15,390 research outputs found
Spline approximation of a random process with singularity
Let a continuous random process defined on be -smooth,
and have an isolated
singularity point at . In addition, let be locally like a -fold
integrated -fractional Brownian motion for all non-singular points. We
consider approximation of by piecewise Hermite interpolation splines with
free knots (i.e., a sampling design, a mesh). The approximation performance
is measured by mean errors (e.g., integrated or maximal quadratic mean errors).
We construct a sequence of sampling designs with asymptotic approximation rate
for the whole interval.Comment: 16 pages, 2 figure typos and references corrected, revised classes
definition, results unchange
Anti-aliasing with stratified B-spline filters of arbitrary degree
A simple and elegant method is presented to perform anti-aliasing in raytraced images. The method uses stratified
sampling to reduce the occurrence of artefacts in an image and features a B-spline filter to compute the final
luminous intensity at each pixel. The method is scalable through the specification of the filter degree. A B-spline
filter of degree one amounts to a simple anti-aliasing scheme with box filtering. Increasing the degree of the B-spline generates progressively smoother filters. Computation of the filter values is done in a recursive way, as part of a sequence of Newton-Raphson iterations, to obtain the optimal sample positions in screen space. The proposed method can perform both anti-aliasing in space and in time, the latter being more commonly known as motion blur. We show an application of the method to the ray casting of implicit procedural surfaces
Error analysis for quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of bounded rectangular domains
Given a non-uniform criss-cross partition of a rectangular domain ,
we analyse the error between a function defined on and two types
of -quadratic spline quasi-interpolants (QIs) obtained as linear
combinations of B-splines with discrete functionals as coefficients. The main
novelties are the facts that supports of B-splines are contained in
and that data sites also lie inside or on the boundary of . Moreover,
the infinity norms of these QIs are small and do not depend on the
triangulation: as the two QIs are exact on quadratic polynomials, they give the
optimal approximation order for smooth functions. Our analysis is done for
and its partial derivatives of the first and second orders and a particular
effort has been made in order to give the best possible error bounds in terms
of the smoothness of and of the mesh ratios of the triangulation
Fast adaptive elliptical filtering using box splines
We demonstrate that it is possible to filter an image with an elliptic window
of varying size, elongation and orientation with a fixed computational cost per
pixel. Our method involves the application of a suitable global pre-integrator
followed by a pointwise-adaptive localization mesh. We present the basic theory
for the 1D case using a B-spline formalism and then appropriately extend it to
2D using radially-uniform box splines. The size and ellipticity of these
radially-uniform box splines is adaptively controlled. Moreover, they converge
to Gaussians as the order increases. Finally, we present a fast and practical
directional filtering algorithm that has the capability of adapting to the
local image features.Comment: 9 pages, 1 figur
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