Laurent D. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. International Conference on Computer Vision, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... the problem of scalar image regularization (particularly within the function framework) We can cite for instance reference papers from Alvarez [3] Charbonnier Aubert [16] Chambolle Lions [10] This article was published in IJCV Special Issue VLSM, 2001 Strong Chan [56, 57] Cohen [18], Cottet Germain [19] Kornprobst Deriche [28, 29, 30] Malladi Sethian [33] Mumford Shah [35, 53] Morel [34] Nordstr om [38] Osher Rudin [47] Proesman [45] Sapiro [8, 51] Weickert [68, 67, 68] and You [71] More recently and thanks to the increase of computer performances, the ....

L. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. International Conference on Computer Vision, 1995.

....the restrictions imposed by linear ltering techniques. Since then, many authors have proposed and studied well posedness PDE s that tackle the problem of scalar image regularization. We can cite for instance papers from Alvarez [2, 1] Aubert [10] Chambolle Lions [7] Chan [5] Cohen [13], Cottet Germain [14] Hamza Krim [18] Kornprobst Deriche [22, 23, 24] Malladi Sethian [27] Mumford Shah [29, 46] Morel [28] Nordstr om [30] Osher Rudin [39] Perona Malik [34] Polyak [35] Proesman [38] Sapiro [6, 43, 44, 45] We This article was published in IEEE SPM ....

L. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. Journal of Mathematical Imaging and Vision, vol.6, p.59-83, 1996.

....like the rst ICP criterion (Eq. 1) has no closed form solution. We construct in this section an auxiliary criterion, depending explicitly on the matching matrix, and we use an alternated optimisation scheme. This construction follows EM principles [4, 12] and the auxiliary variables framework of [5]. 2.3.1 Designing the auxiliary criterion For any matching matrix A, we can write the criterion using Bayes rule: C(T ) Gamma log p(A; SjM;T) log p(AjS; M;T ) where p(AjS; M;T ) is the a posteriori likelihood of matches. Since this is valid for any matching matrix A, it is still for the ....

L. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. J. of Mathematical Imaging and Vision (JMIV), (6):5983, 1996.

.... framework, based on functional minimizations via diffusion PDE s evolutions has proved its efficiency for scalar data regularization (in particular, within the functions theory) We can cite for instance, Alvarez et al. 1, 2] Aubert et al. 10] Chambolle Lions [7] Chan [5] Cohen [11], Kornprobst Deriche [15, 16, 17] Malladi Sethian [18] Mumford Shah [20, 31] Morel [19] Nordstrom [21] Osher Rudin [26] Perona Malik [23] Proesman et al. 25] Sapiro [6, 28, 29, 30] Weickert [38, 39] You [41] More recently, vector field regularization with vector ....

Laurent D. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. International Conference on Computer Vision, 1995.

....is not guaranteed. In practice, a small number of iterations (typically less than 10) is enough to achieve stationary estimates for centroids. The two step optimization algorithm is an example of a larger class of optimization methods based on a set of auxiliary variables studied by Cohen in [14]. The two step 160 British Machine Vision Conference Figure 3: Dynamic clustering of synthetic data with a global motion model approach is also similar to the EM method proposed by Dempster et al. in the context of statistic inference based on Fisher ML principle [15] 7Results To illustrate ....

L. Cohen, Auxiliary Variables and Two-Step Iterative Algorithms in Computer Vision Problems, Journal of Mathematical Imaging and Vision, 58-83, 1996.

....removal, image enhancement and image restoration in real images. These methods are based on evolving nonlinear partial differential equations (PDE s) e.g. Perona Malik [9] Nordstrom [8] Shah [14] Osher Rudin [11, 12] Proesman et al. 10] Cottet and Germain, Alvarez et al. [1, 2] Cohen [4], Weickert, Malladi Sethian [7] Aubert et al. 3] You et al. 15] Sapiro et al. 13] Hence, due this large number of approaches, it clearly appears that there is a strong need to better compare and quantify the experimental results of the most promising techniques. In this article, ....

L. D. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. ICCV, 1995.

....superquadric model to the data. Both of these methods are slow since only a small time step can be taken in each iteration in order to get a stable solution. A relatively faster and more stable integration scheme based on the Runge Kutta method was developed by DeCarlo and Metaxas [3] In [15], Cohen introduced a two step iterative scheme based on auxiliary variables. This scheme is very general and can be applied to most optimization problems. The auxiliary variables may be interpreted as representing intermediary reconstruction steps. Minimization of several non convex potentials ....

L.D. Cohen, \Auxiliary variables and two-step iterative algorithms in computer vision problems," Journal of Mathematical Imaging and Vision, vol. 6, pp. 59-83, 1996.

....addressed. In a related context, namely rigid registration of 3D points without given correspondence, iterative approaches have been reported [1, 15] that may be adopted to solve our task. However, since we also wish to apply robust estimators, we address the question of (local) convergence. Cohen [3] presented a general framework for a broad class of vision problems that has been solved by two step methods. The key idea is to introduce auxiliary variables to guarantee at least local convergence. In the present paper, we adopt this framework to the problem of robustly estimating planar ....

....be improved by using robust estimators. But convergence is only assured if the distance measure used in the local step is the same as that induced by the norm used in the global estimation step, since then the same energy term is minimized in both steps. Therefore, convergence is not guaranteed [3] if a robust technique is used in one step. In the next section, we address this problem and apply the general framework developed in [3] to our stereoscopic setting. To enable the quantative evaluation of point matches in a variety of ways, a potential P (to be specified later on) is introduced ....

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L. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. JMIV, 6:59--83, 1996.

....level discontinuities during the enhancement restoration process. These methods, which have been proved to be very eOEcient, are based on evolving nonlinear partial dioeerential equations (PDE s) See the work of Alvarez et al. [4] Aubert et al. 8] Chambolle Lions [21] Chan [14, 67] Cohen [23], Cottet Germain [24] Kornprobst Deriche [44, 43, 42] Malladi Sethian [46] Mumford Shah [65, 53] Morel [3, 51] Nordstr#m [54] Osher Rudin [60] Perona Malik [58] Proesman et al. 59] Sapiro et al. 20, 61, 62, 12, 63] Weickert [71, 72] You et al. 74] This ....

Laurent D. Cohen. Auxiliary variables and twostep iterative algorithms in computer vision problems. ICCV, 1995.

....models are active contours of Kass et al. also known as snakes [27] and balloons, the extension of snakes to three dimensions [15] The main drawback of the snake model is that it may converge towards some local minima in the image. The snake model has been improved to make it more robust [14], but its application to real medical images is still limited because of their high complexity. In Section 4, we provide an example where snakes are used to determine the input data for 2 D registration. Other examples of the application of the snake model for obtaining point correspondences have ....

L.D. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. Journal of Mathematical Imaging and Vision, 6:59--83, 1996.

....continuous functions ae( Delta) in (3) 4.2. Decoupled auxiliary variables A different usage of auxiliary variables was proposed by Geman and Yang under the notion half quadratic regularization [15] Cohen extended this approach to a class of two step algorithms for computer vision problems [16]. For these different auxiliary variables the functional J(v; w) 1 2 Z Omega (v Gamma g) 2 ffjrv Gamma wj 2 (w) dx (12) is introduced. Under certain conditions (see [16] one can find a function ( Delta) for which J is convex in w and J(v) min w J(v; w) holds true. ....

....[15] Cohen extended this approach to a class of two step algorithms for computer vision problems [16] For these different auxiliary variables the functional J(v; w) 1 2 Z Omega (v Gamma g) 2 ffjrv Gamma wj 2 (w) dx (12) is introduced. Under certain conditions (see [16]) one can find a function ( Delta) for which J is convex in w and J(v) min w J(v; w) holds true. Thus, the two step minimization w n = arg min w J(u n ; w) 13) u n 1 = arg min u J(u; w n ) 14) converges. Furthermore, with the auxiliary variables w n = 1 Gamma 1 ....

L.D. Cohen. Auxiliary variables and two--step iterative algorithms in computer vision problems. J. of Math. Imag. Vision, 6(1):59--83, 1996.

....proposed by Kass et al. 13] as a general energy minimizing model which can be applied to numerous problems in computer vision (edge detection, tracking of moving objects, etc. Since the time of introduction of snakes, several improvements have been made in the model (see, for example, 8] [7]) In this paper, we use the original model from [13] Snakes are parametrically defined curves v(s) x(s) y(s) The snake model minimizes the following energy: E snake = Z Omega fff(s)kv 0 (s)k 2 fi(s)kv 00 (s)k 2 P (v(s) g ds (1) consisting of the internal energy of the curve ....

L.D. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. Journal of Mathematical Imaging and Vision, 6:59--83, 1996.

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Cohen, L. D., Auxiliary variables and two-step iterative algorithms in computer vision problems, in Journal of Mathematical Imaging and Vision 6(1) (1996), 61--86. See also Proc. IEEE ICCV'95, Boston.

....error kX n 1 Gamma X n k P 0 is dened as a uniformly spaced parallelepiped box of control points and X 0 = BP 0 represents the set of points of the initial discretized superellipsoid. This algorithm is similar to the formulation of the B splines snakes with auxiliary variables, as described in [18]. Size of the FFD Computation time Table 1: Typical computation times (in seconds) with an increasing number of control points of the FFD for 20 iterations. In practice, we use boxes of size 5 Theta 5 Theta 5 for data composed of about 6,000 points, and the number of iterations is between ....

L.D. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. Technical report, Ceremade, F#vrier 1995. Cahiers de Math#matiques de la D#cision 9511, to appear in Journal of Mathematical Imaging and Vision and Proceedings ICCV'95, Boston.

....and the superellipsoid. Bottom right: nal model after minimization of the displacement eld (see [11] for details) represents the set of points of the initial discretized superellipsoid. This algorithm is similar to the formulation of the B splines snakes with auxiliary variables, as described in [18]. Size of the FFD Computation time Table 1: Typical computation times (in seconds) with an increasing number of control points of the FFD for 20 iterations. Figure 2: Computation time vs. number of control points (number of columns of the matrix B) In practice, we use boxes of size 5 ....

L.D. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. Technical report, Ceremade, F#vrier 1995. Cahiers de Math#matiques de la D#cision 9511, to appear in Journal of Mathematical Imaging and Vision and Proceedings ICCV'95, Boston.

....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(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 defining a potential P : D IR is as a function of the distance map [15] where each point p is ....

....becomes jj sup d f 0 (d) w d : 25) L. Cohen, R. Kimmel, March 26, 1996. 19 where d ranges from 0 to the maximal distance in the image. The bound in (25) can be easily found for the functions f(d) ffd 2 or f(d) 1 Gamma e Gammaffd 2 which corresponds to robust statistics (see [13]) A synthetic example is presented in Figure 5 where the potential used is obtained from a distance map to the edge points. Observe the way the level curves propagate faster along the line. Figure 5: Line image. From left to right: original, potential, minimal action (random look up table to show ....

Laurent D. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. Journal of Mathematical Imaging and Vision, 6(1):61--86, January 1996. See also ICCV'95.

....when the solution v(t) stabilizes. 2.2 Attraction Potential As introduced and used in [3, 10, 4] the potential is derived from a set S of already extracted edge points. The difficulty is then to solve the problems of segmentation and reconstruction from this unstructured set of points (see [4, 11]) The potential is a function of the distance d (computed from [12, 13] to the closest edge point. For example, a gaussian function models weak strings that break when too long : P (v) Gammae Gammaoed(v) 2 Remark that for the potential defined by P (v) g(d(v) the force becomes F (v) ....

....to the algorithm used in [17] 3 Convexity of the discrete energy The convexity of the energy was also studied in [18] but with a different kind of potential to detect thick curves. A different approach to make convex our minimization problem using auxiliary variables is introduced in [11] giving interpretation of many two step algorithms. We now present our approach which is explained in more details in [19] 3.1 Discretization of the Active contour We first formulate the discretization of the equation by finite differences. Representing the curve by a set of N nodes v i = x i ....

L. D. Cohen. Auxiliary variables and two-step iterative algoritms in computer vision problems. Technical report, Ceremade, F'evrier 1995. MD 9511, Proc. ICCV'95.

No context found.

Laurent D. Cohen. Auxiliary variables and two-step iterative algorithms in computer vision problems. International Conference on Computer Vision, 1995.

No context found.

L. Cohen, "Auxiliary variables and two-steps iterative algorithms in computer vision problems," Journal Mathematical Imaging and Vision, 6, 1996, 59--83.

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Laurent D. Cohen. Auxiliary variables and twostep iterative algorithms in computer vision problems. ICCV, 1995.

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L. Cohen, "Auxiliary variables and two-step iterative algorithms in computer vision problems," Int. J. Comput. Vis., vol. 6, no. 1, pp. 59--83, 1996.

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