This master’s thesis deals with the restoration of audio signals damaged by quantization. The theoretical part starts with a description of quantization and dequantization in general, few existing methods of dequantization of audio signals and theory of sparse representations of signals are also presented. The next part introduces algorithms suitable for dequantization, specifically Douglas–Rachford, Chambolle–Pock, SPADEQ and implementation of these algorithms in MATLAB application in the next chapter. In the last part of this thesis, testing of reconstructed signals using the algorithms takes place and results are evaluated by objective measures SDR, PEMO-Q, PEAQ and subjective listening test MUSHRA.

The goal of this thesis is to demonstrate practical application of sparse data representation in the processing of sparse signals. For solving several example problems - denoising, dequantization, and sparse signal decomposition - convex optimization was used. The solutions were implemented in the Matlab environment. For each of the problems, there are two solutions - one for one-dimensional, and one for two-dimensional signal.