Fast numerically stable computation of orthogonalFourier–Mellin moments

G.A. Papakostas, Y.S. Boutalis, D.A. Karras and B.G. Mertzios

Abstract: An efficient algorithm for the computation of the orthogonal Fourier–Mellin moments(OFMMs) is presented. The proposed method computes the fractional parts of the orthogonalpolynomials, which consist of fractional terms, recursively, by eliminating the number of factorialcalculations. The recursive computation of the fractional terms makes the overall computation ofthe OFMMs a very fast procedure in comparison with the conventional direct method. Actually, thecomputational complexity of the proposed method is linear O(p) in multiplications, with p beingthe moment order, while the corresponding complexity of the direct method is O(p2). Moreover,this recursive algorithm has better numerical behaviour, as it arrives at an overflow situationmuch later than the original one and does not introduce any finite precision errors. These are thetwo major advantages of the algorithm introduced in the current work, establishing thecomputation of the OFMMs to a very high order as a quite easy and achievable task.Appropriate simulations on images of different sizes justify the superiority of the proposedalgorithm over the conventional algorithm currently used.

1 Introduction

Although there is an increasing number of published papersthat introduce a new set of features for image representationand pattern classification purposes, the traditional momentfeature sets are still drawing the attention of the scientificcommunity.Among the various moment types, such as the geometric,

central, normalised, statistical moments [1], there is a verypowerful one, the orthogonal moments [2, 3]. The most sig-nificant property of the orthogonal moments, is their abilityto fully describe an object, with minimum redundant infor-mation and thus the reconstruction of an object by a finitenumber of moments, is possible.The orthogonal moments are categorised in families

according to the type of the orthogonal polynomials thatare making use, as kernel functions. The most utilisedorthogonal families are the Zernike (ZM), pseudo-Zernike(PZM) and Fourier–Mellin (FMM) moments, which arewidely used as image descriptors in image-processingtasks. Moreover, their property to stay invariant to anyrotation of the object presented in a scene, in addition totheir ability to describe spatial frequency components ofan image, make them appropriate for pattern classificationapplications [4–6].Although, the previous orthogonal moments have useful

attributes, over the other moment types, the presence of

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-cvi:20060130

Paper first received 7th May and in revised form 27th November 2006

G.A. Papakostas and Y.S. Boutalis are with the Department of Electrical andComputer Engineering, Democritus University of Thrace, Xanthi 67100, Greece

D.A. Karras is with the Automation Department, Chalkis Institute ofTechnology, Chalkida, Greece

B.G. Mertzios is with the Department of Automation, Laboratory of ControlSys. and Comp. Intell., Thessaloniki Institute of Technology, Thessaloniki,Greece

E-mail: gpapakos@ee.duth.gr

IET Comput. Vis., 2007, 1, (1), pp. 11–16

many factorial terms in their polynomial definitions, maketheir computation a very time-consuming task. While thecomputational capabilities of the modern computers arealways increasing, the factorial of a big number remains avery demanding process.

For this reason, many researchers have introduced recur-sive algorithms for the computation of Zernike [7–11] andpseudo-Zernike [12] moments, by eliminating the factorialcalculations. However, these algorithms have the possibilityto generate and propagate finite precision errors, as classicalsignal-processing algorithms do [13–15]. Additionally,there is not any recursive algorithm for the computationof the Fourier–Mellin moments and the usage of thedirect method that includes many factorial terms is theonly choice.

This work comes to cover the need of factorial-freerecursive algorithm for the case orthogonal Fourier–Mellin moments of OFMMs, by introducing a fast algorithmfor computing the OFMMs. Moreover, the algorithm issuitable for computing higher moment orders, than the con-ventional direct method, as it is driven in overflow con-ditions slower.

The paper is organised by describing the fundamentaltheory of the OFMMs in Section 2, introducing theproposed recursive algorithm in Section 3, and finally byjustifying the efficiency of the algorithm throughappropriate simulations in Section 4.

2 Orthogonal Fourier–Mellin moments

In [16], Sheng and Shen introduced a set of orthogonalmoments, for pattern recognition purposes, by using a setof complex polynomials Upq(x, y), which form a completeorthogonal set over the interior of the unit circlex2 þ y2 ¼ 1. These polynomials in polar coordinates havethe form

Upq(r, u) ¼ Qp(r)exp(iqu) (1)

11

where p (the order of the Mellin radial transform) is anon-negative integer, q ¼ 0, +1, +2, . . . (the circular har-monic order), r is the length of the vector from the origin(�x, �y) to the pixel (x, y) and u is the angle between vectorr and x axis in counter-clockwise direction. The kernel ofthe complex polynomials of (1), is a set of orthogonalradial polynomials [16] in (r, u) polar coordinates defined as

Qp(r) ¼Xpk¼0

(�1) pþk (pþ k þ 1)!

(p� k)!k!(k þ 1)!rk (2)

Equation (2), which calculates the orthogonal polynomialsQp, referred to as the ‘direct method’, for the rest of thepresent paper.

The polynomials of (1) are orthogonal and satisfy theorthognality principle

ð ð

x2þy2�1

U�nm(x, y) � Upq(x, y)dxdy ¼

p

nþ 1dnpdmq (3)

where dab ¼ 1 for a ¼ b and dab ¼ 0 otherwise, is theKronecker delta.

The OFMM of order p and repetition q for a continuousimage function f(x, y), that vanishes outside the unit disk is

Opq ¼pþ 1

p

ð ð

x2þy2�1

f (x, y)U�pq(r, u)dxdy (4)

For a digital image, the integrals are replaced by sum-mations, to get

Opq ¼pþ 1

p

Xx

Xy

f (x, y)U�pq(r, u), x2 þ y2 � 1 (5)

Suppose that one knows all moments Opq of f(x, y) up to agiven order pmax. It is desired to reconstruct a discrete func-tion f̂ (x, y), whose moments exactly match those of f (x, y)up to the given order pmax. The OFMMs are the coefficientsof the image expansion into the orthogonal polynomials (1),as it can be seen in the following reconstruction equation

f̂ (x, y) ¼Xpmax

p¼0

Xpmax

q¼�pmax

OpqUpq(r, u) (6)

Note that as pmax approaches infinity, f̂(x, y) will approachf (x, y).

It has been noted in [16, 17] that the Fourier–Mellinmoments are more appropriate to describe images ofsmall size in terms of image reconstruction errors andsignal-to-noise ratios, than the Zernike and pseudo-Zernike ones. Additionally, as the radial polynomials (2)used in OFMMs have much more zeros than those of theother orthogonal moments, the OFMMs have the capabilityto describe high spatial frequency components of an image.Therefore the order of OFMMs required to represent animage can be much lower than that of ZMs or PZMs, andthus they are less sensitive to variation and noise [16, 17].These properties make the OFMMs very useful in imagerepresentation [18] of various sizes and in pattern recog-nition applications [17, 19].

3 Proposed recursive method

Although, OFMMs have considerable properties, as havealready been discussed in the previous section, the presenceof many factorial computations in (2), which are operations

12

that may consume too much computer time, makes theircomputation a very time-consuming task.The same problem also exists in the case of ZMs and

PZMs. In these cases, recursive algorithms that reduceor even eliminate the factorial calculations, have alreadybeen presented [7–12]. However, a recursive algorithm forthe computation of the OFMMs has not been proposed yet.In this section, a factorial-free recursive algorithm, for

the computation of the OFMMs, without having possiblenumerical instabilities, as in the case of ZMs [20], andwith considerably better behaviour than the direct method,is introduced.A recursive algorithm for the OFMMs computation has to

satisfy some very critical demands in order to be applicable:

1. The algorithm should not exhibit any quantisation errors,such as finite precision errors, as some classical algorithmsin signal processing and recently a recursive algorithm forthe ZMs computation [20], do. This is very crucial, as thegeneration of a numerical error in one step of the algorithmand its propagation to subsequent steps, may cause thealgorithm to ‘destroy’ by resulting in unreliable quantities.2. The recursive algorithm must be faster than the directmethod and of course it should give the same values forthe computed OFMMs.3. The algorithm should permit the computation ofOFMMs up to very high orders in order to capture, asmuch as possible image information.4. The computation of a single moment of order p and rep-etition q, should be allowed without computing intermediatemoments. This characteristic is very useful in pattern recog-nition applications, where feature vectors consisting ofmoments of non-consecutive orders might be desirable [21].

Keeping in mind that these four demands have to besatisfied by an efficient recursive algorithm, we introducethe following equation for computing the radial polynomialof order p

Qp(r) ¼ (�1) p � Tp0 þXpk¼1

(�1) pþk� Tpk � r

k (7)

where

Tp0 ¼ pþ 1 (8a)

Tpk ¼(pþ k þ 1)(p� k þ 1)

k(k þ 1)� Tp(k�1) (8b)

A detailed proof of the derivation of these formulae is givenin the appendix.As it can be seen from (8a) and (8b), the calculation of the

fractional terms (pþ k þ 1)!=(p� k)!k!(k þ 1)! in (2), cannow be evaluated by avoiding the factorial computations,through the use of the recurrence (8b).This fact is very impressive, as the time-consuming task

of computing the fractional terms, because of the factorials,is transformed in easier operations. The absence of the fac-torial terms in the proposed recursive algorithm makes itsuperior to the direct method (2).The first requirement of avoiding the generation and

propagation of any numerical error through the recursivecomputations is totally satisfied. The main reason ofcausing finite precision errors in the recursive algorithms isthe presence of subtractions between real numbers of oppo-site sign and common number of digits [13]. A thoroughstudy of (8a) and (8b), can lead to the conclusion that theproposed algorithm does not generate finite precisionerrors, as the subtractions are between integer numbers,

IET Comput. Vis., Vol. 1, No. 1, March 2007

which will be exactly the same or completely different. Thus,there is no possibility to subtract numbers that have the oppo-site sign and common digits, which could eventually lead tothe generation of finite precision errors.Furthermore, the introduced algorithm recursively

computes the fractional terms of the radial polynomials. Norecursive computation of consecutive polynomials isperformed. It is therefore clear that it permits the recursivecomputation of an individual moment, having order p andrepetition q. Consequently, there is no need to computeintermediate moments and thus the fourth requirement issatisfied, too.The rest demands to be satisfied are studied in the next

sections, where appropriate experiments take place.

3.1 Computational complexity

For the sake of simplicity of the computational complexityof the proposed method, it is decided to take into accountonly the number of the required multiplications, in ourstudy. The number of additions is of less importance asthey are executed in a short time. In the following, a detailedcomparison between the proposed and the direct methods,with respect to the number of multiplications the fractionalterms of the two methods need to execute in order a radialpolynomial of order p, Qp be computed.The computational complexity of our recursive algor-

ithm, against that of the direct method is being discussedin the current section. From (2), it is obvious that thecomputational complexity of the direct method is veryhigh, because of the presence of the factorial terms.Specifically, the number of multiplications that have to beexecuted, in order to compute a fractional term of (2) is(4p2 1), in the worst case. As, the order p increases, thesummation in (2) consists of many fractional terms thathave factorials, therefore the number of required multipli-cations to compute a single polynomial of order p, Qp is(pþ 1) � (4p2 1) in the worst case. Thus, the compu-tational complexity of the direct method, for computing asingle radial polynomial Qp of order p, is O(p

2).The proposed algorithm is of less computational com-

plexity, as its recursive formula, eliminates the factorialterms, by having only three multiplications in every frac-tional term. The number of required multiplications tocompute a single radial polynomial of order p is (3p), inthe case of the proposed algorithm and its computationalcomplexity is linear, that is, O(p).This complexity is signifi-cant smaller than that of the original direct method andmakes the algorithm suitable for the computation ofOFMMs, up to very high orders.Therefore the recursive computation of the fractional

terms, as proposed in the present work, is faster than thedirect method and permits the computation of moments ofhigher orders.The efficiency of the algorithm in computing the

OFMMs, will be studied in Section 4, where themoments of images of several sizes and types are beingcomputed.

3.2 Numerical behaviour

The direct method for the computation of OFMMs needsmany factorial calculations, in each evaluation of radialpolynomial Qp. These calculations represent a very signifi-cant part of the overall computation procedure and influencethe procedure with numerical instabilities. This happens inhigh-order cases, where the need for computing factorialsof big numbers leads to overflow situations.

IET Comput. Vis., Vol. 1, No. 1, March 2007

Overflow is the situation in which a quantity takes a highervalue, from the range of its data type. For example, the float(seven-digit precision) data type has a valid range[1.18 � 10245, 3.40 � 1045], while the double (15-digit pre-cision) data type has a valid range [2.23 � 102308,1.79 � 10308], in the case of IBM PC compatible computers.

More precisely, from (2) it is obvious that as the order pand the index k increase, the numerator’s factorial(pþ kþ 1)!, tends to an overflow state, more quickly thanthe other ones (p2 k)!, k!, (kþ 1)!. For example, whenp ¼ 85 and k ¼ 85, we have to calculate the factorial171!, which is a number greater than the range of adouble data type. In this case, an overflow occurs and theresult takes the 0 value.

This situation is very crucial, when the moments up to ahigh order need to be computed. The proposed recursivealgorithm does not show this weakness, as it does notinclude any factorial computation.

In Fig. 1, the values of the fractional terms of radial poly-nomials for the direct and recursive methods, for variousorders p, have been plotted. This figure shows, that whilethe direct method starts to calculate the fractional terms forp ¼ 85 order and index k ¼ 85 with an overflow, the pro-posed algorithm does not fall in overflow situation. Theproposed algorithm presents an overflow situation onlyowing to the factor r p, which happens in the direct methodtoo. In Fig. 1, the overflowed fractional terms are denotedwith asterisks as markers, and the non-overflowed oneswith dots.

4 Experimental study

In order to study the effectiveness of the proposed recursivealgorithm, a set of experiments have taken place. Twodifferent sizes of the well known benchmark ‘Lena’ imagehave been selected, to be used as test images, with64 � 64 and 128 � 128 pixels, grey-level images, asdepicted in Fig. 2.

The FMMs up to various maximum orders pmax, are com-puted using (2) in the direct method and (7) in the proposedmethod. The CPU elapsed time for each one of the exper-iments has been measured and the results for both images’sizes are illustrated in Fig. 3.

From Fig. 3, it can be seen that as the maximum momentorder increases, the time needed to compute the moments, inthe direct method is exponentially increased. The recursivealgorithm, introduced in the current work, needs less CPUtime, to compute the moments of the same order than theoriginal method. This observation, justifies experimentally,what has already been stated in previously, where thecomputational complexity of the algorithm proved to beof very low order.

Although, one may expect the evolution of thecomputational time be linear, as the computationalcomplexity of the recursive algorithm is linear O(p), withrespect to the number of the required multiplications, thisis not justified in Fig. 3. This happens because of the factthat Fig. 3 shows the computation time of computingan entire set of moments where a lot of additions areincluded. Additionally, the timer used to measure thecomputation time does not have high accuracy because ofsoftware limitations of the computer being used (IBM PCcompatible 2.8 GHz, with Cþþ Builder). These are thereasons why the computational curve of the proposedmethod is not linear but it slightly diverges as the orderincreases.

Apart from these observations, Fig. 3 shows that ouralgorithm performs well, although the image size varies

13

Fig. 1 Computed fractional terms

a By using direct methodb By using recursive method

and thus its complexity is independent of the image sizebeing processed.

More precisely, the moments for the two images andfor maximum orders 10, 20, 30, 40, 50 have been computedand the CPU elapsed time in each case is presented inTable 1.

The outperformance of the recursive method can also bedefined if we calculate the percentage of the time reductiontaking place, as maximum order increases, by using thismethod as successor of the direct one.

For this reason, we define the computation timereduction, CTR (%) as follows

Computation time reduction CTR(%)

¼TimeDirect � TimeRecursive

TimeDirect� 100 (9)

By computing the CTR using (9), for the same experimentsof Fig. 3, Fig. 4 can be drawn, Fig. 4, verifies the benefits ofthe newly introduced algorithm, in terms of the CTR, whenbeing used instead of the direct method, for computing theFMMs of any image size.

The time reduction curves depicted in Fig. 4, for the twoimage sizes and for moment orders smaller than ten, are notidentical, as it was expected, because of the accuracy of the

Fig. 2 ‘Lena’ grey-level images

a 128 � 128b 64 � 64 pixels

14

Fig. 3 CPU elapsed time (ms) for various maximum momentorders

a 64 � 64 ‘Lena’ imageb 128 � 128 ‘Lena’ image

IET Comput. Vis., Vol. 1, No. 1, March 2007

timer used to measure the computational time. Whenthe computational time is quite short the accuracy of thetimer significantly influences it and as the time increasesthis impact is negligible.From this figure, it is impressive to conclude that we

have a very significant CTR, which for high orders goesup to almost over 80%, in comparison with the originalmethod.This is a major advantage of the recursive algorithm, as in

image representation one has to compute the moments of animage up to high orders, in order to optimally reconstruct it,with minimum reconstruction error.Conclusively, we can claim that by computing the FMMs

up to an order pmax of any sized image, using the proposedmethod, the four requirements declared in Section 3, areentirely satisfied.

5 Conclusion

A novel recursive algorithm has been proposed in this work,which computes in a fast way the OFMMs. The structure ofthe algorithm prevents also, overflow conditions to occur.The computational complexity of the proposed algorithmis linear O(p) in multiplications while the original directmethod is of O(p2) complexity. Therefore the computation

Table 1: The CPU elapsed time (ms) for computation ofOFMMs up to pmax order

Maximum order Lena image

64 � 64

Lena image

128 � 128

Direct

method

Recursive

method

Direct

method

Recursive

method

pmax ¼ 10 1760 640 7160 2570

pmax ¼ 20 14 260 3880 58 140 15 700

pmax ¼ 30 53 930 12 000 219 710 48 730

pmax ¼ 40 144 130 27 250 588 200 110 710

pmax ¼ 50 315 590 52 100 1286 920 211 970

Fig. 4 Percent of the computation time reduction

a 64 � 64 ‘Lena’ imageb 128 � 128 ‘Lena’ image

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time required to compute the moments of a high order, canbe reduced to almost over the 80% of the time needed by thedirect method to do the same work, for high-moment orders.Additionally, the algorithm does not generate and propagatefinite precision errors as some traditional recursive algor-ithms do. Also, it is capable to compute an individualmoment without needing the computation of intermediatemoments, for pattern classification purposes. The exper-imental results justify the effectiveness of this new algor-ithm and establish it, as an appropriate successor of thedirect method, used until now.

6 References

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2 The, C.-H., and Chin, R.T.: ‘On image analysis by the methods ofmoments’, IEEE Trans. Pattern Anal. Mach. Intell., 1998, 10, (4),pp. 496–513

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5 Khotanzad, A., and Hong, Y.H.: ‘Invariant image recognition byZernike moments’, IEEE Trans. Pattern Anal. Mach. Intell., 1990,12, (5), pp. 489–497

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7 Chong, C.W., Raveendran, P., and Mukundan, R.: ‘A comparativeanalysis of algorithms for fast computation of Zernike moments’,Pattern Recognit., 2003, 36, (3), pp. 731–742

8 Belkasim, S.O., Ahmadi, M., and Shridhar, M.: ‘Efficient algorithmfor fast computation of Zernike moments’, J. Franklin Inst., 1996,333(B), (4), pp. 577–581

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10 Prata, A., and Rusch, W.V.T.: ‘Algorithm for computation ofZernike polynomials expansion coefficients’, Appl. Opt., 1989, 28,pp. 749–754

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12 Chong, C.-W., Mukundan, R., and Raveendran, P.: ‘An efficientalgorithm for fast computation of pseudo-Zernike moments’. Proc.Int. Conf. Image and Vision Computing, New Zealand, 2001,pp. 237–242

13 Papaodysseus, C.N., Koukoutsis, E.B., and Triantafyllou, C.N.:‘Error sources and error propagation in the Levinson-Durbin algorithm’, IEEE Trans. Signal Process., 1993, 41, (4),pp. 1635–1651

14 Papaodysseus, C.N., Carayannis, G., Koukoutsis, E.B., andKayafas, E.: ‘Comparing LS FIR filtering and l-step aheadlinear prediction’, IEEE Trans. Signal Process., 1993, 41, (2),pp. 768–780

15 Papaodysseus, C., Koukoutsis, E., and Vassilatos, C.: ‘Errorpropagation and methods of error correction in LS FIR’, IEEETrans. Signal Process., 1994, 42, (5), pp. 1097–1108

16 Sheng, Y., and Shen, L.: ‘Orthogonal Fourier–Mellin moments forinvariant pattern recognition’, J. Opt. Soc. Am., 1994, 11,pp. 1748–1757

17 Kan, C., and Srinath, M.D.: ‘Invariant character recognitionwith Zernike and orthogonal Fourier–Mellin moments’, PatternRecognit., 2002, 35, (1), pp. 143–154

18 Papakostas, G.A., Boutalis, Y.S., Karras, D.A., and Mertzios, B.G.:‘On the reconstruction performance of compressed orthogonalmoments’. Proc. Int. Conf. Informatics in Control, Automation andRobotics, Setubal, Portugal, pp. 468–474

19 Terrillon, J.C., McReynolds, D., Sadek, M., Sheng, Y., andAkamatsu, S.: ‘Invariant neural-network based face detection withorthogonal Fourier–Mellin moments’. Proc. Int. Conf. PatternRecognition, 2000

20 Papakostas, G.A., Boutalis, Y.S., Papaodysseus, C.N., and Fragoulis,D.K.: ‘Numerical error analysis in Zernike moments computation’,Image Vis. Comput., 2006, 24, (9), pp. 960–969

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21 Papakostas, G.A., Boutalis, Y.S., and Mertzios, B.G.: ‘Evolutionaryselection of Zernike moment sets in image processing’. Proc. Int.Workshop Systems, Signals and Image Processing, Prague, CzechRepublic, September 2003

7 Appendix: Derivation of the recursive formulae

Let us define the fractional term of order p and index k,which is used in summation for computing the radialpolynomial Qp of (2) as follows

Tpk ¼( pþ k þ 1)!

( p� k)!k!(k þ 1)!(10)

The fractional term for the previous index of the summation(k2 1), has the form

Tp(k�1) ¼( pþ k)!

( p� k þ 1)!(k � 1)!k!(11)

Now, if we restrict our study for indices k = 0, we can usethe property of the factorials

n! ¼ (n� 1)! � n, n = 0 (12)

16

for transforming (11), as follows

Tp(k�1) ¼( pþ k)!

( p� k þ 1)!(k � 1)!k!

¼( pþ k þ 1)!k(k þ 1)

( pþ k þ 1)( p� k)!k!( p� k þ 1)k!(k þ 1)!

¼(pþ k þ 1)!

( p� k)!k!(k þ 1)!�

k(k þ 1)

( pþ k þ 1)( p� k þ 1)

¼ Tpk �k(k þ 1)

( pþ k þ 1)( p� k þ 1)(13)

Equation (13) can be written, in a more suitable form, as

Tpk ¼( pþ k þ 1)( p� k þ 1)

k(k þ 1)� Tp(k�1) (14)

which holds for k ¼ 1, 2, 3, . . .p, and is identical to (8a)introduced in Section 3.Finally, for k ¼ 0, (10) gives

Tp0 ¼( pþ 1)!

p!0!1!¼

p!( pþ 1)

p!¼ pþ 1 (15)

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