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Multi-model Estimation with J-linkage
Jeongkyun Lee
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How do we find parameters of a model that contains outliers?
Application in vision: geometric figure fitting, planar surface detection, mo-tion segmentation, etc.
Motivation
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Least Squares Least Median of Squares (LMedS) Random Sample Consensus (RANSAC) M-SAC MLESAC PROSAC Etc.
Single-model Estimation
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Least Squares
Calculate parameters of model function Overdetermined data set Minimized sum of squared residuals
with a matrix form,
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Least Squares
With outliersWithout outliers
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Iterative method Non-deterministic Robust fitting in the presence of outliers Simple algorithm
RANSAC
M. A. Fischler, R. C. Bolles. Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography. Comm. of the ACM, Vol 24, pp 381-395, 1981.
1. selects N data items at random 2. estimates parameter 3. finds how many data items (of M) fit the model with parameter vector
within a user given tolerance. Call this K. 4. if K is big enough, accept fit and exit with success. 5. repeat 1..4 L times 6. fail if you get here
Algorithm
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RANSAC
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Residual histogram analysis (RHA) Sequential RANSAC Multi-RANSAC J-linkage Kernel fitting (KF) Mean-shift (MS) Etc.
Multi-model Estimation
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Fit multiple structures simultaneously Require no initial parameters: # of models, model
parameters
Multi-model Estimation with J-Linkage
Algorithm
Given N points,
1. Generate M model hypothesis (Random sampling)2. Build a N x M matrix, comprised of Preference Sets of points3. J-linkage clustering
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Multi-model Estimation with J-Linkage
Preference Set
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Random Sampling– A minimal sample set (MSS) is constructed in a way that
neighbouring points are selected with higher probability.
1. One sample is selected with uniform probability
2. If a point is given, then has the following probability:
Multi-model Estimation with J-Linkage
Z is a normalized constant, is chosen heuristically.
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J-linkage Clustering– Starting from all singletons– Each sweep of the algorithm merges the two clusters
with the smallest distance
Multi-model Estimation with J-Linkage
Measure the degree of overlap of the two sets and ranges from 0 (identical sets) to 1 (disjoint sets)
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J-linkage Clustering
Multi-model Estimation with J-Linkage
Algorithm
Assumption
One-to-one matching between a point and a model
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Example
Multi-model Estimation with J-Linkage
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Results
Multi-model Estimation with J-Linkage
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Results
Multi-model Estimation with J-Linkage
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Results
Multi-model Estimation with J-Linkage
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Results
Multi-model Estimation with J-Linkage
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Other Results 1
Multi-model Estimation with J-Linkage
David F. Fouhey, “Multi-model Estimation in the Presence of Outliers”
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Other Results 1
Multi-model Estimation with J-Linkage
David F. Fouhey, “Multi-model Estimation in the Presence of Outliers”
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Other Results 2
Multi-model Estimation with J-Linkage
Hanzi Wang, “Robust Multi-Structure Fitting”, A tutorial in ACCV 2012.
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Other Results 2
Multi-model Estimation with J-Linkage
Hanzi Wang, “Robust Multi-Structure Fitting”, A tutorial in ACCV 2012.
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Reference David F. Fouhey, “Multi-model Estimation in the Presence of Out-
liers” Stefano Branco, “RANSAC/MLESAC, Estimating parameters of
models with outliers” Hanzi Wang, “Robust Multi-Structure Fitting”, A tutorial in ACCV
2012.
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Thank you!
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Appendix
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Appendix