1-Bit compressive sensing

TLDR

Compressive sensing reduces the number of measurements needed for sparse signals by taking random linear projections and reconstructing via convex optimization, and it remains robust even when measurements are quantized. This study investigates the extreme case of 1‑bit measurements that retain only the sign of each random projection. The authors reformulate the reconstruction as a convex program that enforces sign constraints and restricts the solution to the unit sphere, rather than treating the 1‑bit data as ±1 values. They show that this sign‑constraint approach recovers the sparse signal up to a scaling factor and outperforms classical compressive sensing methods, even when sparsity decreases and the number of measurements grows.

Abstract

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 plusmn 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.