SIEMENS Leo Grady and Ali Kemal Sinop [email protected], [email protected] partment of Imaging and Visualization – Siemens Corporate Research, Princeton mputer Science Department – Carnegie Mellon University, Pittsburgh Fast Approximate Random Walker Segmentation Using Eigenvector Precomputation Main Idea Perform an offline computation (without knowledge of seed locations) so that interactive segmentations are very fast. Algorithm summary Relationship to Normalized Cut If we measure distances using spectral coordinates How? Precompute a small set of eigenvectors from the graph Laplacian matrix Recall Random walker segmentation solves the linear system 0 f x x L B B L U S U T S for Laplacian matrix, L, potential function, x, and set of seeds, S, for which foreground seeds are fixed to x i = 1 and background seeds are fixed to x i = 0. S T U U x B x L dervived from the full problem In the case of a single foreground.background seed, f, is equal to ±ρ, where ρ represents the effective conductance between seeds. Given more seeds, f is more complicated. Idea If we can find f and precompute some eigenvectors of L, we can find a K-approximation of x. T K K K T Q Q Q Q L Apply the pseudoinverse to both sides to yield f Q Q x gg I T K K K T * 1 Where g is the 0-eigenvector of L. Without knowing seed locations, precomputed eigenvectors give a O(n) online approximation to the solution x! Offline 1. Generate image weights for Laplacian matrix and precompute a set of K eigenvectors from the Laplacian matrix Online 1. Obtain seeds interactively from a user 2. Estimate f from precomputed eigenvectors (see paper for details – Requires solving a small linear system) 3. Using precomputed eigenvectors, apply pseudoinverse to f to obtain x plus a factor of g 4. Solve for factor of g to obtain final solution (see paper for details – The factor may be determined very efficiently) Approximation quality 5 eigs – Off: 55.9s, On: 0.62s 20 eigs – Off: 89.9s, On: 0.64s 40 eigs – Off: 157s, On: 0.7s 100 eigs – Off: 555s, On: 0.79s Exact PotentialsSegmentation ) ( ) ( ) , ( dist * 1 j i T j i j i Y Y Y Y v v where Y i is the vector of entries for node v i across all generalized eigenvectors 2 2 1 ) , ( dist j jk i ik N k k j i d q d q v v Written in terms of normalized Laplacian eigenvector q and node degree d Equals effective conductance, which is used by RW to classify nodes to seeds Comparison Origi na l Exact RW Precompu ted RW NCut s
SIEMENS
Jan 25, 2016
Fast Approximate Random Walker Segmentation Using Eigenvector Precomputation. Department of Imaging and Visualization – Siemens Corporate Research, Princeton Computer Science Department – Carnegie Mellon University, Pittsburgh. Leo Grady and Ali Kemal Sinop. SIEMENS. - PowerPoint PPT Presentation
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SIEMENSLeo Grady and Ali Kemal Sinop
[email protected], [email protected]
Department of Imaging and Visualization – Siemens Corporate Research, Princeton
Computer Science Department – Carnegie Mellon University, Pittsburgh
Fast Approximate Random Walker Segmentation Using Eigenvector Precomputation
Main IdeaPerform an offline computation (without knowledge of seed locations) so that interactive segmentations are very fast.
Algorithm summary Relationship to Normalized Cuts
If we measure distances using spectral coordinates
How?Precompute a small set of eigenvectors from the graph Laplacian matrix
RecallRandom walker segmentation solves the linear system
0
f
x
x
LB
BL
U
S
UTS
for Laplacian matrix, L, potential function, x, and set of seeds, S, for which foreground seeds are fixed to xi = 1 and background seeds arefixed to xi = 0.
ST
UU xBxL dervived from the full problem
In the case of a single foreground.background seed, f, is equal to ±ρ, where ρ represents the effective conductance between seeds. Given more seeds, f is more complicated.
IdeaIf we can find f and precompute some eigenvectors of L, we can find a K-approximation of x.
TKKK
T QQQQL Apply the pseudoinverse to both sides to yield
fQQxggI TKKK
T *1Where g is the 0-eigenvector of L.
Without knowing seed locations, precomputed eigenvectors give a O(n) online approximation to the solution x!
Offline1. Generate image weights for Laplacian matrixand precompute a set of K eigenvectors from theLaplacian matrix
Online1. Obtain seeds interactively from a user2. Estimate f from precomputed eigenvectors (see paper for details – Requires solving a small linear system)3. Using precomputed eigenvectors, apply pseudoinverse to f to obtain x plus a factor of g4. Solve for factor of g to obtain final solution (see paper for details – The factor may be determined very efficiently)
Approximation quality
5 eigs – Off: 55.9s, On: 0.62s
20 eigs – Off: 89.9s, On: 0.64s
40 eigs – Off: 157s, On: 0.7s
100 eigs – Off: 555s, On: 0.79s
Exact
Potentials Segmentation
)()(),(dist *1ji
Tjiji YYYYvv
where Yi is the vector of entries for node vi across all generalized eigenvectors
2
2
1),(dist
j
jk
i
ikN
k kji
d
q
d
qvv
Written in terms of normalized Laplacian eigenvector q and node degree d
Equals effective conductance, which is used by RW to classify nodes to seeds
Comparison
Original
Exact RW
Precomputed RW
NCuts
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