Compression for Smooth Shape Analysis

Most 3D shape analysis methods use triangular meshes to discretize both the shape and functions on it as piecewise linear functions. With this representation, shape analysis requires fine meshes to represent smooth shapes and geometric operators like normals, curvatures, or Laplace-Beltrami eigenfunctions at large computational and memory costs. We avoid this bottleneck with a compression technique that represents a smooth shape as subdivision surfaces and exploits the subdivision scheme to parametrize smooth functions on that shape with a few control parameters. This compression does not affect the accuracy of the Laplace-Beltrami operator and its eigenfunctions and allow us to compute shape descriptors and shape matchings at an accuracy comparable to triangular meshes but a fraction of the computational cost. Our framework can also compress surfaces represented by point clouds to do shape analysis of 3D scanning data

Statistical Shape Analysis using Kernel PCA

©2006 SPIE--The International Society for Optical Engineering. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. The electronic version of this article is the complete one and can be found online at: http://dx.doi.org/10.1117/12.641417DOI:10.1117/12.641417Presented at Image Processing Algorithms and Systems, Neural Networks, and Machine Learning, 16-18 January 2006, San Jose, California, USA.Mercer kernels are used for a wide range of image and signal processing tasks like de-noising, clustering, discriminant analysis etc. These algorithms construct their solutions in terms of the expansions in a high-dimensional feature space F. However, many applications like kernel PCA (principal component analysis) can be used more effectively if a pre-image of the projection in the feature space is available. In this paper, we propose a novel method to reconstruct a unique approximate pre-image of a feature vector and apply it for statistical shape analysis. We provide some experimental results to demonstrate the advantages of kernel PCA over linear PCA for shape learning, which include, but are not limited to, ability to learn and distinguish multiple geometries of shapes and robustness to occlusions

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

Line-shape analysis of charmonium resonances

We discuss weather the new enhancements found by BES, alias the $Y(4220)$, $Y(4260)$, $Y(4360)$, and $Y(4390)$ are true resonances. We argue that the nearby thresholds $D_s^*\bar{D}_s^*$, $D\bar{D}_1+\bar{D}D_1$, $D_s\bar{D}_{s1}+\bar{D_s}D_{s1}$ and $D^*\bar{D}_1+\bar{D}^*D_1$, as well as the $\psi(4160)$ and $\psi(4415)$ states have a strong influence over the observed $J/\psi \pi^+\pi^-$ and $h_c \pi^+\pi^-$ line-shapes. We propose an unitarized effective Lagrangian model to study the dynamical effect of the interaction between each known $\psi$ state and its closest thresholds. In addition, we present some of our recent motivating results, using the same model, for the $\psi(3770)$ resonance, where the distortion from a Breit-Wigner line-shape is shown to result not only from the kinematic interference, but also from the influence of the $D^0\bar{D}^0+D^+D^-$ one-loops. Moreover, two poles were found, at about 3.78 GeV and at 3.74 GeV, the second one generated dynamically, yet contributing to the distortion of the line-shape.Comment: Proceedings of the Conference "Hadron 17", held on 25-29 September, 2017, in Salamanca, Spai

Event Shape Analysis in ALICE

The jets are the final state manifestation of the hard parton scattering. Since at LHC energies the production of hard processes in proton-proton collisions will be copious and varied, it is important to develop methods to identify them through the study of their final states. In the present work we describe a method based on the use of some shape variables to discriminate events according their topologies. A very attractive feature of this analysis is the possibility of using the tracking information of the TPC+ITS in order to identify specific events like jets. Through the correlation between the quantities: thrust and recoil, calculated in minimum bias simulations of proton-proton collisions at 10 TeV, we show the sensitivity of the method to select specific topologies and high multiplicity. The presented results were obtained both at level generator and after reconstruction. It remains that with any kind of jet reconstruction algorithm one will confronted in general with overlapping jets. The present method determines areas where one does encounter special topologies of jets in an event. The aim is not to supplant the usual jet reconstruction algorithms, but rather to allow an easy selection of events allowing then the application of algorithms.Comment: 24 pages, ALICE Not

Shape deformation analysis from the optimal control viewpoint

A crucial problem in shape deformation analysis is to determine a deformation of a given shape into another one, which is optimal for a certain cost. It has a number of applications in particular in medical imaging. In this article we provide a new general approach to shape deformation analysis, within the framework of optimal control theory, in which a deformation is represented as the flow of diffeomorphisms generated by time-dependent vector fields. Using reproducing kernel Hilbert spaces of vector fields, the general shape deformation analysis problem is specified as an infinite-dimensional optimal control problem with state and control constraints. In this problem, the states are diffeomorphisms and the controls are vector fields, both of them being subject to some constraints. The functional to be minimized is the sum of a first term defined as geometric norm of the control (kinetic energy of the deformation) and of a data attachment term providing a geometric distance to the target shape. This point of view has several advantages. First, it allows one to model general constrained shape analysis problems, which opens new issues in this field. Second, using an extension of the Pontryagin maximum principle, one can characterize the optimal solutions of the shape deformation problem in a very general way as the solutions of constrained geodesic equations. Finally, recasting general algorithms of optimal control into shape analysis yields new efficient numerical methods in shape deformation analysis. Overall, the optimal control point of view unifies and generalizes different theoretical and numerical approaches to shape deformation problems, and also allows us to design new approaches. The optimal control problems that result from this construction are infinite dimensional and involve some constraints, and thus are nonstandard. In this article we also provide a rigorous and complete analysis of the infinite-dimensional shape space problem with constraints and of its finite-dimensional approximations

Steklov Spectral Geometry for Extrinsic Shape Analysis

We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde

Shape sensitivity analysis of the Hardy constant

We consider the Hardy constant associated with a domain in the $n$-dimensional Euclidean space and we study its variation upon perturbation of the domain. We prove a Fr\'{e}chet differentiability result and establish a Hadamard-type formula for the corresponding derivatives. We also prove a stability result for the minimizers of the Hardy quotient. Finally, we prove stability estimates in terms of the Lebesgue measure of the symmetric difference of domains.Comment: 23 pages; showkeys command remove