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