Fast Splitting$alpha$-Rooting Method of Image Enhancement: Tensor Representation

In the tensor representation, a two-dimensional (2-D) image is represented uniquely by a set of one-dimensional (1-D) signals, so-called splitting-signals, that carry the spectral information of the image at frequency-points of specific sets that cover the whole domain of frequencies. The image enhancement is thus reduced to processing splitting-signals and such process requires a modification of only a few spectral components of the image, for each signal. For instance, the alpha-rooting method of image enhancement can be fulfilled through processing separately a maximum of 3N/2 splitting-signals of an image (NtimesN), where N is a power of two. In this paper, we propose a fast implementation of the alpha-rooting method by using one splitting-signal of the tensor representation with respect to the discrete Fourier transform (DFT). The implementation is described in the frequency and spatial domains. As a result, the proposed algorithms for image enhancement use two 1-D N-point DFTs instead of two 2-D NtimesN-point DFTs in the traditional method of alpha-rooting