# Signal Reconstruction from Modulo Observations

## TL;DR

This paper introduces a novel algorithm for reconstructing signals from under-determined modulo observations, inspired by phase retrieval techniques, demonstrating perfect recovery with enough measurements and superior performance over existing methods.

## Contribution

The paper presents a new approach for signal reconstruction from modulo measurements limited to two periods, extending phase retrieval algorithms to this challenging setting.

## Key findings

- Algorithm achieves perfect signal recovery with sufficient measurements.
- Proposed method outperforms existing algorithms in experiments.
- Validated on synthetic and real data showing superior performance.

## Abstract

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.