Browsing by Subject "Nonparametric regression"

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    Edge detection and image restoration of blurred noisy images using jump regression analysis
    (2013-08) Kang, Yicheng
    We consider the problem of edge-preserving image restoration when images are degraded by spatial blur and pointwise noise. When the spatial blur described by a point spread function (psf) is not completely specified beforehand, this is a challenging <&ldquo>ill-posed< &rdquo> problem, because (i) theoretically, the true image can not be uniquely determined by the observed image when the psf is unknown, even in cases when the observed image contains no noise, and (ii) practically, besides blurring, observed images often contain noise, which can cause numerical instability in many existing image deblurring procedures. In the literature, most existing deblurring procedures are developed under the assumption that the psf is completely specified, or that the psf follows a parametric form with one or more unknown parameters. In this dissertation, we propose blind image deblurring (BID) methodologies that do not require such restrictive conditions on the psf. They even allow the psf to change over location. This dissertation has three chapters. Chapter 1 introduces some motivating applications for image processing along with presenting the overall scope of the dissertation. In Chapter 2, the problem of step edge detection in blurred noisy images is studied. In Chapter 3, a BID procedure based on edge detection is proposed. In Chapter 4, an efficient BID procedure without explicitly detecting edges is presented. Both theoretical justifications and numerical studies show that our proposed procedures work well in applications.
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    Searching, Clustering and Regression on non-Euclidean Spaces
    (2015-08) Wang, Xu
    This dissertation considers three common tasks (e.g., searching, clustering, regression) over Riemannian spaces. The first task considers the problem of efficiently deciding which of a database of subspaces is most similar to a given input query. Motivated by applications in recognition, image retrieval and optimization, there has been significant recent interest in this problem. Current approaches to this problem have poor scaling in high dimensions, and may not guarantee sublinear query complexity. We present a new approach to approximate nearest subspace search, based on a simple, new locality sensitive hash for subspaces. For the second task, we advocates a novel framework for segmenting a dataset in a Riemannian manifold into clusters lying around low-dimensional submanifolds. This clustering problem is useful for applications such as action identification, dynamic texture clustering, brain fiber segmentation, and clustering of deformed images. The proposed clustering algorithm constructs an affinity matrix by exploiting the geometry and then applies spectral clustering. Theoretical guarantees are established for a variant of the algorithm. To avoid complication, these guarantees assume that the submanifolds are geodesic. Extensive validation on synthetic and real data demonstrates the resiliency of the proposed method against deviations from the theoretical model as well as its superior performance over state-of-the-art techniques. In the third task, we proposes a novel framework for manifold-valued regression and establishes its consistency as well as its contraction rate for a particular setting. Our setting assumes a predictor with values in the interval [0,1] and response with values in a compact Riemannian manifold. This setting is useful for applications such as modeling dynamic scenes or shape deformations, where the visual scene or the deformed objects can be modeled by a manifold. The proposed framework uses the heat kernel on manifolds as an averaging procedure. It directly generalizes the use of the Gaussian kernel in vector-valued regression problems. In order to avoid explicit dependence on estimates of the heat kernel, we follow a Bayesian setting, where Brownian motion induces a prior distribution on the space of continuous functions. We study the posterior consistency and contraction rate of the discrete and continuous Brownian motion priors.