Signal Reconstruction from Modulo Observations

We consider the problem of reconstructing a signal from under-determined modulo observations (or measurements). This observation model is inspired by a (relatively) less well-known imaging mechanism called modulo imaging, which can be used to extend the dynamic range of imaging systems; variations of this model have also been studied under the category of phase unwrapping. Signal reconstruction in the under-determined regime with modulo observations is a challenging ill-posed problem, and existing reconstruction methods cannot be used directly. In this paper, we propose a novel approach to solving the inverse problem limited to two modulo periods, inspired by recent advances in algorithms for phase retrieval under sparsity constraints. We show that given a sufficient number of measurements, our algorithm perfectly recovers the underlying signal and provides improved performance over other existing algorithms. We also provide experiments validating our approach on both synthetic and real data to depict its superior performance

Folded Graph Signals: Sensing with Unlimited Dynamic Range

Self-reset analog-to-digital converters (ADCs) are used to sample high dynamic range signals resulting in modulo-operation based folded signal samples. We consider the case where each vertex of a graph (e.g., sensors in a network) is equipped with a self-reset ADC and senses a time series. Graph sampling allows the graph time series to be represented by the signals at a subset of sampled vertices and time instances. We investigate the problem of recovering bandlimited continuous-time graph signals from folded signal samples. We derive sufficient conditions to achieve successful recovery of the graph signal from the folded signal samples, which can be achieved via integer programming. To resolve the scalability issue of integer programming, we propose a sparse optimization recovery method for graph signals satisfying certain technical conditions. Such an approach requires a novel graph sampling scheme that selects vertices with small signal variation. The proposed algorithm exploits the inherent relationship among the graph vertices in both the vertex and time domains to recover the graph signal. Simulations and experiments on images validate the feasibility of our proposed approach

Blind Unwrapping of Modulo Reduced Gaussian Vectors: Recovering MSBs from LSBs

We consider the problem of recovering $n$ i.i.d samples from a zero mean multivariate Gaussian distribution with an unknown covariance matrix, from their modulo wrapped measurements, i.e., measurement where each coordinate is reduced modulo $\Delta$, for some $\Delta>0$. For this setup, which is motivated by quantization and analog-to-digital conversion, we develop a low-complexity iterative decoding algorithm. We show that if a benchmark informed decoder that knows the covariance matrix can recover each sample with small error probability, and $n$ is large enough, the performance of the proposed blind recovery algorithm closely follows that of the informed one. We complement the analysis with numeric results that show that the algorithm performs well even in non-asymptotic conditions