Numerical stability of fast computation algorithms of Zernike moments

Transcription

1 Available online at Alied Mathematics and Comutation 195 (2008) Numerical stability of fast comutation algorithms of Zernike moments G.A. Paakostas a, *, Y.S. Boutalis a, C.N. Paaodysseus b, D.K. Fragoulis b a Democritus University of Thrace, Deartment of Electrical and Comuter Engineering, Xanthi, Greece b National Technical University of Athens, School of Electrical and Comuter Engineering, Athens, Greece Abstract A detailed, comarative study of the numerical stability of the recursive algorithms, widely used to calculate the Zernike moments of an image, is resented in this aer. While many aers, introducing fast algorithms for the comutation of Zernike moments have been resented in the literature, there is not any work studying the numerical behaviour of these methods. These algorithms have been in the ast comared to each other only according to their comutational comlexity, without been given the aroriate attention, as far as their numerical stability is concerned, being the most significant art of the algorithms reliability. The resent contribution attemts to fill this ga in the literature, since it mainly demonstrates that the usefulness of a recursive algorithm is defined not only by its low comutational comlexity, but most of all by its numerical robustness. This aer exhaustively comares some well known recursive algorithms for the comutation of Zernike moments and sets the aroriate conditions in which each algorithm may fall in an unstable state. The exeriments show that any of these algorithms can be unstable under some conditions and thus the need to develo more stable algorithms is of major imortance. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Zernike moments; Recursive algorithm; Finite recision error; Numerical stability 1. Introduction Although a long time has assed, since the first introduction of orthogonal moments in image rocessing by Teague [1], the scientific interest in the usage of moments in engineering life is still increased. This haens due to their ability to uniquely describe a signal and thus they are used as discriminative features in image reresentation and attern recognition alications. There are some research toics about orthogonal moments in which scientists focus their attention lately. Many researchers all over the world try to develo fast algorithms [2] that accelerate the comutation of orthogonal moments and make their hardware imlementation an easy task. Most of the fast algorithms * Corresonding author. addresses: gaakos@ee.duth.gr (G.A. Paakostas), ybout@ee.duth.gr (Y.S. Boutalis), caaod@cs.ntua.gr (C.N. Paaodysseus), dfrag@mail.ntua.gr (D.K. Fragoulis) /$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi: /j.amc

2 resented until now, are making use of recursive equations that ermit the comutation of a high order moment kernels by using kernels of lower orders. This methodology has roved to be efficient, esecially when an entire set of moments is necessary to be calculated. Moreover, since orthogonal moments have as kernel functions orthogonal olynomials, which construct orthogonal basis, are very useful to uniquely reresent atterns in attern recognition tasks. For these alications some very romising attemts [3 5] to embody scale, rotation and translation invariances in the comutation of orthogonal moments have been erformed. Recently, the disadvantages of orthogonal moments having continuous orthogonal olynomials as kernel functions have been studied and more numerically accurate orthogonal moments, with discrete olynomials [6 8] are introduced as alternative to the continuous ones. It has been roved that the moments with discrete olynomials are more suitable in image rocessing, since the domain of an image is discrete. However, although a brief study about the accuracy of the orthogonal moments in resect to some aroximation errors, due to the transformation from the continuous sace to the discrete [9] and the accuracy of the moments into scale, translation and rotation transformations [10,11] have been done, there is not any work which deals with the finite recision errors resented in the comutation of orthogonal moments. The occurrence of finite recision errors is of major imortance, esecially in the fast recursive algorithms since a ossible error in a ste of the algorithm may be accumulated iteration by iteration, by resulting to unreliable outcomes. A first investigation of the mechanisms that roduce and roagate finite recision errors in Zernike moments comutation, using the well known q-recursive algorithm, has been successfully erformed and resented by the authors in [12]. In that work a detailed numerical analysis of the way the finite recision errors are being generated and the aroriate conditions under which these errors occurred were discussed. The resent aer comes to comlete the revious work [12] by comaring some very oular recursive algorithms widely used in Zernike moments comutation, in terms of finite recision errors. This comarative study from a different oint of view, the algorithms numerical robustness, in conjunction with the comarison already been done in [2] in resect to their comutational comlexity, constitutes a comlete study of the comutational behaviour that each recursive algorithm resents. By keeing in mind the above objectives the aer is organized by resenting the descrition of each recursive algorithm in section one, by analyzing the algorithms erformance, according to the finite recision errors being generated and finally by discussing the erformance of each algorithm by means of their numerical stability. 2. Comuting the Zernike moments G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Zernike moments (ZMs) are the most widely used family of orthogonal moments due to their roerties, of being invariant to an arbitrary rotation of the object that they describe and that a erfect reconstruction of an image from its moments is ossible. They are used, after making them invariant to scale and translation, as object descritors in attern recognition alications [13 19] and in image retrieval tasks [20,21] with considerable results. The introduction of ZMs in image analysis was made by Teague [1], using a set of comlex olynomials, which form a comlete orthogonal set over the interior of the unit circle x 2 þ y 2 ¼ 1. These olynomials [13,14] have the form V q ðx; yþ ¼V q ðr; hþ ¼R q ðrþ exðjqhþ; ð1þ where is a non-negative integer and q ositive and negative integers subject to the constraints jqj even and jqj 6, r is the length of vector from the origin ðx; yþ to the ixel ðx; yþ and h the angle between vector r and x axis in counter-clockwise direction. R q ðrþ, are the Zernike radial olynomials [22] in ðr; hþ olar coordinates defined as X jqj 2 R q ðrþ ¼ s¼0 ð 1Þ s s! Note that R ; q ðrþ ¼R q ðrþ ð sþ! r 2s : jqj s! s! 2 2 þjqj ð2þ

3 328 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) The olynomials of Eq. (1) are orthogonal and satisfy the orthogonality rincile Z Z V nm ðx; yþv qðx; yþdxdy ¼ n þ 1 d nd mq ; x 2 þy 2 61 where d ab ¼ 1 for a ¼ b and d ab ¼ 0 otherwise, is the Kronecker symbol. The Zernike moment of order with reetition q for a continuous image function f ðx; yþ, that vanishes outside the unit disk is Z q ¼ þ 1 Z Z x 2 þy 2 61 f ðx; yþv qðr; hþr dr dh: ð3þ For a digital image, the integrals are relaced by summations [13,14] to get Z q ¼ þ 1 X X f ðx; yþv q ðr; hþ; x2 þ y 2 6 1: ð4þ x y Suose that one knows all moments Z q of f ðx; yþ u to a given order max. It is desired to reconstruct a discrete function ^f ðx; yþ whose moments exactly match those of f ðx; yþ u to the given order max. Zernike moments are the coefficients of the image exansion into orthogonal Zernike olynomials as can be seen in the following reconstruction equation: ^f ðx; yþ ¼ X max ¼0 X Z q V q ðr; hþ q with q having similar constraints as in (1). Note that as max aroaches infinity ^f ðx; yþ will aroach f ðx; yþ. The method that comutes the Zernike moments and uses Eq. (2) to evaluate the Zernike olynomials, is called Direct Method. As can be seen from Eq. (2) there are a lot of factorial comutations, oerations that consume too much comuter time. For this reason, as it has already been discussed in the introduction, recursive algorithms for the comutation of the radial olynomials (2) have been develoed [2]. The most well known recursive algorithms for Zernike moments comutation, are described in the next sections and their main features are discussed Kintner s algorithm Kintner was the first who studied the roerties of the Zernike olynomials [23] and introduced a recurrent relation [24]. In the following the recursive algorithm for Zernike olynomials comutation, roosed by Kintner, is described. Algorithm ¼ q Direct Method using Eq: ð2þ q ¼ 2 Direct Method using Eq: ð2þ ð otherwise R q ðrþ ¼ K 2r 2 þ K 3 ÞR ð 2Þq ðrþþk 4 R ð 4Þq ðrþ ; with ð6aþ K 1 ð þ qþð qþð 2Þ K 1 ¼ ; K 2 ¼ 2ð 1Þð 2Þ; 2 K 3 ¼ q 2 ð þ q 2Þð q 2Þ ð 1Þ ð 1Þð 2Þ; K 4 ¼ : ð6bþ 2 As it can be seen in the above equations, Kintner s algorithm cannot be alied in cases where ¼ q and q ¼ 2, thus the Direct Method has to be used. This algorithm is faster than the Direct Method but it still includes factorial calculations which take too comutational time. ð5þ

4 2.2. Modified Kintner s algorithm A modified version of Kintner s recursive algorithm described in Section 2.1 has been introduced in [2]. This algorithm eliminates the usage of the Direct Method in ill-osed cases where the recursive equation can not be alied. For these cases other factorial free equations can be used, in order to accelerate the overall comutation time. Algorithm ¼ q R ðrþ ¼r ; ð7aþ q ¼ 2 R ð 2Þ ðrþ ¼R ðrþ ð 1ÞR ð 2Þð 2Þ ðrþ; ð7bþ ð otherwise R q ðrþ ¼ K 2r 2 þ K 3 ÞR ð 2Þq ðrþþk 4 R ð 4Þq ðrþ ; with ð7cþ K 1 K 1 ¼ ð þ qþð qþð 2Þ ; K 2 ¼ 2ð 1Þð 2Þ; 2 K 3 ¼ q 2 ð 1Þ ð 1Þð 2Þ; K 4 ¼ ð þ q 2Þð q 2Þ : ð7dþ 2 This algorithm has been roved to be of significant efficiency in comuting an entire set of Zernike olynomials u to a secific order or a Zernike olynomial of an individual order [2] Prata s algorithm G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Prata [25] roosed, a recursive algorithm that enables the higher order olynomials to be derived from the lower order ones. This algorithm is slower than Kintner s algorithm [2] and it is not ossible to use it in cases where q =0and ¼ q, as shown below, where the Direct Method has to be used. Algorithm q ¼ 0 Direct Method using Eq: ð2þ ¼ qr ðrþ ¼r ; otherwise R q ðrþ ¼L 1 R ð 1Þðq 1Þ ðrþþl 2 R ð 2Þq ðrþ; with; ð8aþ ð8bþ L 1 ¼ 2r þ q ; L 2 ¼ q þ q : ð8cþ Prata s recursive algorithm is of the same comlexity Oð 2 Þ for secific order, as Kintner s one, but its overall erformance is quite low [2] q-recursive algorithm Recently, a novel recursive algorithm with remarkable erformance, called q-recursive algorithm, has been introduced in [2]. Algorithm ¼ qr ðrþ ¼r ; q ¼ 2 R ð 2Þ ðrþ ¼R ðrþ ð 1ÞR ð 2Þð 2Þ ðrþ; ð9aþ ð9bþ

5 330 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) otherwise R ðq 4Þ ¼ H 1 R q ðr þ H 2 þ H 3 R r 2 ðq 2Þ ; with ð9cþ qðq 1Þ H 1 ¼ qh 2 2 þ H 3ð þ q þ 2Þð qþ ; 8 H 2 ¼ H 3ð þ qþð q þ 2Þ þðq 2Þ; ð9dþ 4ðq 1Þ 4ðq 2Þðq 3Þ H 3 ¼ ð þ q 2Þð q þ 4Þ : A thorough study [2] of the comutational erformance of the recursive algorithms resented above has shown that q-recursive algorithm significantly outerforms the other methods in all test cases. These results establish this algorithm as the most efficient recursive algorithm for Zernike olynomials comutation, in terms of CPU execution time. 3. Numerical stability analysis Although a detailed study of the comutational seed of the reviously resented recursive algorithms has been resented in [2], there is not any work dealing with the analysis of their numerical behaviour. The numerical stability of a recursive algorithm seems to be more imortant than its comutation seed. For examle, the aearance of finite recision errors is of major imortance, since a ossible error in a ste of the algorithm may be accumulated iteration by iteration, by resulting to unreliable quantities. So, an efficient recursive algorithm must quickly comute the desirable quantities by ensuring the algorithm s stability. The results of a fast but unstable algorithm can not be safely used, no matter how fast they have been derived. A first investigation about the ossible finite recision errors generation and roagation, in the q-recursive algorithm, has been done by the authors in [12]. In this section a comlementary analysis of the numerical stability of the rest recursive algorithms is taking lace. The algorithms are comared in resect to their numerical behaviour and the cases where these algorithms fall in unstable situations are defined. An efficient methodology, which exlores the way the finite recision errors are generated and roagated, during a recursive algorithm, has been introduced and used in very oular signal rocessing algorithms [26 31]. This methodology emloys a number of fundamental roositions demonstrating the way the four oerations addition, multilication, division and subtraction, influence the generation and transmission of the quantization error. These roositions are described in the following General remarks The roositions stated in this aer hold true indeendently of the radix of the arithmetic system. However, the numerical error generation and roagation will be studied in the decimal reresentation, because the decimal arithmetic system is far more familiar and clear to users. In this arithmetic system, recision comarison between two numbers will be made in accordance with the following: Definition 1. Consider two numbers, n 1 and n 2, written in the canonical exonential form, with the same number, n, of decimal digits in the mantissa, i.e. n 1 ¼ d 1 d 2 d 3 ;...; d n 10 s ; n 2 ¼ d 1 d 2 d 3 ;...; d n 10 q ; with s P q: Then, these two numbers differ by K decimal digits, if and only if jjn 1 j jn 2 jj ¼ d 10 s ðn KÞ ; 1 6 d < 10: For examle, according to this definition, the two numbers and differ indeed by 3 decimal digits, but the following two and differ by 1 decimal digit, as one might intuitively exect.

6 Definition 2. Let all quantities be written in the canonical exonential form, with n decimal digits in the mantissa. Suose that the correct value of an arbitrary quantity a is a c, if all calculations were made with infinite recision. Then, one may define that the quantity a has been comuted with recisely the last k decimal digits erroneous, if and only if, a and a c differ k digits according to Definition 1. As will be evident from the subsequent analysis, all the formulas, that constitute a certain iterative algorithm, are not equivalent from the oint of view of the finite recision error generation and roagation. Notation. For any quantity a exressed in the canonical exonential form, we shall write: (i) man(a) for the mantissa of a, and (ii) EðaÞ for the exonent of a Proosition 1. Let all the involved quantities be comuted with finite recision of n decimal digits in the mantissa, and consider any quantity comuted by means of a formula of the tye Multilication x ¼ y z: Suose that, due to the revious finite recision calculations, the quantity y has been comuted with recisely the last k decimal digits erroneous, while z has been comuted with u to k decimal digits erroneous. Then, (i) ifjman(y) Æ man(z)j P 10, then x is comuted with recisely the last k or k 1 decimal digits erroneous. (ii) if jman(y) Æ man(z)j < 10, then x is comuted with recisely the last k or k + 1 decimal digits erroneous. Proosition 2. Let all the following quantities be comuted with finite recision of n decimal digits in the mantissa, and consider any quantity comuted through a formula of the tye: Division x ¼ y z : G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Suose that, due to the revious finite recision calculations, the quantity y has been comuted with recisely the last k decimal digits erroneous, while z has been comuted with u to k decimal digits erroneous. Then, (iii) ifjman(y)/man(z)j P 1, then x is comuted with recisely the last k or k 1 decimal digits erroneous. (iv) if j man(y)/man(z)j < 1, then x is comuted with recisely the last k or k + 1 decimal digits erroneous. Proosition 3. Let all the involved quantities be comuted with finite recision of n decimal digits in the mantissa, and consider any quantity calculated through a formula of the tye Subtraction x ¼ y z; with y z > 0; where x ¼ x 1 ; x 2 ; x 3 ;...; x n 10 d ; y ¼ y 1 ; y 2 ; y 3 ;...; y n 10 s ; z ¼ z 1 z 2 z 3...z n 10 q are such that d < maxfs; qg ()EðxÞ < maxfeðyþ; EðzÞg: Let d ¼jmaxfs; qg dj: Moreover, suose that, due to the revious finite recision calculations, the higher order quantity say y has been comuted with recisely the last k decimal digits erroneous, while z has been comuted with a number of erroneous decimal digits equal to, or smaller than k. Then x is comuted with the last (k + d) decimal digits erroneous. Proosition 4 (The numerical error relaxation shift). Let all quantities be comuted with finite recision of n decimal digits in the mantissa, and consider any quantity x comuted through a sum of two quantities, that is,

7 332 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Addition x ¼ y þ z: Suose, moreover, that z has its last k decimal digits erroneous and that the exonent of z is by v smaller than the exonent of y, that is, v ¼ EðyÞ EðzÞ > 0: Then z transfers to x only k v erroneous decimal digits if k v > 0 or it does not transfer finite recision error at all, if k v Finite recision errors of the recursive algorithms In order to analyze the numerical behaviour of the recursive algorithms resented in Section 2, we roceed according to the following stes [31], for each algorithm: Ste 1: We execute the algorithm with n digits recision for the mantissa Ste 2: In arallel, we execute the algorithm with 2n digits recision for the mantissa Ste 3: We cast any quantity z 2n comuted by 2n decimal digits recision to a quantity z n of n decimal digits recision. Ste 4: We comare quantities z n and z n, according to Definition 1 and in this way we obtain the exact number of erroneous decimal digits with which each quantity z n is comuted. By alying the above methodology and keeing in mind the definitions and roositions reviously discussed, each recursive algorithm is studied and the corresonding results are resented next. In the forthcoming tables the finite recision error, in erroneous digits, is being measured according to Definition 1 and is deicted in bold face Kintner s algorithm Kintner s algorithm, resents two kind of numerical errors. The first one is the overflow error due to range limitations of data tye used. For examle, the float (seven digit recision) data tye has a valid range ( , ), while the double (15 digit recision) data tye has a valid range ( , ), in the case of IBM PC comatible comuters. The second tye of error is the finite recision error (FPE), Case 1: ¼ q or q ¼ 2 In this case, the comutation of the radial olynomial R q ðrþ, is erformed using the direct method (2). The direct method resents overflow errors due to the limitations on reresenting the factorials of a big numbers (n!þ and the radius in the owers of big numbers (r a ). Table 1 illustrates this behaviour, of the direct method, where the values corresonds to an overflowed quantities. Case 2: 6¼ q or q 6¼ 2 In this case the radial olynomials R q ðrþ are comuted by using Eq. (6a). By alying the algorithm described in Section 3.2, it is concluded that the quantities K 1 ; K 2 ; K 3 ; K 4, do not resent any finite recision error. However, the term K 2 r 2 generates a finite recision error of 2 erroneous digits. This error is quite small and it cannot lead the whole algorithm to destroy, as it is shown in Table 2. The term K 2 r 2 þ K 3, seems to be a candidate source of generating finite recision error, since there are cases where the erroneous digits are many, e.g. 7 digits. The mathematical oeration that is resonsible for the generation of this error is the subtraction between numbers with common digits, as Table 3 resents. The finite recision error will be significantly increased by leading the algorithm to unreliable results, if the following condition is satisfied: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2 r 2 þ K 3 0; () r 1 2 þ q 2 : 2ð 2Þ ð10þ

8 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Table 1 Comutation accuracy R q (r) quantity, for = q or q =2 q r R q bit bit bit bit bit bit bit Table 2 Comutation accuracy of K 2 r 2 quantity q r K 2 K 2 r bit digits error 2 digits error bit digits error 2 digits error bit digits error 2 digits error bit digits error 2 digits error bit digits error 2 digits error bit digits error 2 digits error bit digits error 2 digits error

9 334 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) However, condition (10) can t be satisfied for any values of and q, since the resulted radius r takes values greater than 1, which is outside the region of the unit disk and where the radial olynomials are undefined. Thus, we can conclude that while K 2 r 2 þ K 3 generates finite recision error, this error can t destroy the algorithm. The next term of (6a), ðk 2 r 2 þ K 3 Þ=K 1, behaves similar to the K 2 r 2 þ K 3 term. The error roduced in the revious state is carried out to the next quantity, as Table 4 shows. In the case of ðk 2 r 2 þ K 3 Þ=K 1 R ð 2Þq ðrþ term, the roduced finite recision error, is very critical, about 8 erroneous digits, since the comuted quantities differ significantly. This error is not roduced by the multilication but it is rather carried by the revious state, due to the subtraction ðk 2 r 2 þ K 3 Þ or the overflow of R ð 2Þq ðrþ term, as resented in Table 5. Table 6 illustrates the finite recision errors generated when the K 4 =K 1 quantity is being comuted. As it can be seen, this error is of small magnitude, 2 digits error. As it can be seen in Table 7, the K 4 =K 1 R ð 4Þq ðrþ term, is comuted with the same number of erroneous digits as R ð 4Þq ðrþ and thus the multilication oeration in this case doesn t constitute a finite recision error source. Finally, the radial olynomial of th order and qth reetition, which is comuted by using Eq. (6a), resents high finite recision errors, in some cases this error is of 8 erroneous digits. From Table 8, one realizes that the error increases when the subtraction of two values with common digits is erformed. This ill-osed subtraction affects the final result and an investigation of the aroriate conditions according to which this oeration may be an imortant source of errors, has to be erformed. The subtraction that is resonsible for generating high finite recision errors and affects the stability of the algorithm is erformed between two values having common digits. Table 3 Comutation accuracy of K 2 r 2 + K 3 quantity q r K 2 r 2 K 3 K 2 r 2 + K bit digits error 0 digits error 5 digits error bit digits error 0 digits error 7 digits error bit digits error 0 digits error 5 digits error bit digits error 0 digits error 5 digits error bit digits error 0 digits error 5 digits error bit digits error 0 digits error 6 digits error bit digits error 0 digits error 6 digits error

10 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Table 4 Comutation accuracy of (K 2 r 2 + K 3 )/K 1 quantity q r K 2 r 2 + K 3 K 1 (K 2 r 2 + K 3 )/K bit digits error 0 digits error 5 digits error bit digits error 0 digits error 7 digits error bit digits error 0 digits error 5 digits error bit digits error 0 digits error 5 digits error bit digits error 0 digits error 5 digits error bit digits error 0 digits error 6 digits error bit digits error 0 digits error 5 digits error Table 5 Comutation accuracy of (K 2 r 2 + K 3 )/K 1 Æ R ( 2)q (r) quantity q r (K 2 r 2 + K 3 )/K 1 R ( 2)q (r) (K 2 r 2 + K 3 )/K 1 Æ R ( 2)q (r) bit digits error 5 digits error 6 digits error bit digits error 2 digits error 7 digits error bit digits error 5 digits error 6 digits error bit digits error 8 digits error 8 digits error bit digits error 8 digits error 8 digits error bit digits error 6 digits error 7 digits error bit digits error 7 digits error 7 digits error

11 336 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Table 6 Comutation accuracy of K 4 /K 1 quantity q K 4 K 1 K 4 /K bit digits error 0 digits error 2 digits error bit digits error 0 digits error 2 digits error bit digits error 0 digits error 2 digits error bit digits error 0 digits error 2 digits error bit digits error 0 digits error 2 digits error bit digits error 0 digits error 2 digits error bit digits error 0 digits error 2 digits error Table 7 Comutation accuracy of K 4 /K 1 Æ R ( 4)q (r) quantity q r K 4 /K 1 R ( 4)q (r) K 4 /K 1 Æ R ( 4)q (r) bit digits error 6 digits error 6 digits error bit digits error 6 digits error 6 digits error bit digits error 8 digits error 8 digits error bit digits error 6 digits error 6 digits error bit digits error 7 digits error 8 digits error bit digits error 8 digits error 8 digits error bit digits error 8 digits error 8 digits error

12 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) These values are the ðk 2 r 2 þ K 3 Þ=K 1 R ð 2Þq ðrþ and K 4 =K 1 R ð 4Þq ðrþ terms. By analyzing the conditions where these two values are made almost equal, for secific order and reetition, we have For =4,q = 0, by using (6) we have, R 40 ¼ K 2r 2 þ K 3 R 20 þ K 4 R 00 ; K 1 K 1 where K 1 ¼ 16; K 2 ¼ 48; K 3 ¼ 24; K 4 ¼ 8 and R 20 ¼ 2r 2 1; R 00 ¼ 1: The error is resented when K 2 r 2 þ K 3 R 20 ¼ K 4 R 00 () K 2 r 2 þ K 3 R20 ¼ K 4 R 00 () K 1 K 1 ð48r 2 24Þð2r 2 1Þ ¼8 () 6r 4 6r 2 þ 1 0 () sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ffiffi 3 r : 6 By roceeding in the same way, the radius values for which the subtraction of two common numbers may generate high finite recision errors, for several orders and reetitions, have been derived and resented in Table 9. From the above analysis is concluded that for every usage of the recursive Eq. (6a), there are combinations between the moment order, the reetition q and the radial, for which a significant finite recision error is being generated. This error increases when a subtraction of common real numbers is resented, by resulting to totally unreliable radial olynomial values. Table 8 Comutation accuracy of R q (r) quantity, for 5 q and q 5 2 q r (K 2 r 2 + K 3 )/K 1 Æ R ( 2)q (r) K 4 /K 1 Æ R ( 4)q (r) R q (r) bit digits error 3 digits error 6 digits error bit digits error 4 digits error 6 digits error bit digits error 2 digits error 8 digits error bit digits error 2 digits error 6 digits error bit digits error 4 digits error 6 digits error bit digits error overflow 8 digits error bit digits error 8 digits error 7 digits error

13 338 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) An imortant observation that is derived from the above Table 9, is that as the moment order increases, the number of the radial values for which the aroriate conditions are satisfied, is also dramatically increased. This unstable behaviour constitutes a major drawback of this algorithm, since its numerical stability can be easily altered, even for low moment orders Modified Kintner s algorithm The modified Kintner s algorithm, as roosed in [2], differs from the original one in using more simle equations instead of the direct method (2), in cases of ¼ q (7a) and q ¼ 2 (7b). Case 1: ¼ q The usage of (7a) instead of the direct method for ¼ q, avoids the overflows which occurred due to the comutation of the factorials of big numbers. However, overflow can be still resented when comuting the radius to the ower of big numbers, as it can be seen in Table 10. Case 2: q ¼ 2 In this case it is ossible to generate significant finite recision errors, when real numbers with common digits are being subtracted according to the recursive (7b). By alying the methodology of Section 3.2, the finite recision error for secific radius, moment order and reetition are extracted and illustrated in Table 11. The condition under which the subtraction would be erformed between common real numbers is obtained as, Table 9 High finite recision errors in R q (r) quantity, of Eq. (6a) q r (K 2 r 2 + K 3 )/K 1 Æ R ( 2)q (r) K 4 /K 1 Æ R ( 4)q (r) R q (r) ffiffiffiffiffiffiffi ffiffi 3þ 3 6 ffiffiffiffiffiffiffi ffiffi ffiffiffiffiffiffiffi ffiffi 6þ 6 10 ffiffiffiffiffiffiffi ffiffi ffiffiffiffiffiffiffiffiffi 5þ ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi 10þ ffiffiffiffiffiffiffiffiffiffi bit digits error 0 digits error 17 digits error 64bit digits error 0 digits error 18 digits error 64bit digits error 1 digits error 18 digits error 64bit digits error 2 digits error 17 digits error 64bit digits error 1 digits error 18 digits error 64bit digits error 2 digits error 18 digits error 64bit digits error 1 digits error 17 digits error 64bit digits error 1 digits error 17 digits error 64bit digits error 1 digits error 17 digits error

14 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Table 10 Comutation accuracy of R (r) quantity, for = q r R (r) bit bit bit bit bit bit bit Table 11 High finite recision errors in R ( 2) (r) quantity, of Eq. (7b) q r R (r) ( 1)R ( 2)( 2) (r) R ( 2) bit digits error 0 digits error 18 digits error bit digits error 1 digits error 18 digits error 64bit digits error 0 digits error 18 digits error 64bit digits error 2 digits error 18 digits error 64bit digits error 2 digits error 18 digits error 64bit digits error 2 digits error 17 digits error 64bit digits error 2 digits error 18 digits error

15 340 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi R ðrþ ð 1ÞR ð 2Þð 2Þ ðrþ ¼r ð 1Þr ð 2Þ 1 q þ 1 0 () r ; or r : ð11þ The above analysis shows that while the modified Kitner s method ermits the calculation of the radial olynomials by avoiding any factorial comutations, there is an increased ossibility the algorithm lost its stability. The modified Kintner s algorithm can be unstable more often than the conventional one, since extra conditions, when q ¼ 2, where ill-osed subtractions can exist for many combinations of, q and r, as shown in Table Prata s algorithm Prata s algorithm can be considered a combination of Kintner s and modified Kintner s algorithms, since it uses the Direct method (2) to comute the radial olynomials for q = 0 as Kintner s do and Eq. (7a) in the case of ¼ q, as modified Kintner s one. In the cases, where q 6¼ 0 and 6¼ q, a recursive formula (8b) is used. By analyzing the way the L 1 and L 2 (8c) quantities are being comuted, it is concluded that these terms do not generate significant finite recision errors, as resented in Tables 12 and 13. However, the revious analysis on the Kintner s like algorithms, where finite recision errors are ossible to be generated, due to ill-osed subtractions, focuses the attention to the study of the subtraction in Eq. (8b). As Tables 14 and 15 show, while the terms being subtracted do not generate finite recision errors greater than 3 digits, their subtraction can lead to high finite recision errors, under certain conditions (Table 16). The finite recision error is extremely increased when for secific moment order and reetition q, the radial olynomial is comuted for a ixel having radius r, for which the following holds: Table 12 Comutation accuracy of L 1 quantity q r L bit digits error bit digits error bit digits error bit digits error bit digits error bit digits error bit digits error

16 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Table 13 Comutation accuracy of L 2 quantity q L bit R 31 ¼ L 1 R 20 þ L 2 R 11 ¼ 3 2 r ð2r2 1Þþ 1 rffiffiffi r ¼ 3r 3 2r 0 () rð3r 2 2 2Þ 0 () r ; 2 3 R 42 ¼ L 1 R 31 þ L 2 R 22 ¼ 4 3 r ð3r3 2rÞþð 1 rffiffi 3 Þr2 ¼ 4r 4 3r 2 0 () r 2 ð4r 2 3 3Þ 0 () r ; 4 R 53 ¼ L 1 R 42 þ L 2 R 33 ¼ 5 4 r ð4r4 3r 2 Þþ 1 rffiffi r 3 ¼ 5r 5 4r 3 0 () r 3 ð5r 2 4 4Þ 0 () r ; 4 5 R 64 ¼ L 1 R 53 þ L 2 R 44 ¼ 6 5 r ð5r5 4r 3 Þþ 1 rffiffi r 4 ¼ 6r 6 5r 4 0 () r 4 ð6r 2 5 5Þ 0 () r : 5 6 From the above, it is obvious that the radius for which the subtraction generates finite recision errors, satisfies the following generic equation: sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi 1 q þ 1 r ¼ ; or r ¼ ð12þ for a given moment order and reetition q. The above Table 16 resents some ossible combinations of ; q and r, which satisfy Eq. (12) and for which the finite recision errors generated are significant and dro the algorithm to unstable situations q-recursive algorithm This algorithm constitutes the most recently introduced [2], among the algorithms which are studied in this work and it rovides a fast way to comute the radial olynomials of order and reetition q. It is roved that this algorithm outerforms the other ones, by a significant factor [2] digits error bit digits error bit digits error bit digits error bit digits error bit digits error bit digits error

17 342 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Table 14 Comutation accuracy of L 1 Æ R ( 1)(q 1) quantity q r L 1 R ( 1)(q 1) L 1 Æ R ( 1)(q 1) bit digits error 1 digits error 3 digits error bit digits error 1 digits error 3 digits error bit digits error 2 digits error 3 digits error bit digits error 1 digits error 3 digits error bit digits error 2 digits error 3 digits error bit digits error 1 digits error 3 digits error bit digits error 2 digits error 3 digits error Recently, this algorithm has been studied by the authors [12], for its numerical stability in terms of finite recision errors generation and roagation. More recisely, in [12] has been roved that Eq. (9b), generates significant amount of erroneous digits when the following condition is satisfied sffiffiffiffiffiffiffiffiffiffiffi 1 r ; for P 2: ð13þ Additionally, the H 2 þ H 3 =r 2 quantity of (9c) is ossible to generate considerable finite recision error, when ixels having radius satisfying the following equation exist. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ðq 1Þðq 3Þ r : ð14þ ð3 qþð þ qþð q þ 2Þþð þ q 2Þð q þ 4Þðq 1Þ 4. Discussion The stability analysis of the fast algorithms used for comuting the Zernike moments resented reviously, leads u to significant conclusions about the numerical behaviour and aroriateness of each algorithm. The four algorithms studied in this work, Kintner s, Modified Kintner s, Prata s and q-recursive algorithms, roduce overflow but more imortant finite recision errors, under certain conditions. The mathematical oeration, which is resonsible for the generation of finite recision errors in all cases, is the subtraction erformed between real numbers with common digits, as defined by Proosition 3.

18 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) Table 15 Comutation accuracy of L 2 Æ R ( 2)q quantity Q r L 2 R ( 2)q L 2 Æ R ( 2)q bit digits error 1 digits error 2 digits error bit digits error 2 digits error 2 digits error bit digits error 2 digits error 2 digits error bit digits error 1 digits error 2 digits error bit e e e e e e digits error 2 digits error 3 digits error bit e e e e e e digits error 1 digits error 3 digits error bit e e e e e e digits error 1 digits error 3 digits error Table 16 High finite recision errors in R q (r) quantity, of Eq. (8b) q r L 1 Æ R ( 1)(q 1) L 2 Æ R ( 2)q R q bit digits error 2 digits error 18 digits error bit digits error 0 digits error 17 digits error 64bit digits error 1 digits error 18 digits error 64bit digits error 1 digits error 18 digits error Secifically Modified Kintner s algorithm, is highly unstable as comared to the other methods, since it resents overflow errors (for ¼ qþ and for any other combination of, q, finite recision errors, that can destroy the algorithm by resulting to unreliable quantities. While the modified Kintner s algorithm has some interesting roerties, according to its comutational comlexity comared with the conventional one [2], it is numerically more unstable. It is referable to use the standard Kintner s method than the modified one, since the

19 344 G.A. Paakostas et al. / Alied Mathematics and Comutation 195 (2008) benefits in comutation seed are of less imortance than the stability and reliability the radial olynomials being comuted. Prata s and q-recursive methods are more stable than the Kintner s tye algorithms, by introducing less illosed subtractions for fewer conditions. Particularly, Prata s algorithm behaves unstably only for one condition (12), while q-recursive for the conditions (13) and (14). If one is to select between these more stable algorithms and by taking into account that q-recursive algorithm is very fast, its comlexity is of an order lower than the other ones [2], the q-recursive algorithm is a good choice. While the overflow errors can be handled, by restricting to low moment orders u to 20, the finite recision errors could be robably overcome by modifying the algorithm definitions and introducing algorithms that tackle both the comutational efficiency and the stability ensuring. 5. Conclusions A detailed study of the numerical stability of some very oular recursive algorithms for the fast comutation of Zernike moments has taken lace in the revious sections. The aroriate conditions in which each algorithm falls in unstable states, where the comuted radial olynomials take unreliable values were defined. The above investigation draws the useful conclusion that all the algorithms analysed, resent the ossibility to alter their numerical stability, under certain conditions. The analysis resented in this work constitutes a first attemt to comare these algorithms not according to their comutational comlexity, as this task has been already done, but in resect to their numerical behaviour. Conclusively, one should take under consideration not only the mathematical roerties of the algorithm, but also the numerical imlementation restrictions, in each develoment of any recursive algorithm. References [1] M. Teague, Image analysis via the general theory of moments, J. Ot. Soc. Am. 70 (8) (1980) [2] C.W. Chong, P. Raveendran, R. Mukundan, A comarative analysis of algorithms for fast comutation of Zernike moments, Pattern Recogn. 36 (2003) [3] C.W. Chong, P. Raveendran, R. Mukundan, Translation invariants of Zernike moments, Pattern Recogn. 36 (2003) [4] C.W. Chong, P. Raveendran, R. Mukundan, Translation and scale invariants of Legendre moments, Pattern Recogn. 37 (2004) [5] C.W. Chong, P. Raveendran, R. Mukundan, The scale invariants of seudo-zernike moments, Pattern Anal. Al. 6 (2003) [6] R. Mukundan, S.H. Ong, P.A. Lee, Image analysis by Tchebichef moments, IEEE Trans. Image Process. 10 (9) (2001) [7] R. Mukundan, S.H. Ong, P.A. Lee, Discrete vs. continuous orthogonal moments for image analysis, in: Proceedings of International Conference on Imaging Science Systems and Technology, vol. 1, 2001, [8] P.T. Ya, P. Raveendran, S.H. Ong, Image analysis by Krawtchouk moments, IEEE Trans. Image Process. 12 (11) (2003) [9] S.X. Liao, M. Pawlak, On the accuracy of Zernike moments for image analysis, IEEE PAMI 20 (12) (1998) [10] S. Rodtook, S.S. Makhanov, Numerical exeriments on the accuracy of rotation moment invariants, Image Vis. Comut. 23 (2005) [11] N.K. Kamila, S. Mahaatra, S. Nanda, Invariance image analysis using modified Zernike moments, Pattern Recogn. Lett. 26 (2005) [12] G.A. Paakostas, Y.S. Boutalis, C.N. Paaodysseus, D.K. Fragoulis, Numerical error analysis in Zernike moments comutation, Image Vis. Comut. 24 (2006) [13] A. Khotanzad, J.-H. Lu, Classification of invariant image reresentations using a neural network, IEEE Trans. Acoust., Seech Sign. Process. 38 (6) (1990) [14] A. Khotanzad, Y.H. Hong, Invariant image recognition by Zernike moments, IEEE Trans. Pattern Anal. Machine Intell. PAMI-12 (5) (1990) [15] G.A. Paakostas, D.A. Karras, B.G. Mertzios, Image coding using a wavelet based Zernike moments comression technique, in: 14th International Conference on Digital Signal Processing (DSP2002), vol. II, 1 3 July 2002, Santorini-Hellas, Greece, [16] G.A. Paakostas, Y.S. Boutalis, B.G. Mertzios, Evolutionary selection of Zernike moment sets in image rocessing, in: 10th International Worksho on Systems, Signals and Image Processing (IWSSIP 03), Setember 2003, Prague, Czech Reublic. [17] G.A. Paakostas, D.A. Karras, B.G. Mertzios, Y.S. Boutalis, An efficient feature extraction methodology for comuter vision alications using wavelet comressed Zernike moments, ICGST International Journal on Grahics, Vision and Image Processing, Secial Issue: Wavelets and Their Alications SI1 (2005) [18] M. Zhenjiang, Zernike moment-based image shae analysis and its alication, Pattern Recogn. Lett. 21 (2) (2000)