1-Bit compressive sensing

Compressive sensing is a new signal acquisition technology with the potential to reduce the number of measurements required to acquire signals that are sparse or compressible in some basis. Rather than uniformly sampling the signal, compressive sensing computes inner products with a randomized dictionary of test functions. The signal is then recovered by a convex optimization that ensures the recovered signal is both consistent with the measurements and sparse. Compressive sensing reconstruction has been shown to be robust to multi-level quantization of the measurements, in which the reconstruction algorithm is modified to recover a sparse signal consistent to the quantization measurements. In this paper we consider the limiting case of 1-bit measurements, which preserve only the sign information of the random measurements. Although it is possible to reconstruct using the classical compressive sensing approach by treating the 1-bit measurements as ±1 measurement values, in this paper we reformulate the problem by treating the 1-bit measurements as sign constraints and further constraining the optimization to recover a signal on the unit sphere. Thus the sparse signal is recovered within a scaling factor. We demonstrate that this approach performs significantly better compared to the classical compressive sensing reconstruction methods, even as the signal becomes less sparse and as the number of measurements increases.