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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 !




                          So look for probabilistic answers.


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
 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




Bad estimates in noisy
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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