A part aware surface metric for shape analysis
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A Part-aware Surface Metric for Shape Analysis. Rong Liu 1 , Hao Zhang 1 , Ariel Shamir 2 , and Daniel Cohen-Or 3. 1 Simon Fraser University, Canada 2 The Interdisciplinary Center, Israel 3 Tel Aviv University, Israel. Motivation – Shape Parts.

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A Part-aware Surface Metric for Shape Analysis

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A part aware surface metric for shape analysis

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


Motivation shape parts

Motivation – Shape Parts

  • Parts are useful to many geometry processing applications:

    • Shape retrieval

      • [Berretti99, Dey03, Funkhouser06, Shalom08]

    • Shape modeling

      • [Funkhouser04, Kraevoy07]

    • Animation

      • [Katz03, Lien05]

Part-aware Surface Metric


Shape analysis

Shape Analysis

  • 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


Distances

Distances

  • Euclidean

  • Geodesic [Carmo76]

  • Isophotic [Pottmann04]

  • Diffusion distance [deGoes08]

Surface based!

Part-aware Surface Metric


Missing connection to volume

Missing: Connection to Volume

Part-aware Surface Metric


Shape diameter function shapira08

a

b

Shape Diameter Function [Shapira08]

  • Not a metric

  • Same value on different parts

    • Distance(a,b) = SDF(a)-SDF(b) = 0

Part-aware Surface Metric


Contribution

Contribution

  • 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


Overview of part aware metric

= shortest graph distance between and

distance( , )

Overview of Part-aware Metric

  • 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 angular distance

Geodesic Distance & Angular Distance

  • 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


Geodesic distance angular distance1

No angular difference!

Geodesic Distance & Angular Distance

  • Geodesic distance: insensitive to parts

  • Angular distance: subject to leakage problem

Part-aware Surface Metric


Volumetric shape image

Volumetric Shape Image

  • Look at the object from inside

Part-aware Surface Metric


Volume based distance measure

Volume-based Distance Measure

  • Visibility can capture part information:

    Significant visibility changes across part boundaries

Part-aware Surface Metric


Vsi distance step 1

VSI-distance: Step 1

Connect to surface: find reference points

Part-aware Surface Metric


Vsi distance step 2

VSI-distance: Step 2

Sample visible regions from ref. points

The Volumetric Shape Image (VSI)

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

Part-aware Surface Metric


Vsi distance step 3

VSI-distance: Step 3

Compute VSI difference

Difference is based on the reach of

local volume along sampling direction

Part-aware Surface Metric


Moving along a path on the surface

Moving Along a Path on the Surface

Part-aware Surface Metric


Moving along a path on the surface1

Moving Along a Path on the Surface

Part-aware Surface Metric


Moving along a path on the surface2

Moving Along a Path on the Surface

Part-aware Surface Metric


Moving along a path on the surface3

Moving Along a Path on the Surface

Part-aware Surface Metric


Moving along a path on the surface4

Moving Along a Path on the Surface

Part-aware Surface Metric


Vsi differences along the path

VSI Differences Along the Path

d

VSI Diff

c

b

e

a

Part-aware Surface Metric


No leakage problem

No “Leakage” Problem

Angular distance fields

VSI distance fields

Part-aware Surface Metric


Combined distance graph

Combined Distance Graph

  • 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


Comparison with other metrics

Comparison with Other Metrics

Geodesic

Diffusion[deGoes08]

Part-aware

Angular[Katz03]

Part-aware Surface Metric


Practical issues

Practical Issues

  • 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


Applications

Applications

  • Segmentation

  • Registration

  • Part-aware sampling

  • Retrieval

Part-aware Surface Metric


Segmentation

Segmentation

  • Test algorithm: spectral clustering [Liu04]

    • Use distances to derive a spectral embedding of input mesh faces

    • Cluster in embedding space

Part-aware Surface Metric


Segmentation cont d

Segmentation (Cont’d)

With geodesic+angular distance

With part-aware distance

Part-aware Surface Metric


Registration

Registration

  • 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


Embedding

Embedding

geodesic

part-aware

homer

stretched

Part-aware Surface Metric


Registration1

Registration

geodesic

part-aware

Part-aware Surface Metric


Part aware sampling

Part-aware Sampling

  • 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


Object retrieval

Object Retrieval

  • 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


Retrieval cont d

Retrieval (Cont’d)

Geodesic distance: invariant to articulation

Part-aware distance: stronger discriminating capability

Part-aware Surface Metric


Retrieval cont d1

Retrieval (Cont’d)

Part Aware

Geodesic

D2

Part-aware Surface Metric


Summary

Summary

  • 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


Future work

Future Work

  • 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


Thank you

Thank You!

Part-aware Surface Metric


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