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Chapter 6: 2-D Quaternion and Octonion Discrete Fourier Transforms
There were many attempts to generalize the known concept of the 2-D DFT, including the 2-D quaternion discrete Fourier transform tailored to color images. The two-side quaternion discrete Fourier transformation (QDFT) was introduced in  for the analysis of 2-D linear time-invariant partial-differential systems. The classical fast algorithms are based on representation of the QDFT by a combinations of a few classical DFT transforms. This allows us to obtain QDFT fast numerical implementation with the standard FFT algorithms. In this chapter, we first describe the concept of the 2-D DFT and then the two-side, right, and left-side QDFTs with MATLAB®-based scripts. We start with the effective tensor transform-based algorithm that reduces the 2-D DFT transform to calculation of the separate 1-D DFTs. Then, we describe the 2-D QDFTs, which include the tensor algorithm for calculating the 2-D QDFT and 2-D octonion DFT (ODFT).
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