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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