A New Framework for Reconstructing One-and Multi-Dimensional Signals from Partial Information

Abstract

This thesis addresses a class of inverse problems in signal and image reconstruction using partial information such as zero crossings, Fourier sign data, extrema, and modulus maxima. The focus is on reconstructing images from the locations of modulus maxima of directional derivatives, supplemented by derivative values as auxiliary data. The work investigates the completeness and stability of representation schemes for one- and two-dimensional signals, aiming to establish conditions under which unique reconstruction is possible. A new representation framework is proposed, using sampled points near image edges and their directional derivatives. This approach allows for practical sampling and supports theoretical guarantees of uniqueness and stability. A constructive algorithm is provided to demonstrate the existence of a complete representation scheme, though practical implementation relies on a digital version optimized for computational efficiency. Experimental results illustrate the feasibility of the proposed reconstruction method and highlight the geometric significance of the data. The thesis concludes with a discussion on the limitations of digital reconstruction, the role of regularization, and directions for future research. The contributions offer a mathematically grounded framework for solving inverse problems in computer vision using boundary-based representations.