Gwangju Institute of Science and Technology Intelligent Design and Graphics Laboratory Multi-scale tensor voting for feature extraction from unstructured

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


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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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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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]


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]


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]


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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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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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?


Overview of the algorithm

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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!


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‖


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


Tensor voting in 3D -Voting analysis

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where


Tensor voting in 3D -Voting analysis

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On a face On a curve

Randomly scattered


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


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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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 )


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


Optimal scale of a point

• Fine model

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Large varia-tion

Keep large values

Keep small values


Optimal scale of a point

• Noisy model

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Large varia-tion

Gradual In-crease

Gradual de-crease


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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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 ?


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


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


Results

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Input model Color-coded The result The result by polylines feature weight


Result - poorly sampled point models

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jagged sparse

5k 10k 5k


Result -Robustness to noise

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PCA-based method

Our method


Results -Computational time

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• Only tensor addition and eigen analysis• Multi-scale?

– Asymptotically identical to the single scale


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


Real scanned data

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Processing time: 15 secs for 173k ver-tices


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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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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Documents

Gwangju Institute of Science and Technology Intelligent Design and Graphics Laboratory Multi-scale tensor voting for feature extraction from unstructured