On a Method of Paired Representation: Enhancement and Decomposition by Series Direction Images

Abstract

In the paired representation, a two-dimensional (2-D) image is represented uniquely by a complete set of 1-D signals, so-called splitting-signals, that carry the spectral information of the image at frequency-points of specific subsets that divide the whole domain of frequencies. Image processing can thus be reduced to processing of splitting-signals and such process requires a modification of only a few spectral components of the image, for each signal. For instance, the α-rooting method of image enhancement can be fulfilled through processing one or a few splitting-signals. Such process can even be accomplished without computing the 2-D Fourier transforms of the original and enhanced images. To show that, we present an effective formula for inverse 2-D N×N-point paired transform, where N is a power of 2. The representation of the image and 2-D DFT by paired splitting-signals leads to the new concepts of direction and series images, that define the resolution and periodic structures of the image components, which can be packed in the form of the “resolution map” of the size of the image. Simple method of image enhancement by series images is described.