An Additive Approximation to Multiplicative Noise

TL;DR

This paper introduces an additive approximation method for marginalizing multiplicative noise in high-dimensional inverse problems, enabling more computationally feasible Bayesian inference especially with correlated noise.

Contribution

It proposes a novel additive approximation technique for multiplicative noise marginalization, extending Bayesian methods to correlated noise and large-scale inverse problems.

Findings

Provides feasible error estimates supporting the true image.

Handles correlated multiplicative noise effectively.

Applicable to large-dimensional deconvolution problems.

Abstract

Multiplicative noise models are often used instead of additive noise models in cases in which the noise variance depends on the state. Furthermore, when Poisson distributions with relatively small counts are approximated with normal distributions, multiplicative noise approximations are straightforward to implement. There are a number of limitations in existing approaches to marginalize over multiplicative errors, such as positivity of the multiplicative noise term. The focus in this paper is in large dimensional (inverse) problems for which sampling type approaches have too high computational complexity. In this paper, we propose an alternative approach to carry out approximative marginalization over the multiplicative error by embedding the statistics in an additive error term. The approach is essentially a Bayesian one in that the statistics of the additive error is induced by the statistics of the other unknowns. As an example, we consider a deconvolution problem on random fields with different statistics of the multiplicative noise. Furthermore, the approach allows for correlated multiplicative noise. We show that the proposed approach provides feasible error estimates in the sense that the posterior models support the actual image.