Theorem 5.1, we derive a 2-D robust deconvolution filter in the form (2.3) and (2.4) with where is the measured output. It is assumed that the input signal is the image as shown in Fig. 2 and that the measurement noise is white with zero mean. The designed deconvolution filter is applied to the transmitted image from output of the 2-D system. The reconstruction SNRs are shown in Table I for different levels of measurement noise. It is clear that the deconvolution filter indeed improves the SNR of the received image signal. For example, when the SNR of the measurement is 16.6 db, the images before and after the deconvolution processing are shown in Figs. 3 and 4, respectively, which clearly demonstrates that the deconvolution filter indeed enhances the image quality. Fig. 5 shows the image after robust deconvolution filtering when the SNR of the measurement is 16.6 db.

2326 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 9, SEPTEMBER 2002 Fig. 2. Input image. TABLE I COMPARISON BETWEEN SNRS OF THE SYSTEM WITH AND WITHOUT DECONVOLUTION PROCESSING Fig. 4. Image after deconvolution processing when the measurement SNR is 16.6 db. Fig. 5. Image after robust deconvolution filtering when the measurement SNR is 16.6 db. Fig. 3. Image before deconvolution processing when the measurement SNR is 16.6 db. Let the actual 2-D system be obtained from the above twovertex polytope with and and the measurement noise be white with zero mean. The comparison between the nominal deconvolution filter (without considering uncer- tainty) and the robust deconvolution filter under various levels of measurement noise is shown in Table II. For the same input image as in Fig. 2, the reconstructed output image by the robust deconvolution filter is shown in Fig. 4 when the SNR of the measurement is 16.6 db. Again, it reveals that the robust deconvolution filter performs very well under the channel modeling uncertainties and the noise. VII. CONCLUSION This paper has presented solutions to the deconvolution problems for 2-D digital filters described by the FM LSS model

XIE et al.: DECONVOLUTION FILTERING OF 2-D DIGITAL SYSTEMS 2327 TABLE II COMPARISON OF SNR BETWEEN ROBUST DECONVOLUTION FILTER AND NOMINAL DECONVOLUTION FILTER using the improved 2-D system bounded realness property. It is shown that these solutions can be recast into a convex optimization under the LMI constraints and that it is straightforward to compute the solution by using the recently developed LMI Toolbox. Thus, we have presented efficient design and computational procedures for the 2-D deconvolution filtering and robust deconvolution filtering problems for reconstruction of 2-D signals. Simulation results have demonstrated satisfactory performance of the designed 2-D deconvolution filters for image processing applications. Let and. By pre- and postmultiplying the matrix inequality (7.1) with diag,, we obtain APPENDIX Proof of Theorem 4.1: By applying Theorem 3.1 to the filtering error system (2.5) and (2.6), we have that the filter (2.3) and (2.4) solves the 2-D deconvolution problem for the system (2.1) and (2.2) if the LMI Now, partition and as where,. It is easy to know that. Let (7.2) (7.3) (7.4) (7.1) Pre and post-multiplying (7.2) by diag, and diag,, respectively, we have (7.5), shown at the bottom of the page. To linearize the matrix inequality (7.5), we introduce the following change of variables: (7.6) holds for some and. (7.7) (7.5)

2328 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 9, SEPTEMBER 2002 where, and and are nonsingular matrices satisfying. Using the above change of variables and taking into account,,,,, and in (2.7), (2.8), (7.3), and (7.4), it can be trivially verified that that we have (7.10), shown at the bottom of the page, where,,, Since.,,, and,wehave Then, (7.10) leads to (7.11), shown at the bottom of the next page. Pre- and post-multiplying (7.11) with diag, and diag,, respectively, yields (7.8) Thus, by denoting, (4.1) follows directly from (7.5). This completes the proof. Proof of Theorem 5.1: Following from the line of [16], we consider the modified LMI of (5.5) in (7.9), shown at the bottom of the page, where is any nonsingular matrix in. It is clear that (7.9) reduces to (5.5) if,. To show that any filter satisfying (7.9) solves the robust deconvolution filtering problem, we first note that. Thus, is nonsingular. By noting the convexity in (5.3) and taking into account (5.6) and (5.7), it follows from (7.9) (7.12) Hence, it follows from Theorem 3.1 that any filter satisfying the LMI (7.9) solves the deconvolution problem. (7.9) (7.10)

XIE et al.: DECONVOLUTION FILTERING OF 2-D DIGITAL SYSTEMS 2329 We still need to show that (7.9) is equivalent to this aim, we decompose and as and let. To (7.13) Pre- and post-multiplying by diag, and diag,, respectively, we obtain, where we have (7.20) (7.22), shown at the top of the page after the next page. Thus, we can obtain (5.8) from (7.20) by letting and diag Pre- and post-multiplying (7.9) by and, respectively, and incorporating with (5.6) and (5.7), it can be shown that (7.9) is equivalent to (7.14), shown at the bottom of the page, where we have (7.15) and (7.16), shown at the top of the next page, and Finally, we show that if (5.8) admits a solution, then is nonsingular. Let, and multiply (5.8) from the left and right by diag,,, and its transpose, respectively. It can be shown via the (5,5)th block of the resulting LMI that (7.23) Let and (7.17) This implies that is nonsingular. Since,, and are both nonsingular, it follows that the transfer function matrix from to is (7.18) (7.19) (7.11) (7.14)

2330 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 9, SEPTEMBER 2002 (7.15) (7.16) where,,, and are given in (5.11). REFERENCES [1] B. S. Chen and S. C. Peng, Optimal deconvolution filter design based on orthogonal principle, Signal Process., vol. 25, pp. 361 372, 1991. [2], A deconvolution filter for multichannel nonminimum phase systems via mini max approach, Signal Process., vol. 36, no. 1, pp. 71 91, 1994. [3] L. Chucu and E. Mosca, MMSE deconvolution via polynomial methods and its dual LQG regulation, Automatica, vol. 30, pp. 1197 1201, 1994. [4] D. J. N. Limebeer, U. Shaked, and I. Yaesh, Parametric BRL s for discrete-time linear systems with polytopic uncertainties, Imperial College, London, U.K., Tech. Rep. EEE-CP-99/2, 1999. [5] J. M. Mendel, Optimal Seismic Deconvolution: An Estimation-Based Approach. New York: Academic, 1983. [6] M. T. Silvia and E. A. Robinson, Deconvolution of Geophysical Time Series in the Exploration for Oil and Natural Gas. New York: Elsevier, 1979. [7] J. M. Mendel and C. Y. Chi, Performance of minimal-variance deconvolution filter, IEEE Trans. Acoust., Speech, Signal Processing, vol. 35, pp. 217 226, Feb. 1989. [8] Y. L. Chen and B. S. Chen, Minimax, robust deconvolution filters under stochastic parametric and noise uncertainties, IEEE Trans. Acoust., Speech, Signal Processing, vol. 42, pp. 32 45, Jan. 1994. [9] S. Wang, L. Xie, and C. Zhang, H optimal inverse of periodic FIR digital filters, IEEE Trans. Signal Processing, vol. 48, pp. 2696 2700, Sept. 2000. [10] K. R. Castleman, Digital Image Processing. Englewood Cliffs, NJ: Prentice-Hall, 1996. [11] C. Du, L. Xie, and Y.C Soh, H filtering of 2-D discrete systems, IEEE Trans. Signal Processing, vol. 48, pp. 1760 1768, June 2000. [12] E. Fornasini and G. Marchesini, Doubly indexed dynamical systems: State-space models and structural properties, Math. Syst. Theory, vol. 12, pp. 59 72, 1978. [13] E. Fornasini and G. Marchesini, Finite memory realization of 2D FIR filters, in Proc. IEEE Int. Symp. Circuits Syst., vol. 3, 1992, pp. 1444 1447. [14] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox For Use With Matlab. Natick, MA: The MATH Works Inc., 1995. [15] P. Gahinet and P. Apkarian, A linear matrix inequality approach to H control, Int. J. Robust Nonlinear Contr., vol. 4, pp. 421 448, 1994. [16] J. C. Geromel, M. C. de Oliveira, and J. Bernussou, Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions, in Proc. 38th IEEE Conf. Dec. Contr., Phoenix, AZ, Dec. 1999. [17] J. L. Harris, Sr., Image evaluation and restoration, J. Opt. Soc. Amer., vol. 56, pp. 569 574, May 1966. [18] T. Hinamoto, 2-D Lyapunov equation and filter design based on the Fornasini Marchesini second model, IEEE Trans. Circuits Syst. I, vol. 40, pp. 102 109, Feb. 1993. [19] T. Kaczorek, Two-Dimensional Linear Systems. Berlin, Germany: Springer-Verlag, 1985. [20] W. S. Lu and A. Antoniou, Two-Dimensional Digital Filters. New York: Marcel Dekker, 1992. [21] W. S. Lu, On a Lyapunov approach to stability analysis of 2-D digital filters, IEEE Trans. Circuits Syst. I, vol. 41, pp. 665 669, Oct. 1994. [22] C. A. Lin and C. W. King, Inverting periodic filters, IEEE Trans. Signal Processing, vol. 42, pp. 196 200, Jan. 1994. [23] R. Nathan, Digital video handling, Jet Propulsion Lab., Pasadena, CA, Tech. Rep. 32-877, January 5, 1966. [24] G. M. Robbins and T. S. Huang, Inverse filtering for linear shift-variant imaging systems, Proc. IEEE, vol. 60, pp. 862 872, July 1972. [25] S. Rajan, K. S. Joo, and T. Bose, Analysis of 2-D state-space periodically shift-variant discrete systems, Circuits Syst. Signal Process., vol. 15, no. 3, pp. 395 413, 1996. [26] C. E. de Souza, On stability properties of solutions of the Riccati difference equation, IEEE Trans. Automat. Contr., vol. 34, pp. 1313 1316, Dec. 1989. [27] T. R. Stockham, T. M. Cannon, and R. B. Ingebretsen, Blind deconvolution by digital signal processing, Proc. IEEE, vol. 63, pp. 679 692, Apr. 1975.

XIE et al.: DECONVOLUTION FILTERING OF 2-D DIGITAL SYSTEMS 2331 (7.20) (7.21) (7.22) [28] L. Xie, S. Wang, C. Du, and C. Zhang, H deconvolution of periodic channels, Signal Process., vol. 80, pp. 2365 2378, 2000. [29] L. Xie and Y. C. Soh, Guaranteed cost control of uncertain discrete-time systems, Contr. Theory Advanced Technol., vol. 10, pp. 1235 125, 1995. Lihua Xie (SM 97) received the B.E. and M.E. degrees in electrical engineering from Nanjing University of Science and Technology, Nanjing, China, in 1983 and 1986, respectively, and the Ph.D. degree in electrical engineering from the University of Newcastle, Callaghan, Australia, in 1992. From April 1986 to January 1989, he was a Teaching Assistant and then a Lecturer with the Department of Automatic Control, Nanjing University of Science and Technology. He joined the Nanyang Technological University, Singapore, in 1992, where he is currently an Associate Professor with the School of Electrical and Electronic Engineering. From November 1998 to July 1999, he was a Visiting Fellow with the Department of Electrical and Electronic Engineering, the University of Melbourne, Parkville, Australia. He has been a Guest Professor with Xiamen University and Dalian University of Science and Technology of China. He has authored and coauthored one book and over 90 refereed journal articles. His current research interests include optimal and robust estimation, signal processing, two-dimensional systems, robust control, and active noise control. Dr. Xie is an Associate Editor of the Conference Editorial Board of the IEEE Control Systems Society. Chunling Du was born in China in 1970. She received the B.E. and M.E. degrees in electrical and electronic engineering from the Nanjing University of Science and Technology, Nanjing, China, in 1992, and 1995, respectively, and the Ph.D. degree from the Nanyang Technological University, Singapore, in 2000. She is now with Data Storage Institute, Singapore, where she is involved in advanced disk drive servo control systems. Her research interests include optimal and robust control, image and signal processing, model reduction, multidimensional systems, and periodical systems. Cishen Zhang received the B.Eng. degree from Tsinghua University, Beijing, China, in 1982 and the Ph.D. degree in electrical engineering from Newcastle University, Callaghan, Australia, in 1990. Between 1971 and 1978, he was an Electrician with Changxindian (February Seven) Locomotive Manufactory, Beijing. He carried out research work on control systems at Delft University of Technology, Eindhoven, The Netherlands, from 1983 to 1985. Since 1989, he has been with the Department of Electrical and Electronic Engineering, the University of Melbourne, Parkville, Australia and is currently an Associate Professor and Reader. In 2002, he is visiting the School Electrical and Electronic Engineering, Nanyang Technological University, Singapore for one year. His research interests include time-varying systems and adaptive systems for signal processing and control.

2332 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 9, SEPTEMBER 2002 Yeng Chai Soh received the B.Eng. (Hons. I) degree in electrical and electronic engineering from the University of Canterbury, Christchurch, New Zealand, in 1983 and the Ph.D. degree in electrical engineering from the University of Newcastle, Callaghan, Australia, in 1987. From 1986 to 1987, he was a Research Assistant with the Department of Electrical and Computer Engineering, University of Newcastle. He joined the Nanyang Technological University, Singapore, in 1987, where he is currently a Professor and the Head of the Control and Instrumentation Division, School of Electrical and Electronic Engineering. His current research interests are in the areas of robust system theory and applications, estimation and filtering, model reduction, hybrid systems, and secure communications.