Bayesian detection of image boundaries

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 for 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 optimal rate adapting to the smoothness level, convenience for joint inference and computational efficiency. 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. Several new estimates on modified Bessel functions of the first kind are given. Simulations confirm excellent performance and robustness of the proposed method.