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A Part-aware Surface Metric for Shape Analysis

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A Part-aware Surface Metric for Shape Analysis

Rong Liu1, Hao Zhang1, Ariel Shamir2, and Daniel Cohen-Or3

1Simon Fraser University, Canada

2The Interdisciplinary Center, Israel

3Tel Aviv University, Israel

- Parts are useful to many geometry processing applications:
- Shape retrieval
- [Berretti99, Dey03, Funkhouser06, Shalom08]

- Shape modeling
- [Funkhouser04, Kraevoy07]

- Animation
- [Katz03, Lien05]

- Shape retrieval

Part-aware Surface Metric

- Many problems involving the analysis and understanding of a 3D object utilize a metric, which prescribes a distance function between points on the boundary surface of the object…

a

b

Part-aware Surface Metric

- Euclidean
- Geodesic [Carmo76]
- Isophotic [Pottmann04]
- Diffusion distance [deGoes08]

Surface based!

Part-aware Surface Metric

Part-aware Surface Metric

a

b

- Not a metric
- Same value on different parts
- Distance(a,b) = SDF(a)-SDF(b) = 0

Part-aware Surface Metric

- A novel part-aware surface distance metric
- Able to effectively capture part information of a shape
- Based on volumetric considerations

- Applications:
- Segmentation
- Shape registration
- Part-aware sampling
- Shape retrieval

Part-aware Surface Metric

= shortest graph distance between and

distance( , )

- Derived as graph distance on primal/dual graph
- Geodesic distance
- Angular distance
- VSI distance: Captures part information!
- Large distance between faces from different parts and vice versa

Part-aware Surface Metric

- Geodesic distance (approximate)
- Distant faces tend to belong to different parts (Gestalt principle of proximity)

- Angular distance
- Faces separated by concave regions tend to belong to different parts (Minima rule)

Part-aware Surface Metric

No angular difference!

- Geodesic distance: insensitive to parts
- Angular distance: subject to leakage problem

Part-aware Surface Metric

- Look at the object from inside

Part-aware Surface Metric

- Visibility can capture part information:
Significant visibility changes across part boundaries

Part-aware Surface Metric

Connect to surface: find reference points

Part-aware Surface Metric

Sample visible regions from ref. points

The Volumetric Shape Image (VSI)

stores the normalized intersection points (|S|=100)

Part-aware Surface Metric

Compute VSI difference

Difference is based on the reach of

local volume along sampling direction

Part-aware Surface Metric

Part-aware Surface Metric

Part-aware Surface Metric

Part-aware Surface Metric

Part-aware Surface Metric

Part-aware Surface Metric

d

VSI Diff

c

b

e

a

Part-aware Surface Metric

Angular distance fields

VSI distance fields

Part-aware Surface Metric

- Metric derived as the graph distance on a combined weighted graph

geodesic graph

angular graph

VSI graph

edge weight normalization

edge weight normalization

=

combined graph

Part-aware Surface Metric

Geodesic

Diffusion[deGoes08]

Part-aware

Angular[Katz03]

Part-aware Surface Metric

- Able to handle open meshes with reasonably well-defined volume
- Speed-up
- Space voxelization (100*100*100) for ray-mesh intersection detection
- Use Sampling of VSI and interpolation

- Efficiency
- Empirical complexity: O(rn + n2/3r3)=O(n2) where r = 100 is grid resolution, n is number of triangles
- A mesh with 50K faces: 15 seconds

Part-aware Surface Metric

- Segmentation
- Registration
- Part-aware sampling
- Retrieval

Part-aware Surface Metric

- Test algorithm: spectral clustering [Liu04]
- Use distances to derive a spectral embedding of input mesh faces
- Cluster in embedding space

Part-aware Surface Metric

With geodesic+angular distance

With part-aware distance

Part-aware Surface Metric

- Test algorithm: spectral embedding [Jain07] and iterative closest point [Besl92] alignment
- Use distances to derive a spectral embedding
- Register in the embedding

- Geodesic distance is usually used as it is intrinsic (invariant to articulation)
- Geodesic distance is not invariant to stretch! But VSI distance is (although not to articulations)
- Part aware: best to combines the two

Part-aware Surface Metric

geodesic

part-aware

homer

stretched

Part-aware Surface Metric

geodesic

part-aware

Part-aware Surface Metric

- Algorithm: max-min (farthest point) sampling
- Add samples iteratively
- Each sample maximizes the minimum distance to previously chosen samples

With isophotic distance With part-aware distance

Part-aware Surface Metric

- Test algorithm: probability distribution of shape function [Osada02]
- Use the histogram of a shape function as signature and chi-square to measure histogram distances

histogram of pair-wise geodesic distances between vertices

Part-aware Surface Metric

Geodesic distance: invariant to articulation

Part-aware distance: stronger discriminating capability

Part-aware Surface Metric

Part Aware

Geodesic

D2

Part-aware Surface Metric

- Use the Volume!
- A novel part-aware metric
- Based on volumetric considerations
- Able to capture part information effectively
- Improved upon previous metrics

- Demonstrated effectiveness for a variety of geometry processing and analysis applications
- Mesh segmentation
- Shape registration
- Part-sensitive sampling
- Shape Registration

Part-aware Surface Metric

- Intelligent ways to tune the weights for geodesic, angular, and VSI distances
- Application dependent?
- Training?

- More systematic test on shape retrieval
- Expand database
- Compare with more algorithms

- More applications

Part-aware Surface Metric

Part-aware Surface Metric