Supplement to “ Dead - End Elimination as a Heuristic for Min - Cut Image Segmentation ”

This article is a supplement to our 2006 ICIP paper, “Dead-End Elimination as a Heuristic for Min-Cut Image Segmentation” [1], which we assume the reader has read. We summarize the proof of the dead-end elimination theorem (due to Desmet et al. [2] and Goldstein [3]), briefly discuss the performance of our current implementation of DEE pairs, show the input images for which timings are referenced in the main paper, and present examples of processed images to show that DEE does not affect the resulting segmentation. 1 Proof of DEE Theorem The original dead-end elimination theorem is due to Desmet et al. [2]. Goldstein’s DEE theorem [3] is closely related but more powerful: Let ia and ir be two specific assignments at a particular position i. Then, if E(ia)−E(ib)+∑ j min f [E(ia, j f )−E(ib, j f )] > 0, the assignment ia cannot possibly be in the global minimum configuration and can therefore be eliminated from the space. ia cannot be in the global minimum energy assignment if there exists another assignment at the same position, ib, such that the total energy with ia is higher than the total energy with ib even when we choose every other position to give ia the best pairwise energies relative to ib. We now summarize proofs due to Desmet and Goldstein. Proof. Given two possible assignments, ia and ib at position i, let us assume that E(ia)−E(ib)+∑ j min f [E(ia, j f )−E(ib, j f )] > 0 This is the premise of the DEE theorem. Let the global minimum energy assignment (GMEA) at each position be represented by the subscript g. Define