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# Uncertainty and Variability in Point Cloud Surface Data - PowerPoint by qpY3i8

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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
Point Cloud Data (PCD)

To model some underlying curve/surface

Uncertainty and Variability in PCD
Sources of Uncertainty

    Discrete sampling of a manifold
    Sampling density
    Features of the underlying curve/surface

    Noise
    Noise characteristics

Uncertainty and Variability in PCD
Uncertainty in PCD

Reconstruction
algorithm
PCD                                         curve/
surface
But is this unique?

Uncertainty and Variability in PCD
Motivation

A possible reconstruction

Uncertainty and Variability in PCD
Motivation

or this one,

Uncertainty and Variability in PCD
Motivation

or this …..

Uncertainty and Variability in PCD
Motivation
priors !

Uncertainty and Variability in PCD
What are our Goals?
• Try to evaluate properties of the set of
(interpolating) curves/surfaces.

• Capture the uncertainty introduced by point
representation.

Uncertainty and Variability in PCD
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
Uncertainty and Variability in PCD
Notations

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

MP               Set of all interpolating surfaces for PCD P

p(S)            Prior for a surface S in MP

Uncertainty and Variability in PCD
Expected Value
Conceptually we can define likelihood as

FP (x)                     S ( x )p(S)dS
SMP

Surface prior ?
Set of all interpolating surfaces ?

1 x  S
Characteristic function              S (x)  
0 x  S

Uncertainty and Variability in PCD
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.

Uncertainty and Variability in PCD
Estimates for x

x

x

Interpolating curve more
likely to pass through x

Prior : preference to linear interpolation

Uncertainty and Variability in PCD
Estimates for x

x
qi(x)
qi(x)
x

pi pj                            pi   pj

T      2
(p qi (x))
ij

Uncertainty and Variability in PCD
Likelihood Estimate by pi

High if x agrees with                                 Distance
weighing
neighbors of pi

(p qi (x)) i
T
ij
2
p 
ij

Uncertainty and Variability in PCD
Likelihood Estimates

1 N T
c i j1

Fi (x )   (pij qi (x )) 2 i pij   

Normalization constant

Uncertainty and Variability in PCD
Finally…
1 N T
Fi (x )   (pij qi (x ))2 i pij
c i j1
                     O(N)

1 N
  qi (x ) T pijpij qi (x )i pij
c i j1
T
      
p p   p 
1                    N                     
 qi (x ) 
T
                ij
T
ij i   ij
qi (x )

ci                  j1                    
1
 qi (x ) T Ciqi (x )                                   O(1)
ci

Covariance matrix (independent of x !)

Uncertainty and Variability in PCD
Likelihood Map: Fi(x)

Fi (x)

likelihood

Estimates by point pi

Uncertainty and Variability in PCD
Likelihood Map: Fi(x)

Pinch point is pi

High likelihood

Estimates by point pi

Uncertainty and Variability in PCD
Likelihood Map: Fi(x)

Fi (x )

Distance
weighting

Fi (x)i  x  pi   

Uncertainty and Variability in PCD
Likelihood Map: FP(x)

likelihood

O(N)
FP (x )   Fi (x )i  x  pi   
N

i 1

Uncertainty and Variability in PCD
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

Uncertainty and Variability in PCD
Confidence Map

1  i2  i3 denote the eigenvalues of Ci.
i

3
i  1 /  li
i                  Low value denotes high confidence
l 1   (similar to sampling criteria proposed by Alexa et al. )

CP (x )   i (x )i  x  pi               
N

i 1

Uncertainty and Variability in PCD
Confidence Map

confidence

Red indicates regions with bad normal estimates

Uncertainty and Variability in PCD
Maps in 2d

Likelihood Map   Confidence Map

Uncertainty and Variability in PCD
Maps in 3d

Likelihood Map   Confidence Map

Uncertainty and Variability in PCD
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

Uncertainty and Variability in PCD
Noise
Expected likelihood map simplifies to a convolution.

Modified
covariance matrix
N
1 T  
FP (x )     qi Ci qi gi ( )d


i 1 c i
N
  Fi (x )  gi (x )


i 1

convolution

Uncertainty and Variability in PCD
Likelihood Map for Noisy PCD

gi                No noise   With noise

Uncertainty and Variability in PCD
Scale Space

Proportional to
local sampling
density

Uncertainty and Variability in PCD
Scale Space
Good separation

section

Uncertainty and Variability in PCD
Scale Space
Cannot detect separation

Better estimates in
noisy section

Uncertainty and Variability in PCD
Application 1: Most Likely Surface

Noisy PCD             Likelihood Map

Uncertainty and Variability in PCD
Application 1: Most Likely Surface

Active Contour

Sharp features missed?

Uncertainty and Variability in PCD
Application 2: Re-sampling

Given the shape !!

Confidence map

Add points in low confidence areas

Uncertainty and Variability in PCD
Application 2: Re-sampling

Add points in low confidence areas
Uncertainty and Variability in PCD
Application 2: Re-sampling

Uncertainty and Variability in PCD
Application 3: Weighted PCD

PCD 1      PCD 2

Uncertainty and Variability in PCD
Application 3: Weighted PCD

Merged PCD

Uncertainty and Variability in PCD
Application 3: Weighted PCD

Merged PCD                        Too noisy   Too smooth

Uncertainty and Variability in PCD
Application 3: Weighted PCD

Likelihood Map   Confidence Map

Uncertainty and Variability in PCD
Application 3: Weighted PCD

Weighted PCD

Uncertainty and Variability in PCD
Application 3: Weighted PCD
Merged PCD   Weighted PCD

Uncertainty and Variability in PCD
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

Uncertainty and Variability in PCD
Questions ?

Uncertainty and Variability in PCD

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