A Convex Formulation of Continuous Multi-label Problems

Pock, Thomas; Schoenemann, Thomas; Graber, Gottfried; Bischof, Horst; Cremers, Daniel

doi:10.1007/978-3-540-88690-7_59

Cited by 156 publications

(228 citation statements)

References 14 publications

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“…This confirms the existence of at least one saddle point, see [29,30]. It also follows that the min and max operators of the primal-dual model (19) …”

Section: Equivalent Primal-dual Formulationsupporting

confidence: 72%

“…For the primal-dual model (19), the conditions of the minimax theorem (see e.g., [29] Chapter 6, Proposition 2.4) are all satisfied. That is, the constraints of flows are convex, and the energy function is linear in both the multiplier u and the flow functions p s , p and q, hence convex l.s.c.…”

Section: Equivalent Primal-dual Formulationmentioning

confidence: 99%

“…'unsaturated', we must have u i (x) = 0, i.e. u i (x)p i (x) = 0, which means that at the position x, the flow p i (x) has no contribution to the energy function and the flow p i (x), from x ∈ Ω i to the sink t, can be 'cut' off from the energy function of (19). On the other hand, in view of (8), the indicator function u i (x) = 0 definitely means the position x is not labeled as l i .…”

Section: Variational Perspective Of Flows and Cutsmentioning

confidence: 99%

“…Observe that the energy function of the primal-dual model (19) just gives the Lagrangian function of (16) where u i (x), i = 1 . .…”

Section: Multiplier-based Max-flow Algorithmmentioning

confidence: 99%

“…Recently, convex relaxation approaches were proposed, e.g. [3,[18][19][20][21][22]. Comparing to level set methods, Great advantages in numerics can be achieved, e.g.…”

Section: Introductionmentioning

confidence: 99%

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A Continuous Max-Flow Approach to Potts Model

Yuan

Bae

Tai

et al. 2010

Lecture Notes in Computer Science

140

209

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Abstract. We address the continuous problem of assigning multiple (unordered) labels with the minimum perimeter. The corresponding discrete Potts model is typically addressed with a-expansion which can generate metrication artifacts. Existing convex continuous formulations of the Potts model use TV-based functionals directly encoding perimeter costs. Such formulations are analogous to 'min-cut' problems on graphs. We propose a novel convex formulation with a continous 'max-flow' functional. This approach is dual to the standard TV-based formulations of the Potts model. Our continous max-flow approach has significant numerical advantages; it avoids extra computational load in enforcing the simplex constraints and naturally allows parallel computations over different labels. Numerical experiments show competitive performance in terms of quality and significantly reduced number of iterations compared to the previous state of the art convex methods for the continuous Potts model.

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“…This confirms the existence of at least one saddle point, see [29,30]. It also follows that the min and max operators of the primal-dual model (19) …”

Section: Equivalent Primal-dual Formulationsupporting

confidence: 72%

Section: Equivalent Primal-dual Formulationmentioning

confidence: 99%

Section: Variational Perspective Of Flows and Cutsmentioning

confidence: 99%

“…Observe that the energy function of the primal-dual model (19) just gives the Lagrangian function of (16) where u i (x), i = 1 . .…”

Section: Multiplier-based Max-flow Algorithmmentioning

confidence: 99%

“…Recently, convex relaxation approaches were proposed, e.g. [3,[18][19][20][21][22]. Comparing to level set methods, Great advantages in numerics can be achieved, e.g.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations