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Shape Matching via Quotient Spaces

Maks Ovsjanikov, Quentin Mérigot, Viorica Pătrăucean, Leonidas Guibas
2013 Computer graphics forum (Print)  
quotient space, where the symmetry has been identified and factored out.  ...  Here, we first estimate a single map in an appropriate quotient space and then use it to generate 8 different point-to-point maps between two octopus models.  ...  Guibas / Shape Matching via Quotient Spaces  ... 
doi:10.1111/cgf.12167 fatcat:egxhg7eg2vdolouo7ywtm4q6xi

Manifold Learning in Quotient Spaces

Eloi Mehr, Andre Lieutier, Fernando Sanchez Bermudez, Vincent Guitteny, Nicolas Thome, Matthieu Cord
2018 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition  
To get rid of undesirable input variability our model learns a manifold in a quotient space of the input space. Typically, we propose to quotient the space of 3D models by the action of rotations.  ...  Thus, our quotient autoencoder allows to directly learn in the space of interest, ignoring side information.  ...  Once the data is aligned in the reconstructed space, we can match the input shapes by morphing the reconstructed shapes to the original ones.  ... 
doi:10.1109/cvpr.2018.00955 dblp:conf/cvpr/MehrLBGTC18 fatcat:5ib5i2txtfdfxkljuvjmrlnv64

Geometries on Spaces of Treelike Shapes [chapter]

Aasa Feragen, Francois Lauze, Pechin Lo, Marleen de Bruijne, Mads Nielsen
2011 Lecture Notes in Computer Science  
The QED is a quotient euclidean distance arising from the new shape space formulation, while TED is essentially the classical tree edit distance.  ...  In order to develop statistical methods for shapes with a tree-structure, we construct a shape space framework for treelike shapes and study metrics on the shape space.  ...  The quotient spaceX = (X/ ∼) = {x|x ∈ X} of equivalence classesx is the space of treelike shapes.  ... 
doi:10.1007/978-3-642-19309-5_13 fatcat:edvnhjxcsnaivhskijpsn5tgu4

Shape Theories. II. Compactness Selection Principles [article]

Edward Anderson
2019 arXiv   pre-print
Shape(-and-scale) spaces - configuration spaces for generalized Kendall-type Shape(-and-Scale) Theories - are usually not manifolds but stratified manifolds.  ...  While in Kendall's own case - similarity shapes - the shape spaces are analytically nice - Hausdorff - for the Image Analysis and Computer Vision cases - affine and projective shapes - they are not: merely  ...  I also pay my respects to Professor Graham Allan, who first brought subtleties with quotient spaces to my attention when I was an undergraduate and subsequently passed away, and also thank Professor Timothy  ... 
arXiv:1811.06528v4 fatcat:fc6vbuehrfbfnglgbf7hjwy2jm

Bayesian Registration of Functions and Curves

Wen Cheng, Ian L. Dryden, Xianzheng Huang
2016 Bayesian Analysis  
We also compare ambient and quotient space estimators for mean shape, and explain their frequent similarity in many practical problems using a Laplace approximation.  ...  We distinguish between various spaces of interest: the original space, the ambient space after standardizing, and the quotient space after removing a group of transformations.  ...  A.1 Initialization step Let q 1 (t) and q 2 (t) be the q(t) functions to be matched via dynamic programming.  ... 
doi:10.1214/15-ba957 fatcat:2jwfz5ab2fe3piizvl3ztbgsfe

Varifold-Based Matching of Curves via Sobolev-Type Riemannian Metrics [chapter]

Martin Bauer, Martins Bruveris, Nicolas Charon, Jakob Møller-Andersen
2017 Lecture Notes in Computer Science  
We describe the numerical method used for solving the inexact matching problem, apply it to study the shape of mosquito wings and compare our method to curve matching in the LDDMM framework.  ...  Second order Sobolev metrics are a useful tool in the shape analysis of curves.  ...  Inexact matching on the shape space of curves In this section we combine Sobolev metrics and varifold distances to compute geodesics on shape space via a relaxed optimization problem.  ... 
doi:10.1007/978-3-319-67675-3_14 fatcat:5v6iej4im5bwfltqpw6ymxnnei

A relaxed approach for curve matching with elastic metrics [article]

Martin Bauer, Martins Bruveris, Nicolas Charon, Jakob Møller-Andersen
2018 arXiv   pre-print
Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups.  ...  In this paper we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions.  ...  Acknowledgments We would like to thank Philipp Harms, Eric Klassen, Sebastian Kurtek, Peter Michor, Tom Needham, Anuj Srivastava and the Shape Group at FSU for helpful comments and discussions.  ... 
arXiv:1803.10893v2 fatcat:tyy7l5uvp5dsvcpcgyvo6jhyuu

An inexact matching approach for the comparison of plane curves with general elastic metrics [article]

Yashil Sukurdeep, Martin Bauer, Nicolas Charon
2020 arXiv   pre-print
These benefits are illustrated via a few preliminary numerical results.  ...  Our approach combines the general simplifying transform for first-order elastic metrics that was recently introduced by Kurtek and Needham, together with a relaxation of the matching constraint using parametrization-invariant  ...  Mathematically, we model the space of geometric curves as a quotient space of infinite dimensional manifolds.  ... 
arXiv:2001.02858v1 fatcat:ea3weplqkfhmhnvaf4uyiekub4

Shape analysis of framed space curves [article]

Tom Needham
2018 arXiv   pre-print
In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is  ...  Averages of collections of plane and space curves are computed via a novel algorithm utilizing flag means.  ...  Next I would like to thank Michael Tychonievich for his help in developing a GUI for the framed curves matching program used to produce the numerical experiments.  ... 
arXiv:1807.03477v1 fatcat:lkcjdiyhbvce5dmkv6slh7wz2m

Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework [article]

Emmanuel Hartman, Yashil Sukurdeep, Eric Klassen, Nicolas Charon, Martin Bauer
2022 arXiv   pre-print
Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence  ...  Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing  ...  it as shape space.  ... 
arXiv:2204.04238v1 fatcat:jngsaitp2jdvvm3osgui3gutmy

A Quotient Space Formulation for Generative Statistical Analysis of Graphical Data [article]

Xiaoyang Guo, Anuj Srivastava, Sudeep Sarkar
2021 arXiv   pre-print
To develop statistical analyses for graphical data, especially towards generative modeling, one needs mathematical representations and metrics for matching and comparing graphs, and subsequent tools, such  ...  This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis, statistical testing  ...  To remove this nuisance group, we form a quotient space and inherit a metric on the quotient space from the original set of matrices.  ... 
arXiv:1909.12907v2 fatcat:g6gy6s4y7rhfnfmxyoa25ck7xy

Topography-Based Registration of Developing Cortical Surfaces in Infants Using Multidirectional Varifold Representation [chapter]

Islem Rekik, Gang Li, Weili Lin, Dinggang Shen
2015 Lecture Notes in Computer Science  
state-of-the-art methods: (1) diffeomorphic spectral matching, (2) current-based surface matching and (3) original varifold-based surface matching.  ...  Cortical surface registration or matching facilitates atlasing, cortical morphology-function comparison and statistical analysis.  ...  This quotient space G d (E) contains elements u that are equivalent to u/|u| and −u/|u|, denoted as .  ... 
doi:10.1007/978-3-319-24571-3_28 pmid:27169137 pmcid:PMC4860272 fatcat:qwgqypkw3ragjon7xww2yy35ry

A numerical framework for elastic surface matching, comparison, and interpolation [article]

Martin Bauer, Nicolas Charon, Philipp Harms, Hsi-Wei Hsieh
2020 arXiv   pre-print
Surface comparison and matching is a challenging problem in computer vision.  ...  While reparametrization-invariant Sobolev metrics provide meaningful elastic distances and point correspondences via the geodesic boundary value problem, solving this problem numerically tends to be difficult  ...  Shape analysis provides a mathematical description of shape spaces as quotient spaces in the above sense, as well as a computational toolbox for statistics and machine learning thereon.  ... 
arXiv:2006.11652v1 fatcat:3zoehcz66fchjlqebukwwpax34

Intrinsic Riemannian metrics on spaces of curves: theory and computation [article]

Martin Bauer, Nicolas Charon, Eric Klassen, Alice Le Brigant
2020 arXiv   pre-print
This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curves modulo shape-preserving  ...  We summarize the general construction and theoretical properties of quotient elastic metrics for Euclidean as well as non-Euclidean curves before considering the special case of the square root velocity  ...  The induced geodesic distance on the quotient shape space S(D, M ) can now be calculated via dist S ([c 0 ], [c 1 ]) = inf γ∈Diff+(D) g∈Isom(M ) dist(c 0 , g • c 1 • γ) = inf γ∈Diff+(D) g∈Isom(M ) dist  ... 
arXiv:2003.05590v2 fatcat:nrequrvwebfebfcmoyklcfuuxa

Representations, Metrics and Statistics For Shape Analysis of Elastic Graphs [article]

Xiaoyang Guo, Anuj Srivastava
2020 arXiv   pre-print
testing and modeling of graphical shapes.  ...  This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis and analytical statistical  ...  Instead, one can apply TPCA in the quotient space G, as described in Algorithm 2.  ... 
arXiv:2003.00287v2 fatcat:wlruzehigbgf3cdoiaq7wk2za4
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