## Global Minimum for Active Contour Models: A Minimal Path Approach (1997)

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Citations: | 200 - 64 self |

### BibTeX

@MISC{Cohen97globalminimum,

author = {Laurent D. Cohen and Ron Kimmel},

title = {Global Minimum for Active Contour Models: A Minimal Path Approach},

year = {1997}

}

### Years of Citing Articles

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### Abstract

A new boundary detection approach for shape modeling is presented. It detects the global minimum of an active contour model’s energy between two end points. Initialization is made easier and the curve is not trapped at a local minimum by spurious edges. We modify the “snake” energy by including the internal regularization term in the external potential term. Our method is based on finding a path of minimal length in a Riemannian metric. We then make use of a new efficient numerical method to find this shortest path. It is shown that the proposed energy, though based only on a potential integrated along the curve, imposes a regularization effect like snakes. We explore the relation between the maximum curvature along the resulting contour and the potential generated from the image. The method is capable to close contours, given only one point on the objects’ boundary by using a topology-based saddle search routine. We show examples of our method applied to real aerial and medical images.

### Citations

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Citation Context ...Closed contour: Medical Image In this third example, we want to extract the left ventricle in an MR image of the heart area. The potential is a function of the distance to the closest edge in a Canny =-=[7]-=- edge detection image (see Figure 14). Since it is a closed contour we use the saddle points classification in closing the boundaries of a single object in the heart image (see Figures 15 and 16). Giv... |

3198 | Snakes: active contour models
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Citation Context ...l Image : : : : : : : : : : : : : 29 7 Concluding RemarkssL. Cohen, R. Kimmel, March 26, 1996. 1 1 Introduction An active contour model for boundary integration and features extraction, introduced in =-=[26]-=-, has been considerably used and studied during the last decade. Most of the approaches that were introduced since then try to overcome the main drawbacks of this model: initialization, minimization a... |

1128 | Geodesic active contours
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Citation Context ...btained in the previous section with paths starting at p 0 and p 1 respectively. A natural combination is to use the above method in order to locate the minimal set, and then let the model defined in =-=[9]-=- take over and refine the result. However, we recommend an easier way to compute the path by back propagation. 3.5.2 Back propagation from p 1 In order to determine the minimal path between p 0 and p ... |

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Citation Context ...same looking two initial contours with different control points may lead to completely different solutions. One may argue that using previously mentioned graph search algorithms like the A , Dijkstra =-=[19, 48]-=-, or F as proposed in [22] for road tracking, might be sufficient. These algorithms are indeed efficient, yet suffer from `metrication errors'. The graph based algorithms consider the image as a graph... |

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Citation Context ...n [12] and detailed in Section 4.2, previous local edge detection might be taken into account as data for defining the potential. A geometric approach for deformable models was recently introduced in =-=[8, 37]. A leve-=-l set approach for curve evolution [44, 49] is used to implement a planar curve evolution of the form: @C(s; �� ) @�� = P (C)(C ss + w ~n); (5) where s is the arc-length parameter of the curve... |

480 | On active contour models and balloons
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Citation Context ... initialization is made for the first image, it is possible to use the resulting contour of the first image as initial condition for the second and so on, as proposed in [26]. Using the balloon model =-=[12]-=- allows a less demanding initialization since any initial closed curve inside an object may be used to obtain its complete boundary. It enables in some cases, to obtain a completely automatic initiali... |

451 |
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Citation Context ...same looking two initial contours with different control points may lead to completely different solutions. One may argue that using previously mentioned graph search algorithms like the A , Dijkstra =-=[19, 48]-=-, or F as proposed in [22] for road tracking, might be sufficient. These algorithms are indeed efficient, yet suffer from `metrication errors'. The graph based algorithms consider the image as a graph... |

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Citation Context ...ution scheme that provides at each image pixel an output of the energy along the path of minimal integrated energy joining that pixel to the given start point. We use the Sethian fast marching method =-=[51, 50, 1]-=-. The search for a global minimum is then done efficiently. While this minimum is restricted to connect two given points, we also present a topology--based saddle search that helps in automatically cl... |

394 |
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Citation Context ...ub-pixel estimation of the distance using a parallel algorithm was presented in [30]. It gives a high sub-pixel precision of the distance. This is one possible application of shortest path estimation =-=[28, 54]-=- presented briefly in Section 3.2. Note also that the distance potential selection P may be also considered as the normalized force introduced in [12] for stabilizing the results (i.e. for P = d E we ... |

333 | A Fast Level Set Method for Propagating Interfaces
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Citation Context ...ctive contours without the constant term and fixed end points is the right flow for this case. Observe that it should operate only within the "fat" set, which can be considered as a fixed na=-=rrow band [2]-=-, thereby reducing drastically the computational complexity of this refining. When there are two or more minimal paths, as will be seen in Section 5, the destination point p 1 is a saddle point and ea... |

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298 |
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Citation Context ...each pixel or vertex is considered as a start point and the algorithm should find simultaneously all interesting feature curves. Dynamic programming has also been used for snakes, first by authors of =-=[3]-=-. Although a complete theoretical description of continuous dynamic programming is reviewed, the proposed application to active contours is different from the ones above and our approach. The dynamic ... |

240 | Intelligent scissors for image composition - MORTENSEN, BARRETT - 1995 |

211 | Gradient flows and geometric active contour models
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Citation Context ...v, for the arbitrary parameter v). The curve evolution equation is then reformulated and implemented using the Osher-Sethian [44] numerical algorithm. Similar geometric models were also introduced in =-=[27, 61, 53]-=- and extended to color and texture in [47]. Although our work is related to [9], it is a totally independent approach. Actually, the geodesic active contours may be considered as a natural refinement ... |

208 |
Distance transformations in arbitrary dimensions
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Citation Context ... either when there is no more change in the process (this has to happen in a finite number of iterations) or after a given number of passes. This kind of approach was used to compute distance maps in =-=[5, 17]-=-. It was also used for road detection in [39, 40], using some improvements in the potential definition. The authors also add some constraint on the curvature by taking into account sets of three verti... |

207 |
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Citation Context ... the front and \Deltat is the time step of the scheme, of the above approach we refer to [2]. 3.4.3 Rouy-Tourin Shape from Shading Approach The second approach is based on a shape from shading method =-=[45, 21]-=- and searches for the surface U itself instead of tracking its level sets. In this case the surface may be found according to the following minimization procedure: Given U = 0 at the start point as bo... |

202 | Surface Modelling with Oriented Particle Systems
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Citation Context ...e motivated many other recent works like [9, 10, 61, 27, 57] for 2D and 3D implicit deformable models. Other models that can handle topology changes have also been used for curves in [38] or surfaces =-=[33, 55, 56]-=-. In this paper we present a new approach for finding the global minimum of energy minimizing curves given only one or two end points. Our goal is to help the user in solving the problem in hand by ma... |

182 | Euclidean distance mapping
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(Show Context)
Citation Context ... either when there is no more change in the process (this has to happen in a finite number of iterations) or after a given number of passes. This kind of approach was used to compute distance maps in =-=[5, 17]-=-. It was also used for road detection in [39, 40], using some improvements in the potential definition. The authors also add some constraint on the curvature by taking into account sets of three verti... |

166 | Finite element methods for active contour models and balloons for 2D and 3D images
- Cohen, Cohen
- 1993
(Show Context)
Citation Context ...oundary conditions. These may be free boundaries, as in the original snakes [26], cyclic boundaries by using periodic closed curves [58], or fixed boundaries by giving C(0), C v (0), C(1) and C v (1) =-=[12, 15]-=-. The mechanical properties of the model are controlled by the functions or constants w j . If C is a local minimum of E, it satisfies the associated Euler-Lagrange equation: ( \Gamma(w 1 C v ) v + (w... |

163 | An active testing model for tracking roads in satellite images
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(Show Context)
Citation Context ...ws the middle of the road. This example illustrates the efficiency of our approach compared to classical snakes. We do not claim that this is a road detection algorithm as one can find for example in =-=[22, 25]-=-. For such an application, if the two edges of the road are needed rather than the middle way, our result could be refined using either ribbon snakes [43] or a thick contour model like in [18]. Given ... |

158 |
Tracking deformable objects in the plane using an active contour model
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(Show Context)
Citation Context ... This constant term is thus similar to the pressure force introduced for the balloon model [12]. It is also related to the dilatation transform in mathematical morphology and the grass-fire transform =-=[34]-=-. It was shown that the geometric snakes model performs better than the classical snakes in some cases like topology changes when implemented by the implicit embedding function technique proposed by O... |

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

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139 |
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(Show Context)
Citation Context ...ic initialization. For example, in [12] preprocessing is used to get an initial guess that has to be inside the desired area. The same property can be realized using the geometric model introduced in =-=[8, 36]-=- and recently improved in [9]. In [43], only two end points on the boundary are needed to follow the contour. Although the smoothing effect of the snakes may overcome small defaults in the data, spuri... |

106 |
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Citation Context ...th integration, it also incorporates the regularization of the path like a "snake" model. Qualitatively, the relation between the potential and the smoothness of the result was understood an=-=d used in [22]-=-, long before the age of snakes. Here, we introduce a quantitative bound expressing the connection between the curvature and the generated potential. This bound is useful in many applications. Section... |

75 | Model driven edge detection - Fua, Leclerc - 1990 |

75 | A common framework for curve evolution, segmentation and anisotropic diffusion - Shah - 1996 |

67 |
Randall,Morphing active contours
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Citation Context ...tion equation is then reformulated and implemented using the Osher-Sethian [44] numerical algorithm. Similar geometric models were also introduced in [27, 61, 53] and extended to color and texture in =-=[47]-=-. Although our work is related to [9], it is a totally independent approach. Actually, the geodesic active contours may be considered as a natural refinement procedure to the proposed approach. We not... |

67 |
Recent numerical algorithms for hypersurfaces moving with curvature dependent speed: Hamilton-Jacobi equations and conservation
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(Show Context)
Citation Context ... edge detection might be taken into account as data for defining the potential. A geometric approach for deformable models was recently introduced in [8, 37]. A level set approach for curve evolution =-=[44, 49] is used-=- to implement a planar curve evolution of the form: @C(s; �� ) @�� = P (C)(C ss + w ~n); (5) where s is the arc-length parameter of the curve C in this case. Therefore, C ss j ~n is the curvat... |

58 |
On the optimal detection of curves in noisy pictures
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Citation Context ...ur, by using the surface of minimal action. To find the surface of minimal action, graph search and dynamic programming techniques were often used, considering the image pixels as vertices in a graph =-=[42, 22, 11]-=-. A description of A and F algorithms, applied to road detection, can be found in [22]. The distance image is initialized with value 1 everywhere except at a start point with value zero. At each itera... |

56 |
On shape from shading
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Citation Context ...es L(v; t) construct the level sets of the surface U(x; y) : IR 2 ! IR + defined in (9). The t level set of U is exactly the curve L(:; t). Although a rigorous proof of this statement can be found in =-=[6]-=-, it can be understood simply by the following geometric interpretation. Observe that adding to a path ending at a point of L(:; t) a small segment in the normal direction to L(:; t) and of length 1 ~... |

54 |
Finding shortest paths on surfaces using level sets propagation
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(Show Context)
Citation Context ...oblem to the type of boundary conditions with fixed end points, however, as we will see in Section 5, the proposed approach may also be used for closed contours. Motivated by the ideas put forward in =-=[28, 30]-=- we develop an efficient and consistent method to find the path of minimal cost between two points, using the surface of minimal action [46, 30, 60] and the fact that operating on a given potential (c... |

52 |
Evolutionary fronts for topology-independent shape modeling and recovery
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(Show Context)
Citation Context ...ic initialization. For example, in [12] preprocessing is used to get an initial guess that has to be inside the desired area. The same property can be realized using the geometric model introduced in =-=[8, 36]-=- and recently improved in [9]. In [43], only two end points on the boundary are needed to follow the contour. Although the smoothing effect of the snakes may overcome small defaults in the data, spuri... |

52 | Image segmentation by reaction-diffusion bubbles
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(Show Context)
Citation Context ... with holes, it is possible [8, 36] to model the contour as a level set of a surface, allowing it to change its topology in a natural way. These approaches have motivated many other recent works like =-=[9, 10, 61, 27, 57]-=- for 2D and 3D implicit deformable models. Other models that can handle topology changes have also been used for curves in [38] or surfaces [33, 55, 56]. In this paper we present a new approach for fi... |

48 |
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Citation Context ...rithm (so L. Cohen, R. Kimmel, March 26, 1996. 9 called in [22]) computes the distance with a sequential update of the pixels. It is similar in spirit to the algorithm used in Section 3.4.3 (see also =-=[21]-=-), except that Equation (16) is again consistent. Using the F , the global minimum is reached only after the image is scanned iteratively top to bottom, row by row, left to right followed by right to ... |

40 |
Dynamic Programming and Modern Control Theory
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Citation Context ...nt of the level set L(t) is dt ~ P from (11). So the derivative @U @~n = hrU;~ni = krUk is equal to dt= dt ~ P = ~ P . A rigorous proof of this idea can be found for example in [6]. (see also Bellman =-=[4] for a n-=-ice proof on the orthogonality of the wave fronts and the geodesics). Here, boundary conditions are given in the form of fixing the point C(0) = p 0 , i.e. U(p 0 ; �� ) = 0 for all �� . Author... |

35 | Algorithms for implicit deformable models
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Citation Context ... with holes, it is possible [8, 36] to model the contour as a level set of a surface, allowing it to change its topology in a natural way. These approaches have motivated many other recent works like =-=[9, 10, 61, 27, 57]-=- for 2D and 3D implicit deformable models. Other models that can handle topology changes have also been used for curves in [38] or surfaces [33, 55, 56]. In this paper we present a new approach for fi... |

34 | Medical image segmentation using topologically adaptable surfaces
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(Show Context)
Citation Context ...se approaches have motivated many other recent works like [9, 10, 61, 27, 57] for 2D and 3D implicit deformable models. Other models that can handle topology changes have also been used for curves in =-=[38]-=- or surfaces [33, 55, 56]. In this paper we present a new approach for finding the global minimum of energy minimizing curves given only one or two end points. Our goal is to help the user in solving ... |

32 | Auxiliary Variables and Two-Step Iterative Algorithms
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(Show Context)
Citation Context ...tion [37, 36, 35]. The difficulty here is that there is no order in the set of points and that it is unknown in advance which points belong to the boundary. This is defined as implicit constraints in =-=[13]-=-. Denote by E(x; y) : D ! f0; 1g a binary function representing the result of applying a standard edge detector on the image I, where 1 corresponds to a detected edge point. One possible way of defini... |

32 | ªBrownian Strings: Segmentating Images with Stochastically Deformable Contours,º - Grzeszczuk, Levin - 1997 |

31 | Surface reconstruction using active contour models - Cohen, Bardinet, et al. - 1993 |

30 | Distance maps and weighted distance transforms
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(Show Context)
Citation Context ...oblem to the type of boundary conditions with fixed end points, however, as we will see in Section 5, the proposed approach may also be used for closed contours. Motivated by the ideas put forward in =-=[28, 30]-=- we develop an efficient and consistent method to find the path of minimal cost between two points, using the surface of minimal action [46, 30, 60] and the fact that operating on a given potential (c... |

28 | F.: A Geometric Model for Active Contours. Numerische Mathematik 66 - Caselles, Catte, et al. - 1993 |

27 |
A review of the theory, algorithms, and applications of level set method for propagating surfaces. Acta numerica
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Citation Context ...ution scheme that provides at each image pixel an output of the energy along the path of minimal integrated energy joining that pixel to the given start point. We use the Sethian fast marching method =-=[51, 50, 1]-=-. The search for a global minimum is then done efficiently. While this minimum is restricted to connect two given points, we also present a topology--based saddle search that helps in automatically cl... |

26 |
Shading from shape, the eikonal equation solved by greyweighted distance transform, Pattern Recogn
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(Show Context)
Citation Context ... closed contours. Motivated by the ideas put forward in [28, 30] we develop an efficient and consistent method to find the path of minimal cost between two points, using the surface of minimal action =-=[46, 30, 60]-=- and the fact that operating on a given potential (cost) function helps in finding the solution for our path of minimal action (also known as minimal geodesic, or path of minimal potential). Thereby, ... |

26 | Level Set Methods: Evolving - Sethian - 1996 |

23 | Minimal surfaces: A three dimensional segmen- tation approach
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(Show Context)
Citation Context ... with holes, it is possible [8, 36] to model the contour as a level set of a surface, allowing it to change its topology in a natural way. These approaches have motivated many other recent works like =-=[9, 10, 61, 27, 57]-=- for 2D and 3D implicit deformable models. Other models that can handle topology changes have also been used for curves in [38] or surfaces [33, 55, 56]. In this paper we present a new approach for fi... |

21 |
Estimating shortest paths and minimal distances on digitized three-dimension surfaces
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(Show Context)
Citation Context ... to the true one as the grid is refined. This is known not to be the case in general graph search algorithms that suffer from digitization bias due to the metrication error when implemented on a grid =-=[41, 32]-=-. This gives a clear advantage to our method over minimal path estimation using graph search. Before introducing the proposed method, let us review the graph search based methods that try to minimize ... |