A Comparative Study of MAP and LMMSE Estimators for Blind Inverse Problems

TL;DR

This paper investigates blind 2D image deconvolution by comparing maximum-a-posteriori (MAP) and linear MMSE (LMMSE) estimators under a controlled setting with known signal and kernel distributions. It derives explicit MAP and LMMSE formulations and evaluates reconstruction quality, computation, and hyperparameter sensitivity, revealing that MAP is unstable and highly tuned, whereas LMMSE provides a robust baseline and can effectively initialize MAP to improve performance. The study shows that LMMSE initialization enhances MAP convergence and reduces sensitivity to regularization, with boosted MAP variants achieving superior results in many cases. These findings suggest that LMMSE-based strategies can stabilize blind inverse problems and guide future theoretical and practical developments in kernel and image recovery.

Abstract

Maximum-a-posteriori (MAP) approaches are an effective framework for inverse problems with known forward operators, particularly when combined with expressive priors and careful parameter selection. In blind settings, however, their use becomes significantly less stable due to the inherent non-convexity of the problem and the potential non-identifiability of the solutions. (Linear) minimum mean square error (MMSE) estimators provide a compelling alternative that can circumvent these limitations. In this work, we study synthetic two-dimensional blind deconvolution problems under fully controlled conditions, with complete prior knowledge of both the signal and kernel distributions. We compare tailored MAP algorithms with simple LMMSE estimators whose functional form is closely related to that of an optimal Tikhonov estimator. Our results show that, even in these highly controlled settings, MAP methods remain unstable and require extensive parameter tuning, whereas the LMMSE estimator yields a robust and reliable baseline. Moreover, we demonstrate empirically that the LMMSE solution can serve as an effective initialization for MAP approaches, improving their performance and reducing sensitivity to regularization parameters, thereby opening the door to future theoretical and practical developments.

Figures (3)

• Figure 1: Average MSE $\|\hat{\mathbf{x}}-\overline{\mathbf{x}}\|$ on the generated dataset for the four MAP methods considered under various combinations of $\lambda_X$ and $\lambda_H$. LLMSE initialization (MSE=$0.171$) allows for more stable performance as shown by the flatter error maps. The MSE of the reconstructio obtained by MAP methods in correspondence with the choices of $\lambda_X$ and $\lambda_H$ corresponding to the hyperparameters of the prior functions correspond to a MSE of $0.202$ for $\text{MAP}_\sigma$.
• Figure 2: Evolution of the MSE values for both the signal and kernel for the $\text{MAP}_\sigma$ and $\text{MAP}_\mathbf{h}$ approaches and their boosted variants, under different sets of regularization parameters. For the signal, we show the difference image with the ground truth. The LMMSE outperforms MAP approaches, unless regularization parameters are carefully optimized. Using the $\text{MAP}_\sigma^{\text{boost}}$ and $\text{MAP}_\mathbf{h}^{\text{boost}}$ variants allows for better performances for optimal/non-optimal sets of parameters, thus suggesting that LLMSE solution can be used as a suitable initialization for MAP approaches also when MAP parameters are not properly optimized.
• Figure 3: MSE of the empirical/theoretical LMMSE constructed for different numbers of samples for the signal (top) and the kernel (bottom). Average of 10 runs.