Bayesian Detection of Image Boundaries

TL;DR

This paper introduces a nonparametric Bayesian method for detecting image boundaries in noisy data, achieving optimal rates and adaptivity, with extensive theoretical and simulation validation for 2D and 3D images.

Contribution

It develops a novel Bayesian framework using priors on the sphere for boundary detection, providing theoretical guarantees and practical efficiency.

Findings

Achieves near-minimax rate optimal boundary detection.

Method is adaptive to boundary smoothness levels.

Simulation results show robustness and high performance.

Abstract

Detecting boundary of an image based on noisy observations is a fundamental problem of image processing and image segmentation. For a $d$-dimensional image ($d = 2, 3, \ldots$), the boundary can often be described by a closed smooth $(d - 1)$-dimensional manifold. In this paper, we propose a nonparametric Bayesian approach based on priors indexed by $\mathbb{S}^{d - 1}$, the unit sphere in $\mathbb{R}^d$. We derive optimal posterior contraction rates using Gaussian processes or finite random series priors using basis functions such as trigonometric polynomials for 2-dimensional images and spherical harmonics for 3-dimensional images. For 2-dimensional images, we show a rescaled squared exponential Gaussian process on $\mathbb{S}^1$ achieves four goals of guaranteed geometric restriction, (nearly) minimax rate optimal and adaptive to the smoothness level, convenient for joint inference and computationally efficient. We conduct an extensive study of its reproducing kernel Hilbert space, which may be of interest by its own and can also be used in other contexts. Simulations confirm excellent performance of the proposed method and indicate its robustness under model misspecification at least under the simulated settings.