Compactly-supported smooth interpolators for shape modeling with varying resolution

Abstract

A B S T R A C TIn applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape itself. Additionally, models for shape representation often need to provide local shape control and they need to be able to reproduce common shape primitives such as ellipsoids, spheres, cylinders, or tori. We present a general framework to construct families of compactly-supported interpolators that are piecewise-exponential polynomial. They can be designed to satisfy regularity constraints of any order and they enable one to build parametric deformable shape models by suitable linear combinations of interpolators. They allow to change the resolution of shapes based on the refinability of B-splines. We illustrate their use on examples to construct shape models that involve curves and surfaces with applications to interactive modeling and character design.

Digital Circuit for Seamless Resampling ADC Output Streams

“…In the most common resampling applications both the input and output sampling grid are uniform and the circuit has to deal with samples that are streamed at regular time instants, such that a sampling rate is defined. Also, resampling has to be performed real-time seamlessly on the input stream, which is very challenging, especially in the presence of high-rate data streams [24][25][26][27].…”

Local cardinal interpolation by C2 cubic B2-splines with a tunable shape parameter

“…More precisely, in Section 2 we define the new class of fundamental functions, we show that each member of the class can be written as a linear combination of shifted cubic B-splines on the half integer grid, and that the coefficients of the linear combination depend on a free parameter that can be used to globally modify the shape of the fundamental function. Following the line of reasoning in [7,Prop.1], we prove that the integer shifts of each fundamental function form a Riesz basis and, depending on the value of the free parameter, reproduce either linear or cubic polynomials. Then, in Section 3 we exploit the newly derived fundamental functions for 2D data interpolation, show how the resulting C 2 interpolating curve can be efficiently generated via a two-phase subdivision scheme and develop an algorithm for the automatic selection of the shape parameter.…”

Creating a bridge between cardinal Br-spline fundamental functions for interpolation and subdivision

“…-polynomial and exponential B-splines (see, e.g., [5,8,17] and [9,20,23,28,36,37], respectively), -polynomial and exponential B2-splines (see [29] and [35], respectively), -polynomial and exponential interpolating Hermite splines (see [21,27,33] and [13], respectively). Thus, our general result (which includes as a special case the subdivision method recovering the cubic polynomial Br-splines for cardinal interpolation investigated in [1,14,26,29] and the order-4 exponential B2-spline for cardinal interpolation presented in [35]) fills a knowledge gap in the related fields of subdivision and spline theory.…”

“…Subdivision schemes are efficient computational methods for generating functions (as well as curves, surfaces and volumes) from discrete data by repeated refinements. Their applications are indeed very broad and their usefulness is already well established in contexts like geometric modeling and computer graphics (see, e.g., [17,34,37]), biomedical imaging (see, e.g., [2,13,35]) and isogeometric analysis (see, e.g., [7,25,38]). In order to construct smooth curves and surfaces passing through a given set of data points, two different types of subdivision schemes can be used: the so-called "natural" interpolatory subdivision schemes and the ones known as interpolatory schemes "in the limit".…”