In this thesis, signal and image processing algorithms based on sparsity and interval convex programming are developed for inverse problems. Inverse signal processing problems are solved by minimizing the ℓ1 norm or the Total Variation (TV) based cost functions in the literature. A modified entropy functional approximating the absolute value function is defined. This functional is also used to approximate the ℓ1 norm, which is the most widely used cost function in sparse signal processing problems. The modified entropy functional is continuously differentiable, and convex. As a result, it is possible to develop iterative, globally convergent algorithms for compressive sensing, denoising and restoration problems using the modified entropy functional. Iterative interval convex programming algorithms are constructed using Bregman’s D-Projection operator. In sparse signal processing, it is assumed that the signal can be represented using a sparse set of coefficients in some transform domain. Therefore, by minimizing the total variation of the signal, it is expected to realize sparse representations of signals. Another cost function that is introduced for inverse problems is the Filtered Variation (FV) function, which is the generalized version of the Total Variation (VR) function. The TV function uses the differences between the pixels of an image or samples of a signal. This is essentially simple Haar filtering. In FV, high-pass filter outputs are used instead of differences. This leads to flexibility in algorithm design adapting to the local variations of the signal. Extensive simulation studies using the new cost functions are carried out. Better experimental restoration, and reconstructions results are obtained compared to the algorithms in the literature