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Fitting multiple structures to geometric data: the J-linkage approach

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### Fitting multiple structures to geometric data: the J-linkage approach

Roberto Toldo and

Andrea Fusiello

University of Verona

University of Udine

Big Data - Udine 5/6/2013

The problem

- Fitting multiple instancesof a modelto data corruptedbynoise and outliers

Twotypesofoutliers

- Grossoutlliers
- Pseudo-outliers

Previous work

- Sequential RANSAC: theoretically wrong
- MultiRANSAC [Zuliani et al. ICIP05]: problems with intersecting models
- Residual Histogram Analysis [Zhang&Koseka ECCV06]: peaks finding is unreliable
- Mode finding in parameter space.
- Randomized HT: discretization is critical
- Mean Shift clustering: not rubust enough

Randomsamplingconsensus

- Draw minimal sample sets (MSS) from data points
- Fit a model to each MSS
- Build the consensus set of the model: the set of points such that their distance to the model is below a given threshold (inlier band)
- Select the model with the highest consensus

Randomsamplingconsensus

- The number of MSS to be drawn must be large enough to guarantee that at least an outlier-free MSS is selected with high probability.
- The assumption is that an outliers-free MSS will achieve the highest consensus, because inliers are structured whereas outliers are random
- RANSAC looks at the problem from the model’s viewpoint

The preference set

- Let us look at the problem from the perspective of the points.
- The Preference Set (PS) of a point is the set of models it belongs to.

CS of model j

PS of point i

Conceptualrepresentation

- The PS (or its characteristic function) is a conceptual representationof a point.
- In Pattern Recognition, the conceptual representation of an object x given C classes is: [ P(x | class 1) · · · P(x| class C ) ].
- Conjecture: points belonging to the same model have “similar” conceptual representations.
- In other words, they cluster in the conceptual space.

Jaccarddistance

- Models are extracted by agglomerative clustering in the conceptual space using the Jaccard distance:

A

B

A ∪ B

A ∩ B

∑=8

∑=2

dj=6/8=0.75

J-linkage

- Define the PS of a cluster as the intersection of the PS of all its points.
- Start with one cluster for each point
- Pick the two cluster with the smallest J-distance and merge them
- Repeat 2 while the smallest J-distance < 1
- Postcondition: all the clusters have distance 1 (their PS do not overlap)

J-linkage: properties

model that fits all the points of cluster 2

model that fits all the points of clusters 1 & 2

model that fits all the points of cluster 1

- Foreach cluster, thereis a modelthatfitsall the point, otherwisetheywouldhavedistance = 1
- A modelcannotfitall the pointsoftwoclusters, otherwisetheywouldhavedistance < 1

cluster 2

points

cluster 1

How many clusters?

- Outliersemergesassmallclusters
- If the numberMofmodelsisknown, the largestMclusters are retained
- If the overallnumberofinlier can beestimated, the largestclusters up to the numbetofinliers are retained
- Modelselectiontools can help to solve thisissue

What about the inlier threshold?

- We presented [ICIAP 09] a technique based on clustering validation that is able to automatically select the “just right” threshold.

Continousrelaxation

- The votingfunctionin J-linkageis a stepfunction (indicatorfunctionof the inlier band)
- Idea: choose a soft votingfunctionswithvalues in [0,1]
- ris the residual
- the time constant τplays the role of the inlier theshold

Continousrelaxation

- Instead of the characteristic function of the preference set now the preference vector of a point has entries in [0,1] as produced by the soft voting function.
- The Jaccard distance is generalized by the Tanimoto distance:
- where p,q are the preference vectors

Continous relaxation

- The preferencevectorof a cluster isobtainedas the componentwise minimum among the preferencevectorsof the cluster (generalizes the logical AND ofpreferencevectors)
- The soft J-linkageproceedsasits discrete version
- Post-conditions:
- Atleastonemodel in a cluster hasvotes >0fromall the points in the cluster (itfitsall the points)
- A modelcannothavevotes >0fromall the points in two separate cluster

Experiments

- Each model consists of 50 inliers, corrupted by variable Gaussian noise and variable outliers percentage.
- Compared to: sequential RANSAC, multiRANSAC, residual histogram analysis (RHA) and Mean-Shift.
- Same samples.
- Same inlier threshold
- Parameters needed by MS (bandwidth) and by RHA have been optimized manually.
- Number of models is given.

Real data

- The motivation for this work is fitting 3D primitives (planes, cylinders) to cloud of 3D points provided by a SaM reconstruction pipeline.

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