Abstract
The discrete Fourier series (DFS) is called “the Fourier transform for periodic sequences,” in that it plays the same role for them that the Fourier transform plays for nonperiodic (ordinary) sequences. The DFS also provides a theoretical stepping stone toward the DFT, which has great practical significance in signal and image processing as a Fourier transform for finite support sequences. This chapter examines discrete-space transforms such as discrete Fourier series, discrete Fourier transform (DFT), and discrete cosine transform (DCT) in two dimensions. The DFT is a heavily used tool in image and multidimensional signal processing. Block transforms can be obtained from scanning the data into small blocks and then performing the DFT or DCT on each block. The block DCT is used extensively in image and video compression for transmission and storage. Furthermore, this chapter considers the subband/wavelet transform (SWT), which can be considered as a generalization of the block DCT transform wherein the basis functions are allowed to overlap from block to block. These SWTs can also be considered as a generalization of the Fourier transform wherein resolution in space can be traded off versus resolution in frequency.
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