Comparative analysis of sensing matrices for compressed sensed thermal images

In the conventional sampling process, in order to reconstruct the signal perfectly Nyquist-Shannon sampling theorem needs to be satisfied. Nyquist-Shannon theorem is a sufficient condition but not a necessary condition for perfect reconstruction. The field of compressive sensing provides a stricter sampling condition when the signal is known to be sparse or compressible. Compressive sensing contains three main problems: sparse representation, measurement matrix and reconstruction algorithm. This paper describes and implements 14 different sensing matrices for thermal image reconstruction using Basis Pursuit algorithm available in the YALL1 package. The sensing matrices include Gaussian random with and without orthogonal rows, Bernoulli random with bipolar entries and binary entries, Fourier with and without dc basis vector, Toeplitz with Gaussian and Bernoulli entries, Circulant with Gaussian and Bernoulli entries, Hadamard with and without dc basis vector, Normalised Hadamard with and without dc basis vector. Orthogonalization of the rows of the Gaussian sensing matrix and normalisation of Hadamard matrix greatly improves the speed of reconstruction. Semi-deterministic Toeplitz and Circulant matrices provide lower PSNR and require more iteration for reconstruction. The Fourier and Hadamard deterministic sensing matrices without dc basis vector worked well in preserving the object of interest, thus paving the way for object specific image reconstruction based on sensing matrices. The sparsifying basis used in this paper was Discrete Cosine Transform and Fourier Transform.