Two-dimensional images in frequency-time representation: direction images and resolution map

Abstract

We discuss the concept of the direction image multiresolution, which is derived as a property of the 2-D discrete Fourier transform, when it splits by 1-D transforms. The N×N image, where N is a power of 2, is considered as a unique set of splitting-signals in paired representation, which is the unitary 2-D frequency and 1-D time representation. The number of splitting-signals is 3N−2, and they have different durations, carry the spectral information of the image in disjoint subsets of frequency points, and can be calculated from the projection data along one of 3N/2 angles. The paired representation leads to the image composition by a set of 3N−2 direction images, which defines the directed multiresolution and contains periodic components of the image. We also introduce the concept of the resolution map, as a result of uniting all direction images into log2 N series. In the resolution map, all different periodic components (or structures) of the image are packed into a N×N matrix, which can be used for image processing in enhancement, filtration, and compression