Uncertainty and Variability in Point Cloud Surface Data

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

1 Stanford University 2 ETH, Zurich

Uncertainty and Variability in PCD

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

PCD curve/ surface

Reconstruction algorithm

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

So look for probabilistic answers.

priors !

Uncertainty and Variability in PCD

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.

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 though a point x in space

)(xPF

Prior for a surface S in MPp(S)

Set of all interpolating surfaces for PCD PPM

Uncertainty and Variability in PCD

Expected Value

Surface prior ?

Characteristic function

Set of all interpolating surfaces ?

PMSxx )p(S)dS(F SP )(

S

S )(S x

xx

0

1

Conceptually we can define likelihood as

Uncertainty and Variability in PCD

How to get FP(x) ?

• input : set of points P

• implicitly assume some priors (geometric)

General idea:

Each point piP 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

2))(( xqp iTij

x

x

pipipj

pj

qi(x)qi(x)

Uncertainty and Variability in PCD

Likelihood Estimate by pi

p ijiiTij xqp 2))((

High if x agrees with neighbors of pi

Distance weighing

Uncertainty and Variability in PCD

Likelihood Estimates

Normalization constant

N

jijii

Tij pxqpxF

1

2))(()( c

1

ii

Uncertainty and Variability in PCD

Finally…

Covariance matrix (independent of x !)

)()(

)()(

)()(

))(()(

1

1

1

2

xqCxq

xqpppxq

pxqppxq

pxqpxF

iiT

i

i

N

jiji

Tijij

Ti

N

jijii

Tijij

Ti

N

jijii

Tij

i

i

i

ii

c

1

c

1

c

1

c

1

O(N)

O(1)

Uncertainty and Variability in PCD

Likelihood Map: Fi(x)

Estimates by point pi

likelihood

)(xFi

Uncertainty and Variability in PCD

Likelihood Map: Fi(x)

Estimates by point pi

High likelihood

Pinch point is pi

Uncertainty and Variability in PCD

Likelihood Map: Fi(x)

Distance weighting

iii pxxF )(

)(xFi

Uncertainty and Variability in PCD

Likelihood Map: FP(x)

likelihood

N

iiiiP pxxFxF

1

)()( O(N)

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

denote the eigenvalues of Ci.

3

1

1 /l

liii Low value denotes high confidence

N

iiiiP pxxxC

1

)()(

321iii

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

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

NoiseExpected likelihood map simplifies to a convolution.

Modified covariance matrix

convolution

)()(

)(1

)(

1

1

xgxF

dgqCqc

xF

i

N

ii

ii

N

ii

T

ii

P

Uncertainty and Variability in PCD

Likelihood Map for Noisy PCD

No noise With noisegi

Uncertainty and Variability in PCD

Scale Space

Proportional to local sampling density

Uncertainty and Variability in PCD

Scale Space

Bad estimates in noisy section

Good separation

Uncertainty and Variability in PCD

Scale Space

Better estimates in noisy section

Cannot detect separation

Uncertainty and Variability in PCD

Application 1: Most Likely Surface

Noisy PCD Likelihood Map

Uncertainty and Variability in PCD

Application 1: Most Likely Surface

Sharp features missed?

Active Contour

Uncertainty and Variability in PCD

Application 2: Re-sampling

Add points in low confidence areas

Given the shape !!

Confidence map

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

Too noisy Too smooth Merged PCD

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 ?