Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image

TL;DR

This paper introduces a robust Gibbs posterior method for image boundary detection in noisy images, avoiding pixel intensity modeling and achieving minimax optimal convergence rates, with improved accuracy demonstrated through simulations.

Contribution

The paper presents a novel Gibbs posterior approach that directly estimates image boundaries without modeling pixel intensities, ensuring robustness and optimal convergence.

Findings

Gibbs posterior concentrates at minimax optimal rate

Method is adaptive to boundary smoothness

Simulation shows improved accuracy over existing Bayesian methods

Abstract

Detection of an image boundary when the pixel intensities are measured with noise is an important problem in image segmentation, with numerous applications in medical imaging and engineering. From a statistical point of view, the challenge is that likelihood-based methods require modeling the pixel intensities inside and outside the image boundary, even though these are typically of no practical interest. Since misspecification of the pixel intensity models can negatively affect inference on the image boundary, it would be desirable to avoid this modeling step altogether. Towards this, we develop a robust Gibbs approach that constructs a posterior distribution for the image boundary directly, without modeling the pixel intensities. We prove that, for a suitable prior on the image boundary, the Gibbs posterior concentrates asymptotically at the minimax optimal rate, adaptive to the boundary smoothness. Monte Carlo computation of the Gibbs posterior is straightforward, and simulation experiments show that the corresponding inference is more accurate than that based on existing Bayesian methodology.