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Uncertainty and Variability in Point Cloud Surface Data. Mark Pauly 1,2 , Niloy J. Mitra 1 , Leonidas J. Guibas 1. 1 Stanford University. 2 ETH, Zurich. Point Cloud Data (PCD). To model some underlying curve/surface. Sources of Uncertainty. Discrete sampling of a manifold

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Uncertainty and Variability in Point Cloud Surface Data


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uncertainty and variability in point cloud surface data

Uncertainty and Variability in Point Cloud Surface Data

Mark Pauly1,2, Niloy J. Mitra1, Leonidas J. Guibas1

1 Stanford University

2 ETH, Zurich

point cloud data pcd
Point Cloud Data (PCD)

To model some underlying curve/surface

sources of uncertainty
Sources of Uncertainty
  • Discrete sampling of a manifold
    • Sampling density
    • Features of the underlying curve/surface
  • Noise
    • Noise characteristics
uncertainty in pcd
Uncertainty in PCD

Reconstruction algorithm

PCD

curve/ surface

But is this unique?

what are our goals
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.
related work
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
notations

Likelihood that a surface interpolating P passes though a point x in space

Set of all interpolating surfaces for PCD P

Prior for a surface S in MP

Notations
expected value
Expected Value

Conceptually we can define likelihood as

Surface prior ?

Set of all interpolating surfaces ?

Characteristic function

how to get f p x
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 piagrees with x.
    • 2. Weight of vote depends on distance of pi from x.
estimates for x

x

x

Estimates for x

Interpolating curve more likely to pass through x

Prior : preference to linear interpolation

estimates for x15

x

qi(x)

qi(x)

x

pi

pj

pi

pj

Estimates for x
likelihood estimate by p i
Likelihood Estimate by pi

Distance weighing

High if x agrees with neighbors of pi

likelihood estimates
Likelihood Estimates

Normalization constant

finally
Finally…

O(N)

O(1)

Covariance matrix (independent of x !)

likelihood map f i x
Likelihood Map: Fi(x)

likelihood

Estimates by point pi

likelihood map f i x20
Likelihood Map: Fi(x)

Pinch point is pi

High likelihood

Estimates by point pi

likelihood map f i x21
Likelihood Map: Fi(x)

Distance weighting

likelihood map f p x
Likelihood Map: FP(x)

likelihood

O(N)

confidence map
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
confidence map24
Confidence Map

denote the eigenvalues of Ci.

Low value denotes high confidence

(similar to sampling criteria proposed by Alexa et al. )

confidence map25
Confidence Map

confidence

Red indicates regions with bad normal estimates

maps in 2d
Maps in 2d

Likelihood Map

Confidence Map

noise model
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
noise
Noise

Expected likelihood map simplifies to a convolution.

Modified covariance matrix

convolution

likelihood map for noisy pcd
Likelihood Map for Noisy PCD

gi

No noise

With noise

scale space
Scale Space

Proportional to local sampling density

scale space32
Scale Space

Good separation

Bad estimates in noisy section

scale space33
Scale Space

Cannot detect separation

Better estimates in noisy section

application 1 most likely surface
Application 1: Most Likely Surface

Noisy PCD

Likelihood Map

application 1 most likely surface35
Application 1: Most Likely Surface

Active Contour

Sharp features missed?

application 2 re sampling
Application 2: Re-sampling

Given the shape !!

Confidence map

Add points in low confidence areas

application 2 re sampling37
Application 2: Re-sampling

Add points in low confidence areas

application 3 weighted pcd41
Application 3: Weighted PCD

Merged PCD

Too noisy

Too smooth

application 3 weighted pcd42
Application 3: Weighted PCD

Confidence Map

Likelihood Map

application 3 weighted pcd44
Application 3: Weighted PCD

Weighted PCD

Merged PCD

future work
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