Efficient Computation of Zernike and Pseudo-Zernike Moments for Pattern Classification Applications 1

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1 REPRESENTATION, PROCESSING, ANALYSIS, AND UNDERSTANDING OF IMAGES Efficient Computation of Zernike and Pseudo-Zernike Moments for Pattern Classification Applications 1 G. A. Papakostas a, Y. S. Boutalis a, D. A. Karras b, and B. G. Mertzios a a Democritus University of Thrace, Department of Electrical and Computer Engineering, Xanthi, Hellas b Chalkis Institute of Technology, Automation Department Chalkida, Hellas (gpapakos, ybout, mertzios}@ee.duth.gr, dakarras@teihal.gr Abstract Two novel algorithms for the fast computation of the Zernike and Pseudo-Zernike moments are presented in this paper. The proposed algorithms are very useful, particularly in the case of using the computed moments, as discriminative features in pattern classification applications, where the computation of single moments of several orders is reuired. The derivation of the algorithms is based on the elimination of the factorial computations, by computing recursively the fractional terms of the orthogonal polynomials being used. The newly introduced algorithms are compared to the direct s, which are the only s that permit the computation of single moments of any order. The computational complexity of the proposed is O(p ) in multiplications, with p being the moment order, while the corresponding complexity of the direct is O(p 3 ). Appropriate experiments justify the superiority of the proposed recursive algorithms over the direct ones, establishing them as alternative to the original algorithms, for the fast computation of the Zernike and Pseudo-Zernike moments. Keywords: Orthogonal moments, Zernike moments, Pseudo-Zernike moments, direct, recursive algorithm. DOI: /S INTRODUCTION Image moments have been widely used as image descriptors in image processing and pattern classification applications. Chronologically, the firstly introduced moment type in image processing was the geometric moments [1]. Afterwards, since the geometric moments are not invariant to any image transformations (translation, rotation, scaling) the central and normalized moments [1 4] were introduced. However, since these types of moments are projections to a monomial, their bases are not orthogonal and thus there is a lot of information redundant and the reconstruction of the image by its moments is not possible. This deficiency of the conventional moments is not present in the orthogonal moments [1], which are projections of the image density function to orthogonal bases. The most significant property of the orthogonal moments is their ability to fully describe an object, with minimum redundant information and thus the reconstruction of an object by a finite number of moments, is possible. The orthogonal moments are categorized in families according to the type of the orthogonal polynomials that are making use, as kernel functions. The most utilized orthogonal families are the Zernike and 1 The article is published in the original. Received November 14, 007 Pseudo-Zernike moments [1], which are widely used as image descriptors in image processing tasks. Moreover, their property to remain invariant to any rotation of the object presented in a scene, in addition to their ability to describe spatial freuency components of an image, make them appropriate for pattern classification applications [5 8]. Although the previous orthogonal moments have useful attributes over the other moment types, the presence of many factorial terms in their polynomial definitions, make their computation a very time consuming task. While the computational capabilities of the modern computers are always increasing, the factorial of a big number remains a very demanding process. For this reason, many researchers have introduced recursive algorithms for the computation of Zernike [9 13] and Pseudo-Zernike [14] moments, by eliminating the factorial calculations. However, these algorithms have the possibility to generate and propagate finite precision errors, as classical signal processing algorithms do [15 17]. Moreover, the recursive algorithms that have been introduced until now, need the computation of intermediate polynomials of lower orders or repetitions. While this feature is an advantage for image representation and reconstruction [8 13] that accelerates their computation, in the case of pattern classification applications, where the computation of distinct moments of any order might be reuired, it decelerates the feature extraction process. ISSN , Pattern Recognition and Image Analysis, 010, Vol. 0, No. 1, pp Pleiades Publishing, Ltd., 010.

2 EFFICIENT COMPUTATION OF ZERNIKE AND PSEUDO-ZERNIKE MOMENTS 57 In the present paper, two novel algorithms that permit the computation of Zernike and Pseudo-Zernike moments of arbitrary orders and repetitions are introduced. These algorithms are recursive, do not generate and propagate finite precision errors and finally, allow single moments to be independently computed. The paper is organized as follows: in Section the fundamental theory about the orthogonal moments is presented. The discussion is focused on the Zernike and Pseudo-Zernike orthogonal families. In Section 3 the newly developed recursive algorithms are introduced while in Section 4 appropriate simulations establish the proposed algorithms as a general framework for computing the Zernike and Pseudo-Zernike moments. Some very useful conclusions are presented in Section 5.. ORTHOGONAL MOMENTS In this section a detailed description of the fundamental theory of the most utilized orthogonal families that are widely used in pattern recognition tasks, the Zernike and Pseudo-Zernike moments, is being given..1. Zernike Moments Zernike moments (ZMs) are the most widely used family of orthogonal moments due to their extra property of being invariant to an arbitrary rotation of the object they describe. They are used, after making them invariant to scale and translation, as object descriptors in pattern recognition applications [5 8] and in image retrieval tasks [18 0]. The introduction of ZMs in image analysis was made by Teague [1], using a set of complex polynomials, which form a complete orthogonal set over the interior of the unit circle x + y = 1. These polynomials [5, 6] have the form V p ( x, y) = V p ( r, θ) = R p () r exp( jθ), (1) where p is a non-negative integer and is a non zero integer subject to the constraints p even and p, r is the length of vector from the origin ( x, y ) to the pixel (x, y) and θ the angle between vector r and x axis in counter-clockwise direction. R nm (r), are the Zernike radial polynomials [] in (r, θ) polar coordinates defined as R p p () r = ( 1) s r ( s)! p s! p s! s = s! Note that R p, (r) = R p (r). s. () The polynomials of E. () are orthogonal and satisfy the orthogonality principle x + y 1 * ( x, y)v p ( x, y) dxdy = V nm π n δ + 1 δ, np m (3) where δ αβ = 1 for α = β and δ αβ = 0 otherwise, is the Kronecker symbol. The Zernike moment of order p with repetition for a continuous image function f(x, y), that vanishes outside the unit disk is n + 1 Z p = f( x, y)v p * ( ρθ, ) dxdy. π x + y 1 (4) For a digital image, the integrals are replaced by summations [5, 6] to get p + 1 Z p = f( x, y)v p * r, θ π ( ), x + y 1. x y (5) Suppose that one knows all moments Z p of f(x, y) up to a given order p max. It is desired to reconstruct a discrete function ˆ f (x, y) whose moments exactly match those of f(x, y) up to the given order p max. Zernike moments are the coefficients of the image expansion into orthogonal Zernike polynomials as can be seen in the following reconstruction euation, p max ˆ f ( x, y) = Zp V p ( r, θ), p = 0 (6) with having similar constraints as in (1). Note that as p max approaches infinity ˆ f (x, y) will approach f(x, y). As it can be seen from E. () there are a lot of factorial computations, operations that may consume too much computer time. For this reason, as it has already been discussed in the introduction, recursive algorithms for the computation of the radial polynomials () have been developed [9 13]. In the next section, a novel recursive algorithm for computing the radial polynomials () is presented. This algorithm doesn t include any factorial term that could cause any numerical instability and it computes the radial polynomials in a fast way... Pseudo-Zernike Moments Pseudo-Zernike moments (PZMs) are used in many pattern recognition [3, 4] and image processing applications as alternatives to the traditional ZMs. It has been proved that they have better feature representation capabilities and are more robust to image noise [5] than Zernike moments.

3 58 PAPAKOSTAS et al. The kernel of these moments is the orthogonal set of Pseudo-Zernike polynomials defined inside the unit circle. These polynomials have the form of (1) with the Zernike radial polynomials replaced by the Pseudo-Zernike radial polynomials () r S p = ( 1) s r ( p + 1 s)! p s! ( p+ + 1 s)! ( s)! s = 0 (7) with additional constraints 0 p, p = 0, 1,.,, (8) The corresponding PZMs are computed using the same formulas (4) and (5) as in the case of ZMs, since the only difference is in the form of the polynomial being used. Due to the above constraints (8), the set of Pseudo- Zernike polynomials of order p, contain (p + 1) linearly independent polynomials of degree p. On the other hand, the set of Zernike polynomials contain only (p + 1)(p + )/ linearly independent polynomials of degree p, due to the additional condition that p is even. Thus, PZMs offer more feature vectors than the Zernike moments of the same order. As can be seen from (7) the computation of Pseudo-Zernike moments, involves the calculation of some factorial terms, an operation that adds an extra computational effort. Conseuently, as in the case of Zernike moments, there is a demand for developing fast and numerical robust algorithms, able to calculate accurate moments in a short time. Similarly to ZMs, an image described by a finite number of PZMs, can be reconstructed by using E. (6). In the next section, recursive algorithms for the computation of ZMs and PZMs, which satisfy the previous reuirements, are presented and their computational complexity is being examined, in comparison to their original direct s. 3. THE PROPOSED ALGORITHMS From the previous sections, it is obvious that there is a need for developing algorithms that fast compute the Zernike and Pseudo-Zernike moments of arbitrary orders and repetitions, without generating and propagating numerical uantization errors. Such kind of algorithms would be very useful for pattern classification applications, where moments of any order and repetition may be used as discriminative features. Let us summarize some of the features that an efficient algorithm must have in order to be useful for pattern classification tasks: (1) Firstly, an algorithm for the computation of Zernike type moments, have to be numerically stable. This stability can be achieved by avoiding finite preci- s sion errors to be presented. This property is of major importance, since a possible numerical error generated in a step of the algorithm and propagated in subseuent steps, may cause the collapse of the algorithm. () Secondly, the algorithm has to be faster than its original direct version. This can be achieved by eliminating the need to compute factorial calculations, which adds a high computation effort. (3) Thirdly, the algorithm has to allow the computation of individual moments and radial polynomials, without intermediate calculations. The recursive algorithms introduced until now, compute the radial polynomials recursively. Thus, in order to compute a moment of a specific order and repetition, the calculation of radial polynomials of lower order or repetition is needed. This mechanism is desirable when the moments are used to reconstruct an image. However, for classification purposes, it is more useful to directly compute the radial polynomial of the order and repetition actually needed. This demand has been originated in the last years, where more sophisticated, evolutionary feature selection algorithms are applied, so that the most appropriate features for any application, is selected. A first attempt to use a genetic algorithm for the selection of the appropriate Zernike moments needed to be participant in a pattern recognition task as feature vectors, has been presented in [6]. Taking under consideration the above three reuirements, which should be satisfied by an algorithm for computing the ZMs and PZMs, in order to be useful for pattern classification applications, the following recursive algorithms are presented for both cases. The nature of the radial polynomials used in both cases, help us to manipulate them in the same way. The proof of the new algorithms is given in the Appendix. A. Zernike Moments We propose, the following formula (9) for the computation of the radial polynomial of order p and repetition, where the fractional terms are recursively computed using (10a) and (10b), where R p p, () r = T p0 r p + ( 1) k T pk r p k k = 1 T p! p0 = , ! p ! T k + 1 p k + 1 pk = T k( k+ 1) p k 1 ( ). (9) (10a) (10b)

4 EFFICIENT COMPUTATION OF ZERNIKE AND PSEUDO-ZERNIKE MOMENTS 59 Euation (10b), holds for k = 1,, 3,, (p )/, while for k = 0, one has to use (10a). B. Pseudo-Zernike Moments In the same way as ZMs, the radial polynomials used as kernel functions for the computation of PZMs, can take the form (a) (b) S p () r = Y p0 r p + ( 1) k Y pk r p k k = 1 (11) where Y p0 = , ( p + 1) (1a) ( p+ + 1)! ( )! ( p+ k+ ) ( k + 1) Y pk = Y (1b) k( k+ ) p ( k 1 ). Euation (1b), holds for k = 1,, 3, (p ), while for k = 0, one has to use (1a). The proposed algorithms are numerically robust in terms of finite precision errors, since the fundamental operations, subtraction, addition, multiplication and division are being made between integer and not real values. Therefore, according to [15 17], there is no possibility to produce finite precision errors, as a well known recursive algorithm for the computation of the Zernike moments [9] does, due to the subtraction operation [7]., 3.1. Computational Complexity The computational complexity of the proposed recursive algorithms, versus that of the direct s for the computation of the ZMs and PZMs, is being discussed in the current section. From Es. () and (7), owing to many factorial terms, it is obvious that the computational complexity of the direct s is very high Zernike moments. In the case of Zernike moments, the number of multiplications that have to be executed, in order to compute the radial polynomial using E. (), is almost (p/ + 1)*(p 3)*(p 1), in the worst case. The computational complexity of the direct, for computing a single radial polynomial R p of order p and repetition, is O(p 3 ). The proposed algorithm is of less computational complexity, since its recursive formula for the computation of the fractional terms, eliminates the factorials, by having only 6 multiplications in every fractional term, except the first one. The number of multiplications needed for the computation of a single radial polynomial using the proposed algorithm is approximately (3p 1) + p / + 4p, in the worst case. Thus, the computational complexity of the recursive algorithm, is O(p ). This complexity is smaller Fig. 1. Lena s grey level images of (a) and (b) sizes. than that of the original and makes the algorithm suitable for the computation of ZMs, up to very high orders. The efficiency of the algorithm in computing the ZMs, will be studied in section 4, where the moments of images of several sizes are being computed Pseudo-Zernike moments. Similarly to the Zernike moments, the Pseudo-Zernike moments have O(p ) computational complexity, as it can be observed by (7), where it is clear that the number of the reuired multiplications for computing the radial polynomial R p is approximately (p + 1)*(4p)*(p 1), where p is the moment order. The proposed recursive algorithm in the case of PZMs, has computational complexity O(p ), since the number of the multiplications needed to compute a radial polynomial of order p and repetition, R p is (5p + 1) + 6p + p, in the worst case. Conclusively, taking into account the previous observations about the computational complexity, we can claim that by applying a straightforward ology (see Appendix), very fast algorithms that permit the computation of individual moments of order p and repetition, can be achieved. In the following, the above detailed study of the computational complexity of the proposed recursive algorithms is justified, through appropriate experiments. 4. EXPERIMENTAL STUDY In order to study the effectiveness of the proposed recursive algorithms, a set of experiments has taken place. Two different sizes of the well known benchmark Lena s image have been selected, to be used as test images, with and pixels, in grey level format, as depicted in Fig. 1. The Zernike moments up to various maximum orders p max, are computed using E. () in the direct and E. (9) in the proposed. The CPU elapsed time for each one of the experiments has been measured and the results for both images' sizes are illustrated in Fig..

5 60 PAPAKOSTAS et al. ms Direct (a) Proposed Direct Proposed (b) Maximum moment otder Fig.. The CPU elapsed time (ms) for various maximum Zernike moment orders of (a) and (b) Lena s images. ms (a) Direct 4.0 Direct 10 Proposed 3.5 Proposed (b) Maximum moment otder Fig. 3. The CPU elapsed time (ms) for various maximum Pseudo-Zernike moment orders of (a) and (b) Lena s images. Figure 3 illustrates the CPU elapsed time needed to compute the Pseudo-Zernike moments of various maximum orders p max, for the two images when the direct and recursive algorithms are used. The above figures show that as the maximum moment order increases, the time needed to compute the moments, in the direct s is exponentially increased. The recursive algorithms, introduced in the current paper, need less CPU time, to compute the moments of the same order than the original ones. This observation, justifies experimentally, what has already been stated in Section 3, where the computational complexity of the algorithms has been proved to be of very low order. Apart from the above observations, Fig. and Fig. 3 show that our algorithms perform well, although the image size varies and thus their complexity is independent of the image size being processed. More precisely, the Zernike and Pseudo-Zernike moments for the two images and for maximum orders 10, 0, 30, 40, 50 have been computed and the CPU elapsed time in each case is presented in the following Table 1 and Table, respectively. The outperforming of the recursive s can also be defined if we calculate the % percentage of the time reduction taking place, as maximum order increases, by using the proposed s as alternatives of the direct ones. For this reason, we define the Computation Time Reduction (CTR) % as follows, Computation Time Reduction CRT % = Time Direct Time Direct Time Recursive (13)

6 EFFICIENT COMPUTATION OF ZERNIKE AND PSEUDO-ZERNIKE MOMENTS 61 Table 1. The CPU elapsed time (ms) for computation of Zernike moments up to p max order maximum order Lena Image direct recursive Lena Image direct recursive p max = p max = p max = p max = p max = Table. The CPU elapsed time (in seconds) for computation of Pseudo-Zernike moments up to p max order maximum order Lena Image direct recursive Lena Image direct recursive p max = p max = p max = p max = p max = By computing the computation time reduction using (13), for the same experiments of Fig. and Fig. 3, the following Fig. 4 can be drawn, Figure 4, verifies the benefits of the newly introduced algorithms, in terms of the computation time reduction, when being used instead of the direct s, for computing the Zernike and Pseudo-Zernike moments of any image size. The time reduction curves depicted in Fig. 4, for the two image sizes and for small moment orders, are not identical, as it was expected, due to the accuracy of the timer used to measure the computational time. When the computational time is uite short the accuracy of the timer significantly influences it and as the time increases this impact is negligible. From the above figure, it is impressive to conclude that we have a very significant computation time reduction, for both cases, which for high orders goes up to almost over 70%, in comparison with the original. This is a major advantage of the recursive algorithms, since in image representation one has to compute the moments of an image up to high orders, in order to optimally reconstruct it, with minimum reconstruction error. 5. CONCLUSIONS Two novel recursive algorithms were proposed in this paper, which compute in a fast way the orthogonal Zernike and Pseudo-Zernike moments. The computational complexity of the proposed algorithms is O(p ) in multiplications, while the original direct s are of O(p 3 ) complexity. Therefore, the computation time reuired to compute the moments of a high order, can be reduced by almost over 70% of the time needed by the direct s to do the same work. Additionally, the algorithms do not generate and propagate finite precision errors as some traditional recursive algorithms do. Also, they are capable to compute an individual moments for pattern classification purposes without needing the computation of intermediate moments. The experimental results justify the effectiveness of these new algorithms and establish them, as alternatives of the direct s, used until now. Percent of time reduction-% Lena image Lena image Maximum moment order (a) (b) Lena image Lena image Fig. 4. The CPU elapsed time (ms) for various maximum (a) Zernike and (b) Pseudo-Zernike moment orders of and Lena s images.

7 6 PAPAKOSTAS et al. APPENDIX DERIVATION OF THE RECURSIVE FORMULAS Let us define the fractional terms of order p, repetition and index k, which are used in the summations for computing the radial polynomials R p and S p of () and (7) respectively as follows, ( k)! T pk = (A.1) k! p k! p k! in the case of Zernike moments, and ( p + 1 k)! Y pk = (A.) k! ( p+ + 1 k)! ( k)! in the case of Pseudo-Zernike moments. The above euations (A.1) and (A.), may be combined to a more generic form as, ( a G 1 k)! pk = , (A.3) k! ( a k)! ( a 3 k)! where Zernike a 1 = p a = p a 3 = p Pseudo-Zernike a 1 = p + 1 a = p+ + 1 a 3 =. (A.4) The fractional term of (A.3) for the previous index of the summation (k 1), has the form, ( a 1 k + 1)! ( ) = ( k 1)! ( a k + 1)! ( a 3 k + 1)! G p k 1 (A.5) Now, if we restrict our study for indices k 0, we can use the property of the factorials n! = ( n 1)!n, n 0 (A.6) for transforming euation (A.5), as follows G p( k 1) = a 1 ( k)! ( a 1 k + 1)k k! ( a k)! ( a k + 1) ( a 3 k)! ( a 3 k + 1) ( a = 1 k)! k( a k + 1) (A.7) k! ( a k)! ( a k)! ( k + 1) ( a 3 k + 1) The above euation can be written, in a more suitable form, as ( a G k + 1) ( a 3 k + 1) pk = G (A.8) k( a 1 k + 1) p ( k 1 ). a k( a = T 1 k + 1) pk ( k + 1) ( a 3 k + 1) a By substituting the a 1, a, a 3 parameters in (A.8) using (A.4), we have in the case of Zernike Moments, p T k + 1 p k + 1 pk = T (A.9) k( k+ 1) p ( k 1 ), which holds for k = 1,, 3,, , and is identical to (10b) introduced in Section 3, and in the case of Pseudo-Zernike moments, Y pk = Y ( p+ k+ ) ( k + 1) (A.10) k( k+ ) p ( k 1 ), which holds for k = 1,, 3,, p, and is identical to (11b) introduced in Section 3. Finally, for k = 0, E. (A.3), gives a G 1! p0 = (A.11) a!a 3! In the same way, by substituting the a 1, a, a 3 parameters in (A.11) using (A.4), we have p! T p0 = (A.1) ! p ! and ( p + 1)! Y p0 = , (A.13) ( p+ + 1)! ( )! for the Zernike and Pseudo-Zernike moments, respectively. REFERENCES 1. R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis (World Sci. Publ., 1998).. K. Tsirikolias and B. G. Mertizos, Statistical Pattern Recognition Using Efficient -D Moments with Applications to Character Recognition, Pattern Recognition 6, No. 6, (1993). 3. B. G. Mertzios and K. Tsirikolias, Statistical Shape Discrimination and Clustering Using an Efficient Set of Moments, Pattern Recognition Lett. 14, (June 1993). 4. B. G. Mertzios, Shape Discrimination in Robotic Vision Using Scaled Normalized Central Moments, in Proc. IFAC Workshop on Mutual Impact of Computing Power and Control Theory (Prague, Sept ), pp A. Khotanzad and J.-H. Lu, Classification of Invariant Image Representations Using a Neural Network, IEEE Trans. on Acoustics, Speech and Sign. Processing ASSP 38, No. 6, (1990). 6. G. A. Papakostas, Y. S. Boutalis, D. A. Karras, and B. G. Mertzios, A New Class of Zernike Moments for Computer Vision Applications, Information Sci. 177, No. 13, (007). 7. G. A. Papakostas, D. A. Karras, B. G. Mertzios, and Y. S. Boutalis, An Efficient Feature Extraction Methology for Computer Vision Applications Using Wavelet

8 EFFICIENT COMPUTATION OF ZERNIKE AND PSEUDO-ZERNIKE MOMENTS 63 Compressed Zernike Moments, ICGST Int. J. on Graphics, Vision and Image Processing, Special Issue: Wavelets and Their Applications SI1, 5 15 (005). 8. C. Kan and M. D. Srinath, Invariant Character Recognition with Zernike and Orthogonal Fourie Mellin Moments, Pattern Recognition 35 (1), (00). 9. C. W. Chong, P. Raveendran, and R. Mukundan, A Comparative Analysis of Algorithms for Fast Computation of Zernike Moments, Pattern Recognition 36 (3), (003). 10. S. O. Belkasim, M. Ahmadi, and M. Shridhar, Efficient Algorithm for Fast Computation of Zernike Moments, J. Franklin Inst. 333(B) (4), (1996). 11. J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, A Novel Algorithm for Fast Computation of Zernike Moments, Pattern Recognition 35 (1), (00). 1. A. Prata and W. V. T. Rusch, Algorithm for Computation of Zernike Polynomials Expansion Coefficients, Appl. Opt. 8, (1989). 13. E. C. 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Chou, A Comparative Study of Zernike Moments for Image Retrieval, in Proc. 16th IPPR Conf. on Computer Vision, Graphics and Image Processing (CVGIP 003) (Taiwan, 003), pp M. Zhenjiang, Zernike Moment-Based Image Shape Analysis and its Application, Pattern Recognition Lett. 1 (), (000). 0. D. G. Sim, H. K. Kim, and R. H. Park, Invariant Texture Retrieval Using Modified Zernike Moments, Image and Vision Comp. (4), (004). 1. M. Teague, Image Analysis via the General Theory of Moments, J. Opt. Soc. Amer. 70 (8), (1980).. F. Zernike, Beugunstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkonstraste, Physica 1, (1934). 3. J. Haddadnia, M. Ahmadi, and K. Faez, An Efficient Feature Extraction Method with Pseudo-Zernike Moment in RBF Neural Network-Based Human Face Recognition System, EURASIP J. on Applied Signal Processing 9, (003). 4. J. Haddadnia, K. Faez, and M. Ahmadi, A Neural Based Human Face Recognition System Using an Efficient Feature Extraction Method with Pseudo Zernike Moment, J. Circuits, Systems, and Computers 11 (3), (00). 5. C.-H. The and R. T. Chin, On Image Analysis by the Methods of Moments, IEEE Trans. on Pattern Anal. Machine Intell. PAMI-10, No. 4, (1988). 6. G. A. Papakostas, Y. S. Boutalis, and B. G. Mertzios, Evolutionary Selection of Zernike Moment Sets in Image Processing, in Proc. 10th Intern. Workshop on Systems, Signals and Image Processing (IWSSIP 03) (Prague, Sept ). 7. G. A. Papakostas, Y. S. Boutalis, C. N. Papaodysseus, and D. K. Fragoulis, Numerical Error Analysis in Zernike Moments Computation, Image and Vision Comp. 4, (006). George A. Papakostas received the diploma of Electrical and Computer in Engineering in 1999 and the Msc. and PhD. Degree in Electrical and Computer Engineering (topic in Feature Extraction and Pattern Recognition) in 00 and 007 respectively, from Democritus University of Thrace (DUTH), Greece. He is the author of thirty publications in international scientific journals and conferences. He research interests are focused on the field of pattern recognition, neural networks, feature extraction, optimization, signal and image processing. Dr. Papakostas served as reviewer in numerous scientific journals and conferences and his is member of Technical Chamber of Greece. Yiannis S. Boutalis received the diploma of Electrical Engineer in 1983 from the Democritus University of Thrace (DUTH), Greece and the PhD degree in Electrical and Computer Engineering (topic Image Processing) in 1988 from the Computer Science Division of National Technical University of Athens, Greece. Since 1996, he serves as a faculty member, at the Department of Electrical and Computer Engineering, DUTH, Greece, where he is currently an Associate Professor and director of the automatic control systems lab. Currently, he is also a Visiting Professor for research cooperation at Erlangen-Nuremberg University of Germany, chair of Automatic Control. He served as an assistant visiting professor at University of Thessaly, Greece, and as a visiting professor in Air Defence Academy of General Staff of airforces of Greece. He also served as a researcher in the Institute of Language and Speech Processing (ILSP), Greece, and as a managing director of the R&D SME Ideatech S.A, Greece, specializing in pattern recognition and signal processing applications. His current research interests are focused in the development of Computational Intelligence techniues with applications in Control, Pattern Recognition, Signal and Image Processing Problems.

9 64 PAPAKOSTAS et al. Dimitrios A. Karras received his Diploma and M.Sc. Degree in Electrical and Electronic Engineering from the National Technical University of Athens, Greece in 1985 and the Ph. Degree in Electrical Engineering, from the National Technical University of Athens, Greece in 1995, with honors. In 1990 he received, also, his Diploma Degree in Mathematics from the University of Athens. Since 004, after his election, he has been with the Chalkis Institute of Technology, Automation Dept., Greece as full professor in Digital Systems as well as with the Hellenic Open University, Dept. Informatics as a visiting professor. He has published more than 40 research Journal papers in various areas of pattern recognition, image/signal processing and neural networks and more than 80 research papers in International Scientific Conferences. His research interests span the fields of pattern recognition and neural networks, multidimensional digital signal processing, image processing and analysis, communications and security, as well as parallel algorithms and fast processing. Basil G. Mertzios was born in Kavala, Greece on August 10, He received the Diploma Degree in Electrical and Mechanical Engineering from the Aristotle University of Thessaloniki, Greece in 1979 and the Ph. Degree in Electrical Engineering, from the Democritus University of Thrace, Greece in 198, both with honors. Part of his Ph. D. work has been done in University of Toronto. From he was an Associate Professor at the Democritus University of Thrace, Department of Electrical Engineering and Director of the Laboratory of Automatic Control Systems and from 199 a Professor at the same University. Also he has been invited and served as Professor of Control Systems and Computational Intelligence at the Department of Automation of the Technological Institute of Thessaloniki. During various periods since 1983, he visited extensively University of Toronto and Ruhr-Universitat Bochum, Germany as a research fellow. During part of 1986 he visited Georgia Institute of Technology, Atlanta, U.S.A., as an Adjunct Associate Professor. The academic year he was a Visiting Associate Professor at Georgia Institute of Technology, School of Electrical Engineering, in Fall Semester of 1991 he was a Visiting Professor at the University of Toronto and in 1996 he was a Visiting Professor at the Tokyo Institute of Technology, Japan. He was awarded in 1986 the Senior Fulbright Research Award, in 1987 the Alexander von Humboldt Fellowship and in 1988 the Natural Sciences and Engineering Research Council of Canada International Scientific Exchange Award. He published more than 100 research Journal papers in various areas of control system theory, multidimensional digital signal processing and robotics and more than 140 research papers in International Scientific Conferences. His research interests span the fields of automatic control systems, parallel algorithms and fast processing, robotics, multidimensional digital signal processing, image processing and analysis, pattern recognition and neural networks.