Computation of Hahn Moments for Large Size Images

Transcription

1 Joural of Computer Sciece 6 (9): 37-4, ISSN Sciece Publicatios Computatio of Ha Momets for Large Size Images A. Vekataramaa ad P. Aat Raj Departmet of Electroics ad Commuicatio Egieerig, Quli Qutub Sa Govermet Polytecic College, Old City, Hyderabad, Adra Prades, Idia Departmet of Electroics ad commuicatio Egieerig, College of Egieerig, Osmaia Uiversity, Hyderabad-5 7, Adra Prades, Idia Abstract: Problem statemet: Direct formulas used for Ha momets computatio available i te literature are ot applicable for large size images. Tis is due to te eed for calculatio of factorial of large values as well as umerical istability i te calculatio of polyomial values. Tis study presets a metod to correct tese errors. Approac: I tis researc, recursive formulas for weigt fuctio, squared orm ad weigted Ha polyomials tat were used for Ha momets computatio were derived. Results: Simulatio results were reported for image recostructio ad te quality of recostructed image was assessed by usig uiversal image quality idex. Coclusio: Tis study preseted a procedure for calculatio of Ha momets for ay size images usig te recursive formulas. From te simulatio results, it was observed tat te derived recursive formulas work well for ay size images. Key words: Weigted Ha polyomials, Ha momets, recursive formula INTRODUCTION Ortogoal momets wic are extesively used for image recostructio ad classificatio problems are grouped ito eiter cotiuous or discrete. Geometric (Hu, 96), Legedre (Teague, 98) ad Zerike (Teague, 98) momets are cotiuous were as Tcebicef (Mukuda et al., ), Krawtcouk (Yap et al., 3) ad Ha momets (Yap et al., 7) are discrete. Discrete momets ave umber of advatages over cotiuous momets like o eed for coordiate trasformatio, better image recostructio ad o discretizatio error. Recetly, Yap et al. (7) proposed Ha momets usig discrete weigted Ha polyomials. Tey sowed tat Tcebicef ad Krawtcouk momets are special cases of Ha momets wit te appropriate parameter settigs. I order to compute Ha momets, te autors used direct formulas for obtaiig Ha polyomials, weigt fuctio ad squared orm as well as weigted Ha polyomials tat are applicable for small size images oly. We tese equatios are used for large size images, oe ecouters te followig problems (i) Need for calculatio of factorial of large values (ii) Numerical istability i te calculatio of polyomial values. Tese problems ca be solved by usig recursive formulas. Hece, tis study derives te recursive formulas for weigt fuctio, squared orm ad weigted Ha polyomials. Furter, simulatio results are reported for image recostructio usig te derived recursive formulas. MATERIALS AND METHODS Direct formulas for weigt fuctio, squared orm ad Ha polyomials as give i Table of (Yap et al., 7) are ot suitable for large size images sice it requires to calculate te factorial of large values. Furter, umerical istability occur i te calculatio of polyomial values ad terefore, umerical fluctuatios occur i te computatio of momet values. To overcome tese problems, we derive te recursive relatio for weigt fuctio, squared orm ad discrete weigted Ha polyomials. Table : Computed UIQI for Lea image UIQI wit Ha momets UIQI wit Ha momets Order for (α α β β ) for (α α β β ) (, ).3.4 (4, 4) (6, 6) (8, 8) (,) (,).8.8 Correspodig Autor: A. Vekataramaa, Departmet of Electroics ad Commuicatio Egieerig, Quli Qutub Sa Govermet Polytecic College, Old City, Hyderabad, Adra Prades, Idia 37

2 Weigted Ha polyomials (Yap et al., 7) are give by: ( β + N)! w(; α, β,n) () β!n! (x;,, N) α β (x; α, β,n) ρ(; α, β, N) () For α β, w(; α, β, N) is give by w(; α, β, N). i wic (x; α, β, N) are Ha polyomials wic are defied i terms of ypergeometric fuctio as: (x; α, β, N) 3F (,( + α + β + ), x;( α + ), N;) () were, 3F (.) is te geeralized ypergeometric fuctio give by: F (a,a,a ;b b ;z) 3 3, (a ) (a ) (a ) z k k k 3 k (3) k (b ) k (b ) k k! ad k is te Pocammer symbol give by: k a(a + )(a + )...(a + k ) (4) w(x, α, β, N) is te weigt fuctio give by: α + x β + N x w(x; α, β, N) x N x ad te squared orm ρ(; α, β, N) is give by: ( ) ( + α + β + ) N+ ( β + )! ρ(; α, β, N) ( + α + β + )( α + ) ( N) N! (5) (6) Recursive formula for weigt fuctio: Te weigt fuctio as give i (5) ca be writte as: ( α + x)!( β + N x)! w(x; α, β, N) x! α! β!(n x)! (7) Recursive formula for squared orm: By substitutig + i (6), we obtai: + ( ) ( + α + β + ) N+ ( β + ) + ( + )! ρ ( + ; α, β, N) ( + α + β + 3)( α + ) ( N) N! + + () Dividig () wit (6), we obtai te recursive formula for squared orm as: ρ ( + ; α, β, N) ( )( + )( + α + β + ) \ ( β + + )( + α + β + N + ) ρ(; α, β, N) ( N)( α + + )( α + β + + ) ( + α + β + 3) () Te startig value for te above recursive relatio ca be obtaiig by substitutig i (6) as: ( α + β + ) (;,, N) N + ρ α β (3) ( α + β + ) N! For α β, ρ(; α, β, N) is give by ρ(; α, β, N) N+. Recursive formula for weigted Ha polyomials: Te recursive formula for Ha polyomials wit respect to x is give by: Substitutig x x+ i te above equatio: ( α + x + )!( β + N x )! w(x + ; α, β, N) (x + )! α! β!(n x )! (8) (x + N)(x + α + ) (x + ; α, β, N) (4x β β x + x + + 3N By dividig (8) wit (7), we obtai: for x,,,n- wit (; α, β,n) ad ( α + x + )(N x) w(x + ; α, β, N) (9) ( + α + β + ) (; α, β, N) ( β + N x)(x + ) N( α + ) as iitial values. I order to derive te recursive formula for Te startig value for te above recursio ca be weigted Ha polyomials, we multiply te above obtaiig by substitutig x i (7) as: equatio by: 38 Nx α N + β + + α + α + α x) (x + ; α, β, N) (x + )(x β N) (x; α, β, N) (4)

3 (x + N)(x + α + ) ρ(; α, β, N) (x + ; α, β, N) (4x β β x + x ρ(; α, β, N) 3N Nx N x) + + α + β + + α + α + α (x + ; α, β, N) (x + )(x β N) ρ(; α, β, N) (x; α, β, N) (4x β β x + x ρ(; α, β, N) + + α + β + + α + α + α 3N Nx N x) (x + ; α, β, N) w(x + ; α, β, N)w(x +, α, β,n) ρ(; α, β, N)w(x +, α, β,n) (x + )(x β N) (x; α, β, N) (x N)(x ) (x ;,, N) ρ(; α, β, N) + + α + + α β (4x β β x + x + + 3N Nx α N + β + x) (x ;,, N) + α + α + α + α β (x )(x N) (x;,,n) + β α β w(x +, α, β, N) w(x + ; α, β,n) w(x; α, β,n) (5) By usig te recursive formula for weigt fuctio as give i (9): ( α + x + )(N x ) ad w(x + ; α, β, N) ( β + N x )(x + ) w(x + ; α, β,n) ( α + x + )(N x )( α + x + )(N x) ( β + N x )(x + )( β + N x)(x + ) Substitutig te above values i (5), we obtai te recursive formula for weigted Ha polyomials as: (x N)(x ) (x ;,,N) + + α + + α β (4x x x 3N Nx β β α N + β + + α + α + αx) ( α + x + )(N x ) (x + ; α, β, N) (x + ) ( β + N x )(x + ) ( α + x + )(N x ) ( α + x + )(N x) ( β + N x )(x + ) (x β N) (x; α, β,n) ( β + N x)(x + ) (6) Te iitial values for te above recursive formula ca be obtaied usig () as: w(; α, β,n) α β (; α, β,n) ρ(; α, β,n) (7) w(; α, β, N) ρ (; α, β, N) (;,,N) ad (; α, β, N) (; α, β, N) (; α, β, N) Usig (9), we obtai: ad we ave: w(; α, β,n) ρ(; α, β, N) w(; α, β, N)w(; α, β,n) w(; α, β,n) ρ(; α, β, N) w(; α, β, N) ( α + )N w(; α, β, N) β + N ( + α + β + ) (; α, β, N) N( α + ) Substitutig te above values i (8), we get: ( + α + β + ) N( α + ) (; α, β, N) ( ) α + β + N ( )N (; α, β,n) RESULTS (8) (9) I order to test te validity of te derived recursive formulas, we calculate Ha momets (α α β β ) up to order (,) for a Lea image of size as well as Circbw image of size 7 7. To compute Ha momets, we use te derived recursive formulas for weigt fuctio, squared orm ad Weigted Ha polyomials. Furter, we also use te symmetry property of weigted Ha polyomials ad matrix form represetatio of momets to reduce te computatioal complexity. Te matematical expressio correspodig to symmetry property is give by (x; α, β,n ) ( ) (N x; α, β, N ). To compute te weigted Ha polyomials for a image of size N N, oe eed to compute weigted Ha polyomials for oe quadrat x,y (N/-) ad 39

4 apply te symmetry property to get te weigted Ha polyomials for oter quadrats. Te matrix form represetatio for Ha momets is give by: H AFB T ad image recostructio usig Ha momets is give by: F T A HB I tese expressios, te matrices H, A, B ad F are give by: H i N max, j Mmax H ij i, j [ ] F f (i, j) i N, j M i, j A i (j; α, β, N ) B i (j; α, β,m ) i N max, j N i, j i M max, j M i, j Fig. : Recostructed Lea images usig up to various orders of Ha momets (α α β β ) order (4, 4) order (8, 8) order (, ) T deotes te traspose of matrix. Te images are recostructed usig Ha momets up to order (4, 4), (8, 8) ad (,) ad te recostructed images are sow i Fig. ad. I order to assess te quality of recostructed image, we also compute Uiversal Image Quality Idex (UIQI) (Wag ad Bovik, ) ad te results are etered i Table ad. DISCUSSION It is observed from te UIQI results tat tis value approaces to uity as te order of momets used for image recostructio icreases. Te image recostructio is repeated usig Ha momets wit parameters (α α β β ) ad te results are sow i Fig. 3 ad 4, Table ad. It is observed tat te recostructed results deped ot oly o te umber of momets selected but also o te parameter values. For te give order of momets, as te value of parameters α, α, β, β deviates o eiter side from uity, te quality of recostructed image decreases. Tis is due to te variatios i te amplitude of te polyomial values. Fig. : Recostructed Circbw images usig up to various orders of Ha momets (α α β β ) order (4, 4) order (8, 8) order (, ) 4 Table : Computed UIQI for Circbw image UIQI wit Ha momets UIQI wit Ha momets Order for (α α β β ) for (α α β β ) (, ).6.6 (4, 4).5.5 (6, 6).. (8, 8).9.3 (,) (,).4.4

5 CONCLUSION Tis study preseted a procedure for calculatio of Ha momets for ay size image usig te recursive equatios. Tese equatios avoid te eed for calculatio of factorial of large values ad also avoids te umerical istability, wic occur i te polyomial calculatios. REFERENCES Fig. 3: Recostructed Lea images usig up to various orders of Ha momets (α α β β ) order (4, 4) order (8, 8) order (, ) Hu, M.K., 96. Visual patter recogitio by momet ivariats. IRE Tras. Iform. Teory, 8: DOI:.34/ Mukuda, R., S.H. Og ad P.A. Lee,. Image aalysis by Tcebicef momets. IEEE Tras. Image Proc., : DOI:.9/ Teague, M.R., 98. Image aalysis via te geeral teory of momets. J. Optic. Soc. Am. 7: DOI:.364/JOSA.7.9 Wag, Z. ad A.C. Bovik,. A uiversal image quality idex. IEEE Sig. Proc. Lett., 9: DOI:.75/JHM563. Yap, P.T., R. Paramesa ad S.H. Og, 3. Image aalysis by Krawtcouk momets. IEEE Tras. Image Proc., : DOI:.9/3.43 Yap, P.T., R. Paramesa ad S.H. Og, 7. Image aalysis usig Ha momets. IEEE Tras. Patt. Aal. Mac. Itel., 9: DOI:.9/TPAMI Fig. 4: Recostructed Circbw images usig up to various orders of Ha momets (α α β β ) order (4, 4) order (8, 8) order (, ) 4