Fourier transform is a significant tool in image processing and pattern recognition. By introducing a hypercomplex number, hypercomplex Fourier transform treats a signal as a vector field and generalizes the conventional Fourier transform. Inspired from that, hypercomplex polar Fourier analysis that extends conventional polar Fourier analysis is proposed in this paper. The proposed method can handle signals represented by hypercomplex numbers as color images. The hypercomplex polar Fourier analysis is reversible that means it can be used to reconstruct image. The hypercomplex polar Fourier descriptor has rotation invariance property that can be used for feature extraction. Due to the noncommutative property of quaternion multiplication, both left-side and right-side hypercomplex polar Fourier analysis are discussed and their relationships are also established in this paper. The experimental results on image reconstruction, rotation invariance, color plate test and image retrieval are given to illustrate the usefulness of the proposed method as an image analysis tool.

Polar Fourier Descriptor(PFD) and Spherical Fourier Descriptor(SFD) are rotation invariant feature descriptors for two dimensional(2D) and three dimensional(3D) image retrieval and pattern recognition tasks. They are demonstrated to show superiorities compared with other methods on describing rotation invariant features of 2D and 3D images. However in order to increase the computation speed, fast computation method is needed especially for machine vision applications like realtime systems, limited computing environments and large image databases. This paper presents fast computation method for PFD and SFD that are deduced based on mathematical properties of trigonometric functions and associated Legendre polynomials. Proposed fast PFD and SFD are 8 and 16 times faster than direct calculation that significantly boost computation process. Furthermore, the proposed methods are also compact for memory requirements for storing PFD and SFD basis in lookup tables. The experimental results on both synthetic and real data are given to illustrate the efficiency of the proposed method.

Polar and Spherical Fourier analysis can be used to extract rotation invariant features for image retrieval and pattern recognition tasks. They are demonstrated to show superiorities comparing with other methods on describing rotation invariant features of two and three dimensional images. Based on mathematical properties of trigonometric functions and associated Legendre polynomials, fast algorithms are proposed for multimedia applications like real time systems and large multimedia databases in order to increase the computation speed. The symmetric points are computed simultaneously. Inspired by relative prime number theory, systematic analysis are given in this paper. Novel algorithm is deduced that provide even faster speed. Proposed method are 9–15% faster than previous work. The experimental results on two and three dimensional images are given to illustrate the effectiveness of the proposed method. Multimedia signal processing applications that need real time polar and spherical Fourier analysis can be benefit from this work.

Heat map is an important tool for eye tracking data analysis and visualization. It is very intuitive to express the area watched by observer, but ignores saccade information that expresses gaze shift. Based on conventional heat map generation method, this paper presents a novel heat map generation method for eye tracking data. The proposed method introduces a mixed data structure of fixation points and saccades, and considers heat map deformation for saccade type data. The proposed method has advantages on indicating gaze transition direction while visualizing gaze region.

Hypercomplex polar Fourier analysis treats a signal as a vector field and generalizes the conventional polar Fourier analysis. It can handle signals represented by hypercomplex numbers such as color images. Hypercomplex polar Fourier analysis is reversible that means it can reconstruct image. Its coefficient has rotation invariance property that can be used for feature extraction. However in order to increase the computation speed, fast algorithm is needed especially for image processing applications like realtime systems and limited resource platforms. This paper presents fast hypercomplex polar Fourier analysis based on symmetric properties and mathematical properties of trigonometric functions. Proposed fast hypercomplex polar Fourier analysis computes symmetric points simultaneously, which significantly reduce the computation time.