An Additive Approximation to Multiplicative Noise
This work addresses a computational bottleneck in inverse problems for fields like signal processing or imaging, offering an incremental improvement over existing sampling-based methods.
The paper tackles the computational complexity of marginalizing over multiplicative noise in large-dimensional inverse problems by proposing an alternative Bayesian approach that embeds multiplicative noise statistics into an additive error term. The result is a method that provides feasible error estimates, supporting the actual image in posterior models, as demonstrated in a deconvolution problem with various multiplicative noise statistics.
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.