Growing Least Squares for the Analysis of Manifolds in Scale-Space Nicolas Mellado † , Gaël Guennebaud, Pascal Barla, Patrick Reuter, Christophe Schlick Inria - Univ. Bordeaux - IOGS - CNRS † nicolas.mellado@inria.fr
Context Shape matching [ HFGHP06 ,IT11] Focus on geometric properties Describe shape features Find their pertinent scales Requires a multi-scale analysis 2
Problem statement Find pertinent structures in scale-space Process Ignore noise Extract points Detect similarity Scale Arc-length 3
Previous work - 1/3 Intrinsic characterization of the shape: Heat diffusion (HKS) [ SOG09 ,BK10, ZRH11] 4
Previous work - 1/3 Intrinsic characterization of the shape: Heat diffusion (HKS) [ SOG09 ,BK10, ZRH11] Medium to global scales We want: • Fine to medium • Pertinent scale extraction 5
Previous work - 2/3 Multi-scale geometric descriptor: curvature Mean Curvatures estimated via smoothing [ ZBVH09,MFK ∗ 10 ] Covariance Analysis [ PKG2003 , LG2005, YLHP06] 6 Growing Least Squares
Previous work - 2/3 Multi-scale geometric descriptor: curvature Mean Curvatures estimated via smoothing [ ZBVH09,MFK ∗ 10 ] Curvature is not enough… Covariance Analysis [ PKG2003 , LG2005, YLHP06] 7 Growing Least Squares
Previous work - 3/3 Scale-space analysis [ ZBVH09,MFK ∗ 10 ,DK11] Curvature extrema extraction 8 Growing Least Squares
Previous work - 3/3 Scale-space analysis [ ZBVH09,MFK ∗ 10 ,DK11] Curvature extrema extraction Not exhaustive pertinence detection 9 Growing Least Squares
Our approach: Growing Least Squares Regression method: fit hyper-sphere Multi-scale Support huge point-sets Characterization by 2 nd order proxy Analytic detection of pertinent scales Dense description Bonus 2D curves 3D surfaces 10
Our approach: Growing Least Squares 11
Pipeline Geometric descriptor based on 2 nd -order regression, 1. Continuous & analytic geometric variation in scale. 2. 2 1 12
Geometric Descriptor – 1/3 Algebraic hyper-sphere fitting procedure [GG2007] w 1 τ κ η {x ; S u (x) = 0} 13
Geometric Descriptor – 1/3 Algebraic hyper-sphere fitting procedure [GG2007] w Normalization 1 τ κ η {x ; S u (x) = 0} 14
Geometric Descriptor – 1/3 Algebraic hyper-sphere fitting procedure [GG2007] w Normalization 1 τ κ η Re-parametrization {x ; S u (x) = 0} û = [ τ /t η t κ ] T 15
Geometric Descriptor – 1/3 Algebraic hyper-sphere fitting procedure [GG2007] w Normalization 1 τ κ η Re-parametrization {x ; S u (x) = 0} Scale Invariance û = [ τ /t η t κ ] T 16
Geometric Descriptor – 2/3 Impact of τ Impact of η 17
Geometric Descriptor – 3/3 An example in 3D 18
Geometric Descriptor – 3/3 An example in 3D 19
Geometric Variation Given our descriptor [ τ η κ ] T d τ d η d κ Variation is related to [ ] T dt dt dt Geometric Variation 20
Summary Geometric Descriptor Geometric Variation 21
Results/Applications – 1/3 Descriptor is robust to noise Can guide surface reconstruction Gargoyle - 250k points 5% random noise 20 scales, 6.6 s . 22
Results/Applications – 1/3 Descriptor is robust to noise Can guide surface reconstruction 23
Results/Applications – 2/3 Continuous detection of the feature area 24
Results/Applications – 2/3 Continuous detection of the feature area 2d curve- 5k points Armadillo - 173k points 1000 scales, 3 s . 20 scales, 6.3 s . 25
Results/Applications – 3/3 Similarity detection, Multi-scale profile 26
Results/Applications – 3/3 Similarity detection, Multi-scale profile Specifics scale range Torus- 500k points 20 scales, 42 s . 27
Discussion – 1/2 Main limitation isotropic, Analytic fitness measure Squared Pratt Norm [GG2007] φ = || u l || 2 - 4u c u q 28
Discussion – 2/2 Use multi-scale profile to characterize shape 29
Conclusions Contributions Stable second order descriptor for 2D/3D manifolds Continuous in scale and space from point-set input Continuous analysis to detect pertinent scales Future work: Use non-linear kernel to deal w/ additional attributes Analyze spatial variations to characterize anisotropy 30
Growing Least Squares for the Analysis of Manifolds in Scale-Space Thank you Nicolas Mellado Gaël Guennebaud Pascal Barla Patrick Reuter Christophe Schlick www.labri.fr/perso/mellado Inria - Univ. Bordeaux - IOGS – CNRS http://manao.inria.fr Project ANR SeARCH European Consortium v-must.net 31
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