16 March 2015 New 2D discrete Fourier transforms in image processing
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In this paper, the concept of the two-dimensional discrete Fourier transformation (2-D DFT) is defined in the general case, when the form of relation between the spatial-points (x, y) and frequency-points (ω1, ω2) is defined in the exponential kernel of the transformation by a nonlinear form L(x, y; ω1, ω2). The traditional concept of the 2-D DFT uses the Diaphanous form 1 +yω2 and this 2-D DFT is the particular case of the Fourier transform described by the form L(x, y; ω1, ω2). Properties of the general 2-D discrete Fourier transform are described and examples are given. The special case of the N × N-point 2-D Fourier transforms, when N = 2r, r > 1, is analyzed and effective representation of these transforms is proposed. The proposed concept of nonlinear forms can be also applied for other transformations such as Hartley, Hadamard, and cosine transformations.
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Artyom M. Grigoryan, Artyom M. Grigoryan, Sos S. Agaian, Sos S. Agaian, "New 2D discrete Fourier transforms in image processing", Proc. SPIE 9399, Image Processing: Algorithms and Systems XIII, 939910 (16 March 2015); doi: 10.1117/12.2083389; https://doi.org/10.1117/12.2083389

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