Resumen
RESUMEN
The problem of reconstructing images from boundaries has been studied in the literature chiefly as one based on the information from zero crossings. In contrast, we formulate the problem within the framework of approximation theory, and we propose a representation scheme involving (i) the points arbitrarily near the boundaries and (ii) the gradient information at these points. We show that the representation scheme is complete and stable under reasonable conditions. Further, we develop an efficient reconstruction mechanism, using the concept of a regularized solution, in which we penalize a candidate solution in proportion to its departure from a model which resembles the original in shape but is simple to construct. Finally, we demonstrate the effectiveness of the cost functional with the help of experimental results, using a representation in which the number of data points is much less than the number of pixels in the original image.
Multidimensional signal representation by zero crossings arises in many problems in physics and engineering. Its applications span several areas such as signal modulation, computer vision, and data compression. In this paper, mathematical conditions under which signals are uniquely specified in terms of their real zero crossings are presented. These results apply to multidimensional signals (i.e., the dimension n≥2) modeled by polynomials. This model is appropriate for the z-transform of discrete signals and can also be used for certain two-dimensional images. The central theorems shown in this paper establish conditions under which a multivariate polynomial P is uniquely determined from its real zero crossings. Furthermore, the regularity of a real zero of P will be shown equivalent to the existence of zero crossings. This property is interesting since it shows a deep relationship between the theory of polynomials and the representations used in practical applications. Finally, this paper generalizes the zero crossing-based signal representation problem by proposing new representations on general algebraic coordinate systems. Some open questions will be posed concerning mathematical conditions under which signals are uniquely determined by zero crossings on arbitrary varieties. A case of special interest is the n-dimensional torus but other interesting varieties can be chosen. The results of this paper show the power of algebraic-geometry tools for multidimensional signal processing problems. In particular, a commonality between the uniqueness conditions proposed in this paper and those arising in other inverse problems such as phase retrieval will be drawn.
