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).
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.
This article is a survey of the current state of the art in vertex-based marching algorithms for solving systems of
nonlinear equations and solving multidimensional intersection problems. It addresses also ongoing research and
future work on the topic. Among the new topics discussed here for the first time is the problem of characterizing
the type of singularities of piecewise affine manifolds, which are the numerical approximations to the solution
manifolds, as generated by the most advanced of the considered vertex-based algorithms: the Marching-Simplex
algorithm. Several approaches are proposed for solving this problem, all of which are related to modifications,
extensions and generalizations of the Morse lemma in differential topology.
In this paper we provide an overview about an orthonormal (multi) wavelet-based method for isometric immersion
of smooth n-variate m-dimensional vector fields onto fractal curves and surfaces. This method was proposed in
an earlier publication by two of the authors, with the purpose of extending the applicability of emerging GPU-programming
to rich diversity of multidimensional problems. Here we propose (in Section 3) several directions
for upgrading the method, with respective new applications.
We present an overview of results obtained in the last 10-15 years in the field of constrained deterministic approximation
and constrained statistical estimation of non-parametric regression functions, cumulative distribution
functions and densities. The case of deterministic approximation follows from the case of statistical estimation
of non-parametric regression when the noise variance is zero. Many unpublished results are announced here for
the first time.
This article is a systematic overview of compression, smoothing and denoising techniques based on shrinkage of
wavelet coefficients, and proposes an advanced technique for generating enhanced composite
wavelet shrinkage strategies.
We study the effects of the use of near-degenerate elements in finite and boundary element (multigrid) methods,
and their analogues with wavelet (multiresolution) methods. In the context of these results, a brief comparison
between finite/boundary element methods and wavelet methods is made.
Expo-rational B-splines have been introduced in 2002 and by now have been shown to exhibit certain 'super-properties'
compared to ordinary polynomial B-splines. The Euler Beta-function B-splines, a polynomial version
of the expo-rational B-splines, has been introduced very recently, and has been shown to share some of the
'super-properties' of the expo-rational B-splines. In this paper we discuss several of the ways in which these
'superproperties' can be used to enhance the theory of polynomial spline wavelets and multiwavelets.
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