Bayesian Point Cloud Reconstruction

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Bayesian Point Cloud Reconstruction

• Eurographics 2006
• P.Jenke, M.Bokeloh, A.Schilling, W.Straber
• M.Wand
• University of Tuebingen, Germany
• Stanford University

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Outline

• Introduction
• Bayesian Reconstruction
• The Reconstruction
• Triangulation
• Result
• Conclusions and Future work

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Introduction

• Goal
• Surface reconstruction from unorganized, noisy
  point clouds.

4
Introduction cont.

• Challenges
• The surface topology has to be retrieved.
• The geometry has to be reconstruction from the
  unorganized point cloud.

5
Introduction cont.

• Solution
• Bayesian statistics
• P(SD) posterior distribution
• P(DS) measurement model
• P(S) prior distribution

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Introduction cont.

• Find the most likely reconstruction, SMAP
  (maximum a posteriori solution)
• Defining probabilistic model of the
• 1.measurement process
• 2.prior
• to be reconstruction.

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Bayesian Reconstruction

• Input
• D measured points (m)
• Output
• S original scene consisting of n points
• Assume Measurement process
• P(S) n
• deletes some points D ( )
• add random noise P(DS)

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Bayesian Reconstruction cont.

• Invert the measurement process
• Construct with size
• First m
• n-m points to the deleted points
• Perform a gradient descent on the negative
  log-likelihood

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The Measurement Model

• Must specify the probability of a candidate
  reconstruction agreeing with measured data D.
• witch is mirrored error
  density

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The Measurement Model cont.

• Assume that all measurement errors are
  independent of each other.
• gt
• sum of all per-point error
• (mixture of Gaussians)
  

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Priors

• Define what artifacts are considered noise.
• What the reconstruction scene will look like.
• Assume that the object of piecewise smooth
  patches separated by sharp boundaries.

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Priors cont.

• Priors consists of three main ingredients
• Density priors
• Smoothness priors
• Priors for estimating sharp features
• Z normalization constant
• w(S) a box function 1gtinside 0gtoutside
• limiting the range

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Density priors

• Obtain a reconstruction model which is
  well-sampled with a regular, constant sampling
  density all over the surface.
• Computing the sum of the area of all circles in
  which the k-nearest neighbors of each point lie,
  divided by k.

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Density priors cont.

• Estimate the expected distance between two
  neighboring points
• Define a stochastic potential
• between all pairs of points that constrain
  within a neighborhood radius proportional to the
  expected point to point distance

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Density priors cont.

• Define density prior
• Well in practice is to only consider the 6
  nearest.

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Density priors cont.

• Recomputed at each iteration of the numerical
  optimization to make the approximation more
  accurate.

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Smoothness

• To quantify the smoothness of a surface.
• Fix the size for the local neighbor of diameter
  (about 20-40 points)
• For all points in of a point compute a
  local coordinate frame using PCA
• Fix a set of basis function
• Ex.

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Smoothness cont.normal equation

• is a weighting function that make the
  solution continuous with respect to movement of
  points in S. ( ex.Gaussian)
• being the function that return the
  n-coordinate at their respective u ,v

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Smoothness cont.

• Define two evaluation functions
• How far the points are away
• From the quadratic surface.

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Smoothness cont.

• Fitted patch by computing
• the average squared norm
• of second derivatives of
• the patch.
• Smoothness prior

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Discrete Properties and Sharp Features

• Using mixed discrete-continuous optimization.
• A discrete attribute which determines whether a
  point belongs to
• 1.a region (smooth patch)
• 2.an edge (border curve between regions)
• 3.a corner (two or more edges meet)
• An ID-number to identify the corresponding entity.

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Assignment of attributes

• edge point
• grows with the curvature of the local
  neighborhood ex.
• corner point
• Depends on the number of edge points from
  difference edges or IDs in the neighborhood.

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Sharp Features preserve

• Region point
• paper restrict the neighborhoods used for the
  other priors to points with the same region.
• In case of plane-region. The smoothness priors
  are replaced by a simpler prior that attracts the
  points to a single plane.
• (depended on least-squares error)

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Sharp Features preserve cont.

• Edge point
• defining corresponding pairwise probabilities
  (look at the distance towards the two neighbor
  edge point. )
• smoothness the same in the previous section but
  locally fitting a polynomial curve to the set of
  edge points.

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Sharp Features preserve cont.

• Corner point
• Assume that the number of corner points in such a
  neighborhood is exactly one .
• Assume a probability distribution for its
  position that peaks at the point were regression
  line for all edges are closest to each other.

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The Reconstruction

• Initialization
• Initialize the estimate of with D.
• The additional n-m points are distributed
  randomly in the neighborhood of the points from
  D.
• Estimate the sampling density is the same as used
  for the density priors.
• An exact knowledge of the sampling density is of
  minor importance as points will be redistribution
  by the density prior later on.

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The Reconstruction cont.

• Numerical optimization
• Try to maximize the neglecting the
  discrete components of our model.
• Compute the gradient of the and
  perform a gradient descent.
• would cause significant computational overhead
• measurement model
• Priors restrict the movement of points in each
  neighborhood to the normal direction.

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The Reconstruction cont.

• Discrete Optimization
• Run of the continuous optimization.
• Assigning probabilities for being an edge or a
  corner. (discrete optimization, still noisy)
• Run a second pass of continuous optimization.
  (optimizes the position of edge, region and
  corner points )
• Discrete/continuous estimation process could be
  iterated to refine the model.

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Triangulation

• Similar to the original algorithm Of Surface
  Reconstruction from Unorganized Points
• In order to preserve the reconstructed features.
• gtan individual signed distance function is
  defined for every region

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Result
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Conclusions and Future work

• A surface reconstruction technique from noisy
  point clouds based on Bayesian statistics.
• Adding additional priors on time-dependent scene
  behavior.
• Improve numerical optimization scheme.