## Geodesic Active Contours (1997)

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Citations: | 1150 - 45 self |

### BibTeX

@MISC{Caselles97geodesicactive,

author = {Vicent Caselles and Ron Kimmel and Guillermo Sapiro},

title = {Geodesic Active Contours},

year = {1997}

}

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### Abstract

A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.

### Citations

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(Show Context)
Citation Context ...eral objects by a dynamic model known as the “geodesic active contour” introduced in [4]–[7] (see also [19] and [30]). Geodesic active contours were introduced as a geometric alternative for “snakes” =-=[18]-=-, [32]. Snakes are deformable models that are based on minimizing an energy along a curve. The curve, or snake, deforms its shape so as to minimize an “internal” and “external” energies along its boun... |

1437 | Scale-space and edge detection using anisotropic diffusion
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Citation Context ...itionally stable numerical scheme for nonlinear diffusion in image processing. Let us briefly review its main ingredients and adapt it to our model. The original AOS model deals with the Perona-Malik =-=[26]-=-, nonlinear image evolution equation of the form div with given initial condition as the image . Let us rewrite explicitly the right hand side of the evolution equation div where is an index running o... |

986 | Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations
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(Show Context)
Citation Context ...ward the edges of the object in the image. In [2] and [23], a geometric alternative for the snake model was introduced, in which an evolving curve was formulated by the Osher–Sethian level set method =-=[24]-=-. The method works on a fixed grid, usually the image pixels grid, and automatically handles changes in the topology of the evolving contour. The geodesic active contour model was born latter. It is b... |

916 | The design and use of steerable filters - Freeman, Adelson - 1991 |

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Citation Context ...d “external” energies along its boundary. The internal part causes the boundary curve to become smooth, while the external part leads the curve toward the edges of the object in the image. In [2] and =-=[23]-=-, a geometric alternative for the snake model was introduced, in which an evolving curve was formulated by the Osher–Sethian level set method [24]. The method works on a fixed grid, usually the image ... |

666 | A.: Region competition: unifying snakes, region growing, and Bayes/ MDL for multiband image segmentation
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Citation Context ...ts level set formulation is given by div where is an area element and is the interior of region enclosed by the contour . The Euler Lagrange as steepest descent, following the development in [36] and =-=[37]-=- is Including a weighted area minimization term that yields a constant velocity, modulated by the edge indication function, we have div 1 An early version of a geometric-variational model, in which , ... |

495 | On active contour models and balloons
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Citation Context ...in a way that leads toward a meaningful minimum, which is the minimum of the functional. 1 One may add an additional force that comes from an area minimization term and motivated by the balloon force =-=[10]-=-. This way, the contour may be directed to propagate inwards by minimization of the interior. The functional with the additional area term modulated by an edge indicator function reads This formulatio... |

474 | A marching level set method for monotonically advancing fronts
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Citation Context ...n in the four pixel cells in which changes its sign. The grid points with the exact distance to the zero level set are then used to initialize the fast marching method. Sethian’s fast marching method =-=[29]-=-, [28], is a computationally optimal numerical method for distance computation on rectangular grids. The method keeps a front of updated points sorted in a heap structure and constructs a numerical so... |

354 | A Fast Level Set Method for Propagating Interfaces
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Citation Context ... approach overcomes the last difficulty by limiting the computations to a narrow strip around the zero set. First suggested by Chopp [9], in the context of the level set method and later developed in =-=[1]-=-, the narrow band idea limits the computation to a tight strip of few grid points around the zero set. The rest of the domain serves only as a sign holder. As the curve evolves, the narrow band change... |

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Citation Context ...op of the heap and expanding the solution to its neighboring grid points. This method enjoys a computational complexity bound of , where is the number of grid points in the narrow band. See also [8], =-=[33]-=-, where consistent schemes are used to compute distance maps on rectangular grids. (2)1470 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 10, OCTOBER 2001 III. EDGE INDICATOR FUNCTIONS FOR COLOR... |

268 |
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Citation Context ... as curve shortening flow or geometric heat equation, is a well studied equation in the theory of curve evolution. It is proven to bring every simple closed curve into a circular point in finite time =-=[15]-=-, [17]. Fig. 2 shows an application of the proposed method for a curve evolving by its curvature and vanishes at a point. One can see how the number of iterations needed for the curve to converge to a... |

266 | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations - Chen, Giga, et al. - 1991 |

256 |
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Citation Context ...rve shortening flow or geometric heat equation, is a well studied equation in the theory of curve evolution. It is proven to bring every simple closed curve into a circular point in finite time [15], =-=[17]-=-. Fig. 2 shows an application of the proposed method for a curve evolving by its curvature and vanishes at a point. One can see how the number of iterations needed for the curve to converge to a point... |

246 |
Constraints on deformable models: Recovering 3d shape and nonrigid motions
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Citation Context ...bjects by a dynamic model known as the “geodesic active contour” introduced in [4]–[7] (see also [19] and [30]). Geodesic active contours were introduced as a geometric alternative for “snakes” [18], =-=[32]-=-. Snakes are deformable models that are based on minimizing an energy along a curve. The curve, or snake, deforms its shape so as to minimize an “internal” and “external” energies along its boundary. ... |

216 | Gradient flows and geometric active contour models
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(Show Context)
Citation Context ...at may include other objects and a background. Here, we focus on boundary detection of one or several objects by a dynamic model known as the “geodesic active contour” introduced in [4]–[7] (see also =-=[19]-=- and [30]). Geodesic active contours were introduced as a geometric alternative for “snakes” [18], [32]. Snakes are deformable models that are based on minimizing an energy along a curve. The curve, o... |

215 | A Viscosity Solutions Approach to Shape-from-Shading - Rouy, Tourin - 1992 |

211 |
Level Set Methods: Evolving Interfaces
- Sethian
- 1996
(Show Context)
Citation Context ...he four pixel cells in which changes its sign. The grid points with the exact distance to the zero level set are then used to initialize the fast marching method. Sethian’s fast marching method [29], =-=[28]-=-, is a computationally optimal numerical method for distance computation on rectangular grids. The method keeps a front of updated points sorted in a heap structure and constructs a numerical solution... |

207 | TERZOPOULOS D.: Snakes: Active Contour Models - KASS, WITKIN - 1988 |

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191 | Efficient and reliable scheme for nonlinear diffusion filtering
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Citation Context ... band changes its shape and serves as a dynamic numerical support around the location of the zero level set. B. AOS Scheme Additive operator splitting (AOS) schemes were introduced by Weickert et al. =-=[35]-=- as an unconditionally stable numerical scheme for nonlinear diffusion in image processing. Let us briefly review its main ingredients and adapt it to our model. The original AOS model deals with the ... |

189 | R.: A general framework for low level vision
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(Show Context)
Citation Context ...CATOR FUNCTIONS FOR COLOR AND MOVIES What is a proper edge indicator for color images? Several generalizations for the gradient magnitude of gray level images were proposed, see e.g., [11], [27], and =-=[31]-=-. In [25], Paragios and Deriche introduced a probability-based edge indicator function for movies. In this paper, we have chosen the geometric philosophy to extract an edge indicator. We consider a me... |

161 | Dynamic programming for detecting, tracking, and matching deformable contours - Geiger, Gupta, et al. - 1995 |

152 |
D.L.: Anisotropic diffusion of multivalued images with application to color filtering
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Citation Context ... EDGE INDICATOR FUNCTIONS FOR COLOR AND MOVIES What is a proper edge indicator for color images? Several generalizations for the gradient magnitude of gray level images were proposed, see e.g., [11], =-=[27]-=-, and [31]. In [25], Paragios and Deriche introduced a probability-based edge indicator function for movies. In this paper, we have chosen the geometric philosophy to extract an edge indicator. We con... |

142 |
A geometric model for active contours
- Caselles, Catte, et al.
- 1993
(Show Context)
Citation Context ...rnal” and “external” energies along its boundary. The internal part causes the boundary curve to become smooth, while the external part leads the curve toward the edges of the object in the image. In =-=[2]-=- and [23], a geometric alternative for the snake model was introduced, in which an evolving curve was formulated by the Osher–Sethian level set method [24]. The method works on a fixed grid, usually t... |

121 | S.: A note on the gradient of a multi-image - Zenzo - 1986 |

118 |
Computing minimal surfaces via level set curvature flow
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(Show Context)
Citation Context ...time consuming operation for a sequential computer. The narrow band approach overcomes the last difficulty by limiting the computations to a narrow strip around the zero set. First suggested by Chopp =-=[9]-=-, in the context of the level set method and later developed in [1], the narrow band idea limits the computation to a tight strip of few grid points around the zero set. The rest of the domain serves ... |

106 | Using deformable surfaces to segment 3-D images and infer differential structures - Cohen, Cohen, et al. - 1992 |

85 | A Tannenbaum, Affine invariant scale-space - Sapiro - 1993 |

84 | Detecting and localizing edges composed of steps, peaks and roofs - Perona, Malik |

80 | A common framework for curve evolution, segmentation and anisotropic diffusion
- Shah
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(Show Context)
Citation Context ...clude other objects and a background. Here, we focus on boundary detection of one or several objects by a dynamic model known as the “geodesic active contour” introduced in [4]–[7] (see also [19] and =-=[30]-=-). Geodesic active contours were introduced as a geometric alternative for “snakes” [18], [32]. Snakes are deformable models that are based on minimizing an energy along a curve. The curve, or snake, ... |

77 | G.: Model driven edge detection
- Fua, Leclerc
- 1988
(Show Context)
Citation Context ...erm that yields a constant velocity, modulated by the edge indication function, we have div 1 An early version of a geometric-variational model, in which , that deals with open curves was proposed in =-=[14]-=-. We have yet to determine a numerical scheme and an appropriate edge indication function . An explicit Euler scheme withGOLDENBERG et al.: FAST GEODESIC ACTIVE CONTOURS 1469 forward time derivative,... |

74 | Modeling surfaces of arbitrary topology with dynamic particles - Szeliski, Tonnesen, et al. - 1993 |

72 | Minimal surfaces based object segmentation - Caselles, Kimmel, et al. - 1997 |

72 | On affine plane curve evolution - Sapiro, Tannenbaum |

71 | Tracking level sets by level sets: a method for solving the shape from shading problem - Kimmel, Bruckstein - 1995 |

68 | A review of recent numerical algorithms for hypersurfaces moving with curvature dependent speed - Sethian - 1989 |

64 | Direct method for reconstructing shape from shading - Dupuis, Oliensis - 1992 |

54 | Finding shortest paths on surfaces using level sets propagation - Kimmel, Amir, et al. - 1995 |

48 | Implementing continuous-scale morphology by curve evolution - Sapiro, Kimmel, et al. - 1993 |

44 | Area and length preserving geometric invariant scale-spaces - Sapiro, Tannenbaum - 1995 |

39 |
Snakes in movement
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(Show Context)
Citation Context ...er spatial-temporal steps one should apply in order to cover the same volume. A different approach uses the contour location in frame as an initial condition for the 2-D solution in frame , see e.g., =-=[3]-=- and [25]. The above edge indicator is still valid in this case. Note, that the aspect ratios between the time, the image space and the intensity, should be determined according to the application. Th... |

37 | Differential invariant signatures and flows in computer vision: A symmetry group approach,” Geometry Driven Diffusion in Computer Vision edited by Bart ter Haar Romeny - Olver, Sapiro, et al. - 1994 |

33 | On invariant curve evolution and image analysis - Sapiro, Tannenbaum - 1993 |

32 |
L.D.: Multi-resolution algorithms for active contour models
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(Show Context)
Citation Context ...ike force to the scheme. The resulting AOS scheme with the “balloon” then reads where is the weighted area/balloon coefficient. In order to reduce the computational cost, we use a multiscale approach =-=[20]-=-. We construct a Gaussian pyramid of the original image. The algorithm is first applied at the lower resolution. Next, the zero set is embedded at a higher resolution and the distance function is comp... |