Bayesian Methods for Images and Trees

In the first part of the dissertation, we propose a multiscale model for Gaussian noised images under a Bayesian framework for both 2-dimensional (2D) and 3-dimensional (3D) images. We use a Chinese restaurant process prior to randomly generate ties among intensity values at neighboring pixels in the image. The resulting Bayesian estimator enjoys some desirable asymptotic properties for identifying precise structures in the image. The proposed Bayesian denoising procedure is completely data-driven. A conditional conjugacy property allows analytical computation of the posterior distribution without involving Markov chain Monte Carlo (MCMC) methods, making the method computationally efficient. Simulations on Shepp-Logan phantom and Lena test images confirm that our smoothing method is comparable with the best available methods for light noise and outperforms them for heavier noise both visually and numerically. The proposed method is further extended for 3D images. A simulation study shows that the proposed method is numerically better than most existing denoising approaches for 3D images. A 3D Shepp-Logan phantom image is used to demonstrate the visual and numerical performance of the proposed method, along with the computational time. MATLAB toolboxes are made available online (both 2D and 3D) to implement the proposed method and reproduce the numerical results.;Translation Invariant (TI) cycle spinning is an effective method for removing artifacts from images. In the second part of the dissertation, we propose a Fast Translation Invariant (FTI) algorithm and a more general k-Translation-Invariant (k-TI) algorithm allowing TI for the last k scales of the image, which are applicable to general d-dimensional images (d = 2, 3,...) with either Gaussian or Poisson noise. The proposed FTI leads to the exact TI estimation but only requires O(n log2 n) time. The proposed k-TI can achieve almost the same performance as the exact TI estimation, but requires even less time. We achieve this by exploiting the regularity present in the multiscale structure, which is justified theoretically. The proposed FTI and k-TI are generic in that they are applicable on any smoothing techniques based on the multiscale structure. We demonstrate the FTI and k-TI algorithms on some recently proposed state-of-the-art methods for both Poisson and Gaussian noised images. Both simulations and real data application confirm the appealing performance of the proposed algorithms. Matlab toolboxes are online accessible to reproduce the results and be implemented for general multiscale denoising approaches provided by the users.;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,...), the boundary can often be described by a closed smooth (d - 1)-dimensional manifold. In the third part of the dissertation, we propose a nonparametric Bayesian approach based on priors indexed by Sd -1, the unit sphere in Rd. 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 S1 achieves four goals of guaranteed geometric restriction, (nearly) minimax optimal rate adaptive to the smoothness level, convenient 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 the modified Bessel functions of the first kind are given. Simulations confirm excellent performance of the proposed method and indicate its robustness under model misspecification at least under the simulation settings.;Tree-structured data may be highly non-Euclidean, which makes it challenging for statistical analysis. A tree may have rich information such as nodal attributes in addition to its topological structure. Motivated by the brain artery data set (Aylward and Bullitt, 2002; Aydin et al., 2011), we aim to classify trees by supervised learning based on characteristics extracted from a tree. The Bayes classifier can be described by computing the marginal likelihood of a tree given covariates, and the later can be obtained from regression of trees on covariates. We represent a tree by its three important characteristics, branching probability, thickness and branching length. We model the branching probability using varying coefficient probit regression, which leads to a convenient Gibb's sampler. For the thickness and branching length, we use a vector autoregression (VAR) model with guaranteed causality constraint, which is similar to a VAR model in time series but differs in terms of indexing which conveys the topological structure of a tree instead of time. A general Bayesian classifier on trees is discussed. (Abstract shortened by UMI.).