Image Representation by Sign Information

Abstract

Signal representation by sign information or zero crossings arises in several problems in physics and engineering. Its applications span many areas such as signal modulation, data compression, and computer vision. In this paper, we deal with image representation by sign data in its most general context. Specifically, if f is a two-dimensional signal, we will study conditions under which f is determined by its sign, or in other words, by its quantization to 1 bit of information. This study will be carried out in two different directions. First, we present new theoretical results that set an algebraic condition under which real zero crossings uniquely specify a band-limited image. This condition involves the irreducibility of the entire exponential-type extension of the function to C2. An interesting paradigm arising in theoretical computer vision will then be posed: are the zero crossings of f convolved with a Laplacian-of-a-Gaussian at a single channel enough for unambiguously representing f? Second, we address the fundamental problem of the completeness of the representation when the position of the zero crossings is known only approximately. Specifically, we will show that when sign(f) is sampled, significant ambiguities are introduced in the representation. For the first time in the literature, we will present experimental results obtained from an iterative algorithm devised to reconstruct real images from multiscale sign information. We hope that the theoretical and experimental results reported in this paper will be insightful for many applications where signal representation by zero crossings plays a key role. © 1989 IEEE