IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 16, NO. 3, MARCH 2006
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Comparison of Wavelets for Multiresolution Motion Estimation Jinwen Zan, Member, IEEE, M. Omair Ahmad, Fellow, IEEE, and M. N. S. Swamy, Fellow, IEEE
Abstract—The performance of various wavelets, including those known to be well suited for the coding of still images, has been evaluated for the multiresolution motion estimation of video sequences. The multiresolution motion estimation scheme proposed by Zhang and Zafar, which has been widely cited in the literature, is used as the simulation scheme in this study. In our study, the prediction mean square error in the wavelet transform coefficient domain is used as the measure for prediction performance. In order to show the overall rate distortion performance, the number of bits needed to encode the motion vectors is also calculated. Simulation results show that the 7/9 biorthogonal wavelet, one of the best wavelets for the coding of still images, is the best wavelet for the task of multiresolution motion estimation among the wavelets evaluated in this study. Index Terms—Motion estimation, multiresolution analysis, multiresolution motion estimation, video coding, wavelets.
I. INTRODUCTION
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AVELET theory [1], [2] provides a unified framework for a number of techniques that have been developed independently for various applications such as multiresolution signal processing for computer vision, subband coding for speech and image compression, and wavelet series expansions in applied mathematics. As an alternative to the classical short-time Fourier transform (STFT) [3], [4] or Gabor transform (GT) [5], the wavelet transform (WT) provides an elegant solution for the analysis of nonstationary signals. In contrast to the STFT or GT, both of which use only a single analysis window, the WT uses short windows at high frequencies and long windows at low frequencies. This is in the spirit of so-called constant relative bandwidth frequency or “constant-Q” frequency analysis [2]. This tradeoff in the time/space frequency resolution is very useful for the analysis of image/video signals, which are typically nonstationary. In image compression, the conventional transform coding techniques, such as those using the discrete cosine transform (DCT), decompose images into a representation in which each coefficient corresponds to a fixed-size spatial area and a fixed frequency bandwidth, where the bandwidth and spatial area are essentially the same for all of the coefficients in the representation. As a consequence, an image coder that fully exploits the multiresolution representation, such as the embedded zerotree wavelet (EZW) coder [6]
Manuscript received April 8, 2004; revised March 5, 2005. This work was supported by the Natural Sciences and Engineering Research Council of Canada, the Micronet National Network of Centers of Excellence, and Fonds pour la Formation de Chercheurs et l’Aide à la Recherche (FCAR) of Québec, Canada. This paper was recommended by Associate Editor Dr. L. Onural. The authors are with the Center for Signal Processing and Communications, Department of Electrical and Computer Engineering, Concordia University, Montréal, QC H3G 1M8, Canada (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCSVT.2005.857302
or the set partitioning in hierarchical trees (SPIHT) coder [7], is able to achieve a better performance than that of a DCT-based coder. Although studies have been carried out on the comparative performance of wavelets for the coding of still images [8], [9], the authors are not aware of any such study that evaluates the performance of wavelets for the coding of video signals. However, some work has been done on the multiresolution motion estimation (MRME) of video sequences using wavelets. For example, the Daubechies orthogonal wavelet [1] has been used in [10] and [11] even though it does not possess a linear phase, and the 7/9 biorthogonal wavelets [12] has been used in [13] without any explanation as to why it has been chosen. Since one of the objectives of this study is to apply motion estimation on the multiresolution representation of video signals using wavelets, we make a comparative study of wavelets for the MRME of video signals [14]. This paper is organized as follows. In Section II, we describe the wavelets used in our study and the reasons for their choice. In Section III, we first present a testbench [10] for the study of wavelets in the MRME of video signals and then carry out a simulation study using this testbench. In Section IV, we summarize the contributions of this study. II. WAVELETS FOR MULTIRESOLUTION MOTION ESTIMATION In order to choose the wavelets from the point of view of comparing their performance for MRME of video sequences, there are two characteristics of the wavelets that should be taken into consideration. First, the wavelet should provide a good performance for the coding of still images, since the reference frame for MRME is encoded in the intra mode. Second, the wavelet should have a reasonably small number of coefficients so that the computational load required for the wavelet decomposition of a video frame is not excessively large. It has been shown that there are six biorthogonal wavelets of minimum order satisfying the first property [8], and these are listed in [8, Table I]. They are the 7/9, 3/5, 10/6, 2/6, 11/13, and 3/9 wavelets. Out of these six, the following four wavelets are chosen in our study concerning their performance for MRME of video sequences. The reasons for their choice are also given. Wavelet #1: The 7/9 wavelet has the best overall performance for image coding and has been very popular in image processing, due to its property of high-energy compaction and small shift-variance. For example, in the SPIHT image coder [7], the 7/9 wavelet is considered to be one of the reasons for its success over the EZW image coder [6], where a length-9 quadrature mirror filter (QMF) has been used. In a recent paper on the wavelet-based MRME technique [13], the 7/9 wavelet has been shown to have the capacity to provide a better motion
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TABLE I CANDIDATE BIORTHOGONAL WAVELETS
estimation performance than that of the conventional block matching algorithm (BMA). Wavelet #2: The 3/5 wavelet also has an odd number of coefficients for both the analysis and synthesis filters and has the shortest length. Wavelet #3: The 10/6 wavelet [8] provides a performance comparable to that of wavelet #1 for the coding of still images [8]. It is observed that this wavelet has a total number of 16 coefficients for computing, six from the analysis filter and ten from the synthesis filter. Thus, the computational load of this wavelet is the same as that of the 7/9 wavelet, which has nine analysis and seven synthesis filter coefficients. Another feature of wavelet #3 is that it has an even number of coefficients for both the analysis and the synthesis filters and is favored by the authors of [8] for having a significantly smaller shift variance than that of the odd-length wavelet. Wavelet #4: The 2/6 wavelet also has an even number of coefficients for both the analysis and synthesis filters and has the shortest length. The 11/13 wavelet is not included in our study, since its performance is not comparable to that of the 7/9 and its length is larger than that of the latter. Similarly, the reason for not including the 3/9 wavelet is that its length is not the smallest, in addition to its performance being not as good as that of the 7/9 wavelet. In addition to the above four wavelets, we choose the following two additional wavelets. Wavelet #5: This is an 8/16 wavelet proposed in [15] that outperforms the 7/9 wavelet for coding the image Barbara, which is considered to be one of the most challenging images to be compressed because of its containing a great deal of detailed information, while still providing a comparable coding performance for other test images. Note that it is also a wavelet with an even number of coefficients. Wavelet #6: This is the Haar wavelet [16], which is the only orthogonal wavelet with linear phase. Because it has only four coefficients (two for the analysis filter and two for the synthesis filter) for computing, it is a wavelet that renders the least computational load. The above six wavelets along with their filter coefficients are listed in Table I.
III. SIMULATION RESULTS AND ANALYSIS In video coding, interframe prediction exploiting the BMA in the spatial domain has been widely used to remove the temporal redundancy in a video signal. The mean absolute difference (MAD) is the most widely used criterion for block matching because of its low computational complexity and good matching , the MAD performance. Assuming a motion block of size is defined as MAD
(1)
and denote the relative displacement in the and where directions, and and represent the of the current frame and location pixel values at location of the reference frame, respectively. Compared to the conventional BMA, the advantages of the wavelet-based MRME technique include low computational complexity, ease of generating an embedded video bitstream, and the possibility of taking advantage of the human visual system (HVS) for quantization. Although wavelets have been evaluated for the coding of still images, as discussed in the previous section, the authors are not aware of any such study on the evaluation of wavelets for MRME of video sequences. Since interframe coding mode dominates most of the video compression schemes and the criterion of evaluating the motion estimation performance is different from that of still image coding, the choice of a wavelet for the wavelet-based MRME technique is of crucial importance. In this section, we will first give a brief review of a widelyreferred-to MRME scheme [10], which is the testbench for our study, and then we will present and analyze the outcome of the simulation study. A. Simulation Scheme In an MRME scheme, motion vectors (MVs) are first estimated at the lowest resolution, that is, in the subband(s) at the top of the pyramid, since it is at this resolution that most of the image energy is preserved [10]. The motion information from the top pyramid subband(s) is then manipulated as the prediction at finer resolutions. In the variable block size MRME scheme of
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TABLE II COMPARISON OF VARIOUS WAVELETS
Fig. 1. Motion relationship in a three-level variable-block-size MRME scheme.
[10], based on the pyramidal data structure resulting from the is wavelet analysis, a block size of is the analysis level. used for the th-level subbands, where Thus, the lower the resolution level, the smaller the block size. The constant is the size of the block used at the lowest resolution. With this structure, the number of motion blocks for all of the subbands is the same and the blocks at the same position across all of the resolution levels corresponds to the same global position and the same object. Since its introduction by Zhang and Zafar [10], this variable-block-size MRME technique has been used extensively as a baseline MRME scheme in other studies [17]–[19], because of its simple architecture and good performance. In the present study, this MRME scheme is exploited as the testbench for comparing the performance of different wavelets. A total of ten subbands are obtained with a three-level wavelet analysis, and the motion relationships among the resolution levels are as depicted in Fig. 1. In order to obtain the prediction MV in a bandpass subband at a given resolution level, the corresponding bandpass MV at the lowest resolution level is multiplied by an appropriate power of two. A refinement factor is then added to this prediction after carrying out a motion search around this motion prediction. The MV prediction equation can be written as
for
and
(2)
is the MV at the resolution level with orientation where and is a refinement factor to be determined by a motion search process carried out around the predicted value to estimate the MV .
Fig. 2. Performance on the Flower Garden sequence. (a) PMSE. (b) Bits for coding motion vectors.
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Fig. 3. Performance on the Football sequence. (a) PMSE. (b) Bits for coding motion vectors.
B. Simulation Results and Analysis The performance of the wavelets for MRME is evaluated by carrying out simulations on four test video sequences: 100 frames of each of the Football, Flower Garden, and Mobile & Calendar sequences and 60 frames of the Tennis sequence. In all of the simulations, the motion block size parameter is set . Forward motion estimation is carried out by estias mating the current frame using the preceding one as the reference frame. The search window size at all resolution levels is set as 2. The motion search process can be described as follows. A full search with integer pixel precision is conducted within the search window, and the displacement having the lowest MAD is chosen. This is followed by another full search with onehalf pixel accuracy around the chosen displacement. The pixel values at the half-pixel positions are derived by an arithmetic interpolation of the nearest integer pixels. For example, for such a pixel with two horizontal integer pixels on the left and right, its value is the average of these two pixels; for such a pixel
Fig. 4. Performance on the Mobile & Calendar sequence. (a) PMSE. (b) Bits for coding motion vectors.
with two vertical integer pixels on the top and bottom, its value is the average of these two pixels; for such a pixel with only half-pixel neighbors in the top, bottom, left, and right directions, its value is the average of the four integer pixels from the diagonal directions. After the motion estimation, the MVs at the lowest resolution level are encoded independently, while those at all higher levels are DPCM encoded with respect to their predictions. That is, for all higher levels, only the values in (2) are coded. The entropy coder for the MVs of is an arithmetic coder [20]. It is noted that, although the MVs in the bandpass subbands of the lowest resolution level might be DPCM encoded with respect to the MVs of the LL subband, our simulation study shows that both kinds of coding schemes provide similar performance. In order to provide a more robust performance under an error-prone environment, the MVs in the bandpass subbands at the lowest resolution level are encoded without reference to the MVs of the LL subband. In all of the MRME video coding schemes that have already been proposed
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Fig. 6. Frame 52 of the Tennis sequence. (a) The original and the motion-compensation results using (b) the 7/9 wavelet, (c) the 3/5 wavelet, (d) the 10/6 wavelet, (e) the 6/2 wavelet, (f) the 8/16 wavelet, and (g) the Haar wavelet. Fig. 5. Performance on the Tennis sequence. (a) PMSE. (b) Bits for coding motion vectors.
[10], [18], [19], [21], the motion-compensated wavelet coefficients in the transform domain are quantized and entropy-coded. From the coding point of view, the smaller the energy of the motion-compensated residual, the larger the coding gain. As such, the average prediction mean square error (PMSE) in the WT domain and the average number of bits needed for coding the MVs per frame are used as the performance indices. Once the motion information has been recovered at the decoder side, a motion-compensated video frame can be constructed from the motion information. The MVs [as calculated from (1)] from the lowest resolution level are used directly as displacement to get the wavelet coefficients in the corresponding subbands in the reference frame. For the subbands at finer resolution levels, the MVs are derived by using the corresponding prediction MVs from the lowest resolution level and add the refinement MVs as shown in (2). Once the MVs are obtained, they are used as displacement to get the predicted wavelet coefficients in the corresponding subband in the reference frame. All of the wavelet coefficients thus predicted form
the compensated image in the wavelet transform domain. The PMSE in the wavelet transform domain may be defined as PMSE
(3)
and represent the original WT coefficients where and the motion-compensated WT coefficients, respectively. In our simulation study, the PMSE is calculated in the WT domain between the original video frame and the one generated by multiresolution motion compensation. Table II shows the recorded PMSE and number of coding bits for MVs using the six wavelets on all of the test sequences. It is seen from Table II that, for the Flower Garden sequence that involves the pan camera motion, and the Mobile & Calendar sequence that has only small motions, the 7/9 wavelet provides the smallest PMSE while requiring the least number of coding bits for the MVs at the same time, among the six wavelets in Table I. For the Football sequence that has a lot of fast motion activities, the 7/9 wavelet provides the smallest PMSE, with the
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Fig. 7. Difference images between the original and motion-compensated images for frame 52 of the Tennis sequence: (a) the 7/9 wavelet, (b) the 3/5 wavelet, (c) the 10/6 wavelet, (d) the 6/2 wavelet, (e) the 8/16 wavelet, and (f) the Haar wavelet.
number of coding bits being second best only to that of the Haar wavelet, whose PMSE, however, is much inferior to that of the 7/9 wavelet. For the Tennis sequence that involves a camera zooming motion, the 7/9 wavelet provides a PMSE comparable to that of the 6/2 wavelet which has the smallest PMSE, while at the same time requiring coding bits comparable to that of the Haar wavelet that requires the least number. It is also noted that, for the specific motion activities in the Tennis sequence, all of the wavelets provide good motion estimation results, and the performance difference among the wavelets is very limited. As compared to the amount of PMSE performance of the 7/9 wavelet over any other wavelet in Table I, which could be up to 69%, the amount of coding bits performance of the 7/9 over any other wavelet is limited. Except for the Mobile & Calendar sequence, the amount of performance over any other wavelet is below 5%. This phenomenon could be explained by the fact that the differential motion vectors in the bandpass subbands mostly follow an even distribution, no matter which wavelet has been used for the MRME. In summary, taking into consideration both the PMSE and the number of coding bits required for the MVs, the 7/9 wavelet provides the best overall performance. Figs. 2–5 illustrate the PMSE and the number of coding bits for the MVs as a function of the frame number for all of the four sequences. Fig. 6 shows the motion-compensated images of frame 52 of the Tennis sequence using all of the wavelets listed in Table I, and Fig. 7 shows the corresponding difference images between the original and the motion-compensated ones. The better quality of the motion-compensated image provided by the 7/9
Fig. 8. Frame 2 of the Football sequence. (a) The original and the motion compensation results using (b) the 7/9 wavelet, (c) the 3/5 wavelet, (d) the 10/6 wavelet, (e) the 6/2 wavelet, (f) the 8/16 wavelet, and (g) the Haar wavelet.
wavelet, as compared to any other wavelet shown in Table I, can be observed at the edges of the table and the ball. Figs. 8 and 9 show the corresponding results for frame 2 of the Football sequence. The better quality of the motion-compensated image provided by the 7/9 wavelet can be observed around the contours of the players. It is clear from these images that the 7/9 biorthogonal wavelet yields better subjective quality images than any of the other wavelets. From all of the simulation results shown above, it may be seen that, for a wavelet with an even number of coefficients, the performance is not as good as that with an odd number of coefficients. In this context, it may be noted that Villasenor et al. [8] have shown that the even-length biorthogonal wavelets have a significantly lower shift variance than the ones with oddlength coefficients and that they are very suitable for the coding of still images. This property of the even-length biorthogonal wavelets may lead one to infer that they may also provide a better MRME performance than the odd-length biorthogonal wavelets. However, as seen through our simulation study, this is not true. The 7/9 biorthogonal wavelet, which is an odd-length biorthogonal wavelet, provides a better MRME performance
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lower shift variance than the ones with odd-length coefficients, do not provide a better performance on the multiresolution motion estimation of a video sequence. REFERENCES
Fig. 9. Difference images between the original and motion-compensated images for frame 2 of the Football sequence: (a) the 7/9 wavelet, (b) the 3/5 wavelet, (c) the 10/6 wavelet, (d) the 6/2 wavelet, (e) the 8/16 wavelet, and (f) the Haar wavelet.
than any of the three even-length biorthogonal wavelets investigated in this study, although these even-length biorthogonal wavelets are suitable for the coding of still images. Since the arithmetic coder used in the simulation studies adapts to the entropy of symbols very well, we may use the number of coding bits for the MVs as an indication of the smoothness of the motion field. Therefore, we may conclude that the 7/9 biorthogonal wavelet not only provides a very good PMSE performance, but also results in a smooth motion field. IV. CONCLUSION In this paper, a comparative study on the performance of wavelets for the multiresolution motion estimation of video sequences has been carried out. A widely referred to wavelet-based multiresolution motion estimation scheme has been employed as the testbench. The evaluation criteria used are the PMSE in the wavelet transform domain and the number of coding bits for the motion vectors, which are two important performance indices in a video coding application. A large number of wavelets used in signal processing applications have been investigated, and, among these, six biorthogonal wavelets with good performance have been considered in the simulation study with their performance analyzed. It has been found that the 7/9 biorthogonal wavelet, which is one of the best wavelets for the coding of still images, turns out to be the best wavelet for performing the multiresolution motion estimation among the wavelets evaluated in this study. The even-length biorthogonal wavelets investigated in this study, which have a significantly
[1] I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Comm. Pure Appl. Math., pp. 909–996, 1988. [2] O. Rioul and M. Vetterli, “Wavelet and signal processing,” IEEE Signal Process. Mag., vol. 8, no. 4, pp. 14–38, Oct. 1991. [3] J. B. Allen and L. R. Rabiner, “A unified approach to short-time Fourier analysis and synthesis,” Proc. IEEE, vol. 65, no. 11, pp. 1558–1564, Nov. 1977. [4] M. R. Portnoff, “Time-frequency representation of digital signals and systems based on short-time Fourier analysis,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-28, no. 2, pp. 55–69, Feb. 1980. [5] D. Gabor, “Theory of communication,” J. Inst. Elect. Eng., vol. 93, pp. 429–457, 1946. [6] J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3445–3462, Dec. 1993. [7] A. Said and W. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circuits Syst. Video Technol., vol. 6, no. 3, pp. 243–250, Jun. 1996. [8] J. Villasenor, B. Belzer, and J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process., vol. 4, no. 8, pp. 1053–1060, Aug. 1995. [9] Z. Xiong, K. Ramchandran, and Y.-Q. Zhang, “A comparative study of DCT- and wavelet-based image coding,” IEEE Trans. Circuits Syst. Video Technol., vol. 9, no. 4, pp. 692–695, Jul./Aug. 1999. [10] Y.-Q. Zhang and S. Zafar, “Motion-compensated wavelet transform coding for color video compression,” IEEE Trans. Circuits Syst. Video Technol., vol. 2, no. 3, pp. 285–296, Sep. 1992. [11] S. Zafar, Y.-Q. Zhang, and B. J. Jabbari, “Multiscale video representation using multiresolution motion compensation and wavelet decomposition,” IEEE J. Sel. Areas Commun., vol. 1, no. 1, pp. 24–35, Jan. 1993. [12] M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using the wavelet transform,” IEEE Trans. Image Process., vol. 1, no. 2, pp. 205–220, Apr. 1992. [13] H.-W. Park and H.-S. Kim, “Motion estimation using low-band-shift method for wavelet-based moving-picture coding,” IEEE Trans. Image Process., vol. 9, no. 4, pp. 577–587, Apr. 2000. [14] J. Zan, M. O. Ahmad, and M. N. S. Swamy, “Wavelet filters in multiresolution motion estimation,” in Proc. Canadian Conf. Electrical and Computer Engineering, vol. 2, May 2001, pp. 1321–1326. [15] Y. Zhao and M. N. S. Swamy, “Techniques for designing biorthogonal wavelet filters with an application to image compression,” Electron. Lett., vol. 35, pp. 1530–1532, 1999. [16] A. Haar, “Zur theorie der orthogonalen Funktionen systeme,” Math. Ann., vol. 69, pp. 331–371, 1910. [17] J. Lee and B. W. Dickinson, “Subband video coding with scene-adaptive hierarchical motion estimation,” IEEE Trans. Circuits Syst. Video Technol., vol. 9, no. 3, pp. 459–466, Apr. 1999. [18] T. Naveen and J. W. Woods, “Motion compensated multiresolution transmission of high definition video,” IEEE Trans. Circuits Syst. Video Technol., vol. 4, no. 1, pp. 29–40, Feb. 1994. [19] K. M. Nam, J.-S. Kim, R.-H. Park, and Y. S. Shim, “A fast hierarchical motion vector estimation algorithm using mean pyramid,” IEEE Trans. Circuits Syst. Video Technol., vol. 5, no. 4, pp. 344–351, Aug. 1995. [20] I. H. Witten, R. M. Neal, and J. G. Cleary, “Arithmetic coding for data compression,” Commun. ACM, vol. 30, pp. 520–540, Jun. 1987. [21] S. Martucci, I. Sodagar, T. Chiang, and Y.-Q. Zhang, “A zerotree wavelet video coder,” IEEE Trans. Circuits Syst. Video Technol., vol. 7, no. 1, pp. 109–118, Feb. 1997.
Jinwen Zan (S’00–M’01) received the B.Eng. degree (hons.) from the Harbin Institute of Technology, Harbin, China, the M.Eng. degree from Nanjing University of Posts and Telecommunications, Nanjing, China, and the Ph.D. degree from Concordia University, Montréal, QC, Canada, all in electrical engineering. He is now a consultant to the industry. His research interests include digital image/video processor architecture design, digital image/video processing, and communications.
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M. Omair Ahmad (S’69–M78–SM’83–F’01) received the B.Eng. degree from Sir George Williams University, Montreal, QC, Canada, and the Ph.D. degree from Concordia University, Montreal, QC, Canada, both in electrical engineering. From 1978 to 1979, he was a member of the Faculty of the State University of New York at Buffalo. In September 1979, he joined the Faculty of Concordia University, where he was an Assistant Professor of computer science. Subsequently, he joined the Department of Electrical and Computer Engineering of the same university, where presently he is a Professor and Chair of the department. He has published extensively in the area of signal processing and holds three patents. His current research interests include the areas of multidimensional filter design, image and video signal processing, nonlinear signal processing, communication DSP, artificial neural networks, and VLSI circuits for signal processing. He is a researcher in the Micronet National Network of Centers of Excellence and was previously an Examiner of the Order of Engineers of Quebec. Dr. Ahmad was an Associate Editor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I—FUNDAMENTAL THEORY AND APPLICATIONS from June 1999 to December 2001. He was the Local Arrangements Chairman of the 1984 IEEE International Symposium on Circuits and Systems. During 1988, he was a member of the Admission and Advancement Committee of the IEEE. Presently, he is the Chairman of the IEEE Circuits and Systems Chapter (Montreal Section). He was a recipient of the Wighton Fellowship from the Sandford Fleming Foundation.
M. N. S. Swamy (S’59–M’62–SM’74–F’80) received the B.Sc. (Hons.) degree in mathematics from Mysore University, India, in 1954, the Diploma in electrical communication engineering from the Indian Institute of Science, Bangalore, India, in 1957, the M.Sc. and Ph.D. degrees in electrical engineering from the University of Saskatchewan, SK, Canada, in 1960 and 1963, respectively, and the Doctor of Science degree in Engineering (Honoris Causa) from Ansted University, Penang, Malaysia, in 2001. He is presently a Research Professor and the Director of the Center for Signal Processing and Communications, Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada, where he served as the Chair of the Department of Electrical Engineering from 1970 to 1977 and Dean of Engineering and Computer Science from 1977 to 1993. Since July 2001, he holds the Concordia Chair (Tier I) in Signal Processing. He has also taught in the Electrical Engineering Department, Technical University of Nova Scotia, Halifax, NS, Canada, and the University of Calgary, Calgary, AB, Canada, as well as in the Department of Mathematics, University of Saskatchewan. He has published extensively in the areas of number theory, circuits, systems, and signal processing and holds four patents. He is the coauthor of two book chapters and three books: Graphs, Networks and Algorithms (Wiley, 1981), Graphs: Theory and Algorithms Wiley, 1992), and Switched Capacitor Filters: Theory, Analysis and Design (Prentice-Hall, 1995). A Russian Translation of the first book was published by Mir Publishers, Moscow, in 1984, while a Chinese version was published by the Education Press, Beijing, in 1987. Dr. Swamy is a Fellow of the Institute of Electrical Engineers (U.K.), the Engineering Institute of Canada, the Institution of Engineers (India), and the Institution of Electronic and Telecommunication Engineers (India). He is a member of Micronet, a National Network of Centers of Excellence in Canada, and its coordinator for Concordia University. He has served the IEEE CAS Society in various capacities such as the President-Elect in 2003, President during 2004, Past President during 2005, Vice President (Publications) during 2001–2002, and as Vice-President in 1976, Editor-in-Chief of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS during 1999–2001, Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS during 1985–1987, Program Chair for the 1973 IEEE CAS Symposium, General Chair for the 1984 IEEE CAS Symposium, Vice-Chair for the 1999 IEEE CAS Symposium and a member of the Board of Governors of the IEEE CAS Society. He was the recipient of many IEEE-CAS Society awards including the Education Award in 2000, Golden Jubilee Medal in 2000, and the 1986 Guillemin–Cauer Best Paper Award.
