Signal and Image Recovery Under Multiplicative Noise: Methods and Analysis

Multiplicative noise is commonly encountered in many signal and image processing applications, for example ultrasound imaging, electron microscopy, synthetic aperture radar, and laser images. Mathematically, the observation degraded by multiplicative noise can be modeled as the element-wise multiplication of an original signal and a noise component. Due to the multiplicative degraded mechanism, the distortion caused by multiplicative noise can lead to a substantial loss of information from the original signal, thereby posing greater challenges for its removal compared to additive noise. When dealing with measurements corrupted by multiplicative noise, the actual measurement matrix is noisy and unknown. In literature, the structure of multiplicative noise observations is not considered and studied in the structured total least squares formulation. On the other hand, one-bit quantization of signals or data recently has received much attention in both signal processing and machine learning communities because of power consumption, manufacturing cost and communication cost. Recent years have witnessed increasing literature on signal recovery under one-bit observations. However, it is unknown to the analysis of signal recovery performance under one-bit observations with multiplicative and additive noises.

The main aims of this project are to develop methods and analysis for sparse signal recovery under one-bit observations with multiplicative and additive noises. We propose and study hard thresholding and soft thresholding methods to solve the recovery problem with multiplicative and additive noises and analyze that both methods yield outputs that offer an approximate estimation of the underlying sparse signal. Our objective is to show that the thresholding methods are reliable and comparable with existing results in the absence of any noise. We study regularized structured total least
squares models under multiplicative and additive noises, and their applications in signal recovery and image reconstruction. Our objective is to provide error estimates for solutions of regularized structured total least squares models in terms of sparsity and noise level. For robust image reconstruction, we would like to establish an optimal error estimation up to a logarithmic factor. The methods developed will be applied to the problem of computed tomography image reconstruction and natural image deblurring and the numerical examples will be shown to demonstrate the effectiveness of the proposed recovery models.