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