this paper we describe two adaptive procedures, one based on C p and the other based on crossvalidation. Still, when we have a final adaptive fit in hand, it is critical to subject it to graphical diagnostics to study its performance. The important implication of these statements is that the above choices must be tailored to each data set in practice; that is, the choices represent a modeling of the data. It is widely accepted that in global parametric regression there are a variety of choices that must be made --- for example, the parametric family to be fitted and the form of the distribution of the response --- and that we must rely on our knowledge of the mechanism generating the data, on model selection diagnostics, and on graphical diagnostic methods to make the choices. The same is true for smoothing. Cleveland (1993) presents many examples of this modeling process. For example, in one application, oxides of nitrogen from an automobile engine are fitted to the equivalence ratio, E, of the fuel and the compression ratio, C, of the engine. Coplots show that it is reasonable to use quadratics as the local parametric family but with the added assumption that given E the fitted f

This paper addresses the point-wise estimation of differential properties of a smooth manifold S —a curve in the plane or a surface in 3D — assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, and either interpolation or approximation. A jet is a truncated Taylor expansion, and the incentive for using jets is that they encode all local geometric quantities —such as normal, curvatures, extrema of curvature. On the way to using jets, the question of estimating differential properties is recasted into the more general framework of multivariate interpolation / approximation, a well-studied problem in numerical analysis. On a theoretical perspective, we prove several convergence results when the samples get denser. For curves and surfaces, these results involve asymptotic estimates with convergence rates depending upon the degree of the jet used. For the particular case of curves, an error bound is also derived. To the best of our knowledge, these results are among the first ones providing accurate estimates for differential quantities of order three and more. On the algorithmic side, we solve the interpolation/approximation problem using Vandermonde systems. Experimental results for surfaces of R 3 are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.

This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolution-based interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.

Abstract. Deformations are a powerful tool for shape modeling and design. We present a new model for producing controlled spatial deformations, which we term Simple Constrained Deformations (Scodef). The user defines a set of constraint points, giving a desired displacement and radius of influence for each. Each constraint point determines a local B-spline basis function centered at the constraint point, falling to zero for points beyond the radius. The deformed image of any point in space is a blend of these basis functions, using a projection matrix computed to satisfy the constraints. The deformation operates on the whole space regardless of the representation of the objects embedded inside the space. The constraints directly influence the final shape of the deformed objects, and this shape can be fine-tuned by adjusting the radius of influence of each constraint point. The computations required by the technique can be done very efficiently, and real-time interactive deformation editing on current workstations is possible.

In a general setting, the transfinite interpolation problem requires constructing a single function #### that takes on the prescribed values and/or derivatives on some collection of point sets. The sets of points may contain isolated points, bounded or unbounded curves, as well as surfaces and regions of arbitrary topology. All such closed semi-analytic sets may be represented implicitly by real valued functions with guaranteed differential properties.

We propose a universal approach to the problem of computer modeling of shapes with continuously varying material properties satisfying prescribed material conditions on a finite collection of material features and global constraints. The central notion is a parameterization of space by distances from the material features -- either exactly or approximately. Functions of such distances provide a systematic and intuitive means for modeling of desired material distributions as they arise in design, manufacturing, analysis and optimization of components with varying material properties.

1 Introduction There is a growing interest in the so-called "meshless " methods. It might be partly traced to high costs involved in meshing procedures. Modelling of adapting domain geometry, fracture, fragmentation and similar phenomena requires considerable remeshing efforts, which can easily constitute the largest portion of analysis costs.

This report describes an attempt at capturing segmental transition information for speech recognition tasks. The slowly varying dynamics of spectral trajectories carries much discriminant information that is very crudely modelled by traditional approaches such as HMMs. In approaches such as recurrent neural networks there is the hope, but not the convincing demonstration, that such transitional information could be captured. The method presented here starts from the very different position of explicitly capturing the trajectory of short time spectral parameter vectors on a subspace in which the temporal sequence information is preserved. We approach this by introducing a temporal constraint into the well known technique of Principal Component Analysis. On this subspace, we attempt a parametric modelling of the trajectory, and compute a distance metric to perform classification of diphones. We use the principal curves method of Hastie and Stuetzle and the Generative Topographic map (GTM...

Abstract. Creating models of real scenes is a complex task for which the use of traditional modelling techniques is inappropriate. For this task, laser rangefinders are frequently used to sample the scene from several viewpoints, with the resulting range images integrated into a final model. In practice, due to surface reflectance properties, occlusions and accessibility limitations, certain areas of the scenes are usually not sampled, leading to holes and introducing undesirable artifacts in the resulting models. We present an algorithm for filling holes on surfaces reconstructed from point clouds. The algorithm is based on moving least squares and can recover both geometry and shading information, providing a good alternative when the properties to be reconstructed are locally smooth. The reconstruction process is mostly automatic and the sampling rate in the reconstructed areas follows the given samples. We demonstrate the use of the algorithm on both real and synthetic data sets to obtain complete geometry and reasonable shading. 1

Fellow and leader of InSAR and deformation monitoring