Fast Polar Harmonic Transforms

Paper

† †† † Zhuo YANG , Alireza AHRARY (Member), Sei-ichiro KAMATA † Graduate School of Information, Production and Systems, Waseda University †† Department of Informatics, Nagasaki Institute of Applied Science

〈Summary〉 Polar Harmonic Transform (PHT) is termed to represent a set of transforms those kernels are basic waves and harmonic in nature. PHTs consist of Polar Complex Ex- ponential Transform (PCET), Polar Cosine Transform (PCT) and Polar Sine Transform (PST). They are proposed to represent invariant image patterns for two dimensional image retrieval and pattern recognition tasks. They are demonstrated to show superiorities comparing with other methods on describing rotation invariant patterns for images. Ker- nel computation of PHTs is also simple and has no numerical stability issue. However in order to increase the computation speed, fast computation method is needed especially for real world applications like limited computing environments, large image databases and realtime systems. This paper presents Fast Polar Harmonic Transforms (FPHTs) including Fast Polar Complex Exponential Transform (FPCET), Fast Polar Cosine Transform (FPCT) and Fast Polar Sine Transform (FPST) that are deduced based on mathematical properties of trigonometric functions and number theory. The proposed FPHTs are averagely over 10 times faster than PHTs that signiﬁcantly boost computation process. The experimental results on both synthetic and real data are given to illustrate the eﬀectiveness of the proposed fast transforms. Keywords: Fast Polar Harmonic Transform, Fast Polar Complex Exponential Transform, Fast Polar Cosine Transform, Fast Polar Sine Transform, image retrieval, pattern recognition

tation invariant patterns. These methods are used 1. Introduction in many real world applications like character recog- Rotation invariant pattern representation is one of nition6) and multi-spectral texture analysis 7).But the essential challenges in image retrieval and pat- computations of their kernels involve a number of fac- tern recognition arises from the fact that in many real torial terms and suﬀer from numerical stability. Po- world applications, images should be considered to be lar Complex Exponential Transform (PCET), Polar the same even if they are rotated. There are mainly Cosine Transform (PCT) and Polar Sine Transform two kinds of features to represent rotation invariant (PST) are proposed as a set of Polar Harmonic Trans- patterns. Non-orthogonal rotation invariant features forms (PHTs) to represent two dimensional images like rotational moment 1) and complex moments 2),3) and demonstrated to show superiorities comparing are proposed. But these methods are not orthogo- with other methods 8). With the orthogonal property, nal that means lack of information compactness on PCET, PCT and PST can transform the image func- pattern representation. Orthogonal rotation invari- tion to a set of mutually independent patterns with ant features like Zernike moment 4) and Orthogonal minimum redundancy and maximal discriminant in- Fourier-Mellin moment 5) are proposed to describe ro- formation.

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PHTs are defined on a unit circle. Unfortunately, bases, PHTs can generate patterns that are used for kernel generation of these transforms involves many rotation invariant representation. Applications like trigonometric functions to compute angular and ra- image retrieval and pattern recognition benefit from dial parts that no fast method has been reported to PHTs. Kernel computation of PHTs has no numer- the best of our knowledge. The high computational ical stability issue but suffers from high computa- complexity is the constraint for real world applica- tion complexity problem. This section introduces the tions such as realtime systems, limited computing background of PCET, PCT and PST. environments and large image databases. Therefore, 2.1 Polar Complex Exponential Transform reduction of the computational complexity for PHTs Given a 2D image function f(x, y), it can be trans- is very significant. formed from cartesian coordinate to polar coordinate This paper proposes fast PHTs. Fast and com- f(r, θ),wherer and θ denote radius and azimuth re- pact methods to compute the kernel coefficients of spectively. The following equations transform from PHTs are proposed by using mathematical proper- cartesian coordinate to polar coordinate, ties of trigonometric functions and number theory. r = x2 + y2, (1) The kernel function of PHTs has symmetry properties and y with respect to the x axis, y axis, y = x line, y = −x θ =arctan . (2) x line and origin that can be used for fast computa- The transform involves points within the largest in- tion. The computational complexity of PHTs can ner circle of the image. After normalization(detail in be reduced by calculating half of the first quadrant. subsection 3.3), it is defined on the unit circle that That is only one eighth of the direct transform. An r ≤ 1, and can be expanded with respect to the basis even more efficient kernel generation method based functions Hnl(r, θ)as on relative prime number theory is proposed to com- ∞ ∞ f(r, θ)= M nlHnl(r, θ), (3) pute PHTs after analyzing the point distribution on n=−∞ l=−∞ two dimensional discrete space. Much more symmet- where the coefficient is 2π 1 1 ∗ ric points are involved to compute simultaneously in M nl = f(r, θ)Hnl(r, θ)rdrdθ. (4) π 0 0 order to significantly accelerate the transform speed. The basis function is given by ilθ These fast transforms are named as Fast Polar Har- Hnl(r, θ)=Rn(r)e , (5) monic Transforms (FPHTs) that consist of Fast Polar where 2 i2πnr Complex Exponential Transform (FPCET), Fast Po- Rn(r)=e , (6) lar Cosine Transform (FPCT) and Fast Polar Sine and satisfying orthogonality condition 1 Transform (FPST). ∗ 1 Rn R (r) n (r)rdr = δnn , (7) The organization of this paper is as follows. The 0 2 and basic theories of PCET,PCT and PST including 2π 1 ∗ mathematics descriptions are provided in Section 2. Hnl(r, θ)Hn l (r, θ)rdrdθ = πδnn δll , 0 0 The proposed method is presented in Section 3 af- (8) ter given the mathematical properties of trigonomet- where δij is Kronecker delta. Rewrite (4) with (5) ric functions and sufficient number theory knowledge. and (6), In Section 4, the performance of FPHTs and PHTs 2π 1 1 2 M nl = f(r, θ)(cos (2πnr + lθ) are compared against both synthetic and real images. π 0 0 . 2 The experimental results illustrate the effectiveness −i sin (2πnr + lθ))rdrdθ of our proposed FPHTs. Finally, Section 5 concludes (9) this study. |Mnl| is rotation invariant. Figure 1 show the real

and imaginary parts of the basis functions Hnl(r, θ) 2. Background under diﬀerent n,l values. It takes two trigonometric PHTs are a set of transforms that consist of PCET, operations and one inverse trigonometric operation to PCT and PST. Based on set of orthogonal projection generate PCET kernel coeﬃcient for each point.

400 Paper : Fast Polar Harmonic Transforms

Fig. 1 The basis functions Hnl(r, θ) of PCET (Left is real parts and right is imaginary parts)

S 2 2.2 Polar Cosine Transform and Polar Sine Rn(r)=sin(πnr ). (19) Transform Rewrite (17) with (18) and (19), Polar Cosine Transform is given by 2π 1 ∞ ∞ S 2 C C M nl =Ωn f(r, θ)sin(πnr ) f(r, θ)= M nlHnl(r, θ), (10) 0 0 . n=0 l=−∞ (cos (lθ) − i sin (lθ))rdrdθ where the coefficient is (20) 2π 1 C C∗ M =Ωn f(r, θ)H (r, θ)rdrdθ. (11) C nl nl PCT and PST are defined on unit circle. M and 0 0 nl S The basis function is given by Mnl are rotation invariant. PCT and PST need 3 C C ilθ Hnl(r, θ)=Rn (r)e , (12) trigonometric functions to generate kernel coefficient where for each point. C 2 Rn (r)=cos(πnr ), (13) 3. Fast Polar Harmonic Transforms and ⎧ Polar Harmonic Transforms provide orthogonal ⎪ 1 ⎨⎪ π if n =0 bases for representing rotation invariant patterns. Ωn = (14) ⎪ They mainly need trigonometric functions to gener- ⎩⎪ 2 π if n =0 ate the kernel coefficients. Mathematical properties Rewrite (11) with (12)-(14), of trigonometric functions can be employed to accel- 2π 1 erate the computation process. Fast PHTs are in- M C f 2 nl =Ωn (r, θ)cos(πnr ) troduced in subsection 3.1 and 3.2. The proposed 0 0 . (cos (lθ) − i sin (lθ))rdrdθ method is firstly represented on PCT and PST, and (15) then is deduced for PCET. Inspired by number the- Similarly, Polar Sine Transform is given by ory, faster PHTs are given in subsection 3.3 and fi- ∞ ∞ nally named as FPHTs. f(r, θ)= M S HS (r, θ), (16) nl nl 3.1 Fast PCT and PST n=1 l=−∞ From Polar Cosine Transform equation (15), we can where the coefficient is 2π 1 S S∗ find that for the points on the same radius r, the dif- M nl =Ωn f(r, θ)Hnl (r, θ)rdrdθ. (17) 0 0 ferent integrand part for each point is f(r, θ)(cos lθ − The basis function is given by i sin lθ). As Figure 2 shown, point (x, y)isapoint S RS ilθ Hnl(r, θ)= n(r)e , (18) in first quadrant between y = x and x axis, has other where seven symmetric points with respect to x axis, y axis,

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Similar relationships also exist for cosine function and other l values. For the eight symmetric points on the same radius r, if their PCT coeﬃcients can be calculated simultaneously, then the computation time for trigonometric function can be reduced.

Based on foregoing discussion, fast PCT is given by C 2 2 F astM nl=Ωn cos(πn(x + y )) D , (28) (Gl(x, y) − iHl(x, y))dxdy where

D = {(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ x, 0 ≤ x2 + y2 ≤ 1}, Fig. 2 2D space symmetric points (29) Table 1 (r,θ) and its symmetric points Polar Coordinate Cartesian Coordinate and Gl(x, y)andHl(x, y) are given in equation (30) (r,θ) (x,y) and (31). By using this equation, the whole PCT can π (r, 2 − θ )(y,x) be generated by using part of the basic functions. (r, π + θ )(-y,x) 2 Similarly, fast PST is given by (r,π − θ )(-x,y) S 2 2 F astM nl=Ωn sin(πn(x + y )) (r,π + θ )(-x,-y) D . (32) (r, 3π − θ )(-y,-x) 2 (Gl(x, y) − iHl(x, y))dxdy (r, 3π + θ )(y,-x) 2 By directly computing the PCT/PST kernel, it (r,2π − θ )(x,-y) takes 24 trigonometric functions for symmetric eight points. By using fast PCT/PST equation (28) and y = x, y = −x and origin. (32), computational complexity is reduced, only one Their polar and cartesian coordinates are shown in eighth of the trigonometric is computed. Table 1. 3.2 Fast PCET As known sin(θ)andcos(θ) functions are periodic From equation(9) we can find that for the points functions with period 2π. Periods for sin(lθ)and with the same radius r, the different integrand cos(lθ)are2π/l. Derived from the periodic and sym- part for each point is f(r, θ)(cos (2πnr2 + lθ) − metric properties of trigonometric functions that used i sin (2πnr2 + lθ)). Inspired by fast PCT and PST, in fast Fourier transform 9), there are mathematical we want to simultaneously compute symmetric eight relationships for trigonometric functions respect to points of PCET. different l. For example, if l is divided by 4 with By using mathematical property of trigonometric remainder 3 that means mod(l, 4) = 3, following re- functions, we have lationship for sine function can be deduced cos (2πnr2 + lθ)=cos(2πnr2)cos(lθ) , (33) − sin (2πnr2)sin(lθ) π sin l − θ = − cos(lθ), (21) 2 and π sin (2πnr2 + lθ)=sin(2πnr2)cos(lθ) sin l + θ = − cos(lθ), (22) . (34) 2 +cos(2πnr2)sin(lθ) sin(l(π − θ)) = sin(lθ), (23) Fast PCET is given by − 1 2 2 sin(l(π + θ)) = sin(lθ), (24) F astM nl= (cos(2πn(x + y ))Gl(x, y) π 3π − D sin l θ =cos(lθ), (25) 2 2 2 − sin(2πn(x + y ))Hl(x, y)) . 2 2 3π −i(sin(2πn(x + y ))Gl(x, y) sin l + θ =cos(lθ), (26) 2 2 2 +cos(2πn(x + y ))Hl(x, y))dxdy sin(l(2π − θ)) = − sin(lθ). (27) (35)

402 Paper : Fast Polar Harmonic Transforms ⎧ ⎪ ⎪(f(x, y)+f(y, x)+f(−y, x)+f(−x, y) ⎪ ⎪ ⎪ − − − − − − ⎪+f( x, y)+f( y, x)+f(y, x)+f(x, y)) cos(lθ)ifmod(l, 4) = 0 ⎪ ⎪ ⎪(f(x, y) − f(−x, y) − f(−x, −y)+f(x, −y)) cos(lθ) ⎪ ⎨⎪ +(f(y, x) − f(−y, x) − f(−y, −x)+f(y, −x)) sin(lθ)ifmod(l, 4) = 1 Gl(x, y)=⎪ (30) ⎪(f(x, y) − f(y, x) − f(−y, x)+f(−x, y) ⎪ ⎪ ⎪ − − − − − − − − ⎪+f( x, y) f( y, x) f(y, x)+f(x, y)) cos(lθ)ifmod(l, 4) = 2 ⎪ ⎪ ⎪(f(x, y) − f(−x, y) − f(−x, −y)+f(x, −y)) cos(lθ) ⎪ ⎩⎪ −(f(y, x) − f(−y, x) − f(−y, −x)+f(y, −x)) sin(lθ)ifmod(l, 4) = 3

⎧ ⎪ ⎪(f(x, y) − f(y, x)+f(−y, x) − f(−x, y) ⎪ ⎪ ⎪ − − − − − − − − ⎪+f( x, y) f( y, x)+f(y, x) f(x, y)) sin(lθ)ifmod(l, 4) = 0 ⎪ ⎪ ⎪(f(x, y)+f(−x, y) − f(−x, −y) − f(x, −y)) sin(lθ) ⎪ ⎨⎪ +(f(y, x)+f(−y, x) − f(−y, −x) − f(y, −x)) cos(lθ)ifmod(l, 4) = 1 Hl(x, y)=⎪ (31) ⎪(f(x, y)+f(y, x) − f(−y, x) − f(−x, y) ⎪ ⎪ ⎪ − − − − − − − − ⎪+f( x, y)+f( y, x) f(y, x) f(x, y)) sin(lθ)ifmod(l, 4) = 2 ⎪ ⎪ ⎪(f(x, y)+f(−x, y) − f(−x, −y) − f(x, −y)) sin(lθ) ⎪ ⎩⎪ −(f(y, x)+f(−y, x) − f(−y, −x) − f(y, −x)) cos(lθ)ifmod(l, 4) = 3

Direct PCET transform need 16 trigonometric They are defined 10) by functions to generate the kernel coefficients for the a⊥b, if gcd(a, b)=1. (37) eight symmetric points, but by using fast PCET Conventionally 1 is relative prime to any other posi- equation (35) , only four trigonometric functions is 11) tive integer . needed. 1⊥a, if a ∈ N. (38) 3.3 Relative Prime and FPHTs Given an N × N size image, there are two steps Foregoing proposed algorithm significantly boost needed to transform from image conventional carte- the computation speed. Whether it is possible to sian coordinate to PHTs defined normalized unit co- make PHTs much faster is an interesting question. 10) 12) ordinate. First, move the origin from left upper cor- Inspired by number theory ～ , this subsection ner of image to the center. The transform equation presents even faster PHTs that involve much more of a point P (Xp,Yp) from original coordinate to its symmetric points calculated simultaneously, and fi- corresponding centered coordinate (Xc,Yc)isgiven nally named as Fast Polar Harmonic Transforms by (FPHTs). Given two integers a and b, with at least one of CartesianT oCenter(Xp,Yp) N − 1 N − 1 these being nonzero. The largest positive integer that = Xp − , − Yp =(Xc,Yc). 2 2 divides both a,b is termed as the greatest common (39) divisor of a and b. Second, the centered coordinate is normalized to unit. gcd(a, b). (36) The transform equation from centered coordinates to Here are some examples gcd(2, 6) = 2,gcd(3, 5) = 1 normalized is and gcd(3, 8) = 1. 2Xc 2Yc CenterToUnit(Xc,Yc)= , =(x, y), Given two integers a and b, they are said to be N − 1 N − 1 relative prime if their greatest common divisor is 1. (40)

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Fig. 3 Odd and even number size image mapping in the first quadrant and its reverse transform equation is Table 2 Distribution of Relative Prime Points in Odd Number Size Image UnitToCenter (x, y) (N − 1)x (N − 1)y (41) Radius Number of Relative Prime Points Probability = , =(Xc,Yc). 2 2 1-200 9544 0.614434 For example in 21×21 size image, cartesian coordi- 201-400 28657 0.610321 nates (Xp,Yp)are(12, 9), (14, 8) and (16, 7) . Based 401-600 47746 0.609277 on equation (39), after moving origin to center of im- 601-800 66847 0.608879 age their coordinates (Xc,Yc) equal to (2, 1), (4, 2) 801-1000 85927 0.608544 and (6, 3). Based on equation (40), after normalized 1001-2000 716216 0.608383 to unit their coordinates (x, y)are(0.2, 0.1), (0.4, 0.2) 2001-3000 1193638 0.608184 and (0.6, 0.3). Figure 3 shows the first quadrant of 21 × 21 size image after mapping to unit circle. We Table 3 Distribution of Relative Prime Points in define (x, y) is a⎧ relative prime point if satisfied ⎨ Even Number Size Image Xc⊥Yc, if N is odd rpp(x, y)=⎩ . (42) Radius Number of Relative Prime Points Probability 2Xc⊥2Yc, if N is even 1-199 3186 0.818392 Given a relative prime point (x, y), for odd 201-399 9550 0.813043 number size image, the points set in same angle 401-599 15918 0.811977 can be represented by {(mx, my)|m ∈ N}, for even 601-799 22273 0.811491 number size image, they can be represented by 801-999 28655 0.811412 {((2m − 1)x, (2m − 1)y)|m ∈ N}. The relative prime 1001-1999 238738 0.811066 points distributions of odd number size image and 2001-2999 397882 0.810857 even number size image are different as shown in Fig- ure 3. When generating kernel of PHTs, no need to generate the angular part if a point is not a relative large number of points are not relative prime points, prime point. Table 2 gives a distribution of relative that means their angular part is not needed to com- prime points within a circle with different size radius. puted when generating the PHTs kernel coefficients. Table 3 gives a distribution of relative primes but Based on foregoing discussion, faster PCT is given only for odd numbers. This is useful for even number by size image. K 12) C 2 2 2 There is theoretical proof to show the probabil- F asterM nl=Ωn {cos(πnk (x + y )) k , ity of two randomly given integers A =1 1 6 (Gl(kx, ky) − iHl(kx, ky)}dxdy p(a⊥b)= = ≈ 0.607927102 ≈ 61%, (43) 2 ζ(2) π (44) where ζ(z) refers to the Riemann zeta function. From Table 2, Table 3 and equation (43), we can find that where

404 Paper : Fast Polar Harmonic Transforms

Table 4 Computation Complexity to Generate PHTs Kernel Functions PHTs DM SM RM PCET 16 4 2+2p Trigonometric PCT 24 3 1+2p PST 24 3 1+2p PCET 8 1 p Inverse Trigonometric PCT 8 1 p PST 8 1 p p=probability of relative prime points

coordinates using equation (1) and (2), and trigonometric function is needed to compute the coeﬃcients of PHTs. We use DM to denote direct transform PHT Fig. 4 symmetric relative prime points method (section 2), SM to denote fast PHT method 1 that using symmetric properties (subsection 3.1 and K = , (45) x2 + y2 3.2). and RM to denote the proposed FPHT method and that use relative prime points (subsection 3.3). As A = {(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ x, , (46) a summary , computation complexities in terms of 2 2 0 ≤ x + y ≤ 1,rpp(x, y)} number of trigonometric function and inverse trigono- x is ﬂoor function that return integral part of x. metric function needed to generate kernel for sym- Given a point (x, y) that is a relative prime point, metric eight points is given in Table 4.FPHTwhich then by multiplying a factor k all the coordinates is based on RM is the fastest. (kx, ky) that are in the same angle can be obtained. 4. Experimental Results Fig 4 gives an example to compute 24 points together. Faster PST is given by The performance of the proposed fast transforms K for PCET, PCT and PST in computation reduction S 2 2 2 F asterM nl=Ωn {sin(πnk (x + y )) is validated through comparative experiments using A k=1 . various images. DM, SM and RM based transforms (Gl(kx, ky) − iHl(kx, ky))}dxdy are all evaluated. Both synthetic images and real (47) images are used in the experiments. Images with dif- Similarly faster PCET is given by ferent resolutions and contents are tested to illustrate

K the eﬀectiveness and feasibility of the proposed fast 1 F asterM nl= transforms. PC environment (Celeron 1.86G Hz, 2 G π A k=1 Memory) is used to perform the experiments. Algo- 2 2 2 {(cos(2πnk (x + y ))Gl(kx, ky) . rithms are implemented by C++ and complied by gcc − 2 2 2 sin(2πnk (x + y ))Hl(kx, ky)) 4.3.2 on Linux 2.6.26. − 2 2 2 i(sin(2πnk (x + y ))Gl(kx, ky) 4.1 Synthetic Images 2 2 2 } +cos(2πnk (x + y ))Hl(kx, ky)) dxdy In this experiment, synthetic images which are gray (48) scale in format are used. They are generated using By using equation (44),(47) and (48) much more sym- the equation given below, metric points are involved, and only small number of f(i, j)=round[random(N,N)], (49) computation is needed to generate the kernel. We 0 ≤ f(i, j) ≤ 255, ∀i, j, name (44),(47) and (48) as FPCT, FPST and FPCET where f(i, j) is the image function which its pixels respectively. integer in values, N × N is the image resolution and 3.4 Computation Complexity Analysis i,j are the indices of the image pixels. Given a two dimensional function f(x, y), inverse These synthetic images are varied in resolution and trigonometric function is needed to compute the polar content. The purpose is to demonstrate fast trans-

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Fig. 5 Real images forms are independent from image content and hence take 8% and 9% time compared with PCT/PST re- robust evaluation can be attained from the experi- spectively. FPCET takes about 10% and 12% time mental results. The PCET, PCT and PST are per- for odd and even number size image compared with formed from the synthetic images. 20 coefficients are PCET respectively. computed for PHTs. With same computation result, Fast algorithm is crucial for applications like lim- DM, SM and RM based transforms have different run- ited computing environments, large image databases ning time. Both odd and even number size images are and realtime systems. By using the proposed RM tested because relative prime point distributions for based transform, the computation speed is signifi- them are different. Their computation performances cantly boosted and helps the success of applications in terms of CPU elapsed time are given in Table 5. that need rotation invariant patterns. The results from Table 5 show significant reduc- tions in average CPU elapsed time for PHTs. Com- 5. Conclusions putation time is averagely 90% saved for PHTs and similar with the theoretical analysis in Section 3. In this paper, we propose Fast Polar Harmonic Transforms. By using the symmetric properties 4.2 Real Images and mathematical properties of trigonometric func- The performance tests for DM,SM and RM based tions, the proposed methods only calculate one eighth transforms are also carried out for real images. We trigonometric functions to generate the PHTs kernels. use standard image for test as shown in Figure 5. The proposed methods also employ relative prime The real images are resized to 512×512 and 511×511 points to accelerate the speed. Based on the num- to make up test data sets A and B. Based on PHTs ber theory theorem, large number of trigonometric equations, only points within the largest inner circle functions are saved when calculating points in the of images are involved in transforms. Different num- same angle. FPCT and FPST are about 12.5and11 bers of PHTs coefficients are computed for these two times faster than PCT and PST for odd and even test data sets. The test results for test data sets A number size image respectively. FPCET is about and B are shown in Table 6 and Table 7 respec- 10 and 8 times faster than PCET for odd and even tively. number size image respectively. Comprehensive ex- We can observe that both SM and RM based trans- perimental results are also given on both synthetic form are effective compared with direct transform. and real images to illustrate their effectiveness. Wide FPHT ( RM based transform ) performs best. For range of real world applications that need Fast Polar odd and even number size image, FPCT/FPST only Harmonic Transforms including Fast Polar Complex

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Table 5 CPU Elapsed Time for Synthetic Images PHTs Resolution DM SM RM SM/DM RM/DM 200*200 3.357 0.561 0.406 0.167 0.121 PCET 199*199 3.311 0.539 0.337 0.163 0.102 200*200 3.659 0.468 0.337 0.128 0.092 PCT 199*199 3.639 0.451 0.273 0.124 0.075 200*200 3.622 0.459 0.326 0.127 0.090 PST 199*199 3.671 0.474 0.287 0.129 0.078 DM=Direct Method, SM=Symmetric Method, RM=Relative Prime Method

Table 6 CPU Elapsed Time for Data Set A PHTs Coeﬃcients DM SM RM SM/DM RM/DM 10 21.244 3.505 2.549 0.165 0.120 PCET 20 42.777 7.229 5.219 0.169 0.122 30 63.279 10.631 7.720 0.168 0.122 10 23.298 3.005 2.167 0.129 0.093 PCT 20 46.921 5.912 4.270 0.126 0.091 30 70.179 8.842 6.316 0.126 0.090 10 23.182 2.967 2.133 0.128 0.092 PST 20 46.623 5.874 4.243 0.126 0.091 30 69.598 8.699 6.264 0.125 0.090 DM=Direct Method, SM=Symmetric Method, RM=Relative Prime Method

Table 7 CPU Elapsed Time for Data Set B PHTs Coeﬃcients DM SM RM SM/DM RM/DM 10 21.395 3.573 2.204 0.167 0.103 PCET 20 42.575 7.195 4.470 0.169 0.105 30 63.896 10.606 6.517 0.166 0.102 10 23.313 3.007 1.888 0.129 0.081 PCT 20 46.518 6.000 3.675 0.129 0.079 30 69.798 8.864 5.514 0.127 0.079 10 23.211 2.994 1.857 0.129 0.080 PST 20 46.664 5.973 3.686 0.128 0.079 30 69.404 8.745 5.483 0.126 0.079 DM=Direct Method, SM=Symmetric Method, RM=Relative Prime Method

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ed., Springer Press (2004). Sei-ichiro Kamata received 11) J. Dence and T. Dence: “Elements of the Theory of Num- the M.S. degree in computer bers”, Harcourt Academic Press (1999). science from Kyushu University, 12) G. Hardy and E. Wright: “An Introduction to the Theory Fukuoka, Japan, in 1985, and of Numbers”, 6th ed., Oxford University Press (2008). the Doctor of Computer Science, Kyushu Institute of Technology, (Received Feb. 5, 2010) Kitakyushu, Japan, in 1995. From (Revised May 20, 2010) 1985 to 1988, he was with NEC, Ltd., Kawasaki, Japan. In 1988, he joined the faculty at Kyushu In- Zhuo Yang received the B.E. stitute of Technology. From 1996 and M.E. degree in control engi- to 2001, he has been an Associate neering and software engineering Professor in the Department of In- from Beijing Institute of Technol- telligent System, Graduate School ogy, Beijing, China, in 2005 and of Information Science and Elec- 2008 respectively. From 2006 to trical Engineering, Kyushu Univer- 2009 he worked in IBM China sity. Since 2003, he has been a Development Lab. He is cur- professor in the Graduate School rently a Ph.D student in Gradu- of Information, Production and ate School of Information, Produc- Systems, Waseda University. In tion and Systems, Waseda Univer- 1990 and 1994, he was a Visit- sity, Japan. His current research ing Researcher at the University of interests are mainly content based Maine, Orono. His research inter- image retrieval and pattern recog- ests include image processing, pat- nition. tern recognition, image compres- sion, and space-ﬁlling curve appli- cation. Dr. Kamata is a member of Alireza Ahrary (Member) re- the IEEE, the IEICE and the ITE ceived M.E. in Industrial Sci- in Japan. ence and Technology Engineering and PhD in Electronic Engineering from the Niigata University and Kyushu Institute of Technology, Japan, in 2004 and 2007, respectively. In 2004, he joined GMD- Japan as research assistance. He was a research scientist at FAIS- Robotics Development Support Of- ﬁce, which took over GMD-Japan in April 2004. He has been a postdoctoral research fellow of the Japan Society for the Promotion of Science (JSPS) at Waseda Univer- sity in 2007. From 2008 to 2009, he was a researcher of Fukuoka Indus- try, Science and Technology Foun- dation (Fukuoka IST) and guest researcher at IPSRC, Waseda Uni- versity, Japan. Now, he is an Assis- tant Professor in the Department of Informatics, Nagasaki Institute of Applied Science, Japan. His research interests are Intelligent Robots, Image Processing, Pattern Recognition, Computer Vision and Biometric. He is a member of IEEE, IEICE and IIEEJ.

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