Abstract. Motivated by Tadmor, Nezzar, and Vese‘s work [ 30 ] dedicated to multiscale image representation using hierarchical [Formula: see text] decompositions, we propose transposing their approach to the case of registration, a task which consists in determining a smooth deformation aligning the salient constituents visible in an image into their counterpart in another. The underlying goal is to obtain a hierarchical decomposition of the deformation in the form of a composition of intermediate deformations: the coarser one, computed from versions of the two images capturing the essential features, encodes the main structural/geometrical deformation, while iterating the procedure and refining the versions of the two images yields more accurate deformations that faithfully map small-scale features. The proposed model falls within the framework of variational methods and hyperelasticity by viewing the shapes to be matched as Ogden materials. The material behavior is described by means of a specifically tailored strain energy density function, complemented by [Formula: see text]-penalizations ensuring that the computed deformation is a bi-Lipschitz homeomorphism. Theoretical results emphasizing the mathematical soundness of the model are provided, among which the existence of minimizers/asymptotic results, and a suitable numerical algorithm is supplied, along with numerical simulations demonstrating the ability of the model to produce accurate hierarchical representations of deformations. A very preliminary version of this work has been accepted for publication in the Eighth International Conference on Scale Space and Variational Methods in Computer Vision [Springer, Cham, Switzerland, 2021] but it does not include all the theoretical results, nor the detailed related proofs. A more complete and detailed analysis of the numerical experiments is also provided. The theoretical analysis of the numerical algorithm (introduced in section 3 and which is a result in itself) will be the subject of a separate article in preparation.

In this work, the main goal is to study the evolution of the coastline at a local scale, and to get a better understanding of the erosion process. We propose a methodology based on a new software (PyMERI) in order to build a 3D mesh of the subsoil from a Digital Elevation Model (DEM) on which a Partial Differential Equation-direct problem-allowing to recover the voltage (electrical field) in the ground is solved yielding numerical simulations of Electrical Resistivity Imaging (ERI). This study is complemented by the resolution of an inverse problem of ERI for which, unlike the direct problem, the conductivity is now the searched quantity, and by the introduction of a segmentation approach to segment faults on images.

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 Dmsplines. Possible applications include estimation of wind energy potential and prediction of wind power or marine current simulation.

In medical image analysis, constructing an atlas, i.e., a mean representative of an ensemble of images, is a critical task for practitioners to estimate variability of shapes inside a population, and to characterize and understand how structural shape changes have an impact on health. This involves identifying significant shape constituents of a set of images, a process called segmentation, and mapping this group of images to an unknown mean image, a task called registration, making a statistical analysis of the image population possible. To achieve this goal, we propose treating these operations jointly to leverage their positive mutual influence, in a hyperelasticity setting, by viewing the shapes to be matched as Ogden materials. The approach is complemented by novel hard constraints on the $L^\infty$ norm of both the Jacobian and its inverse, ensuring that the deformation is a bi-Lipschitz homeomorphism. Segmentation is based on the Potts model, which allows for a partition into more than two regions, i.e., more than one shape. The connection to the registration problem is ensured by the dissimilarity measure that aims to align the segmented shapes. A representation of the deformation field in a linear space equipped with a scalar product is then computed in order to perform a geometry-driven Principal Component Analysis (PCA) and to extract the main modes of variations inside the image population. Theoretical results emphasizing the mathematical soundness of the model are provided, among which are existence of minimizers, analysis of a numerical method, asymptotic results, and a PCA analysis, as well as numerical simulations demonstrating the ability of the model to produce an atlas exhibiting sharp edges, high contrast, and a consistent shape.

Segmentation and registration are cornerstone steps of many imaging situations: while segmentation aims to identify relevant constituents of an image for visualization or quantitative analysis, registration consists of mapping salient features of an image onto the corresponding ones in another. Instead of treating these tasks linearly one after another, so without correlating them, we propose a unified variational model, in a hyperelasticity setting, processing these two operations simultaneously. The dissimilarity measure relates local and global (or region-based) information, since it relies on weighted total variation and nonlocal shape descriptors inspired by the piecewise constant Mumford--Shah model. Theoretical results emphasizing the mathematical and practical soundness of the model are provided, including existence of minimizers, connection with the segmentation step, nonlocal characterization of weighted seminorms, asymptotic results, and $\Gamma$-convergence properties. A preliminary version of this work appeared in [N. Debroux and C. Le Guyader, “A unified hyperelastic joint segmentation/registration model based on weighted total variation and nonlocal shape descriptors,” in Sixth International Conference on Scale Space and Variational Methods in Computer Vision, F. Lauze, Y. Dong, and A. B. Dahl, eds., Springer International, Cham, 2017, pp. 614--625], but it contains neither proofs of the proposed material nor details on the numerical treatment (developed nonlocal algorithm and extensive comparisons with related works).

This paper falls within the scope of joint segmentation-registration using nonlinear elasticity principles. Because Saint Venant--Kirchhoff materials are the simplest hyperelastic materials (hyperelasticity being a suitable framework when dealing with large and nonlinear deformations), we propose viewing the shapes to be matched as such materials. Then we 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. Adding a weighted total variation--based criterion enables us to align the edges of the objects even when the modalities are different. We derive a relaxed problem associated to the 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 $\Gamma$-convergence. Applications are illustrated in academic and biological images.

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.

In this paper, we address the issue of designing a theoretically well-motivated registration model capable of handling large deformations. Motivated by the fine properties of the Saint Venant--Kirchhoff material, we propose a variational model combining a measure of dissimilarity and a regularizer based on the stored energy of such a material. We prove the existence of generalized solutions of this problem. We then describe and analyze a numerical method of resolution based on the introduction of an associated decoupled problem in which an auxiliary variable simulates the Jacobian matrix of the deformation field. A theoretical result is established, and we investigate the efficiency of the proposed matching model for the registration of mouse brain gene expression data to a neuroanatomical mouse atlas.

The problem of vector field approximation emerges in a wide range of fields such as: motion control, computer vision, geometrical analysis, geometrical design, analysis of acoustic and/or electromagnetic waves, as well as in geophysics, medical imaging, fluid mechanics and so on... Many different approaches have been introduced to solve each specific problem occuring in the above fields of investigation. One can mention for instance: finite element approximation, PDE-based methods, spline and Rational Basis Function approximations.. The originality of this work consists: - in considering that the vector field derives from a potential (conservative vector field): it occurs for instance in meteorology (winds derive from temperature potentials), oceanography (currents derive from pression potentials), image processing... For in land wind velocity field, we also take into account the topography effects. - of a rigorous study of existence-uniqueness of the solution of the problem phrased as an energy minimization, - in establishing a convergence result (while many approaches only give algorithms without mathematical study) and providing an approximation error estimate. - in using a specific visualization tool.

In this paper, we investigate a new method to enforce topology preservation on deformation fields. The method is composed of two steps. The first one consists in correcting the gradient vector fields 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 Karaçali and Davatzikos, proposes a new approach based on interval analysis. The second one aims to reconstruct the deformation, given its full set of discrete gradient vectors. The problem is phrased as a functional minimization problem on the convex subset K of the Hilbert space V. The 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 . Experimental results demonstrate the efficiency of the method.

The implicit framework of the level-set method has several advantages when tracking propagating fronts. Indeed, the evolving contour is embedded in a higher dimensional level-set function and its evolution can be phrased in terms of a Eulerian formulation. The ability of this intrinsic method to handle topological changes (merging and breaking) makes it useful in a wide range of applications (fluid mechanics, computer vision) and particularly in image segmentation, the main subject of this paper. Nevertheless, in some applications, this topological flexibility turns out to be undesirable: for instance, when the shape to be detected has a known topology, or when the resulting shape must be homeomorphic to the initial one. The necessity of designing topology-preserving processes arises in medical imaging, for example, in the human cortex reconstruction. It is known that the human cortex has a spherical topology so throughout the reconstruction process this topological feature must be preserved. Therefore, we propose in this paper a segmentation model based on an implicit level-set formulation and on the geodesic active contours, in which a topological constraint is enforced.