Peter Zanaty

First Instances of Generalized Expo‐Rational Finite Elements on Triangulations

In this communication we consider a construction of simplicial finite elements on triangulated two‐dimensional polygonal domains. This construction is, in some sense, dual to the construction of generalized expo‐rational B‐splines (GERBS). The main result is in the obtaining of new polynomial simplicial patches of the first several lowest possible total polynomial degrees which exhibit Hermite interpolatory properties. The derivation of these results is based on the theory of piecewise polynomial GERBS called Euler Beta‐function B‐splines. We also provide 3‐dimensional visualization of the graphs of the new polynomial simplicial patches and their control polygons.

Fitting of Discrete Data with GERBS

In this paper, we present a study of fitting discrete data with Generalized Expo-rational B-splines. We investigate different ways to determine interpolation knots and generate GERBS local curves by partitioning the parametric space and solving a corresponding least-squares fitting problem. We apply our technique to discrete evaluations of continuous synthetic benchmark functions and compare the resulting GERBS to the original data with respect to errors and performance.

Beta-function B-spline smoothing on triangulations

In this work we investigate a novel family of Ck-smooth rational basis functions on triangulations for fitting, smoothing, and denoising geometric data. The introduced basis function is closely related to a recently introduced general method introduced in utilizing generalized expo-rational B-splines, which provides Ck-smooth convex resolutions of unity on very general disjoint partitions and overlapping covers of multidimensional domains with complex geometry. One of the major advantages of this new triangular construction is its locality with respect to the star-1 neighborhood of the vertex on which the said base is providing Hermite interpolation. This locality of the basis functions can be in turn utilized in adaptive methods, where, for instance a local refinement of the underlying triangular mesh affects only the refined domain, whereas, in other method one needs to investigate what changes are occurring outside of the refined domain. Both the triangular and the general smooth constructions have the potential to become a new versatile tool of Computer Aided Geometric Design (CAGD), Finite and Boundary Element Analysis (FEA/BEA) and Iso-geometric Analysis (IGA).

On the numerical performance of FEM based on piecewise rational smooth resolutions of unity on triangulations

This work is a comparative study of a number of recently introduced finite element methods based on generalized expo-rational B-splines [7, 9]. In [12] a new construction of piecewise-rational Cm-smooth resolution of unity on triangulated domains in R2, m = 1,2, …, has been introduced. The present work is a first study of the potential for applications of this new construction in the numerical analysis of boundary-value problems for PDEs. It also represents a continuation and complement of certain aspects of the study in [23] which was conducted before the introduction of the new construction in [12] and the introduction of the logistic expo-rational B-spline in [10].

Smooth partition of unity with Hermite interpolation: applications to image processing

We explore the one-to one correspondence between parametric surfaces in 3D and two dimensional color images in the RGB color space. For the case of parametric surfaces defined on general parametric domains recently a new approximate geometric representation has been introduced1 which also works for manifolds in higher dimensions. This new representation has a form which is a generalization to the B´ezier representation of parametric curves and tensorproduct surfaces. The main purpose of the paper is to discuss how the so generated technique for modeling parametric surfaces can be used for respective modification (re-modeling) of images. We briefly consider also some of the possible applications of this technique.

Blending functions for hermite interpolation by beta-function b-splines on triangulations

In the present paper we compute for the first time Beta-function B-splines (BFBS) achieving Hermite interpolation up to third partial derivatives at the vertices of the triangulation. We consider examples of BFBS with uniform and variable order of the Hermite interpolation at the vertices of the triangulation, for possibly non-convex star-1 neighbourhoods of these vertices. We also discuss the conversion of the local functions from Taylor monomial bases to appropriately shifted and scaled Bernstein bases, thereby converting the Hermite interpolatory form of the linear combination of BFBS to a new, Bezier-type, form. This conversion is fully parallelized with respect to the vertices of the triangulation and, for Hermite interpolation of uniform order, the load of the computations for each vertex of the computation is readily balanced.

Cog Framework-3D Visualization for Mobile Robot Teleoperation

A multi-layer mobile robot controller unit has been created and tested successfully to be able to work with different types of mobile robot agents. The paper presents a modular extensible system which is relying on top of modern open source libraries. The system handles a motion capturing suit and adapts it similarly as traditional peripheries. Robust posture recognition has been introduced on top of the motion suit adapter, which is used to instruct a mobile robot agent, while immerse stereographic feedback is provided to the human operator.

Multivariate Hermite interpolation on scattered point sets using tensor‐product expo‐rational B‐splines

At the Seventh International Conference on Mathematical Methods for Curves and Surfaces, To/nsberg, Norway, in 2008, several new constructions for Hermite interpolation on scattered point sets in domains in Rn,n∈N, combined with smooth convex partition of unity for several general types of partitions of these domains were proposed in [1]. All of these constructions were based on a new type of B‐splines, proposed by some of the authors several years earlier: expo‐rational B‐splines (ERBS) [3].In the present communication we shall provide more details about one of these constructions: the one for the most general class of domain partitions considered.This construction is based on the use of two separate families of basis functions: one which has all the necessary Hermite interpolation properties, and another which has the necessary properties of a smooth convex partition of unity. The constructions of both of these two bases are well‐known; the new part of the construction is the combined use of these bases...

Image processing with LERBS

We investigate the performance of image compression using a custom transform, related to the discrete cosine transform, where the shape of the waveform basis function can be adjusted via setting a shape parameter. A strategy for generating quantization tables for various shapes of the basis function, including the cosine function, is proposed.

Smooth GERBS, orthogonal systems and energy minimization

New results are obtained in three mutually related directions of the rapidly developing theory of generalized expo-rational B-splines (GERBS) [7, 6]: closed-form computability of C∞-smooth GERBS in terms of elementary and special functions, Hermite interpolation and least-squares best approximation via smooth GERBS, energy minimizing properties of smooth GERBS similar to those of the classical cubic polynomial B-splines.