Extended EMfor

Planar Approximation of3D Laser Range Data

Rolf Lakaemper, Longin Jan Latecki, Temple University, USA

Topic:

Approximate 3D point clouds using

‘planar patches’

Why ?

Patches represent higher geometric information than raw point data…

Why ?

Why ?

Why ?

…and are therefore a useful representation for

• Robot Mapping• 3D Object recognition (landmarks)• CAD modelling• …

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)

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

Example (2D):

Converged!

• 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

Example : Approximation with a single patch:

Problem

Dynamic adjustment of number

of patches extending EM by

Split & Merge

Solution

Split: insufficiently fitting patches are split

Split & Merge

Merge: sufficiently similar patches are merged

Split & Merge

The extended algorithm

dynamically adjusts the number of model components and solves the

problems of classical EM

Extended EM

EM SPLIT EM MERGE

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

Some Details

patch

support points

supported tiles

1. Determine Split-lines

2. Split, if result would not be merged

How to Split

1. Determine Split-lines

How to Split

How to 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)

Split

EM SPLIT EM MERGE

Split + Single EM step

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

1. Determine candidates

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

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

1. Determine candidates

…the underlying similarity measure takes into account the closeness, coplanarity and angle between normals of two

patches…

D1

1. Determine candidates

…the underlying similarity measure takes into account the closeness, coplanarity and angle between normals of two

patches…

D2

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

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)

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

Results: Wall Test (robustness to noise)

(Init, Ground Truth Model)

Results: Wall Test(Init, Random number and location of patches)

Results: Wall Test

Results: Wall Test

Results: Wall Test(Init, Random number and location of patches)

Results: Berkeley Campus(Init, random number & location of patches)

Results: Berkeley Campus(Iteration 1)

Results: Berkeley Campus(Iteration 3)

Results: Berkeley Campus(final)

Results: Berkeley Campus(final, supporting point sets)

Results: Berkeley Campus

Segmentation into planar elements allows for 2D shape (landmark) recognition

Results: Berkeley Campus

Segmentation into planar elements allows for 2D shape (landmark) recognition

Alternative Applications

Creating CAD Models

Results: Socket

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 !

Thank You !