SIEMENS

1
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
leo.grady_at_siemens.com, asinop_at_cmu.edu
Main Idea
Algorithm summary
Relationship to Normalized Cuts
Approximation quality
Offline
Perform an offline computation (without knowledge
of seed locations) so that interactive
segmentations are very fast.
Potentials
Segmentation
If we measure distances using spectral
coordinates
1. Generate image weights for Laplacian
matrix and precompute a set of K eigenvectors
from the Laplacian matrix
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How?
Online
where Yi is the vector of entries for node vi
across all generalized eigenvectors
Precompute a small set of eigenvectors from the
graph Laplacian matrix
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)
Recall
Written in terms of normalized Laplacian
eigenvector q and node degree d
Random walker segmentation solves the linear
system
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dervived from the full problem
Equals effective conductance, which is used by RW
to classify nodes to seeds
Comparison
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for Laplacian matrix, L, potential function, x,
and set of seeds, S, for which foreground seeds
are fixed to xi 1 and background seeds
are fixed to xi 0.
Original
In the case of a single foreground.background
seed, f, is equal to ?, where ? represents the
effective conductance between seeds. Given more
seeds, f is more complicated.
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Exact RW
Idea
If we can find f and precompute some
eigenvectors of L, we can find a K-approximation
of x.
Precomputed RW
Apply the pseudoinverse to both sides to yield
Exact
Where g is the 0-eigenvector of L.
NCuts
Without knowing seed locations, precomputed
eigenvectors give a O(n) online approximation to
the solution x!