Diffusion-geometric maximally stable component detection in deformable shapes

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Diffusion-geometric maximally stable component detection in deformable 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

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

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

- Add informative descriptor

“Head+Arm”

take a shape

Detect (stable) regions

“Head”

“Arm”

“Upper Body”

“Leg”

“Leg”

(Taken from the TOSCA dataset)

Horse regions +

Human regions

Distance = 0.44

Distance

= 0.34

Distance

= 0.02

Distance

= 0.08

Distance= 0.17

Distance = 0.25

Query

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

Taken from the SCAPE dataset

Query

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

(The “how”)

The Feature Approach

for Images

Deformable Shape Analysis

Shape

MSER

MSER

Maximally Stable ExtremalRegion

Diffusion Geometry

- Threshold image at consecutive gray-levels
- Search regions whose area stay nearly the same through a wide range of thresholds

- Represent as weighted graph

- Component tree

- Stable component detection

- Represent as weighted graph

In images

- Illumination (Gray-scale)
- Color (RGB)
In Shapes

- Mean Curvature (not deformation invariant)
- Diffusion Geometry

- 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

Color-mapped

Level-set animation

- Analysis of diffusion (random walk) processes
- Governed by the heat equation
- Solution is heat distributionat point at time

- Given
- Initial condition
- Boundary condition, if these’s a boundary

- Solve using:
- i.e. - find the “heat-kernel”

The probability density for a transition

by random walk of length ,

from to

- How to calculate ?
- Heat kernel can be calculated directly from eigen-decomposition of the Laplacain
- By spectral decomposition theorem:

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

Color-mapped

Level-set animation

- Represent as weighted graph

- Component tree

- Stable component detection

- 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

- 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

65% at 0.75

- 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?