Extended EM

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Extended EM for Planar Approximation of 3D Laser
Range Data
Rolf Lakaemper, Longin Jan Latecki, Temple
University, USA
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Topic Approximate 3D point clouds using
planar patches
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Why ? Patches represent higher geometric
information than raw point data
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Why ?
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Why ?
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• Why ?
• and are therefore a useful representation for
• Robot Mapping
• 3D Object recognition (landmarks)
• CAD modelling

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How ? The classical approach Expectation
Maximization (EM) Approximating the data (the
points) with a model (the patches) in an
optimal way (maximizing the log-likelihood of
the data given the model)
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• EM
• is used to iteratively
• determine the correspondence between data points
  and patches.
• Relocate the patches using linear regression
  weighted by the (a priori) probability of
  correspondences of points to patches

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Example (2D)
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Converged!
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Problem

• Number of model components must be known ( fixed
  in the classical approach, the reason being the
  log-likelihood, leading to over fitting if
  arbitrary model components are allowed)
• Initial position of model components must be
  close to final solution (since EM converges to a
  local minimum only)

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Problem
Example Approximation with a single patch
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Solution
Dynamic adjustment of number of patches extending
EM by Split Merge
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Split Merge
Split insufficiently fitting patches are split
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Split Merge
Merge sufficiently similar patches are merged
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Extended EM
The extended algorithm dynamically adjusts the
number of model components and solves the
problems of classical EM
EM
SPLIT
EM
MERGE
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Some Details
A patch is a rectangular element subdivided into
a grid of tiles. A tile is supported if a
sufficient number of data points is close enough
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Some Details
supported tiles
support points
patch
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How to Split

1. Determine Split-lines
2. Split, if result would not be merged

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How to Split

• Determine Split-lines

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How to Split

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Split
SPLIT is followed by EM step (Note split
always leads to a better fit by log-likelihood
criterion, but not necessarily to a visually
better result, e.g. over fitting)
EM
SPLIT
EM
MERGE
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Split Single EM step

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How to Merge

1. Determine similarity of pairs of patches
   (candidates)
2. Exit if no candidates are present
3. Compute merged patch of best candidate by linear
   regression
4. Goto 1

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• Determine candidates
• the underlying similarity measure takes into
  account the closeness, coplanarity and angle
  between normals of two patches

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• Determine candidates
• the underlying similarity measure takes into
  account the closeness, coplanarity and angle
  between normals of two patches
• Overlapping bounding boxes
• Sharing support points

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• Determine candidates
• the underlying similarity measure takes into
  account the closeness, coplanarity and angle
  between normals of two patches

D1
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• Determine candidates
• the underlying similarity measure takes into
  account the closeness, coplanarity and angle
  between normals of two patches

D2
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• Determine candidates
• the underlying similarity measure takes into
  account the closeness, coplanarity and angle
  between normals of two patches

Candidate min(D1,D2) lt Threshold
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• Determine Merged Patch
• Simple (unweighted)regression with union of
  point-sets (this equals a single EM step with a
  single model component, i.e. the new patch)

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Merge
Merge is followed by EM step Merge controls
the max. number of patches, it extends the log
likelihood quality criterion to avoid
overfitting
EM
SPLIT
EM
MERGE
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Results Wall Test (robustness to noise) (Init,
Ground Truth Model)
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Results Wall Test (Init, Random number and
location of patches)
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Results Wall Test
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Results Wall Test
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Results Wall Test (Init, Random number and
location of patches)
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Results Berkeley Campus (Init, random number
location of patches)
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Results Berkeley Campus (Iteration 1)
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Results Berkeley Campus (Iteration 3)
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Results Berkeley Campus (final)
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Results Berkeley Campus (final, supporting point
sets)
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Results Berkeley Campus Segmentation into
planar elements allows for 2D shape (landmark)
recognition
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Results Berkeley Campus Segmentation into
planar elements allows for 2D shape (landmark)
recognition
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Alternative Applications Creating CAD Models
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Results Socket
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• Conclusion
• Approximation of 3D point sets by patches to gain
  higher representation
• Classical EM was extended by Split and Merge
• Number of Model Components is dynamically
  adjusted
• Merge avoids overfit
• Works pretty well !

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Thank You !