Diffusion geometric maximally stable component detection in deformable shapes
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Diffusion-geometric maximally stable component detection in deformable shapes. Roee Litman , Dr. Alexander Bronstein, Dr. Michael Bronstein. The “Feature Approach” to Image Analysis. Video tracking Panorama alignment 3D reconstruction Content-based image retrieval. Non-rigid Shapes.

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Diffusion-geometric maximally stable component detection in deformable shapes

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Diffusion-geometricmaximally stable component detection in deformable shapes

Roee Litman, Dr. Alexander Bronstein, Dr. Michael Bronstein


The “Feature Approach”to Image Analysis

  • Video tracking

  • Panorama alignment

  • 3D reconstruction

  • Content-based image retrieval


Non-rigid Shapes


Problem formulation

  • Find a semi-local feature detector

    • High repeatability

    • Invariance to (non-stretching) deformation

    • Robustness to noise, sampling, etc.

  • Add informative descriptor


The Goal

“Head+Arm”

take a shape

Detect (stable) regions

“Head”

“Arm”

“Upper Body”

“Leg”

“Leg”


Deformation Invariant


More Results

(Taken from the TOSCA dataset)

Horse regions +

Human regions


Region Description

Distance = 0.44

Distance

= 0.34

Distance

= 0.02

Distance

= 0.08

Distance= 0.17

Distance = 0.25


Region Matching

Query

1st, 2nd, 4th, 10th, and 15th matches


3D Human Scans

Taken from the SCAPE dataset


Scanned Region Matching

Query

1st, 2nd, 4th, 10th, and 15th matches


(The “how”)

Methodology


In a nutshell…

The Feature Approach

for Images

Deformable Shape Analysis

Shape

MSER

MSER

Maximally Stable ExtremalRegion

Diffusion Geometry


Original MSER (Matas et-al)


MSER – In a nutshell

  • Threshold image at consecutive gray-levels

  • Search regions whose area stay nearly the same through a wide range of thresholds


MSER – In a nutshell


Algorithm overview


Algorithm overview

  • Represent as weighted graph

  • Component tree

  • Stable component detection


Algorithm overview

  • Represent as weighted graph


Weighting the graph

In images

  • Illumination (Gray-scale)

  • Color (RGB)

    In Shapes

  • Mean Curvature (not deformation invariant)

  • Diffusion Geometry


Weighting Option

  • For every point on the shape:

  • Calculate the prob. of a random walk to return to the same point.

  • Similar to Gaussian curvature

  • Intrinsic - i.e. deformation invariant


Weight example

Color-mapped

Level-set animation


Diffusion Geometry

  • Analysis of diffusion (random walk) processes

  • Governed by the heat equation

  • Solution is heat distributionat point at time


Heat-Kernel

  • Given

    • Initial condition

    • Boundary condition, if these’s a boundary

  • Solve using:

  • i.e. - find the “heat-kernel”


Probabilistic Interpretation

The probability density for a transition

by random walk of length ,

from to


Spectral Interpretation

  • How to calculate ?

  • Heat kernel can be calculated directly from eigen-decomposition of the Laplacain

  • By spectral decomposition theorem:


Laplace-Beltrami Eigenfunctions


Deformation Invariance


Auto-diffusivity

  • Special case -

  • The chance of returning to after time

  • Related to Gaussian curvature by

  • Now we can attach scalar value to shapes!


Weight example

Color-mapped

Level-set animation


Algorithm overview

  • Represent as weighted graph

  • Component tree

  • Stable component detection


Performance


Benchmarking The Method

  • Method was tested on SHREC 2010 data-set:

    • 3 basic shapes (human, dog & horse)

    • 9 transformations, applied in 5 different strengths

    • 138 shapes in total

Scale

Original

Deformation

Holes

Noise


Results


Quantitative Results

  • Regions were projected onto “original” shape,and overlap ratio was measured

    • Vertex-wise correspondences were given

  • Overlap ratio between a region and its projected counterpart is

  • Repeatability is the percent of regions with overlap above a threshold


Repeatability

65% at 0.75


Conclusion

  • Stable region detector for deformable shapes

  • Generic detection framework

    • Vertex- and edge-weighted graph representation

    • Surface and volume data

  • Partial matching & retrieval potential

  • Tested quantitatively (on SHREC10)


Thank You

Any Questions?


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