Mathematical Modeling and Analysis Basis pursuit denoising using a primal-dual interior point method

Signals with a sparse representation are more easily analyzed and processed, thus the search for this representation has become the focus of intensive research. We aim to analyze an optimization principle for signal decomposition known as Basis pursuit; write a parametrizable library for its fast and efficient implementation, and test its suitability for practical applications such as signal and image denoising. As a result, a C library was created and successfully tested for 1D signal denoising and a basic extension to 2D signals is presented as proof of concept. Given a dictionary Φ = [φ1,φ2, . . . ,φd ], where φk,1 ≥ k ≤ d are vectors, a signal s = [s1,s2 . . .sN] is represented as a linear combination s = ∑k=1 φkαk with scalar coefficients α = [α1,α2, . . . ,αk]. The dictionary Φ can be seen as a matrix NxL with prototype signals as columns such that s = Φα. For overcomplete dictionaries, L ≫ N and α is non-unique. In general, a signal’s representation whose coefficients have the smallest l1 norm, is a good approximation of the sparsest representation [1]. Basis pursuit (BP) is an optimization principle for signal decomposition whose objective is to find such a representation. The basis pursuit problem is expressed as: