Discrete Orthogonal Moment Features Using Chebyshev Polynomials


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1 Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity Malacca, Malaysia. Istitute of Mathematical Scieces, Uiversity of Malaya Kuala Lumpur, Malaysia. 3 Faculty of Iformatio Techology, Multimedia Uiversity Cyberjaya, Malaysia Abstract This paper itroduces a ew set of momet fuctios based o Chebyshev polyomials which are orthogoal i the discrete domai of the image coordiate space. Chebyshev momets elimiate the problems associated with covetioal orthogoal image momets such as the Legedre momets ad the Zerike momets. The theoretical framework of discrete orthogoal momets is give, ad their superior feature represetatio capability is demostrated. Keywords: Image Momet Fuctios, Orthogoal Momets, Chebyshev Polyomials 1 Itroductio Momet fuctios are used i image aalysis as feature descriptors, i a wide rage of applicatios like object classificatio, ivariat patter recogitio, object idetificatio, robot visio, pose estimatio ad stereopsis. A geeral defiitio of momet fuctios Φ pq of order (p+q), of a image itesity fuctio f(x, ca be give as follows: Φ pq = x y Ψpq(x, f(x, dx dy, p, q = 0,1,,3... (1) where Ψ pq (x, is a cotiuous fuctio of (x, kow as the momet weightig kerel or the basis set. The simplest of the momet fuctios, with Ψ pq (x, = x p y q () were itroduced by Hu [1] to derive shape descriptors that are ivariat with respect to image plae trasformatios. Legedre ad Zerike momets were later itroduced by Teague [] with the correspodig orthogoal fuctios as kerels. These orthogoal momets have bee proved to be less sesitive to image oise as compared to geometric momets, ad possess far better feature represetatio capabilities. The iformatio redudacy measure is miimum i a orthogoal momet set. The computatio of orthogoal momets of images pose two major problems [3, 4, 6]: (i) The image coordiate space must be ormalized to the rage (typically, 1 to +1) where the orthogoal polyomial defiitios are valid. (ii) The cotiuous itegrals i (1) must be approximated by discrete summatios without loosig the essetial properties associated with orthogoality. This paper itroduces a ew set of momet fuctios based o Chebyshev (some times also writte as Tchebichef [5]) polyomials that are orthogoal i the discrete domai of the image coordiate space. Chebyshev momets completely elimiate the two problems referred 0
2 above, ad preserve all the theoretical properties, sice their implemetatio does ot ivolve ay kid of approximatio. The superiority of Chebyshev momets over covetioal orthogoal momets i terms of their feature represetatio capability ca be coclusively established by usig the iverse momet trasform. Images recostructed usig the iverse trasform of Chebyshev momets provide lower recostructio errors compared to those obtaied usig Legedre ad Zerike momets. Chebyshev Momets Give a NxN image, we first seek discrete orthogoal polyomials {t (x)} that satisfy the coditio N 1 tm( x) t( x) = ρ(, N) δm, m, = 0,1,, N 1. (3) x = 0 where ρ(,n) is the squared orm of the polyomial set t. The classical discrete Chebyshev polyomials[5] satisfy the property of orthogoality (3), with ρ( N, ) = N( N 1)( N )...( N + 1 ad have the followig recurrece relatio: + ), =0, 1,, N 1 (4) ( + 1) t 1( x) ( + 1)( x N + 1) t( x) + ( N ) t 1( x) = 0, =1,, N 1. (5) However, the Chebyshev polyomials as defied above together with their orms, become umerically ustable for large values of N. It ca be easily verified that the magitudes of t grow at the rate of N. We therefore further scale the Chebyshev polyomials t (x) by a factor N to make them suitable for image aalysis, ad defie the Chebyshev momets as follows (heceforth t (x) deotes the scaled polyomials): T pq = 1 N 1N 1 tp( x) tq( f ( x,, p, q = 0, 1, N 1. (6) ρ( pn, ) ρ( qn, ) x = 0 y = 0 where ρ( pn, ) = N N... 1 N p + 1 p N (7) ad the scaled Chebyshev polyomials t p (x) are computed usig the followig recurrece relatio: t p (x) = t 0 (x) = 1. (8) t 1 (x) = x + 1 N N (9) ( p 1) ( p 1) t1( x) tp 1( x) ( p 1) 1 t ( ) p x N, p>1 (10) p 1
3 The classical Chebyshev polyomials modified as above do ot lead to umerical overflows for large images. Both the polyomials ad the associated momets do ot show large variatio i the dyamic rage of values, as i the case of geometric momets. Sice the Chebyshev polyomials are exactly orthogoal i the discrete coordiate space of the image, we further have the followig theorem: Theorem: The image itesity fuctio f(x, has a polyomial represetatio give by N 1N 1 f(x, = Tpqtp ( x) tq ( (11) p= 0 q = 0 where the coefficiets T pq are the Chebyshev momets defied i (6). The above result follows whe the lefthad side of (3) is applied as a operator to both sides of equatio (6). 3 Image Feature Represetatio Equatio (11) is the iverse Chebyshev momet trasform, ad provides a image recostructio from a fiite set of its momets. The recostructed image is a measure of image features that are captured by the momet terms. The followig figure shows the origial image of the letter E o a 0x0 pixel grid (N=0), ad the recostructed images usig equatio (11), with the maximum order of momets varied from 5 through 11. Figure 1: Image recostructio usig Chebyshev momets There is also a close relatioship betwee Chebyshev ad Legedre momets arisig from the fact that the Chebyshev polyomial values ted to the values of the Legedre polyomials evaluated at the correspodig poits i the ormalized coordiate space [1,1], as the image size N teds to ifiity. Ideed, the discrete approximatio of Legedre momets[4] is very similar to the expressio for Chebyshev momets. Legedre momets λ pq of order (p+q) are defied as λ pq = N 1N 1 i= 0 j= 0 ( p+ 1)( q+ 1) i N + 1 P p N N 1 j N + 1 Pq N 1 f (, i j). (1) where P () deotes the Legedre polyomial of order. The biary image recostructio usig the above momets is give i Fig.. Figure : Image recostructio usig Legedre momets A compariso plot of rootmeasquare recostructio error obtaied from the above results is i Fig. 3. The superior feature represetatio capability of Chebyshev momets over Legedre momets is evidet from this figure. The higher recostructio error i Legedre momets is also partly due to the approximatio of cotiuous momet itegrals, which result i sigificat discretizatio errors whe the image size is small.
4 Recostructio Error Mea Square Error Chebyshev Legedre Max. Order of Momets Figure 3: Compariso of recostructio errors usig Chebyshev ad Legedre momets. 4 Computatioal Aspects The symmetry property of Chebyshev polyomials ca be made use of, to cosiderably reduce the time required for computig the associated momets. The scaled Chebyshev polyomials have the same symmetry property which the classical Chebyshev polyomials satisfy: t ( N 1 x) = ( 1) t ( x) (13) The above relatio suggests the subdivisio the domai of a NxN image (where N is eve) ito four equal parts, ad performig the computatio of the polyomials oly i the first quadrat where 0 x, y (N/ 1). The expressio for Chebyshev momets i (6) ca be modified with the help of (13), as follows: p f ( x, + ( 1) f ( N 1 x, ( N / ) 1 ( N / ) 1 1 q T pq = t p ( x) tq ( + ( 1) f ( x, N 1 (14) ρ( p, N) ρ( q, N) x= 0 y= 0 p+ q + ( 1) f ( N 1 x, N 1 I additio to reducig the computatio time by a factor of 4, the symmetry property is also useful i miimizig the storage required for the scaled Chebyshev polyomials. The scaled Chebyshev polyomial t (x) ca be expressed as a polyomial of x, as give below. The polyomial expasio is useful i relatig the Chebyshev momets to the discrete approximatio of geometric momets. k 1 ( i) i t (x)= Ck (, N) sk x, (15) β where k= 0 i= 0! 1 + k C k (, N) = ( 1) N k k k! k (16) ad () i s k are the Stirlig umbers of the first kid [7], which satisfies the equatio 3
5 x! = ( x k)! k i= 0 s () i k x i (17) 5 Noise Effects It is well kow that image oise sigificatly affects the recostructio error. Momets of higher orders ca become more sesitive to image oise. The effect of oise was aalyzed usig a 60x60 biary image of a Chiese character show i Fig. 4. The recostructed images with the maximum order of momets varied from 18 to 30 i steps of are also give i Fig. 4. Origial Image With Noise Image Recostructio Usig Chebyshev Momets Image Recostructio Usig Legedre Momets Figure 4: Recostructio of a image after addig oise. Recostructio Error 0.37 Mea Square Error Chebyshev Legedre Maximum Order of Momets Figure 5: Effect of image oise o recostructio error. 4
6 The rootmeasquare error of recostructio for both Chebyshev ad Legedre momets are plotted i Fig. 5. The performace of Chebyshev momets is still better compared to that of Legedre momets. Chebyshev momets of higher orders are also less sesitive to image oise as ca be see from Fig. 5. As the order of Legedre momets are icreased, the oise factor starts domiatig, ad causes a icrease i the recostructio error. 6 Coclusios The paper has preseted the theoretical framework for discrete orthogoal momets based o Chebyshev polyomials for a more accurate represetatio of image features tha those obtaied usig cotiuous momet fuctios. The motivatio for developig discrete orthogoal momets arises from the eed for circumvetig the commoly ecoutered problems of large discrete approximatio errors ad coordiate trasforms associated with Legedre ad Zerike momets. Images ca be accurately recostructed usig the iverse Chebyshev momet trasform. Image recostructio from momets also demostrate the superiority of the feature represetatio capability of Chebyshev momets over Legedre momets. Certai computatioal aspects of Chebyshev polyomials are also discussed. Refereces [1] M.K. Hu: Visual patter recogitio by momet ivariats, IRE Tras. o Iformatio Theory: 8 (196) [] M.R. Teague: Image aalysis via the geeral theory of momets, J. of Opt. Soc. of America 70 (1980), [3] R. Mukuda, K.R. Ramakrisha: Fast computatio of Legedre ad Zerike momets, Patter Recogitio 8 (1995) [4] R. Mukuda, K.R. Ramakrisha: Momet Fuctios i Image AalysisTheory ad Applicatios, World Scietific, Sigapore (1998). [5] Petr Beckma: Orthogoal Polyomials for Egieers ad Physicists, The Golem Press (1973).. [6] Sasoe: Orthogoal Fuctios, Dove Publicatios (1991). [7] Temme N.M, Special Fuctios, Joh Wiley & Sos, NY (1996). 5
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