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Fast Polar Harmonic Transforms Paper Fast Polar Harmonic Transforms † †† † 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 im- age 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) includ- ing 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 aver- agely 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 recogni- tion 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. 399 The Journal of the IIEEJ vol. 39 no. 4(2010) 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, 401 The Journal of the IIEEJ vol. 39 no. 4(2010) 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.
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