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In the seminal paper The Gradient Theory of Phase Transitions and the Minimal Interface Criterion, 1987, L. Modica ( [13]) proves some conjectures related to the Van der Waals-Cahn-Hilliard theory of phase transitions.This theory intends to overcome the issue of lack of uniqueness of the solution of the initially considered minimization problem —which aims to minimize the total energy of a fluid confined to a bounded container Ω ⊂ℝn and with Gibbs free energy per unit volume, a prescribed function W of the density distribution u —by enforcing that the interface has minimal surface. In that purpose, a family of functionals parameterized by e > 0 and including a dependency on the density gradient modelling this interfacial energy is introduced, and the asymptotic behavior as e→0+ of the solutions u e of the related minimization problem is analyzed through a Γ-convergence result. Motivated by this work, this paper addresses the question of extending this result to the weighted case. It is shown that this new model inherits the fine properties of the original unweighted one with in particular, the validity of the Γ-convergence result, useful for the minimization of weighted perimeter, relevant for image segmentation.

We study a spline-based approximation of vector fields in the conservative case. Its motivation comes from for instance when approximating current or wind velocity fields, the data deriving in those cases from a potential (pressure for the wind, etc‥). To model the problem, we introduce a minimization problem on a Hilbert space for which the existence and uniqueness of the solution are provided. A convergence result in Sobolev metric spaces is established using norm equivalences and compactness arguments, as well as an approximation error estimate of the involved smoothing $D^{m}$ splines. Possible applications include estimation of wind energy potential and prediction of wind power or marine current simulation.

International audience; We study a spline-based approximation of vector fields in the conservative case (the gradient vector field derives from a potential function). We introduce a minimization problem on a Hilbert space for which the existence and uniqueness of the solution is given. We apply this approach to a registration process in image processing.

In this paper, we present a new non-parametric combined segmentation and registration method. The shapes to be registered are implicitly modeled with level set functions and the problem is cast as an optimization one, combining a matching criterion based on the active contours without edges for segmentation (Chan and Vese, 2001) [8] and a nonlinear-elasticity-based smoother on the displacement vector field. This modeling is twofold: first, registration is jointly performed with segmentation since guided by the segmentation process; it means that the algorithm produces both a smooth mapping between the two shapes and the segmentation of the object contained in the reference image. Secondly, the use of a nonlinear-elasticity-type regularizer allows large deformations to occur, which makes the model comparable in this point with the viscous fluid registration method. In the theoretical minimization problem we introduce, the shapes to be matched are viewed as Ciarlet-Geymonat materials. We prove the existence of minimizers of the introduced functional and derive an approximated problem based on the Saint Venant-Kirchhoff stored energy for the numerical implementation and solved by an augmented Lagrangian technique. Several applications are proposed to demonstrate the potential of this method to both segmentation of one single image and to registration between two images.

International audience; In this paper, we propose a segmentation method based on the generalized fast marching method (GFMM) developed by Carlini et al. (submitted). The classical fast marching method (FMM) is a very efficient method for front evolution problems with normal velocity (see also Epstein and Gage, The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modelling and Computation, 1997) of constant sign. The GFMM is an extension of the FMM and removes this sign constraint by authorizing time-dependent velocity with no restriction on the sign. In our modelling, the velocity is borrowed from the Chan-Vese model for segmentation (Chan and Vese, IEEE Trans Image Process 10(2):266-277, 2001). The algorithm is presented and analyzed and some numerical experiments are given, showing in particular that the constraints in the initialization stage can be weakened and that the GFMM offers a powerful and computationally efficient algorithm.

In this paper, we address the issue of designing a theoretically well-motivated joint segmentation-registration method capable of handling large deformations. The shapes to be matched are implicitly modeled by level set functions and are evolved in order to minimize a functional containing both a nonlinear-elasticity-based regularizer and a criterion that forces the evolving shape to match intermediate topology-preserving segmentation results. Theoretical results encompassing existence of minimizers, \(\varGamma \)-convergence result and existence of a weak viscosity solution of the related evolution problem are provided.

In this paper, we address the issue of designing a theoretically well-motivated joint segmentation-registration method capable of handling large deformations. Motivated by the fine properties of Saint Venant-Kirchhoff materials, we propose to view the shapes to be matched as such materials and introduce a variational model combining a measure of dissimilarity based on weighted total variation and a regularizer based on the stored energy function of a Saint Venant-Kirchhoff material. We derive a relaxed problem associated to this initial one for which we are able to provide a result of existence of minimizers. A description and analysis of a numerical method of resolution based on a decoupling principle is then provided including a theoretical result of Γ-convergence. Applications on biological images are provided.

Let I :Ω→ℜ be a given bounded image function, where Ω is an open and bounded domain which belongs to ℜn. Let us consider n=2 for the purpose of illustration. Also, let S={xi}i∈Ω be a finite set of given points. We would like to find a contour Γ⊂Ω, such that Γ is an object boundary interpolating the points from S. We combine the ideas of the geodesic active contour (cf. Caselles et al. [7,8]) and of interpolation of points (cf. Zhao et al. [40]) in a level set approach developed by Osher and Sethian [33]. We present modelling of the proposed method, both theoretical results (viscosity solution) and numerical results are given.

In this paper, we propose a new 2D segmentation model including geometric constraints, namely interpolation conditions, to detect objects in a given image. We propose to apply the deformable models to an explicit function using the level set approach (Osher and Sethian [24]); so, we avoid the classical problem of parameterization of both segmentation representation and interpolation conditions. Furthermore, we allow this representation to have topological changes. A problem of energy minimization on a closed subspace of a Hilbert space is defined and introducing Lagrange multipliers enables us to formulate the corresponding variational problem with interpolation conditions. Thus the explicit function evolves, while minimizing the energy and it stops evolving when the desired outlines of the object to detect are reached. The stopping term, as in the classical deformable models, is related to the gradient of the image. Numerical results are given.

In this article, we propose a new segmentation model including geometric constraints, namely interpolation conditions, to detect objects in a given image sequence. We propose to apply the deformable models to an explicit function to avoid the problem of parameterization (see Gout, C. and Vieira-Teste, S. (2003). An algorithm for segmentation under interpolation conditions using deformable models. Int. J. Comput. Math., 80(1), 47–54.). A problem of energy minimization on a closed subspace of a Hilbert space is defined, and introducing Lagrange multipliers enables us to formulate the corresponding variational problem with interpolation conditions. We apply this method in order to ouline the cross-sectional area (CSA) of a great thoracic vessel, namely the main pulmonary artery, in order to non-invasively assess pulmonary arterial hypertension (see Laffon, E., Vallet, C., Bernard, V., Montaudon, M., Ducassou, D., Laurent, F. and Marthan, R. (2003). A computed method for non-invasive MRI assessment of pulmona...

In this paper, we address the issue of designing a theoretically well-motivated registration model capable of handling large deformations and including geometrical constraints, namely landmark points to be matched, in a variational framework. The theory of linear elasticity being unsuitable in this case, since assuming small strains and the validity of Hooke’s law, the introduced functional is based on nonlinear elasticity principles. More precisely, the shapes to be matched are viewed as Ciarlet-Geymonat materials. We demonstrate the existence of minimizers of the related functional minimization problem and prove a convergence result when the number of geometric constraints increases. We then describe and analyze a numerical method of resolution based on the introduction of an associated decoupled problem under inequality constraint in which an auxiliary variable simulates the Jacobian matrix of the deformation field. A theoretical result of 􀀀-convergence is established. We then provide preliminary 2D results of the proposed matching model for the registration of mouse brain gene expression data to a neuroanatomical mouse atlas.

This paper proposes a numerical algorithm for image registration using energy minimization and nonlinear elasticity regularization. Application to the registration of gene expression data to a neuroanatomical mouse atlas in two dimensions is shown. We apply a nonlinear elasticity regularization to allow larger and smoother deformations, and further enforce optimality constraints on the landmark points distance for better feature matching. To overcome the difficulty of minimizing the nonlinear elasticity functional due to the nonlinearity in the derivatives of the displacement vector field, we introduce a matrix variable to approximate the Jacobian matrix and solve for the simplified Euler-Lagrange equations. By comparison with image registration using linear regularization, experimental results show that the proposed nonlinear elasticity model also needs fewer numerical corrections such as regridding steps for binary image registration, it renders better ground truth, and produces larger mutual information; most importantly, the landmark points distance and L 2 dissimilarity measure between the gene expression data and corresponding mouse atlas are smaller compared with the registration model with biharmonic regularization.

In this work, we do not want to compute a potential that could generate the vector field data. We only want to get a global approximation of the vector field dataset on a bounded domain, taking into account in the modeling that this approximation derives from a potential. Furthermore, contrary to interpolation methods, we prefer to fit the vector field dataset in the case of realistic data (when the number of vectors is large or when the data are corrupted by noise). To achieve this, we introduce a minimization problem defined as a regularized least-square problem formulated on a Sobolev space of potentials. Obviously, this problem has an infinite number of solutions, but we derive from it a problem expressed in terms of the gradient vectors. We prove that the associated problem in terms of vectors has a unique solution which is the corresponding approximation of the vector field dataset. Then, we give a convergence result when the number of vectors increases to infinity. We also give the discretization complemented by an approximation error estimate of the involved smoothing splines. We then focus on numerical examples (real data sets from METEO FRANCE).

cited By (since 1996)0; International audience; In this paper, we investigate a new method to enforce topology preservation on two/three-dimensional deformation fields for non-parametric registration problems involving large-magnitude deformations. The method is composed of two steps. The first one consists in correcting the gradient vector field of the deformation at the discrete level, in order to fulfill a set of conditions ensuring topology preservation in the continuous domain after bilinear interpolation. This part, although related to prior works by Karacali and Davatzikos (Estimating Topology Preserving and Smooth Displacement Fields, B. Karacali and C. Davatzikos, IEEE Transactions on Medical Imaging, vol. 23(7), 2004), proposes a new approach based on interval analysis and provides, unlike their method, uniqueness of the correction parameter α at each node of the grid, which is more consistent with the continuous setting. The second one aims to reconstruct the deformation, given its full set of discrete gradient vector field. The problem is phrased as a functional minimization problem on a convex subset K of an Hilbert space V. Existence and uniqueness of the solution of the problem are established, and the use of Lagrange's multipliers allows to obtain the variational formulation of the problem on the Hilbert space V. The discretization of the problem by the finite element method does not require the use of numerical schemes to approximate the partial derivatives of the deformation components and leads to solve two/three uncoupled sparse linear subsystems. Experimental results in brain mapping and comparisons with existing methods demonstrate the efficiency and the competitiveness of the method. © 2012 SPIE.

We propose a unified variational approach for registration of gene expression data to neuroanatomical mouse atlas in two dimensions. The proposed energy (minimized in the unknown displacement u) is composed of three terms: a standard data fidelity term based on L 2 similarity measure, a regularizing term based on nonlinear elasticity (allowing larger smooth deformations), and a geometric penalty constraint for landmark matching. We overcome the difficulty of minimizing the nonlinear elasticity functional by introducing an auxiliary variable v that approximates ∇u, the Jacobian of the unknown displacement u. We therefore minimize now the functional with respect to the unknowns u (a vector-valued function of two dimensions) and v (a two-by-two matrix-valued function). An additional quadratic term is added, to insure good agreement between v and ∇u. In this way, the nonlinearity in the derivatives of the unknown u no longer exists in the obtained Euler-Lagrange equations, producing simpler implementations. Several satisfactory experimental results show that gene expression data are mapped to a mouse atlas with good landmark matching and smooth deformation. We also present comparisons with the biharmonic regularization. An advantage of the proposed nonlinear elasticity model is that usually no numerical correction such as regridding is necessary to keep the deformation smooth, while unifying the data fidelity term, regularization term, and landmark constraints in a single minimization approach.

We propose a new nonlinear image registration model which is based on nonlinear elastic regularization and unbiased registration. The nonlinear elastic and the unbiased regularization terms are simplified using the change of variables by introducing an unknown that approximates the Jacobian matrix of the displacement field. This reduces the minimization to involve linear differential equations. In contrast to recently proposed unbiased fluid registration method, the new model is written in a unified variational form and is minimized using gradient descent. As a result, the new unbiased nonlinear elasticity model is computationally more efficient and easier to implement than the unbiased fluid registration. The unbiased large-deformation nonlinear elasticity method was tested using volumetric serial magnetic resonance images and shown to have some advantages for medical imaging applications.