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Gwangju Institute of Science and TechnologyIntelligent Design and Graphics Laboratory
Multi-scale tensor voting for feature extraction from unstructured point
clouds
Min Ki Park* Seung Joo Lee Kwan H. Lee
Gwangju Institute of Science and Technology (GIST)
Geometric Modeling and Processing 2012
2012. 06. 22
Geometric Modeling and Processing 2012
Contents
• Introduction• Previous work• Method
– Tensor voting of 3D point cloud– Multi-scale tensor voting
• Experimental results• Limitation and Future work
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Geometric Modeling and Processing 2012
Point-based Surface
• Scanning technology– A huge amount of dense point data– Laser scanner, structured-light and Time-of-Flight sensor
• No need to generate triangular meshes
• Difficulties– No connectivity and normal information– Random noise, outliers and non-uniform distributions
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Geometric Modeling and Processing 2012
Why feature extraction?
• Better understanding of underlying surfaces– Insight about crucial characteristics of geometry– A priori knowledge for various geometry processing ap-
plications
e.g.) Adaptive sampling, feature-preserving simplification, geometry segmentation, etc.
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[Demarsin et al. 07]
Geometric Modeling and Processing 2012
Previous work -PCA-based Approach• Differential properties of a surface
– Principal component analysis (PCA) of covariance matrix– Approximation of normal or curvature over local neighbor-
hood
• Multi-scale feature classification– Differential properties at multiple scales– Enhancement of feature recognition in noisy data
• Drawbacks– First- or second-derivative approximation– Wide band of feature points in the vicinity of a sharp edge
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[Pauly et al. 02]
Geometric Modeling and Processing 2012
Previous work -Surface reconstruction • Moving least squares (MLS)
– Local surface approximation fit to neighborhood– Point projection to the approximated surface
• Robust Moving least squares (RMLS)– Feature-preserving noise removal during MLS recon-
struction – More accurate approximations of features
• Drawbacks– Considerable computational cost
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[Daniels et al. 07]
Geometric Modeling and Processing 2012
In this paper,
• GivenAn unstructured point set 1) no connectivity and normal information 2) random noise contained 3) Unknown intrinsic dimensionality
• Goal Extract a set of feature points
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Geometric Modeling and Processing 2012
Contributions
• Extend the tensor voting theory to feature extraction of point set with any intrinsic di-mensionality
• Propose the multi-scale tensor voting scheme for robust shape analysis
• Provide a very high computational efficiency
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Geometric Modeling and Processing 2012
Key Idea
• Tensor voting for shape analysis
• In voting process,
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[P. Mordohai2005]Input image Edge detection By human observer
Scale parameter control how many neighboring points
vote!!
How to determine an optimal scale?
Geometric Modeling and Processing 2012
Overview of the algorithm
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Geometric Modeling and Processing 2012
Tensor voting in 3D -Neighborhood selection
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𝑤𝑖=(𝜃𝑖− 1+𝜃 𝑖)
𝑑𝑖
𝜃𝑖
𝜃𝑖−1
K-nearest neigh-bor
Our neighborhood selection suggested by [Ma et al.
2011]
Non-uniformly distributed
𝑑𝑖
Unbalanced neighbor-
hood!
Geometric Modeling and Processing 2012
Tensor voting in 3D -Normal voting from neighborhood
• Normal space voting for two points
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𝐭�⃗�
ℝ 2
𝐭𝟏
𝐭𝟐
�⃗�
ℝ 3
𝐭
�⃗�𝟏
�⃗�𝟐
𝕋⊕ℕ=ℝ𝑛
ℕ=n n𝑇=ℐ𝑛−t t𝑇
‖t t𝑇‖
Geometric Modeling and Processing 2012
Tensor voting in 3D -Normal voting tensor
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p 𝑗p𝑖𝑣 𝑗
𝜇 𝑗=𝑒−(
𝑠 𝑗2
𝜎 2 )
The size of the vote is attenuated by the Gaussian function
𝑇 𝑖𝑗=𝜇𝑖 (ℐ3−�⃗� 𝑗 �⃗� 𝑗
𝑇
‖�⃗� 𝑗 �⃗� 𝑗𝑇‖)
For every neighbor, integrate the votes
Geometric Modeling and Processing 2012
Tensor voting in 3D -Voting analysis
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where
Geometric Modeling and Processing 2012
Tensor voting in 3D -Voting analysis
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On a face On a curve
Randomly scattered
Geometric Modeling and Processing 2012
Tensor voting in 3D -Feature weight• A point with larger is most likely on a fea-
ture
• Feature confidence value (feature weight)
e.g.,) , is on a plane , is on an edge or corner
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𝜔 𝑖=𝜆2+𝜆3
𝜆1
Feature weight
Geometric Modeling and Processing 2012
In the presence of noise,
• Can you distinguish a feature point from noise?
– A face needs to be smoothed out
– An edge needs to be preserved
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Geometric Modeling and Processing 2012
Revisit - Scale parameter
• It depends on noise level and sampling qual-ities
• How to adjust it?– Control voting neighborhood– Modify attenuation degree
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𝜇 𝑗=𝑒−(
𝑠 𝑗2
𝜎 2 )
Geometric Modeling and Processing 2012
Multi-scale tensor voting
• Adaptive scale in tensor computation
– Small scale for the fine point data
– Large scale for the noisy point data
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Feature weight
Scale
Geometric Modeling and Processing 2012
Optimal scale of a point
• Fine model
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Large varia-tion
Keep large values
Keep small values
Geometric Modeling and Processing 2012
Optimal scale of a point
• Noisy model
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Large varia-tion
Gradual In-crease
Gradual de-crease
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How to determine an optimal scale?• Adaptive scale selection algorithm
1. Initial scale 2. Compute of point at scale 3. Classify using pre-defined threshold 4. Observe the feature weight variation over scale domain
4.1. The large increase tells the optimal scale 4.2. Otherwise, larger scale is likely to be optimal
5. Update the current scale and repeat [2-4] until the ev-ery point is classified or maxIter is reached.
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Geometric Modeling and Processing 2012
Discussion - our multi-scale TV• It allows the tensor voting framework to
deal with both a noisy region and a sharp edge– Feature preserving
• Similar to [Pauly et al. 2003], but, no evalu-ation of the measure over the entire scale space
• Efficient implementation
– Update points newly included in the voting at the current scale
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Geometric Modeling and Processing 201224/34
• Each point has own optimal scale and feature weight– If , is a feature point– If , is a non-feature point
• How to classify the remaining points ?
Geometric Modeling and Processing 2012
Feature classification
• Adaptive thresholding for unclassified points.
If the feature weight is local maximum (30%), add to a fea-ture set
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miss-ing
If largest 30% points in local neighbor-hood
Geometric Modeling and Processing 2012
Feature completion
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Out-liers
• In the presence of severe noise, many outliers exist
• Outlier removal– Make feature clusters– Remove clusters of small size (under
10)
• Misclassified feature set is suc-cessfully removed
Geometric Modeling and Processing 2012
Results
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Input model Color-coded The result The result by polylines feature weight
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Result - poorly sampled point models
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jagged sparse
5k 10k 5k
Geometric Modeling and Processing 2012
Result -Robustness to noise
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PCA-based method
Our method
Geometric Modeling and Processing 2012
Results -Computational time
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• Only tensor addition and eigen analysis• Multi-scale?
– Asymptotically identical to the single scale
Geometric Modeling and Processing 2012
Dimensionality advantage
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PCA-based method
Gauss map Clus-tering
Our method
Non-manifold Space curve Different intrinsic dimension
PCA
Gauss map clustering
Our tensor voting
Plane with one normal
Plane with one normal
Space curve with two normals
Geometric Modeling and Processing 2012
Real scanned data
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Processing time: 15 secs for 173k ver-tices
Geometric Modeling and Processing 2012
Limitation and future work
• Limitations– Sampling quality is very poor– Signal-to-Noise ratio is too low
• Fail to distinguish between a sharp edge and a planar re-gion in the vicinity of a real edge
• In future work, – Improve the reliability for many uncertainties (e.g., poor
sampling quality, extreme noise)– Fit a continuous feature-line to the feature points
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Geometric Modeling and Processing 2012
Thank you for your attention
Q&A
Intelligent Design and Graphics LaboratoryGwangju Institute of Science and Technology (GIST)
http://ideg.gist.ac.kr
Contact info. [email protected]
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