Fast Polar Harmonic Transforms

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 Exponential 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. Kernel 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. The proposed FPHTs are averagely over 6 ~ 8 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.