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 ($d=2,3,\dots$), 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.

References in corpus (6)

• Optimal detection of changepoints with a linear computational cost
• Rates of contraction of posterior distributions based on Gaussian process priors
• Convergence rates of posterior distributions for noniid observations
• Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
• Bayesian inference with rescaled Gaussian process priors
• Bayesian optimal adaptive estimation using a sieve prior

Cited by in corpus (4)

• Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image
• Oracle posterior contraction rates under hierarchical priors
• Equivalence of Convergence Rates of Posterior Distributions and Bayes Estimators for Functions and Nonparametric Functionals
• BayesBD: An R Package for Bayesian Inference on Image Boundaries