Non-equispaced B-spline wavelets

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorisation of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.Comment: 42 pages, 2 figure

Principled Design and Implementation of Steerable Detectors

We provide a complete pipeline for the detection of patterns of interest in an image. In our approach, the patterns are assumed to be adequately modeled by a known template, and are located at unknown position and orientation. We propose a continuous-domain additive image model, where the analyzed image is the sum of the template and an isotropic background signal with self-similar isotropic power-spectrum. The method is able to learn an optimal steerable filter fulfilling the SNR criterion based on one single template and background pair, that therefore strongly responds to the template, while optimally decoupling from the background model. The proposed filter then allows for a fast detection process, with the unknown orientation estimation through the use of steerability properties. In practice, the implementation requires to discretize the continuous-domain formulation on polar grids, which is performed using radial B-splines. We demonstrate the practical usefulness of our method on a variety of template approximation and pattern detection experiments

Observing Non-Gaussian Sources in Heavy-Ion Reactions

We examine the possibility of extracting non-Gaussian sources from two-particle correlations in heavy-ion reactions. Non-Gaussian sources have been predicted in a variety of model calculations and may have been seen in various like-meson pair correlations. As a tool for this investigation, we have developed an improved imaging method that relies on a Basis spline expansion of the source functions with an improved implementation of constraints. We examine under what conditions this improved method can distinguish between Gaussian and non-Gaussian sources. Finally, we investigate pion, kaon, and proton sources from the p-Pb reaction at 450 GeV/nucleon and from the S-Pb reaction at 200 GeV/nucleon studied by the NA44 experiment. Both the pion and kaon sources from the S-Pb correlations seem to exhibit a Gaussian core with an extended, non-Gaussian halo. We also find evidence for a scaling of the source widths with particle mass in the sources from the p-Pb reaction.Comment: 16 pages, 15 figures, 5 tables, uses RevTex3.

Improving convergence in smoothed particle hydrodynamics simulations without pairing instability

The numerical convergence of smoothed particle hydrodynamics (SPH) can be severely restricted by random force errors induced by particle disorder, especially in shear flows, which are ubiquitous in astrophysics. The increase in the number NH of neighbours when switching to more extended smoothing kernels at fixed resolution (using an appropriate definition for the SPH resolution scale) is insufficient to combat these errors. Consequently, trading resolution for better convergence is necessary, but for traditional smoothing kernels this option is limited by the pairing (or clumping) instability. Therefore, we investigate the suitability of the Wendland functions as smoothing kernels and compare them with the traditional B-splines. Linear stability analysis in three dimensions and test simulations demonstrate that the Wendland kernels avoid the pairing instability for all NH, despite having vanishing derivative at the origin (disproving traditional ideas about the origin of this instability; instead, we uncover a relation with the kernel Fourier transform and give an explanation in terms of the SPH density estimator). The Wendland kernels are computationally more convenient than the higher-order B-splines, allowing large NH and hence better numerical convergence (note that computational costs rise sub-linear with NH). Our analysis also shows that at low NH the quartic spline kernel with NH ~= 60 obtains much better convergence then the standard cubic spline.Comment: substantially revised version, accepted for publication in MNRAS, 15 pages, 13 figure