Georges-Pierre Bonneau

Triangular G 1 interpolation by 4-splitting domain triangles

A piecewise quintic G1 spline surface interpolating the vertices of a triangular surface mesh of arbitrary topological type is presented. The surface has an explicit triangular Bezier representation, is affine invariant and has local support. The twist compatibility problem which arises when joining an even number of polynomial patches G1 continuously around a common vertex is solved by constructing C2-consistent boundary curves. Piecewise C1 boundary curves and a regular 4-split of the domain triangle make shape parameters available for controlling locally the boundary curves. A small number of free inner control points can be chosen for some additional local shape effects.

Polynomial surfaces interpolating arbitrary triangulations

Triangular Bezier patches are an important tool for defining smooth surfaces over arbitrary triangular meshes. The previously introduced 4-split method interpolates the vertices of a 2-manifold triangle mesh by a set of tangent plane continuous triangular Bezier patches of degree five. The resulting surface has an explicit closed form representation and is defined locally. In this paper, we introduce a new method for visually smooth interpolation of arbitrary triangle meshes based on a regular 4-split of the domain triangles. Ensuring tangent plane continuity of the surface is not enough for producing an overall fair shape. Interpolation of irregular control-polygons, be that in 1D or in 2D, often yields unwanted undulations. Note that this undulation problem is not particular to parametric interpolation, but also occurs with interpolatory subdivision surfaces. Our new method avoids unwanted undulations by relaxing the constraint of the first derivatives at the input mesh vertices: The tangent directions of the boundary curves at the mesh vertices are now completely free. Irregular triangulations can be handled much better in the sense that unwanted undulations due to flat triangles in the mesh are now avoided.

Hierarchical triangular splines

Smooth parametric surfaces interpolating triangular meshes are very useful for modeling surfaces of arbitrary topology. Several interpolants based on these kind of surfaces have been developed over the last fifteen years. However, with current 3D acquisition equipments, models are becoming more and more complex. Since previous interpolation methods lack a local refinement property, there is no way to locally adapt the level of detail. In this article, we introduce a hierarchical triangular surface model. The surface is overall tangent plane continuous and is defined parametrically as a piecewise quintic polynomial. It can be adaptively refined while preserving the overall tangent plane continuity. This model enables designers to create a complex smooth surface of arbitrary topology composed of a small number of patches to which details can be added by locally refining the patches until an arbitrary small size is reached. It is implemented as a hierarchical data structure where the top layer describes a coarse, smooth base surface and the lower levels encode the details in local frame coordinates.

Sharp feature preserving MLS surface reconstruction based on local feature line approximations

Graphical abstractDisplay Omitted Highlights? Use of local feature line approximation for MLS with sharp feature reconstruction. ? Feature reconstruction by segmentation and up-sampling of local neighborhoods along feature curves. ? Capable to handle sharp line-type features and corner features. ? Full automatic sharp feature detection as preprocess speeds up iterative reconstructions. Sharp features in manufactured and designed objects require particular attention when reconstructing surfaces from unorganized scan point sets using moving least squares (MLS) fitting. It is an inherent property of MLS fitting that sharp features are smoothed out. Instead of searching for appropriate new fitting functions our approach computes a modified local point neighborhood so that a standard MLS fitting can be applied enhanced by sharp features reconstruction.We present a two-stage algorithm. In a pre-processing step sharp feature points are marked first. This algorithm is robust to noise since it is based on Gauss map clustering. In the main phase, the selected feature points are used to locally approximate the feature curve and to segment and enhance the local point neighborhood. The MLS projection thus leads to a piecewise smooth surface preserving all sharp features. The method is simple to implement and able to preserve line-type features as well as corner-type features during reconstruction.

Smooth adaptive fitting of 3D models using hierarchical triangular splines

The recent ability to measure quickly and inexpensively dense sets of points on physical objects has deeply influenced the way engineers represent shapes in CAD systems, animation software or in the game industry. Many researchers advocated to completely bypass smooth surface representations, and to stick to a dense mesh model throughout the design process. Yet smooth analytic representations are still required in standard CAD systems and animation software, for reasons of compactness, control, appearance and manufacturability. In this paper we present a new method for fitting a smooth adoptively refinable triangular spline surface of arbitrary topology to an arbitrary dense triangular mesh. The key ingredient in our solution is that adaptive fitting is achieved by 4-splitting triangular surface patches locally therefore no particular attention has to be paid the validity of an underlying subdivided mesh. Furthermore, the final surface is composed of low-degree polynomial patches that always join with Gl-continuity. The ability to adoptively refine the model allows to achieve a given approximation error with a minimal number of patches.

Visualization of uncertain scalar data fields using color scales and perceptually adapted noise

We present a new method to visualize uncertain scalar data fields by combining color scale visualization techniques with animated, perceptually adapted Perlin noise. The parameters of the Perlin noise are controlled by the uncertainty information to produce animated patterns showing local data value and quality. In order to precisely control the perception of the noise patterns, we perform a psychophysical evaluation of contrast sensitivity thresholds for a set of Perlin noise stimuli. We validate and extend this evaluation using an existing computational model. This allows us to predict the perception of the uncertainty noise patterns for arbitrary choices of parameters. We demonstrate and discuss the efficiency and the benefits of our method with various settings, color maps and data sets.

Topology-preserving simplification of 2D nonmanifold meshes with embedded structures

Mesh simplification has received tremendous attention over the years. Most of the previous work in this area deals with a proper choice of error measures to guide the simplification. Preserving the topological characteristics of the mesh and possibly of data attached to the mesh is a more recent topic and the subject of this paper. We introduce a new topology-preserving simplification algorithm for triangular meshes, possibly nonmanifold, with embedded polylines. In this context, embedded means that the edges of the polylines are also edges of the mesh. The paper introduces a robust test to detect if the collapse of an edge in the mesh modifies either the topology of the mesh or the topology of the embedded polylines. This validity test is derived using combinatorial topology results. More precisely, we define a so-called extended complex from the input mesh and the embedded polylines. We show that if an edge collapse of the mesh preserves the topology of this extended complex, then it also preserves both the topology of the mesh and the embedded polylines. Our validity test can be used for any 2-complex mesh, including nonmanifold triangular meshes, and can be combined with any previously introduced error measure. Implementation of this validity test is described. We demonstrate the power and versatility of our method with scientific data sets from neuroscience, geology, and CAD/CAM models from mechanical engineering.

Length preserving multiresolution editing of curves

In this paper a method for multiresolution deformation of planar piecewise linear curves that preserves the curve length is presented. In a wavelet based multiresolution editing framework, the curve can be deformed at any level of resolution through its control points. Enforcing the length constraint is carried out in two steps. In a first step the multiresolution decomposition of the curve is used in order to approximate the initial curve length. In a second step the length constraint is satisfied exactly by iteratively smoothing the deformed curve. Wrinkle generation is an application the paper particularly focuses on. It is shown how the multiresolution definition of the curve allows to explicitly and intuitively control the scale of the generated wrinkles.

Variational Surface Design and Surface Interrogation

The generation of technical smooth surfaces from a mesh of three‐dimensional data points is an important problem in geometric modelling. In this publication we give a survey of some new techniques based on a calculus of variation approach. Apart from the pure construction of these surfaces, the analysis of their quality is equally important in the design and manufacturing process. Generalized focal surfaces are presented here as a new surface interrogation tool.

Detail preserving deformation of B-spline surfaces with volume constraint

Geometric constraints have proved to be helpful for shape modeling. Moreover, they are efficient aids in controlling deformations and enhancing animation realism. The present paper addresses the deformation of B-spline surfaces while constraining the volume enclosed by the surface. Both uniform and non-uniform frameworks are considered. The use of level-of-detail (LoD) editing allows the preservation of fine details during coarse deformations of the shape. The key contribution of this paper is the computation of the volume with respect to the appropriate basis for LoD editing: the volume is expressed through all levels of resolution as a trilinear form and recursive formulas are developed to make the computation efficient. The volume constrained is maintained through a minimization process for which we develop closed solutions. Real-time deformations are reached thanks to sparse data structures and efficient algorithms.