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...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...

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...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...

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...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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...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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...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...

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...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...

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...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 ...

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... 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...

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...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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...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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...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 ...

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... 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...

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... 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...

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...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 ...

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... 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...

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...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...

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...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...

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... 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...

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...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 ...

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...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...

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...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...

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...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...

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...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...

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... 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...

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...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...

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...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...

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...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 ~...

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...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...

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... 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...

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...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 ...

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...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 ...

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...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...

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... 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...

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...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...

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...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...

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... 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, ...

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... 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 ...

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... 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...

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... 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, ...

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... from images. The extraction of local features is specified by initial conditions that lead to the selection of one of the local minima. Snakes are a special case of deformable models as presented in =-=[58]-=-. The deformable contour model is a mapping: C(v) :\Omega \Gamma! IR 2 (1) v 7\Gamma! (x(v); y(v)) where\Omega = [0; 1] is the parameterization interval. In some cases v is chosen to be the arclength ...

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... 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 ...

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...ion, which is not the case in the approach we use. Also, some fixes that minimize the average error by modifying the weights along the connections between the pixels were proposed in [32] and used in =-=[29]-=-. We show the example of Figures 1 and 2 for the simplest case of graph search, to clarify the metrication error effect. Our philosophy here is different. We propose to deal with the continuous proble...

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...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...

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... 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 a start point p 0 on the bottom left, the image of minimal action U(x; y) from this point is shown in Figure 8. Observe the way the level curves propagate faster along the road. In the top exa...

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...mma minfU k\Gamma1;l ; U k+1;l g; 0g) 2 +(maxfu \Gamma minfU k;l\Gamma1 ; U k;l+1 g; 0g) 2 = P 2 k;l ;(17) and let U k;l = u. For efficiency, the trial list is kept as min heap structure. We refer to =-=[51, 50, 1, 31]-=- for further details on the above algorithm, as well as a proof of correct construction. Using a min-heap structure for the trial list, the algorithm computational complexity is O(N logN) where N is t...

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...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...

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...reprocessing 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, spurious edges generated by noise or in a c...

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...optimal no matter how much we refine the (4-neighboring) grid. larger neighborhood as structuring element, giving better approximations of the distance in some directions (like p 2 for the diagonals) =-=[5, 59]-=-. These give a different polygonal approximation of the circle, but there will always be an error in some direction that will be invariant to the grid resolution, which is not the case in the approach...

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...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...

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...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 ...

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...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 renement p...

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... path is introduced. This justify the fact that although our approach is a path integration, it also incorporate the regularization of the path like a "snake" model. 2 Deformable Contours We=-= refer to [10] for an ov-=-erview of the different approaches for active contour models. Since the introduction of "snakes" [1], deformable models have been extensively used to integrate boundaries and extract feature...

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... 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 forsnd...

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...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...

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...e domain and serve as the key points in generating a single boundary contour. Finding such a contour is usually referred to as `shape modeling' that is used for object segmentation and classification =-=[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...

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...ess (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 vertices instead of two in the graph search to update ...

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...ess (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 vertices instead of two in the graph search to update ...

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...ignificant local minimum of the energy. The inflation or expansion force [12] may help the contour to avoid being trapped by isolated edges into a local minimum. A region based approach introduced in =-=[14]-=- also makes the solution less sensitive to local minima and initialization. It considers a mixed energy including a snake like term on the boundary and an homogeneous value constraint inside the regio...

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...y the size of the graph, since the vertices are the nodes on the curve and the possible values for these are only the eight neighbors L. Cohen, R. Kimmel, March 26, 1996. 10 of the initial vertex. In =-=[24]-=-, the range of possible local deformations is broadened using a multiscale dynamic programming algorithm. However, in both approaches, this kind of graph search does not avoid undesirable local minima...