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. 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

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
The problem
  • Fitting multiple instancesof a modelto data corruptedbynoise and outliers
two types of outliers
  • Grossoutlliers
  • Pseudo-outliers
previous work
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
random sampling consensus
  • 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
random sampling consensus1
  • 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
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

conceptual representation
  • 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.
jaccard distance
  • Models are extracted by agglomerative clustering in the conceptual space using the Jaccard distance:



A ∪ B

A ∩ B




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
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


cluster 1

how many clusters
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
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.
continous relaxation
  • 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
continous relaxation1
  • 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 relaxation2
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
  • 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
Real data
  • The motivation for this work is fitting 3D primitives (planes, cylinders) to cloud of 3D points provided by a SaM reconstruction pipeline.