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Segmentation into Planar Patches for Recovery of Unmodeled Objects

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Segmentation into Planar Patches for Recovery of Unmodeled Objects

Kok-Lim Low

COMP 290-075

Computer Vision

4/26/2000

- Work by Marjan Trobina
- “From planar patches to grasps: a 3-D robot vision system handling unmodeled objects.” Ph.D. thesis, ETHZ, 1995.

- Overview of whole system

acquire range images from 2 or more views

segment into planar patches

generate object hypotheses

compute grasping points

robot arm

- Objectives
- extract planar patches from range images in a robust way
- just sufficient info for recovery of unmodeled objects for computing grasping points

- Can be viewed as data compression
- transforming range image to just a few parameters

- 1) Edge-based segmentation
- detect surface discontinuities
- find closed edge chains
- not robust against noisy data

- 2) Split-and-merge paradigm
- tessellation of image
- using quadtrees or Delaunay triangles

- 3) Clustering
- map data to feature space
- find clusters
- points in feature space have no image-space connectivity

- 4) Region growing
- grow region until approximation error too large
- order dependent, result usually far from optimal

- 5) Recover-and-select paradigm

input

model recovery

iterate until remaining models are completely grown

model selection

output

- Originally proposed by Leonardis
- Very robust against statistical noise and outliers
- Consists of 2 intertwined stages:

- Model recovery
- regularly place seeds
- grow statistically consistent seeds independently
- fit a plane (model) to each region
- stop growth of region if planar fit is lousy
- stop growth of region if no compatible points can be added

- Model selection
- select some recovered (plane) models
- minimize number of selected models while keeping approximation error low

- Planar patch parameterized by a1, a2, a3 in
f(X, Y) = a1X + a2Y + a3 = Z

- Distance function from range data point r(m) to planar patch
d2(m) = ( r(m) – f(m) )2

- Approximation error for set D of n points is
- For each set of points D, minimize by Linear Least Squares method to obtain plane parameters a1, a2, a3.

- Place seeds in regular grid of 7x7 windows
- Define model acceptance thresholdT
- Grow seed if its < T (statistically consistent)
- Define compatibility constraint C
- Add adjacent point m to patch if d2(m) < C (compatible)
- Stop when T or no compatible point can be added
- Output is a set of overlapping planar patch models

- Select smallest number of models while keeping approximation error small
- Objective function (to be maximized) for model si
F(si) = K1ni – K2i– K3Ni

where ni = | D |

i = approximation error of model si

Ni = number of parameters in model si

- Objective function for M models
wherep = [ p1...pM ] and pi = 0 or 1

cii = K1ni – K2i– K3Ni

cij = ( –K1 | DiDj | + K2ij) / 2

- Use greedy algorithm to find vector p so that F(p) is near to maximum

- In RS, many seeds are grown and then discarded
- MRS uses hierarchical approach to reduce waste
- Basic idea
- build an image pyramid
- apply standard RS on coarsest image
- selected patches are projected to the next finer level and used as seeds for the new level
- start new seeds on the unprojected regions in the next finer level

- Speedup of 10 to 20 times

- Range images from different viewpoints
- A planar patch extracted from different views should have same parameters and error measure
- Modifications to model recovery stage:
- project data points into direct 3-D space prior to the segmentation
- minimizing the orthogonal distance to the plane

- Planar patch now parameterized by a1, a2, a3, a4 in
f(X, Y, Z) = a1X + a2Y + a3Z + a4 = 0

- Distance function from 3-D range data point M to planar patch
d2(M) = f(M)2

- Approximation error
= 1

where 1is the smallest eigenvalue of the covariance matrix of the n points in the patch

- The normal (a1, a2, a3) of the planar patch is the eigenvector with eigenvalue 1

- create explicit patch boundary description
- post-processing to clean edge
- classify patches as “true planes” or “curved patches”, and fit points on curved planes with quadrics
- classify adjacency relation as concave or convex
- e.t.c.

- Objects are unmodeled
- Group planar patches into Single-View Object Hypotheses (SVOHs)
- Combine SVOHs into Global Object Hypotheses (GOHs)
- Prefer oversegmentation to undersegmentation — avoid grasping 2 objects at the same time

- A SVOH is a set of connected patches, such that for any 2 patches, there exists at least one path that does not contain any concave relation

- A GOH is a set of SVOHs, such that for any SVOHi there is at least one SVOHj (from a different view) such that SVOHi and SVOHj have at least one pair of patches sk (from SVOHi) and sl (from SVOHj) which fulfills the same-surface predicate
- Rough idea of “same-surface predicate”
- when 2 patches satisfy the same-surface predicate, they are on the same plane or on the same curved surface and they are intersecting each other

- Marjan Trobina
- “From Planar Patches to Grasps: A 3-D Robot Vision System Handling Unmodeled Objects.” Ph.D. thesis, ETHZ, 1995

- A. Leonardis
- “Image Analysis Using Parametric Models: Model-Recovery and Model-Selection Paradigm.” Ph.D. thesis, University of Ljubljana, 1993

- A. Leonardis
- “Recover-and-Select on Multiple Resolutions.” Technical report LRV-95, Computer Vision Lab, University of Ljubljana, 1995

- Frank Ade, Martin Rutishauser and Marjan Trobina
- “Grasping Unknown Objects.” ETHZ, 1995

- Martin Rutishauser, Markus Stricker and Marjan Trobina
- “Merging Range Images of Arbitrarily Shaped Objects.” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 1994