A New Framework for Reconstruction of Images from Boundaries
D. Ravi, Y.V. Venkatesh · SIAM Journal on Applied Mathematics · 2000
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.