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Diffusion-geometric maximally stable component detection in deformable shapesPowerPoint Presentation

Diffusion-geometric maximally stable component detection in deformable shapes

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### Diffusion-geometricmaximally stable component detection in deformable shapes

### Thank You

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

The “Feature Approach”to Image Analysis

- Video tracking
- Panorama alignment
- 3D reconstruction
- Content-based image retrieval

Problem formulation

- Find a semi-local feature detector
- High repeatability
- Invariance to (non-stretching) deformation
- Robustness to noise, sampling, etc.

- Add informative descriptor

Region Description

Distance = 0.44

Distance

= 0.34

Distance

= 0.02

Distance

= 0.08

Distance= 0.17

Distance = 0.25

3D Human Scans

Taken from the SCAPE dataset

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

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

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

Auto-diffusivity

- Special case -
- The chance of returning to after time
- Related to Gaussian curvature by
- Now we can attach scalar value to shapes!

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

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)

Any Questions?

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