Reconstruction of sparse signals from distorted randomized measurements

In this paper we show that, surprisingly, it is possible to recover sparse signals from nonlinearly distorted measurements, even if the nonlinearity is unknown. Assuming just that the nonlinearity is monotonic, we use the only reliable information in the distorted measurements: their ordering. We demonstrate that this information is sufficient to recover the signal with high precision and present two approaches to do so. The first uses order statistics to compute the minimum mean square (MMSE) estimate of the undistorted measurements and use it with standard compressive sensing (CS) reconstruction algorithms. The second uses the principle of consistent reconstruction to develop a deterministic nonlinear reconstruction algorithm that ensures that measurements of the reconstructed signal have ordering consistent with the ordering of the distorted measurements. Our experiments demonstrate the superior performance of both approaches compared to standard CS methods.