Data-Driven Shape Analysis and Processing

Data-driven methods play an increasingly important role in discovering geometric, structural, and semantic relationships between 3D shapes in collections, and applying this analysis to support intelligent modeling, editing, and visualization of geometric data. In contrast to traditional approaches, a key feature of data-driven approaches is that they aggregate information from a collection of shapes to improve the analysis and processing of individual shapes. In addition, they are able to learn models that reason about properties and relationships of shapes without relying on hard-coded rules or explicitly programmed instructions. We provide an overview of the main concepts and components of these techniques, and discuss their application to shape classification, segmentation, matching, reconstruction, modeling and exploration, as well as scene analysis and synthesis, through reviewing the literature and relating the existing works with both qualitative and numerical comparisons. We conclude our report with ideas that can inspire future research in data-driven shape analysis and processing.Comment: 10 pages, 19 figure

"What was Molyneux's Question A Question About?"

Molyneux asked whether a newly sighted person could distinguish a sphere from a cube by sight alone, given that she was antecedently able to do so by touch. This, we contend, is a question about general ideas. To answer it, we must ask (a) whether spatial locations identified by touch can be identified also by sight, and (b) whether the integration of spatial locations into an idea of shape persists through changes of modality. Posed this way, Molyneux’s Question goes substantially beyond question (a), about spatial locations, alone; for a positive answer to (a) leaves open whether a perceiver might cross-identify locations, but not be able to identify the shapes that collections of locations comprise. We further emphasize that MQ targets general ideas so as to distinguish it from corresponding questions about experiences of shape and about the property of tangible (vs. visual) shape. After proposing a generalized formulation of MQ, we extend earlier work (“Many Molyneux Questions,” Australasian Journal of Philosophy 2020) by showing that MQ does not admit a single answer across the board. Some integrative data-processes transfer across modalities; others do not. Seeing where and how such transfer succeeds and fails in individual cases has much to offer to our understanding of perception and its modalities

What was Molyneux's Question A Question About?

Molyneux asked whether a newly sighted person could distinguish a sphere from a cube by sight alone, given that she was antecedently able to do so by touch. This, we contend, is a question about general ideas. To answer it, we must ask (a) whether spatial locations identified by touch can be identified also by sight, and (b) whether the integration of spatial locations into an idea of shape persists through changes of modality. Posed this way, Molyneux’s Question goes substantially beyond question (a), about spatial locations, alone; for a positive answer to (a) leaves open whether a perceiver might cross-identify locations, but not be able to identify the shapes that collections of locations comprise. We further emphasize that MQ targets general ideas so as to distinguish it from corresponding questions about experiences of shape and about the property of tangible (vs. visual) shape. After proposing a generalized formulation of MQ, we extend earlier work (“Many Molyneux Questions,” Australasian Journal of Philosophy 2020) by showing that MQ does not admit a single answer across the board. Some integrative data-processes transfer across modalities; others do not. Seeing where and how such transfer succeeds and fails in individual cases has much to offer to our understanding of perception and its modalities

Eigenvalue Separation in Some Random Matrix Models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the size of the matrices are fixed and c goes to infinity, and higher rank analogues of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions, and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogues an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.Comment: 25 pages, 9 figures include