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Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces  [PDF]
Vladimir Kadets,Miguel Martin
Mathematics , 2012, DOI: 10.1016/j.jmaa.2012.06.031
Abstract: We prove that an onto isometry between unit spheres of finite-dimensional polyhedral Banach spaces extends to a linear isometry of the corresponding spaces.
Building Statistical Shape Spaces for 3D Human Modeling  [PDF]
Leonid Pishchulin,Stefanie Wuhrer,Thomas Helten,Christian Theobalt,Bernt Schiele
Computer Science , 2015,
Abstract: Statistical models of 3D human shape and pose learned from scan databases have developed into valuable tools to solve a variety of vision and graphics problems. Unfortunately, most publicly available models are of limited expressiveness as they were learned on very small databases that hardly reflect the true variety in human body shapes. In this paper, we contribute by rebuilding a widely used statistical body representation from the largest commercially available scan database, and making the resulting model available to the community (visit http://humanshape.mpi-inf.mpg.de). As preprocessing several thousand scans for learning the model is a challenge in itself, we contribute by developing robust best practice solutions for scan alignment that quantitatively lead to the best learned models. We make implementations of these preprocessing steps also publicly available. We extensively evaluate the improved accuracy and generality of our new model, and show its improved performance for human body reconstruction from sparse input data.
On Low Distortion Embeddings of Statistical Distance Measures into Low Dimensional Spaces  [PDF]
Arnab Bhattacharya,Purushottam Kar,Manjish Pal
Computer Science , 2009, DOI: 10.1007/978-3-642-03573-9_13
Abstract: Statistical distance measures have found wide applicability in information retrieval tasks that typically involve high dimensional datasets. In order to reduce the storage space and ensure efficient performance of queries, dimensionality reduction while preserving the inter-point similarity is highly desirable. In this paper, we investigate various statistical distance measures from the point of view of discovering low distortion embeddings into low-dimensional spaces. More specifically, we consider the Mahalanobis distance measure, the Bhattacharyya class of divergences and the Kullback-Leibler divergence. We present a dimensionality reduction method based on the Johnson-Lindenstrauss Lemma for the Mahalanobis measure that achieves arbitrarily low distortion. By using the Johnson-Lindenstrauss Lemma again, we further demonstrate that the Bhattacharyya distance admits dimensionality reduction with arbitrarily low additive error. We also examine the question of embeddability into metric spaces for these distance measures due to the availability of efficient indexing schemes on metric spaces. We provide explicit constructions of point sets under the Bhattacharyya and the Kullback-Leibler divergences whose embeddings into any metric space incur arbitrarily large distortions. We show that the lower bound presented for Bhattacharyya distance is nearly tight by providing an embedding that approaches the lower bound for relatively small dimensional datasets.
Statistical properties of determinantal point processes in high-dimensional Euclidean spaces  [PDF]
A. Scardicchio,C. E. Zachary,S. Torquato
Physics , 2008, DOI: 10.1103/PhysRevE.79.041108
Abstract: The goal of this paper is to quantitatively describe some statistical properties of higher-dimensional determinantal point processes with a primary focus on the nearest-neighbor distribution functions. Toward this end, we express these functions as determinants of $N\times N$ matrices and then extrapolate to $N\to\infty$. This formulation allows for a quick and accurate numerical evaluation of these quantities for point processes in Euclidean spaces of dimension $d$. We also implement an algorithm due to Hough \emph{et. al.} \cite{hough2006dpa} for generating configurations of determinantal point processes in arbitrary Euclidean spaces, and we utilize this algorithm in conjunction with the aforementioned numerical results to characterize the statistical properties of what we call the Fermi-sphere point process for $d = 1$ to 4. This homogeneous, isotropic determinantal point process, discussed also in a companion paper \cite{ToScZa08}, is the high-dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble (CUE). In addition to the nearest-neighbor probability distribution, we are able to calculate Voronoi cells and nearest-neighbor extrema statistics for the Fermi-sphere point process and discuss these as the dimension $d$ is varied. The results in this paper accompany and complement analytical properties of higher-dimensional determinantal point processes developed in \cite{ToScZa08}.
The Poincare conjecture for digital spaces. Properties of digital n-dimensional disks and spheres  [PDF]
Alexander V. Evako
Mathematics , 2006,
Abstract: Motivated by the Poincare conjecture, we study properties of digital n-dimensional spheres and disks, which are digital models of their continuous counterparts. We introduce homeomorphic transformations of digital manifolds, which retain the connectedness, the dimension, the Euler characteristics and the homology groups of manifolds. We find conditions where an n-dimensional digital manifold is the n-dimensional digital sphere and discuss the link between continuous closed n-manifolds and their digital models.
Metric measure limits of spheres and complex projective spaces  [PDF]
Takashi Shioya
Mathematics , 2014,
Abstract: We study the limits of sequences of spheres and complex projective spaces with unbounded dimensions. A sequence of spheres (resp. complex projective spaces) either is a Levy family, infinitely dissipates, or converges to (resp. the Hopf quotient of) a virtual infinite-dimensional Gaussian space, depending on the size of the spaces. These are the first discovered examples with the property that the limits are drastically different from the spaces in the sequence. For the proof, we introduce a metric on Gromov's compactification of the space of metric measure spaces.
Dynamic shape analysis and comparison of leaf growth  [PDF]
Stephan Huckemann
Mathematics , 2010,
Abstract: In the statistical analysis of shape a goal beyond the analysis of static shapes lies in the quantification of `same' deformation of different shapes. Typically, shape spaces are modelled as Riemannian manifolds on which parallel transport along geodesics naturally qualifies as a measure for the `similarity' of deformation. Since these spaces are usually defined as combinations of Riemannian immersions and submersions, only for few well featured spaces such as spheres or complex projective spaces (which are Kendall's spaces for 2D shapes), parallel transport along geodesics can be computed explicitly. In this contribution a general numerical method to compute parallel transport along geodesics when no explicit formula is available is provided. This method is applied to the shape spaces of closed 2D contours based on angular direction and to Kendall's spaces of shapes of arbitrary dimension. In application to the temporal evolution of leaf shape over a growing period, one leaf's shape-growth dynamics can be applied to another leaf. For a specific poplar tree investigated it is found that leaves of initially and terminally different shape evolve rather parallel, i.e. with comparable dynamics.
Review of Statistical Shape Spaces for 3D Data with Comparative Analysis for Human Faces  [PDF]
Alan Brunton,Augusto Salazar,Timo Bolkart,Stefanie Wuhrer
Computer Science , 2012, DOI: 10.1016/j.cviu.2014.05.005
Abstract: With systems for acquiring 3D surface data being evermore commonplace, it has become important to reliably extract specific shapes from the acquired data. In the presence of noise and occlusions, this can be done through the use of statistical shape models, which are learned from databases of clean examples of the shape in question. In this paper, we review, analyze and compare different statistical models: from those that analyze the variation in geometry globally to those that analyze the variation in geometry locally. We first review how different types of models have been used in the literature, then proceed to define the models and analyze them theoretically, in terms of both their statistical and computational aspects. We then perform extensive experimental comparison on the task of model fitting, and give intuition about which type of model is better for a few applications. Due to the wide availability of databases of high-quality data, we use the human face as the specific shape we wish to extract from corrupted data.
A Statistical Texture Model of the Liver Based on Generalized N-Dimensional Principal Component Analysis (GND-PCA) and 3D Shape Normalization  [PDF]
Xu Qiao,Yen-Wei Chen
International Journal of Biomedical Imaging , 2011, DOI: 10.1155/2011/601672
Abstract: We present a method based on generalized N-dimensional principal component analysis (GND-PCA) and a 3D shape normalization technique for statistical texture modeling of the liver. The 3D shape normalization technique is used for normalizing liver shapes in order to remove the liver shape variability and capture pure texture variations. The GND-PCA is used to overcome overfitting problems when the training samples are too much fewer than the dimension of the data. The preliminary results of leave-one-out experiments show that the statistical texture model of the liver built by our method can represent an untrained liver volume well, even though the mode is trained by fewer samples. We also demonstrate its potential application to classification of normal and abnormal (with tumors) livers. 1. Introduction In the recent years, digital atlases of human anatomy have become popular and important topics in medical image analysis research [1, 2]. For interpretation of images of structures and variations in the organs of the human body, it is important to have a model of the way organ volumes can be represented. The digital atlas can be categorized as a statistical shape atlas (statistical shape model) and a statistical appearance (volume) atlas (statistical appearance (volume) model). The statistical shape model focuses on the shape information, such as feature points and volume surface [3]. It is a useful tool for study of variations in anatomic shape and has been widely used in medical image analysis, for example, medical image segmentation [4–6] and shape registration [7]. The statistical appearance model is focused on both shape and texture (voxel intensity) information. Inspired from the works of active shape models (ASMs) [3], the authors of [5, 8] proposed 3D ASMs for construction of 3D statistical models for segmentation of the left ventricle of the heart. In [9], the authors extended the work on active appearance models (AAMs) [10], and propose the use of 3D AAMs for the segmentation of cardiac MR and ultrasound images. Also, work [11] was done to build the 3D statistical deformation models (SDMs) for 3D MR brain images. Radiologists are mainly depending on the intensity variations (texture information) in livers on medical images to identify modules or tumors and make a diagnostic decision. However, there has been little research on applications of digital atlas to computer-assisted diagnosis (CAD). We have shown the potential application of statistical shape models to the classification of normal and cirrhotic livers [12]. Because many diseases will
Homological Stability For The Moduli Spaces of Products of Spheres  [PDF]
Nathan Perlmutter
Mathematics , 2014,
Abstract: We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres $S^{p}\times S^{q}$, for $p < q < 2p - 2$. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homological stability for the moduli spaces of manifolds of dimension $2n > 4$, with respect to forming connected sums with $S^{n}\times S^{n}$.
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