## An integral solution to surface evolution PDEs via geo-cuts (2006)

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Venue: | In ECCV |

Citations: | 20 - 7 self |

### BibTeX

@INPROCEEDINGS{Boykov06anintegral,

author = {Yuri Boykov and Vladimir Kolmogorov and Daniel Cremers and Andrew Delong},

title = {An integral solution to surface evolution PDEs via geo-cuts},

booktitle = {In ECCV},

year = {2006},

pages = {409--422}

}

### OpenURL

### Abstract

Abstract. We introduce a new approach to modelling gradient flows of contours and surfaces. While standard variational methods (e.g. level sets) compute local interface motion in a differential fashion by estimating local contour velocity via energy derivatives, we propose to solve surface evolution PDEs by explicitly estimating integral motion of the whole surface. We formulate an optimization problem directly based on an integral characterization of gradient flow as an infinitesimal move of the (whole) surface giving the largest energy decrease among all moves of equal size. We show that this problem can be efficiently solved using recent advances in algorithms for global hypersurface optimization [4, 2, 11]. In particular, we employ the geo-cuts method [4] that uses ideas from integral geometry to represent continuous surfaces as cuts on discrete graphs. The resulting interface evolution algorithm is validated on some 2D and 3D examples similar to typical demonstrations of levelset methods. Our method allows for computation of gradient flows for hypersurfaces with respect to a fairly general class of continuous functionals and it is flexible with respect to distance metrics on the space of contours/surfaces. Our preliminary tests for standard L2 distance metric demonstrate numerical stability, topological changes and an absence of any oscillatory motion. Index Terms — front propagation, gradient flows for hypersurfaces, min-cut/max-flow algorithms on graphs, geo-cuts, integral geometry. 2 1

### Citations

3252 | Snakes: Active contour models
- Kass, Witkin, et al.
- 1988
(Show Context)
Citation Context ...low partial differential equation. This technique has become standard in various fields of computer vision ranging from motion estimation [9,3, 17,13], over image enhancement [15, 16] to segmentation =-=[10,6]-=-. While not all partial differential equations are derived from a variational approach, we will in this work focus on the class of partial differential equations which correspond to the gradient flow ... |

1871 | Determining optical flow
- Horn, Schunck
- 1981
(Show Context)
Citation Context ...d by implementing the Euler-Lagrange equations in a gradient flow partial differential equation. This technique has become standard in various fields of computer vision ranging from motion estimation =-=[9,3, 17,13]-=-, over image enhancement [15, 16] to segmentation [10,6]. While not all partial differential equations are derived from a variational approach, we will in this work focus on the class of partial diffe... |

865 | An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision
- Boykov, Kolmogorov
- 2004
(Show Context)
Citation Context ...e at the boundary if it is 4-connected to some pixel q in the other segment). In our tests we compute the first cut C1 and reset t ′ 0 = t1, C ′ 0 = C1. We use an implementation of maxflow graph-cuts =-=[5]-=- 11 (7)s12 as a tool for optimizing the energy. Note that in the figures below we show only selected time frames of the gradient flow motion computed by out method. In Figures 3, 6 and 7 we have inten... |

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- Chan, Vese
(Show Context)
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556 |
The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields
- Black, Anandan
- 1996
(Show Context)
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302 |
Using dynamic programming for solving variational problems in vision
- Amini, Weymouth, et al.
- 1990
(Show Context)
Citation Context ...e algorithms for active contours are special cases of our general approach for some specific distance metric dist(C, C0). Example 2. (DP snakes) A well-known dynamic programming approach to 2D snakes =-=[1]-=- uses control points to represent discrete snake C. Then, a typical snake energy functional F(C) is iteratively minimized by dynamic programming over positions of control points. In order to simulate ... |

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- Weickert
- 1998
(Show Context)
Citation Context ...equations in a gradient flow partial differential equation. This technique has become standard in various fields of computer vision ranging from motion estimation [9,3, 17,13], over image enhancement =-=[15, 16]-=- to segmentation [10,6]. While not all partial differential equations are derived from a variational approach, we will in this work focus on the class of partial differential equations which correspon... |

188 | KOLMOGOROV V.: Computing geodesics and minimal surfaces via graph cuts
- BOYKOV
(Show Context)
Citation Context ... surface giving the largest energy decrease among all moves of equal size. We show that this problem can be efficiently solved using recent advances in algorithms for global hypersurface optimization =-=[4, 2, 11]-=-. In particular, we employ the geo-cuts method [4] that uses ideas from integral geometry to represent continuous surfaces as cuts on discrete graphs. The resulting interface evolution algorithm is va... |

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- Cremers, Tischhäuser, et al.
- 2002
(Show Context)
Citation Context ...nt flow models dynamics of many natural phenomena in physics. In computer vision, ability to track gradual changes in segmentation allowed variational methods to successfully incorporate shape priors =-=[8, 14]-=-. In this paper we propose a new integral approach to computing gradient flow of interfaces with respect to a fairly general class of energy functionals. The proposed algorithm generates a timely sequ... |

89 | Approximations of shape metrics and application to shape warping and empirical shape statistics
- Charpiat, Faugeras, et al.
- 2005
(Show Context)
Citation Context ...by a distance d0(p) to the nearest point on C0. sophisticated contour metrics, focussed either on using the correct metric [19], or on designing novel gradient flows with certain desirable properties =-=[7]-=-. A very similar freedom in the choice of metric on the space of contours will also arise in our novel integral formulation of boundary evolution. Note that motion of a contour C in a differential fra... |

83 | A Theoretical Framework for Convex Regularizers in PDE-Based Computation of Image Motion
- Weickert, Schnörr
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Citation Context ...d by implementing the Euler-Lagrange equations in a gradient flow partial differential equation. This technique has become standard in various fields of computer vision ranging from motion estimation =-=[9,3, 17,13]-=-, over image enhancement [15, 16] to segmentation [10,6]. While not all partial differential equations are derived from a variational approach, we will in this work focus on the class of partial diffe... |

50 | BOYKOV Y.: What metrics can be approximated by geo-cuts, or global optimization of length/area and flux
- KOLMOGOROV
(Show Context)
Citation Context ...step in the geo-cuts approach is to construct a graph whose cut metric approximates that of functional E(C). Such construction exists for a fairly large class of continuous functionals E, as shown in =-=[4, 12]-=-. In general, E(C) can be any submodular (graph-representable) functional over contours/surfaces. In particular, it can include the following terms: – Geometric length/area under a fairly wide class o... |

45 | Image Sequence Analysis via Partial Differential Equations
- Kornprobst, Deriche
- 1999
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39 | Globally Minimal Surfaces by Continuous Maximal Flows
- Appleton, Talbot
- 2006
(Show Context)
Citation Context ...or evolution of an interface C under a gradient flow (descent) with respect to a given energy functional F(C). Our general approach to computing gradient flows will be based on optimization of energy =-=(2)-=-. Theorem 1. Consider a family of contours Ct minimizing energy Et(C) = F(C) + 1 2(t − t0) · dist(C, C0) (2) where metric dist(C, C0) is consistent with some inner product 〈, 〉 according to equation (... |

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- Xu, Bansal, et al.
- 2003
(Show Context)
Citation Context ...ox-based move constraints with a quadratic motion penalty dist(C, C0) = � p dC2 p which can be easily handled by dynamic programming. Example 3. (Fixed Band Graph-Cuts) An approach to segmentation in =-=[18]-=- is very similar to DP-snakes algorithm in [1]. Dynamic programming in DP-snakes is replaced by graph cuts but [18] still uses the idea of a fixed size “box”. Since graph cuts do not use control point... |

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- 2005
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Citation Context ...weighted area of the highlighted region. The weight of each point p is given by a distance d0(p) to the nearest point on C0. sophisticated contour metrics, focussed either on using the correct metric =-=[19]-=-, or on designing novel gradient flows with certain desirable properties [7]. A very similar freedom in the choice of metric on the space of contours will also arise in our novel integral formulation ... |

23 | S.J.: A discrete global minimization algorithm for continuous variational problems. harvard computer science technical report
- Kirsanov, Gortler
- 2004
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Citation Context ... surface giving the largest energy decrease among all moves of equal size. We show that this problem can be efficiently solved using recent advances in algorithms for global hypersurface optimization =-=[4, 2, 11]-=-. In particular, we employ the geo-cuts method [4] that uses ideas from integral geometry to represent continuous surfaces as cuts on discrete graphs. The resulting interface evolution algorithm is va... |

15 | Globally optimal surfaces by continuous maximal flows
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Citation Context ...volution of an interface C under a gradient flow (descent) with respect to a given energy functional F(C). Our general approach to computing gradient flows will be based on optimization of the energy =-=(2)-=- given below. Theorem 1. Consider a family of contours Ct minimizing energy Et(C) = F(C) + 1 2(t − t0) · dist(C, C0) (2) 5s6 where metric dist(C, C0) is consistent with some inner product 〈, 〉 accordi... |

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Shape-based segmentation and tracking in cardiac image analysis
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Citation Context ...nt flow models dynamics of many natural phenomena in physics. In computer vision, ability to track gradual changes in segmentation allowed variational methods to successfully incorporate shape priors =-=[8, 14]-=-. In this paper we propose a new integral approach to computing gradient flow of interfaces with respect to a fairly general class of energy functionals. The proposed algorithm generates a timely sequ... |

4 |
Scale-space and edge-detection
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- 1990
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Citation Context ...equations in a gradient flow partial differential equation. This technique has become standard in various fields of computer vision ranging from motion estimation [9,3, 17,13], over image enhancement =-=[15, 16]-=- to segmentation [10,6]. While not all partial differential equations are derived from a variational approach, we will in this work focus on the class of partial differential equations which correspon... |