Uncertainty and Variability in Point Cloud Surface Data

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Uncertainty and Variability in Point Cloud
Surface Data
Mark Pauly1,2, Niloy J. Mitra1, Leonidas J.
Guibas1
1 Stanford University
2 ETH, Zurich
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Point Cloud Data (PCD)
To model some underlying curve/surface
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Sources of Uncertainty

• Discrete sampling of a manifold
• Sampling density
• Features of the underlying curve/surface
• Noise
• Noise characteristics

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Uncertainty in PCD
Reconstruction algorithm
PCD
curve/ surface
But is this unique?
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Motivation
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Motivation
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Motivation
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Motivation
priors !
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What are our Goals?

• Try to evaluate properties of the set of
  (interpolating) curves/surfaces.
• Answers in probabilistic sense.
• Capture the uncertainty introduced by point
  representation.

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Related Work

• Surface reconstruction
• reconstruct the connectivity
• get a possible mesh representation
• PCD for geometric modeling
• MLS based algorithms
• Kalaiah and Varshney
• PCA based statistical model
• Tensor voting

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Notations
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Expected Value
Conceptually we can define likelihood as
Surface prior ?
Set of all interpolating surfaces ?
Characteristic function
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How to get FP(x) ?

• input set of points P
• implicitly assume some priors (geometric)
• General idea
• Each point pi?P gives a local vote of likelihood
• 1. Local likelihood depends on how well
  neighborhood of pi agrees with x.
• 2. Weight of vote depends on distance of pi from
  x.

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Estimates for x
Interpolating curve more likely to pass through x
Prior preference to linear interpolation
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Estimates for x
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Likelihood Estimate by pi
Distance weighing
High if x agrees with neighbors of pi
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Likelihood Estimates
Normalization constant
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Finally
O(N)
O(1)
Covariance matrix (independent of x !)
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Likelihood Map Fi(x)
likelihood
Estimates by point pi
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Likelihood Map Fi(x)
Pinch point is pi
High likelihood
Estimates by point pi
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Likelihood Map Fi(x)
Distance weighting
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Likelihood Map FP(x)
likelihood
O(N)
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Confidence Map

• How much do we trust the local estimates?
• Eigenvalue based approach

• Likelihood estimates based on covariance
  matrices Ci
• Tangency information implicitly coded in Ci

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Confidence Map
denote the eigenvalues of Ci.
Low value denotes high confidence
(similar to sampling criteria proposed by Alexa
et al. )
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Confidence Map
confidence
Red indicates regions with bad normal estimates
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Maps in 2d
Likelihood Map
Confidence Map
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Maps in 3d
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Noise Model

• Each point pi corrupted with additive noise ?i
• zero mean
• noise distribution gi
• noise covariance matrix ?i
• Noise distributions gi-s are assumed to be
  independent

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Noise
Expected likelihood map simplifies to a
convolution.
Modified covariance matrix
convolution
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Likelihood Map for Noisy PCD
gi
No noise
With noise
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Scale Space
Proportional to local sampling density
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Scale Space
Good separation
Bad estimates in noisy section
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Scale Space
Cannot detect separation
Better estimates in noisy section
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Application 1 Most Likely Surface
Noisy PCD
Likelihood Map
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Application 1 Most Likely Surface
Active Contour
Sharp features missed?
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Application 2 Re-sampling
Given the shape !!
Confidence map
Add points in low confidence areas
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Application 2 Re-sampling
Add points in low confidence areas
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Application 2 Re-sampling
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Application 3 Weighted PCD
PCD 1
PCD 2
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Application 3 Weighted PCD
Merged PCD
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Application 3 Weighted PCD
Too noisy
Too smooth
Merged PCD
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Application 3 Weighted PCD
Confidence Map
Likelihood Map
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Application 3 Weighted PCD
Weighted PCD
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Application 3 Weighted PCD
Weighted PCD
Merged PCD
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Future Work

• Soft classification of medical data
• Analyze variability in family of shapes
• Incorporate context information to get better
  priors
• Statistical modeling of surface topology

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Questions ?