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SIEMENS

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.

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SIEMENS

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  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@siemens.com, asinop@cmu.edu Main Idea Approximation quality Algorithm summary Relationship to Normalized Cuts Offline Potentials Segmentation Perform an offline computation (without knowledge of seed locations) so that interactive segmentations are very fast. 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 5 eigs – Off: 55.9s, On: 0.62s 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 20 eigs – Off: 89.9s, On: 0.64s dervived from the full problem Equals effective conductance, which is used by RW to classify nodes to seeds Comparison 40 eigs – Off: 157s, On: 0.7s 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 ±ρ, where ρ represents the effective conductance between seeds. Given more seeds, f is more complicated. 100 eigs – Off: 555s, On: 0.79s 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!

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