Abstract not found

This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shift-invariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Band-limited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,

This article develops a dynamic generalization of the nonuniform rational B-spline (NURBS) model. NURBS have become a de facto standard in commercial modeling systems because of their power to represent free-form shapes as well as common analytic shapes. To date, however, they have been viewed as purely geometric primitives that require the user to manually adjust multiple control points and associated weights in order to design shapes. Dynamic NURBS, or D-NURBS, are physics-based models that incorporate mass distributions, inertial deformation energies, and other physical quantities into the popular NURBS geometric substrate. Using D-NURBS, a modeler can interactively sculpt curves and surfaces and design complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. D-NURBS move and deform in a physically intuitive manner in response to the user's direct manipulations. Their dynamic behavior results from the numerical integration of a set of nonlinear differential equations that automatically evolve the control points and weights in response to the applied forces and constraints. To derive these equations, we employ Lagrangian mechanics and finite-element-like discretization. Our approach supports the trimming of D-NURBS surfaces using D-NURBS curves. We demonstrate D-NURBS models and constraints in applications including the rounding of solids, optimal surface fitting to unstructured data, surface design from cross-sections, and free-form deformation. We also introduce a new technique for 2D shape metamorphosis using constrained D-NURBS surfaces.

The human face is an elastic object. A natural paradigm for representing facial expressions is to form a complete 3D model of facial muscles and tissues. However, determining the actual parameter values for synthesizing and animating facial expressions is tedious; evaluating these parameters for facial expression analysis out of grey-level images is ahead of the state of the art in computer vision. Using only 2D face images and a small number of anchor points, we show that the method of radial basis functions provides a powerful mechanism for processing facial expressions. Although constructed specifically for facial expressions, our method is applicable to other elastic objects as well.

This paper presents dynamic NURBS, or D-NURBS, a physics-based generalization of Non-Uniform Rational B-Splines. NURBS have become a de facto standard in commercial modeling systems because of their power to represent both free-form shapes and common analytic shapes. Traditionally, however, NURBS have been viewed as purely geometric primitives, which require the designer to interactively adjust many degrees of freedom (DOFs) -- control points and associated weights -- to achieve desired shapes. The conventional shape modi cation process can often be clumsy and laborious. D-NURBS are physics-based models that incorporate mass distributions, internal deformation energies, forces, and other physical quantities into the NURBS geometric substrate. Their dynamic behavior, resulting from the numerical integration of a set of nonlinear differential equations, produces physically meaningful, hence intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and setting weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. We use Lagrangian mechanics to formulate the equations of motion for D-NURBS curves, tensor-product D-NURBS surfaces, swung D-NURBS surfaces, and triangular D-NURBS surfaces. We apply finite element analysis to reduce these equations to eficient numerical algorithms computable at interactive rates on common graphics workstations. We implement a prototype modeling environment based on D-NURBS, and demonstrate that D-NURBS can be effective tools in a wide range of CAGD applications such as shape blending, scattered data fitting, and interactive sculpting.

This article provides a historical account of the major developments in the area of curves and surfaces as they entered the area of CAGD – Computer Aided Geometric Design – until the middle 1980s. We adopt the definition that CAGD deals with the construction and representation of free-form curves, surfaces, or volumes. 1.

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—Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we propose a complete and self-contained signal processing formulation of exponential splines defined on a uniform grid. We specify the corresponding B-spline basis functions and investigate their reproduction properties (Green function and exponential polynomials); we also characterize their stability (Riesz bounds). We show that the exponential B-spline framework allows an exact implementation of continuous-time signal processing operators including convolution, differential operators, and modulation, by simple processing in the discrete B-spline domain. We derive efficient filtering algorithms for multiresolution signal extrapolation and approximation, extending earlier results for polynomial splines. Finally, we present a new asymptotic error formula that predicts the magnitude and the th-order decay of the P-approximation error as a function of the knot spacing. Index Terms—Continuous-time signal processing, convolution, differential operators, Green functions, interpolation, modulation, multiresolution approximation, splines. I.

This paper presents a new triangular B-spline scheme that allows to construct piecewise polynomial surfaces over arbitrary triangulations of the parameter plane. The development of this scheme is based on the study of polar forms [79]. Polar forms have originally been a tool from classical mathematics [88]. They have first been introduced to CAGD by P. de Faget de Casteljau [24, 26] and by L. Ramshaw [65, 66, 67]. The author has subsequently extended this theory to more general surface representations and has used polar forms for the development of B-patches [77, 76, 84]. Further extensions to simplex splines have finally led to the new triangular B-spline scheme described in this paper [18, 38, 39, 49, 78, 81]. While previous approaches to the construction of B-spline like surfaces over irregular domains have been based on subdivision, interpolation, and on the use of multisided patches the new scheme is based on blending functions and control points. The resulting surfaces are defined as linear combinations of the blending functions and are parametric piecewise polynomials over an arbitrary triangulation of the parameter plane, whose shape is determined by their control points. The paper is organized as follows: After a brief introduction to bivariate polynomials and polar forms (Section 2) we discuss triangular B'ezier patches (Section 3) and introduce a new surface representation for bivariate polynomials, the B-patch (Section 4). In connection with simplex splines (Section 5) this finally leads to the construction of the new triangular B-spline scheme (Section 6). We hope that our presentation will provide a thorough unterstanding of the polar form of a polynomial surface, of triangular B'ezier patches, and of some of the main issues that are involved in the constr...

We develop a dynamic, free-form surface model which is useful for representing a broad class of objects with symmetries and topological variability. The new model is based upon swung NURBS surfaces, and it inherits their desirable cross-sectional design properties. It melds these geometric features with the demonstrated conveniences of surface design within a physics-based framework. We demonstrate several applications of dynamic NURBS swung surfaces, including interactive sculpting through the imposition of forces and the adjustment ofphysical parameters such as mass, damping, and elasticity. Additional applications include surface design with geometric and physical constraints, by rounding solids, and through the tting of unstructured data. We derive the equations of motion for the dynamic NURBS swung surface model using Lagrangian mechanics of an elastic surface and the nite element method. We also show that these surfaces are a special case of D-NURBS surfaces, a recently proposed physicsbased generalization of standard geometric NURBS. Our free-form, rational model not only provides a systematic and uni ed approach for a variety ofCAGD problems such as constraint-based optimization, variational design, automatic weight selection, shape approximation, etc., but it also supports interactive sculpting using physics-based manipulation tools.