Compressed sensing for finite-valued signals

Abstract

The need of reconstructing discrete-valued sparse signals from few measurements, that is solving an undetermined system of linear equations, appears frequently in science and engineering. Whereas classical compressed sensing algorithms do not incorporate the additional knowledge of the discrete nature of the signal, classical lattice decoding approaches such as the sphere decoder do not utilize sparsity constraints.In this work, we present an approach that incorporates a discrete values prior into basis pursuit. In particular, we address unipolar binary and bipolar ternary sparse signals, i.e., sparse signals with entries in {0, 1}, respectively in {−1, 0, 1}. We will show that phase transition takes place earlier than when using the classical basis pursuit approach and that, independently of the sparsity of the signal, at most N/2, respectively 3N/4, measurements are necessary to recover a unipolar binary, and a bipolar ternary signal uniquely, where N is the dimension of the ambient space. We will further discuss robustness of the algorithm and generalizations to signals with entries in larger alphabets.

Recovery of Binary Sparse Signals With Biased Measurement Matrices

“…The algorithm then chooses the solution which is closest to being binary-valued. In [11] it has also been shown, that one of the solutions is indeed exactly binary, provided we have sufficiently many measurements. However, this algorithm can only succeed in the case that only one of the solutionsx 1 andx 2 is binary.…”

“…This experiment is specified in Subsection 2.2. (b) In [11,14] theoretically proven phase transition of (P bin ) in comparison to the classical algorithm (P + ) and (P 1 ) for arbitrary N ∈ N. Successful Reconstruction is guaranteed in the area above the curves with high probability. Figure 1: Numerically and theoretically derived phase transition of (P bin ) from Gaussian measurements.…”

“…To ensure robust recovery from noisy measurements b = Ax 0 + n, with n 2 ≤ η, the following adaptation of (P bin ) has been considered (e.g. in [11]):…”

“…to recover x 0 . The authors of [11] proposed to combine (P bin ) and (P Mbin ) to form a new recovery algorithm, Mirrored Binary Basis Pursuit (Algorithm 1).…”

A Novel Convex Relaxation for Non-binary Discrete Tomography

“…More generally, the discrete tomography problem can be regarded as reconstructing a discrete-valued synthesis/analysis-sparse signal from few measurements which is observed by deterministic sensors A. This is, in turn, a special instance of the compressed sensing problem [7], for which it has been shown that discreteness constraints on the possible values of the reconstructed function can significantly reduce the number of required measurements [10]. However, the discreteness constraint leads to great computational challenges.…”