There is an unmet medical need to identify neuroimaging biomarkers that is able to accurately diagnose and monitor Alzheimer's disease (AD) at very early stages and assess the response to AD-modifying therapies. To a certain extent, volumetric and functional magnetic resonance imaging (fMRI) studies can detect changes in structure, cerebral blood flow and blood oxygenation that are able to distinguish AD and mild cognitive impairment (MCI) subjects from normal controls. However, it has been challenging to use fully automated MRI analytic methods to identify potential AD neuroimaging biomarkers. We have thus proposed a method based on independent component analysis (ICA), for studying potential AD-related MR image features, coupled with the use of support vector machine (SVM) for classifying scans into categories of AD, MCI, and normal control (NC) subjects. The MRI data were selected from Open Access Series of Imaging Studies (OASIS) and the Alzheimer's Disease Neuroimaging Initiative (ADNI) databases. The experimental results showed that our ICA-based method can differentiate AD and MCI subjects from normal controls, although further methodological improvement in the analytic method and inclusion of additional variables may be required for optimal classification.

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We prove that there are no stable intersections of regular Cantor sets in the$C$^{1}topology: given any pair ($K$,$K$′) of regular Cantor sets, we can find, arbitrarily close to it in the$C$^{1}topology, pairs $ \left( {\tilde{K},\tilde{K}'} \right) $ of regular Cantor sets with $ \tilde{K} \cap \tilde{K}' = \emptyset $ .

In this paper, we propose a novel iterative convolution-thresholding method (ICTM) that is applicable to a range of variational models for image segmentation. A variational model usually minimizes an energy functional consisting of a fidelity term and a regularization term. In the ICTM, the interface between two different segment domains is implicitly represented by their characteristic functions. The fidelity term is usually written as a linear functional of the characteristic functions and the regularized term is approximated by a functional of characteristic functions in terms of heat kernel convolution. This allows us to design an iterative convolution-thresholding method to minimize the approximate energy. The method is simple, efficient and enjoys the energy-decaying property. Numerical experiments show that the method is easy to implement, robust and applicable to various image segmentation models.

This dissertation splits into two distinct halves. The first half is devoted to the study of the holography of higher spin gauge theory in AdS_3. We present a conjecture that the holographic dual of W_N minimal model in a 't Hooft-like large N limit is an unusual ``semi-local" higher spin gauge theory on AdS_3×S^1. At each point on the S^1 lives a copy of three-dimensional Vasiliev theory, that contains an infinite tower of higher spin gauge fields coupled to a single massive complex scalar propagating in AdS_3. The Vasiliev theories at different points on the S^1 are correlated only through the AdS_3 boundary conditions on the massive scalars. All but one single tower of higher spin symmetries are broken by the boundary conditions. This conjecture is checked by comparing tree-level two- and three-point functions, and also one-loop partition functions on both side of the duality. The second half focuses on the holography of higher spin gauge theory in AdS_4. We demonstrate that a supersymmetric and parity violating version of Vasiliev's higher spin gauge theory in AdS_4 admits boundary conditions that preserve N=0,1,2,3,4 or 6 supersymmetries. In particular, we argue that the Vasiliev theory with U(M) Chan-Paton and N=6 boundary condition is holographically dual to the 2+1 dimensional U(N)_k×U(M)_{−k} ABJ theory in the limit of large N,k and finite M. In this system all bulk higher spin fields transform in the adjoint of the U(M) gauge group, whose bulk t'Hooft coupling is M/N. Our picture suggests that the supersymmetric Vasiliev theory can be obtained as a limit of type IIA string theory in AdS_4×CP^3, and that the non-Abelian Vasiliev theory at strong bulk 't Hooft coupling smoothly turn into a string field theory. The fundamental string is a singlet bound state of Vasiliev's higher spin particles held together by U(M) gauge interactions.