What energy functions can be minimized via graph cuts?

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What energy functions can be minimized via graph cuts? (2004)

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by Vladimir Kolmogorov , Ramin Zabih

Venue:	IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
Citations:	1047 - 23 self

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@ARTICLE{Kolmogorov04whatenergy,
    author = {Vladimir Kolmogorov and Ramin Zabih},
    title = {What energy functions can be minimized via graph cuts?},
    journal = {IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE},
    year = {2004},
    volume = {26},
    number = {2},
    pages = {65--81}
}

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Abstract

In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a general-purpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to create the appropriate graph. A software implementation is freely available.

Keyphrases

graph cut energy function binary variable particular energy function appropriate graph scene reconstruction necessary condition several new algorithm large number computer vision minimum cut precise characterization general-purpose construction limited application software implementation graph construction last year energy minimization problem many previous construction image restoration

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