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Optimal Linear Representations of Images under Diverse Criteria
Title
Optimal Linear Representations of Images under Diverse Criteria.
Creator
Rubinshtein, Evgenia, Srivastava, Anuj, Liu, Xiuwen, Huffer, Fred, Chicken, Eric, Department of Statistics, Florida State University
Abstract/Description
Image analysis often requires dimension reduction before statistical analysis, in order to apply sophisticated procedures. Motivated by eventual applications, a variety of criteria have been proposed: reconstruction error, class separation, non-Gaussianity using kurtosis, sparseness, mutual information, recognition of objects, and their combinations. Although some criteria have analytical solutions, the remaining ones require numerical approaches. We present geometric tools for finding linear...
Show moreImage analysis often requires dimension reduction before statistical analysis, in order to apply sophisticated procedures. Motivated by eventual applications, a variety of criteria have been proposed: reconstruction error, class separation, non-Gaussianity using kurtosis, sparseness, mutual information, recognition of objects, and their combinations. Although some criteria have analytical solutions, the remaining ones require numerical approaches. We present geometric tools for finding linear projections that optimize a given criterion for a given data set. The main idea is to formulate a problem of optimization on a Grassmann or a Stiefel manifold, and to use differential geometry of the underlying space to construct optimization algorithms. Purely deterministic updates lead to local solutions, and addition of random components allows for stochastic gradient searches that eventually lead to global solutions. We demonstrate these results using several image datasets, including natural images and facial images.
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Date Issued
2006
Identifier
FSU_migr_etd-1926
Format
Thesis
Bayesian MRF Framework for Labeling Terrain Using Hyperspectral Imaging
Title
A Bayesian MRF Framework for Labeling Terrain Using Hyperspectral Imaging.
Creator
Neher, Robert E., Srivastava, Anuj, Liu, Xiuwen, Huffer, Fred, Wegkamp, Marten, Department of Statistics, Florida State University
Abstract/Description
We explore the non-Gaussianity of hyperspectral data and present probability models that capture variability of hyperspectral images. In particular, we present a nonparametric probability distribution that models the distribution of the hyperspectral data after reducing the dimension of the data via either principal components or Fisher's discriminant analysis. We also explore the directional differences in observed images and present two parametric distributions, the generalized Laplacian...
Show moreWe explore the non-Gaussianity of hyperspectral data and present probability models that capture variability of hyperspectral images. In particular, we present a nonparametric probability distribution that models the distribution of the hyperspectral data after reducing the dimension of the data via either principal components or Fisher's discriminant analysis. We also explore the directional differences in observed images and present two parametric distributions, the generalized Laplacian and the Bessel K form, that well model the non-Gaussian behavior of the directional differences. We then propose a model that labels each spatial site, using Bayesian inference and Markov random fields, that incorporates the information of the non-parametric distribution of the data, and the parametric distributions of the directional differences, along with a prior distribution that favors smooth labeling. We then test our model on actual hyperspectral data and present the results of our model, using the Washington D.C. Mall and Indian Springs rural area data sets.
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Date Issued
2004
Identifier
FSU_migr_etd-2691
Format
Thesis
Nonparametric Estimation of Three Dimensional Projective Shapes with Applications in Medical Imaging and in Pattern Recognition
Title
Nonparametric Estimation of Three Dimensional Projective Shapes with Applications in Medical Imaging and in Pattern Recognition.
Creator
Crane, Michael, Patrangenaru, Victor, Liu, Xiuwen, Huffer, Fred W., Sinha, Debajyoti, Department of Statistics, Florida State University
Abstract/Description
This dissertation is on analysis of invariants of a 3D configuration from its 2D images in pictures of this configuration, without requiring any restriction on the camera positioning relative to the scene pictured. We briefly review some of the main results found in the literature. The methodology used is nonparametric, manifold based combined with standard computer vision re- construction techniques. More specifically, we use asymptotic results for the extrinsic sample mean and the extrinsic...
Show moreThis dissertation is on analysis of invariants of a 3D configuration from its 2D images in pictures of this configuration, without requiring any restriction on the camera positioning relative to the scene pictured. We briefly review some of the main results found in the literature. The methodology used is nonparametric, manifold based combined with standard computer vision re- construction techniques. More specifically, we use asymptotic results for the extrinsic sample mean and the extrinsic sample covariance to construct boot- strap confidence regions for mean projective shapes of 3D configurations. Chapters 4, 5 and 6 contain new results. In chapter 4, we develop tests for coplanarity. In chapter 5, is on reconstruction of 3D polyhedral scenes, including texture from arbitrary partial views. In chapter 6, we develop a nonparametric methodology for estimating the mean change for matched samples on a Lie group. We then notice that for k '' 4, a manifold of projective shapes of k-ads in general position in 3D has a structure of 3k and #8722; 15 dimensional Lie group (P-Quaternions) that is equivariantly embedded in an Euclidean space, therefore testing for mean 3D projective shape change amounts to a one sample test for extrinsic mean PQuaternion Objects. The Lie group technique leads to a large sample and nonparametric bootstrap test for one population extrinsic mean on a projective shape space, as recently developed by Patrangenaru, Liu and Sughatadasa [1]. On the other hand, in absence of occlusions, the 3D projective shape of a spatial configuration can be recovered from a stereo pair of images, thus allowing to test for mean glaucomatous 3D projective shape change detection from standard stereo pairs of eye images.
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Date Issued
2010
Identifier
FSU_migr_etd-7118
Format
Thesis
Nonparametric Estimation of Three Dimensional Projective Shapes with Applications in Medical Imaging and in Pattern Recognition
Title
Nonparametric Estimation of Three Dimensional Projective Shapes with Applications in Medical Imaging and in Pattern Recognition.
Creator
Crane, Michael, Patrangenaru, Victor, Liu, Xiuwen, Huffer, Fred W., Sinha, Debajyoti, Department of Statistics, Florida State University
Abstract/Description
This dissertation is on analysis of invariants of a 3D configuration from its 2D images in pictures of this configuration, without requiring any restriction on the camera positioning relative to the scene pictured. We briefly review some of the main results found in the literature. The methodology used is nonparametric, manifold based combined with standard computer vision re-construction techniques. More specifically, we use asymptotic results for the extrinsic sample mean and the extrinsic...
Show moreThis dissertation is on analysis of invariants of a 3D configuration from its 2D images in pictures of this configuration, without requiring any restriction on the camera positioning relative to the scene pictured. We briefly review some of the main results found in the literature. The methodology used is nonparametric, manifold based combined with standard computer vision re-construction techniques. More specifically, we use asymptotic results for the extrinsic sample mean and the extrinsic sample covariance to construct boot-strap confidence regions for mean projective shapes of 3D configurations. Chapters 4, 5 and 6 contain new results. In chapter 4, we develop tests for coplanarity. In chapter 5, is on reconstruction of 3D polyhedral scenes, including texture from arbitrary partial views. In chapter 6, we develop a nonparametric methodology for estimating the mean change for matched samples on a Lie group. We then notice that for k ≥ 4, a manifold of projective shapes of k-ads in general position in 3D has a structure of 3k − 15 dimensional Lie group (P-Quaternions) that is equivariantly embedded in an Euclidean space, therefore testing for mean 3D projective shape change amounts to a one sample test for extrinsic mean PQuaternion Objects. The Lie group technique leads to a large sample and nonparametric bootstrap test for one population extrinsic mean on a projective shape space, as recently developed by Patrangenaru, Liu and Sughatadasa. On the other hand, in absence of occlusions, the 3D projective shape of a spatial configuration can be recovered from a stereo pair of images, thus allowing to test for mean glaucomatous 3D projective shape change detection from standard stereo pairs of eye images.
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Date Issued
2010
Identifier
FSU_migr_etd-4607
Format
Thesis
One-and Two-Sample Problem for Data on Hilbert Manifolds with Applications to Shape Analysis
Title
The One-and Two-Sample Problem for Data on Hilbert Manifolds with Applications to Shape Analysis.
Creator
Qiu, Mingfei, Patrangenaru, Victor, Liu, Xiuwen, Slate, Elizabeth H., Barbu, Adrian G. (Adrian Gheorghe), Clickner, Robert Paul, Paige, Robert, Florida State University, College...
Show moreQiu, Mingfei, Patrangenaru, Victor, Liu, Xiuwen, Slate, Elizabeth H., Barbu, Adrian G. (Adrian Gheorghe), Clickner, Robert Paul, Paige, Robert, Florida State University, College of Arts and Sciences, Department of Statistics
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Abstract/Description
This dissertation is concerned with high level imaging analysis. In particular, our focus is on extracting the projective shape information or the similarity shape from digital camera images or Magnetic Resonance Imaging(MRI). The approach is statistical without making any assumptions about the distributions of the random object under investigation. The data is organized as points on a Hilbert manifold. In the case of projective shapes of finite dimensional configuration of points, we...
Show moreThis dissertation is concerned with high level imaging analysis. In particular, our focus is on extracting the projective shape information or the similarity shape from digital camera images or Magnetic Resonance Imaging(MRI). The approach is statistical without making any assumptions about the distributions of the random object under investigation. The data is organized as points on a Hilbert manifold. In the case of projective shapes of finite dimensional configuration of points, we consider testing a one-sample null hypothesis, while in the infinite dimensional case, we considered a neighborhood hypothesis testing methods. For 3D scenes, we retrieve the 3D projective shape, and use the Lie group structure of the projective shape space. We test the equality of two extrinsic means, by introducing the mean projective shape change. For 2D MRI of midsections of Corpus Callosum contours, we use an automatic matching technique that is necessary in pursuing a one-sample neighborhood hypothesis testing for the similarity shapes. We conclude that the mean similarity shape of the Corpus Callosum of average individuals is very far from the shape of Albert Einstein's, which may explain his geniality. Another application of our Hilbert manifold methodology is two-sample testing problem for Veronese-Whitney means of projective shapes of 3D contours. Particularly, our data consisting comparing 3D projective shapes of contours of leaves from the same tree species.
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Date Issued
2015
Identifier
FSU_2015fall_Qiu_fsu_0071E_12922
Format
Thesis
Statistical Analysis on Object Spaces with Applications
Title
Statistical Analysis on Object Spaces with Applications.
Creator
Yao, Kouadio David, Patrangenaru, Victor, Kercheval, Alec N., Liu, Xiuwen, Mio, Washington, Wang, Xiaoming, Florida State University, College of Arts and Sciences, Department of...
Show moreYao, Kouadio David, Patrangenaru, Victor, Kercheval, Alec N., Liu, Xiuwen, Mio, Washington, Wang, Xiaoming, Florida State University, College of Arts and Sciences, Department of Mathematics
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Abstract/Description
Most of the data encountered is bounded nonlinear data. The Universe is bounded, planets are sphere like shaped objects, and life growing on Earth comes in various shapes and colors that can hardly be represented as points on a linear space, and even if the object space they sit on is embedded in a Euclidean space, their mean vector can not be represented as a point on that object space, except for the case when such space is convex. To address this misgiving, since the mean vector is the...
Show moreMost of the data encountered is bounded nonlinear data. The Universe is bounded, planets are sphere like shaped objects, and life growing on Earth comes in various shapes and colors that can hardly be represented as points on a linear space, and even if the object space they sit on is embedded in a Euclidean space, their mean vector can not be represented as a point on that object space, except for the case when such space is convex. To address this misgiving, since the mean vector is the minimizer of the expected square distance, following Fr\'echet (1948), on a compact metric space, one may consider both minimizers and maximizers of the expected square distance to a given point on the object space as mean, respectively {\bf anti-mean} of a given random point. Of all distances on a object space, one considers here the chord distance associated with an embedding of the object space, since for such distances one can give a necessary and sufficient condition for the existence of a unique Fr\'echet mean (respectively Fr\'echet anti-mean). For such distributions these location parameters are called extrinsic mean (respectively extrinsic anti-mean), and the corresponding sample statistics are consistent estimators of their population counterparts. Moreover one derives the limit distribution of such estimators around a mean located at a smooth extrinsic anti-mean. Extrinsic analysis is thus a general framework that allows one to run object data analysis on nonlinear object spaces that can be embedded in a numerical space. In particular one focuses on Veronese-Whitney (VW) means and anti-means of 3D projective shapes of configurations extracted from digital camera images. The 3D data extraction is greatly simplified by an RGB based algorithm followed by the Faugeras-Hartley-Gupta-Chen 3D reconstruction method. In particular one derives two sample tests for face analysis based on projective shapes, and more generally a MANOVA on manifolds method to be used in 3D projective shape analysis. The manifold based approach is also applicable to financial data analysis for exchange rates.
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Date Issued
2016
Identifier
FSU_FA2016_Yao_fsu_0071E_13605
Format
Thesis
High Level Image Analysis on Manifolds via Projective Shapes and 3D Reflection Shapes
Title
High Level Image Analysis on Manifolds via Projective Shapes and 3D Reflection Shapes.
Creator
Lester, David T. (David Thomas), Patrangenaru, Victor, Liu, Xiuwen, Barbu, Adrian G. (Adrian Gheorghe), Tao, Minjing, Florida State University, College of Arts and Sciences,...
Show moreLester, David T. (David Thomas), Patrangenaru, Victor, Liu, Xiuwen, Barbu, Adrian G. (Adrian Gheorghe), Tao, Minjing, Florida State University, College of Arts and Sciences, Department of Statistics
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Abstract/Description
Shape analysis is a widely studied topic in modern Statistics with important applications in areas such as medical imaging. Here we focus on two-sample hypothesis testing for both finite and infinite extrinsic mean shapes of configurations. First, we present a test for equality of mean projective shapes of 2D contours based on rotations. Secondly, we present a test for mean 3D reflection shapes based on the Schoenberg mean. We apply these tests to footprint data (contours), clamshells (3D...
Show moreShape analysis is a widely studied topic in modern Statistics with important applications in areas such as medical imaging. Here we focus on two-sample hypothesis testing for both finite and infinite extrinsic mean shapes of configurations. First, we present a test for equality of mean projective shapes of 2D contours based on rotations. Secondly, we present a test for mean 3D reflection shapes based on the Schoenberg mean. We apply these tests to footprint data (contours), clamshells (3D reflection shape) and human facial configurations extracted from digital camera images. We also present the method of MANOVA on manifolds, and apply it to face data extracted from digital camera images. Finally, we present a new statistical tool called anti-regression.
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Date Issued
2017
Identifier
FSU_2017SP_Lester_fsu_0071E_13856
Format
Thesis