Recovering H\"older smooth functions from noisy modulo samples

TL;DR

This paper proposes a three-stage method for recovering H"older smooth functions from noisy modulo samples, combining denoising, unwrapping, and spline interpolation, with proven uniform error bounds.

Contribution

It introduces a novel three-stage recovery strategy specifically designed for H"older smooth functions from noisy modulo samples, extending previous Lipschitz results.

Findings

Achieves uniform error bounds with high probability

Extends previous Lipschitz results to H"older functions

Demonstrates effectiveness of combined denoising and unwrapping approach

Abstract

In signal processing, several applications involve the recovery of a function given noisy modulo samples. The setting considered in this paper is that the samples corrupted by an additive Gaussian noise are wrapped due to the modulo operation. Typical examples of this problem arise in phase unwrapping problems or in the context of self-reset analog to digital converters. We consider a fixed design setting where the modulo samples are given on a regular grid. Then, a three stage recovery strategy is proposed to recover the ground truth signal up to a global integer shift. The first stage denoises the modulo samples by using local polynomial estimators. In the second stage, an unwrapping algorithm is applied to the denoised modulo samples on the grid. Finally, a spline based quasi-interpolant operator is used to yield an estimate of the ground truth function up to a global integer shift. For a function in H\"older class, uniform error rates are given for recovery performance with high probability. This extends recent results obtained by Fanuel and Tyagi for Lipschitz smooth functions wherein $k$NN regression was used in the denoising step.