Extended EM for Planar Approximation of 3D Laser Range Data
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Transcript of Extended EM for Planar Approximation of 3D Laser Range Data
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Extended EMfor
Planar Approximation of3D 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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• 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)
Problem
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Example : Approximation with a single patch:
Problem
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Dynamic adjustment of number
of patches extending EM by
Split & Merge
Solution
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Split: insufficiently fitting patches are split
Split & Merge
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Merge: sufficiently similar patches are merged
Split & Merge
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The extended algorithm
dynamically adjusts the number of model components and solves the
problems of classical EM
Extended EM
EM SPLIT EM MERGE
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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
Some Details
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Some Details
patch
support points
supported tiles
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1. Determine Split-lines
2. Split, if result would not be merged
How to Split
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1. Determine Split-lines
How to Split
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How to Split
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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)
Split
EM SPLIT EM MERGE
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Split + Single EM step
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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
How to Merge
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1. Determine candidates
…the underlying similarity measure takes into account the closeness, coplanarity and angle between normals of two patches…
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1. 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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1. Determine candidates
…the underlying similarity measure takes into account the closeness, coplanarity and angle between normals of two
patches…
D1
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1. Determine candidates
…the underlying similarity measure takes into account the closeness, coplanarity and angle between normals of two
patches…
D2
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1. Determine candidates
…the underlying similarity measure takes into account the closeness, coplanarity and angle between normals of two
patches…
Candidate: min(D1,D2) < 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 is followed by EM step
Merge controls the max. number of patches, it extends the log likelihood quality criterion to avoid overfitting
Merge
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 !