Geological systems can be represented as unstructured tetrahedral grids where faults are topological discontinuities in the volume and each fault block is a connected component. These grids are built from interpretations of seismic and borehole data. Obtaining additional data or new interpretations can point out errors in the tetrahedralised volume. When the location of a fault is slightly off, a new tetrahedralised volume has to be created from scratch. Directly editing the volume would be more effective and comfortable. A free-form deformation technique was developed to implement this interaction. The deformation is confined to a zone around the fault to be edited. Volume deformation is achieved by dragging control points; the geometry of the volume in the vicinity of the discontinuity is updated in realtime. A constraint maintains model consistency throughout editing by preventing tetrahedrons from flipping over. A smoothness constraint distributes the displacement in the whole active region.

This article introduces new techniques for non-distorted texture mapping on complex triangulated meshes.Texture coordinates are assigned to the vertices of the triangulation by using an iterative optimization algorithm, honoring a set of constraints minimizing the distortions.As compared to other global optimization techniques, our method allows the user to specify the surface zones where distortions should be minimized in order of preference.The modular approach described in this paper results in a highly flexible method, facilitating a customized mapping construction.For instance, it is easy to align the texture on the surface with a set of user defined isoparametric curves.Moreover, the mapping can be made continuous through cuts, allowing to parametrize in one go complex cut surfaces.It is easy to specify other constraints to be honored by the so-constructed mappings, as soon as they can be expressed by linear (or linearizable) relations.This method has been integrated successfully within a widely used C.A.D. software dedicated to geosciences.In this context, applications of the method comprise numerical computations of physical properties stored in fine grids within texture space, unfolding geological layers and generating grids that are suitable for finite element analysis.The impact of the method could be also important for 3D paint systems.

The problem of reconstructing a 3D image of an object from a few number of X-ray projections is highly underdetermined. We propose a flexible method based on the regularization of the inverse linear problem with a general quadratic criterion. The minimization is performed by an iterative algorithm with a Gauss-Seidel behaviour. Due to the discrete smooth interpolation formulation, additional linear constraints are inserted, and the method is ensured to converge to the unique minimum. The application of this method is shown for 3D reconstruction of cerebral blood vessels from six projections, and the effect of various criteria is compared to the result of other algebraic methods.

Interpolation of a function ƒ (.) known at some data points of R P is a common problem. Many computer applications (e.g., automatic contouring) need to perform interpolation only at the nodes of a given grid. Whereas most classical methods solve the problem by finding a function defined everywhere, the proposed method avoids explicitly computing such a function and instead produces values only at the grid points. For two-dimensional regular grids, a special case of this method is identical to the Briggs method (see “Machine Contouring Using Minimum Curvature,” Geophysics 17 , 1 (1974)), while another special case is equivalent to a discrete version of thin plate splines (see J. Duchon, Fonctions Splines du type Plaque Mince en Dimention 2 , Séminaire d'analyse numérique, n 231, U.S.M.G., Grenoble, 1975; and J. Enriquez, J. Thomann, and M. Goupillot, Application of bidimensional spline functions to geophysics, Geophysics 48 , 9 (1983)).