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

Fitting multiple structures to geometric data: the J-linkage approach. Roberto Toldo and Andrea Fusiello University of Verona University of Udine. The problem. Fitting multiple instances of a model to data corrupted by noise and outliers. Two types of outliers. Gross outlliers

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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

• Define the PS of a cluster as the intersection of the PS of all its points.

• 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)

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

• 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

• 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

• 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

• 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.

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