A Unified Algebraic Approach to 2-D and 3-D Motion Segmentation
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A Unified Algebraic Approach to 2-D and 3-D Motion Segmentation
2006. 6. 9 (Fri)Young Ki Baik, Computer Vision Lab.
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• References• A Unified Algebraic Approach to 2-D and 3-
D Motion Segmentation• Rene Vidal and Yi Ma (ECCV 2004)
• Generalized Principal Component Analysis (GPCA)
• Rene Vidal, Yi Ma, et. al. (PAMI 2005)
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• Contents• Introduction• GPCA• 2-D motion segmentation• 3-D motion segmentation• Experimental results• Summary
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• Introduction (Motion segmentation)• Target
• 2D, 3D motion segmentation
• Problem statement• Most previous work
Iterative approach (EM, RANSAC, etc.) Manual or random initial value.
Cause of divergence or bad results
• Proposed algorithm• Good initial value using non-iterative
algebraic method
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• Introduction (This paper)• Contribution
• Applying GPCA to 2D, 3D motion segmentation problem
• Condition• Subspaces are all linear.• Known correspondences• Known number of subspace (class)
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• GPCA (Generalized PCA)• GPCA treats heterogeneous data and
multiple subset with different linear model.
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• GPCA (Generalized PCA)• Find the basis of each subspace which
orthogonal to data x
b
x
SforT xbx 0
SdatabasissubspaceS
:::
xb
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• GPCA (Generalized PCA)• A homogeneous polynomial of degree n
• The mixture of subspaces can be linearly fitting general polynomial to the given data.
• Example• Number of subspace n = 2• Vector size k = 3
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n vp mapVeronesev
polynomialpsetdata
subspaceofnumbern
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• GPCA (Generalized PCA)• Find the basis from derivatives of the
polynomials with y on the S
b
y
by ~nDp
S databasissubspaceS
:::
yb
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• GPCA (Algorithm)• Fitting polynomials to data lying in multiple
subspaces
• Obtaining basis of each subspace by polynomial differentiation
• Choosing data per subspace by using basis
by ~nDp
xcx nT
n vp 0 Ac
SforT xbx 0
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• Motion segmentation• Notation
• Let be a vector in or .z KR KC
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in
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vector ofdimension :subspace ofnumber :data ofnumber :
mapVeronesev
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n
n
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i
::
: :
cb
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• 2D Motion segmentation• Translation case
• Under 2-d translation motion model, the two images are related by on out of n possible 2-d translation .
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• 2D Motion segmentation• Translation case
• Finding coefficient c (fitting polynomial)
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5432122 abjcbacjbcacjccvp zcz
06 cAN
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• 2D Motion segmentation• Translation case
• Finding basis (using polynomial differentiation)
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• 2D Motion segmentation• Translation case
• Segmentation
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end
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• 2D Motion segmentation• Affine motion case
• In this case, we assume that the images are related by a collection on n 2-D affine motion models .
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• 3D Motion segmentation• Epipolar constraint • Homography
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• Experimental results
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A Unified Algebraic Approach to 2D and 3D Motion Segmentation
• Summary• Contribution
• Applying GPCA to 2D, 3D motion segmentation problem
• Good initial value using non-iterative algebraic method
• Limitation• Linear subspace• Known correspondences and number of subspace • If subspace is increased, then computational
complexity will be exponentially increased.