A Useful Bound for Region Merging Algorithms in a Bayesian Model

Tao, T. and Crisp, D.J.

A well known and effective approach to image segmentation is based on the Mumford-Shah model, where finding an optimal segmentation of an image is posed as the minimization of an energy functional. The simplest of these models is mathematically tractable and desirable properties of the resulting segmentations have been rigorously established. An important step in the proofs involves establishing an 'inverse isoperimetric' type of bound. In particular, it can be shown that for segmentations where it is impossible to decrease the functional energy by merging any pair of regions, the length of the boundary between two regions must be bounded above by their areas and variation of the image itself. Recent work of one of the authors has extended the Mumford-Shah segmentation model to a Bayeasian setting. In this report we show that an 'inverse isoperimetric' type of bound also exists in this new setting. The new bound can likewise be used to prove desirable properties o the corresponding segmentations.

Cite as: Tao, T. and Crisp, D.J. (2003). A Useful Bound for Region Merging Algorithms in a Bayesian Model. In Proc. Twenty-Sixth Australasian Computer Science Conference (ACSC2003), Adelaide, Australia. CRPIT, 16. Oudshoorn, M. J., Ed. ACS. 95-100.

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