Robust One-Bit Compressed Sensing via Continuous Sign Approximation and Hybrid Ordinary-Welsch Function

One-bit quantized measurements are commonly encountered in resource-constrained imaging applications such as medical imaging and satellite remote sensing. Although stable signal reconstruction from noise-free one-bit measurements has been widely demonstrated, noise in practical data acquisition systems often leads to unknown sign flips, significantly increasing the difficulty of accurate recovery. While numerous robust methods have been developed for signal reconstruction from noisy one-bit measurements, most of them necessitate prior knowledge of the number of sign flips, which is typically unavailable in practice. In this paper, we propose two novel optimization models based on continuous approximation functions (CAFs) of the sign function. The first model employs the quadratic loss function, specifically designed for scenarios with low sign flipping ratios, while the second incorporates a Hybrid Ordinary-Welsch (HOW) loss function, yielding enhanced robustness under higher flipping ratios without requiring prior knowledge. Then, we propose two algorithmic frameworks, OBCSA CAF and OBCSA CAF HOW, to solve the respective models, and prove that both algorithms generate convergent sequences of objective function values and subsequences that, with high probability, converge to specific accumulation points. Extensive simulations demonstrate the superior performance of our methods over the state-of-the-art algorithms in terms of signal-to-noise ratio (SNR), rate of support recovery, Hamming error, and Hamming distance. In particular, OBCSA CAF achieves 0.5--4 dB gains in SNR and 1\%--3.6\% improvements in support recovery rate at low flipping ratios, whereas OBCSA CAF HOW yields 2--6.7 dB gains and 2\%--8\% improvements, respectively, under high flipping ratios. Experiments on the MNIST and CIFAR-10 datasets further illustrate the superior performance of our methods in image reconstruction.