## Fast Approximate Energy Minimization with Label Costs (2009)

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### BibTeX

@TECHREPORT{Delong09fastapproximate,

author = {Andrew Delong and Anton Osokin and Hossam N. Isack and Yuri Boykov},

title = {Fast Approximate Energy Minimization with Label Costs},

institution = {},

year = {2009}

}

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### Abstract

The α-expansion algorithm [4] has had a significant impact in computer vision due to its generality, effectiveness, and speed. Thus far it can only minimize energies that involve unary, pairwise, and specialized higher-order terms. Our main contribution is to extend α-expansion so that it can simultaneously optimize “label costs ” as well. An energy with label costs can penalize a solution based on the set of labels that appear in it. The simplest special case is to penalize the number of labels in the solution. Our energy is quite general, and we prove optimality bounds for our algorithm. A natural application of label costs is multi-model fitting, and we demonstrate several such applications in vision: homography detection, motion segmentation, and unsupervised image segmentation. Our

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Citation Context ...ion was also discussed by Torr [55] and Li [42] in the context of motion estimation. Another well-known example is the Bayesian information criterion (BIC) [13,44]: min Θ −2log Pr(X |Θ) + |Θ|·log |P| =-=(32)-=- where |P| is the number of observations. The BIC suggests that label costs should be scaled in logarithmic proportion to the number of data points or, in practice, to the estimated number of observat... |

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Citation Context ...combination of label subset cost potentials, but not the other way around. Section 6 elaborates on this point, and mentions a possible extension to our work based on the Robust P n Potts construction =-=[21]-=-. A final detail is how to handle the case when label α was not used in the current labeling f ′ . The corrective term C α in (6) incorporates the label costs for α itself: C α (x) = ∑ ( ∏ ) hL − hL ¯... |

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Citation Context ...ost functions (see [32] for review). Besides the add-facilities-greedily strategy, other greedy moves have been proposed for UFL such as the greedy-interchange and dynamic programming heuristics (see =-=[9, 10]-=- for review). Our C++ library implements the greedy heuristic [23] and, when smooth costs are all zero, it is 5–20 times faster than α-expansion while yielding similar energies. Indeed, “open facility... |

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Citation Context ... in supervised settings as well. The label cost terms are included in energy (⋆) linearly and can thus be learned by max-margin methods [54,57]. This approach was recently used for CRF learning, e.g. =-=[53]-=-. Acknowledgements We would like to thank Fredrik Kahl for referring us to the works of Li and Vidal, and for suggesting motion segmentation as an application. We also wish to thank Lena Gorelick for ... |

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Citation Context ...f parameters in Θ that can vary. This criterion was also discussed by Torr [35] and Li [26] in the context of motion estimation. Another well-known example is the Bayesian information criterion (BIC) =-=[8, 28]-=-: min Θ −2ln Pr(X |Θ) + |Θ|·ln |P| (25) where |P| is the number of observations. The BIC suggests that label costs should be scaled in some proportion (linear or logarithmic) to the number of data poi... |

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Citation Context ...s necessary. We can introduce label costs into E(f) to penalize each unique label that appears in f: E(f) = ∑ Dp(fp) + ∑ hl·δl(f) (1) p∈P l∈L (b) Figure 1. Motion segmentation on the 1RT2RCR sequence =-=[36]-=-. Energy (1) finds 3 dominant motions (a) but labels many points incorrectly. Energy (2) gives coherent segmentations (b) but finds redundant motions. Our energy combines the best of both (c). where h... |

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Citation Context ...me form. For example, in [18] we proposed our subset costs in (⋆) as a form of co-occurrence cost in object recognition. This application was thoroughly and independently developed by Ladick´y et al. =-=[39]-=-, also within an α-expansion framework but with a heuristic extension; see Section 7 for discussion. Others have independently proposed label cost energies for specific applications. For example, we l... |

46 |
High-arity interactions, polyhedral relaxations, and cutting plane algorithm for soft constraint optimisation (MAP-MRF). In: Computer vision and pattern recognition conference
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Citation Context ...neralizes our high-order potentials δL(·) if needed. Related global interactions. Label costs can be viewed as a special case of global interactions recently studied in vision, for example, by Werner =-=[38]-=- and Woodford et al. [40]. Werner proposed a cutting plane algorithm to make highorder potentials tractable in an LP relaxation framework. The algorithm is very slow but much more general, and he demo... |

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Citation Context ...nsion of α-expansion to instance cost potentials in 2004 that only appeared as part of (⋆)Fast Approximate Energy Minimization with Label Costs 3 a supervised part-based object recognition framework =-=[30]-=-, though his approach to deriving an algorithm is quite different from ours 1 . Special case energy (1) corresponds to objective functions studied in vision by Torr [55] and in a number of independent... |

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Citation Context ... Term Cα simply corrects for the case when α /∈ L ′ and is discussed later. Each product term in (6) adds a higher-order clique PL beyond the standard α-expansion energy E α (x). Freedman and Drineas =-=[14]-=- generalized the graph construction of [22] to handle terms c ∏ pxp of arbitrary degree when c ≤ 0. This means we can transform each product seen in (6) into a sum of quadratic and linear terms that g... |

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Citation Context ...∈L As an internal step, EM also computes responsibilities Pr(fp = l | xp,θ,ω) in order to estimate which mixture components could have generated each data point [4]. The elliptical3 K-means algorithm =-=[33]-=- maximizes a different likelihood function on the same probability space Pr(X |f,θ) = ∏ Pr ( ) xp |θfp . (18) p∈P In contrast to EM, this approach directly computes labeling f = {fp | p ∈ P}, while mi... |

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Citation Context ...ent fashion iterating f, M and Ī optimization steps. We initialize Ī = I. Note that f and M steps are analogous to (39) since Ī is fixed. 8 A similar discussion also appears after Proposition 1(b) in =-=[36]-=-. Optimization over Ī for fixed f and M requires some clarification. Since variables Īp appear only in the unary terms in (40), optimization over Īp can be done very efficiently. For example, one can ... |

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Citation Context ...gorithm is quite different from ours 1 . Special case energy (1) corresponds to objective functions studied in vision by Torr [55] and in a number of independent later works for specific applications =-=[42,40, 4]-=-. Our combined energy (⋆) has recently been extended to convex continuous total variation (TV) formulations [61]. Label costs can be viewed as a special case of other global interactions recently stud... |