We consider the problem of reconstructing a signal from under-determined modulo observations (or measurements). This observation model is inspired by a relatively new 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 signal recovery problem under sparsity constraints for the special case to modulo folding limited to two periods. We show that given a sufficient number of measurements, our algorithm perfectly recovers the underlying signal. We also provide experiments validating our approach on toy signal and image data and demonstrate its promising performance.

我们考虑从欠定的模态观测（或测量）重建信号的问题。这个观测模型的灵感来自一种相对较新的成像机制，即模成像，可用于扩展成像系统的动态范围。还已经在相位展开类别下研究了该模型的各种变化形式。在欠定状态下使用模态观测进行信号重建是一个具有挑战性的不适定问题，现有的重建方法无法直接使用。在本文中，我们提出了一种新颖的方法来解决稀疏约束下的信号恢复问题，这种特殊情况适用于模折叠仅限于两个周期的特殊情况。我们表明，如果有足够的测量值，我们的算法就可以完美地恢复基础信号。