B-spline Kernel Research Articles

Characterization of B-Spline Digital Filters

Digital filters arising in the B-spline signal processing are characterized in a unified manner in this paper. The transfer functions of these filters are the z-transforms of the uniformly sampled central B-splines shifted by an arbitrary value. The transfer functions are cascades of an FIR kernel filter and a simple moving average FIR filter. For certain values of the shift parameter, the filters are identical to those referred to as the B-spline digital filters in the literature. The filters thus form a general family of B-spline digital filters. The kernel part of the B-spline filters may be used for transforming a discrete-time signal to a representation based on the B-spline coefficients. The B-spline filters may also be used to convert a sequence of B-spline coefficients to a discrete-time spline signal. The contributions of the paper are as follows. A unifying recurrence relation enabling the computation of the impulse response coefficients of the B-spline kernel filters is derived. An accompanying recurrence relation is also obtained for the entire transfer function of the kernel filters. The recurrences are valid for arbitrary values of the shift parameter. It is proved that the roots of the transfer functions of the kernel filters are distinct, negative and real. We also prove that the roots of the kernel filters of successive orders interlace. The results regarding the location of the zeros are also valid for arbitrary values of the shift parameter. The relation of the kernel filters to the Eulerian polynomials is discussed. It is shown that for certain choices of the parameters the kernel filters are equivalent to the classical Eulerian polynomials that frequently arise in combinatorics. An alternative closed-form expression for the kernel filters in the Bernstein form is also derived. Besides their importance in unifying the existing results on B-spline filters, the generalized family of B-spline filters studied in this paper find applications in fractional delay of B-spline signals.

Scale-space derived from B-splines

This paper proposes a scale-space theory based on B-spline kernels. Our aim is twofold: 1) present a general framework, and show how B-splines provide a flexible tool to design various scale-space representations. In particular, we focus on the design of continuous scale-space and dyadic scale-space frame representations. A general algorithm is presented for fast implementation of continuous scale-space at rational scales. In the dyadic case, efficient frame algorithms are derived using B-spline techniques to analyze the geometry of an image. The relationship between several scale-space approaches is explored. The behavior of edge models, the properties of completeness, causality, and other properties in such a scale-space representation are examined in the framework of B-splines. It is shown that, besides the good properties inherited from the Gaussian kernel, the B-spline derived scale-space exhibits many advantages for modeling visual mechanism including the efficiency, compactness, orientation feature and parallel structure.

Smoothed Particle Hydrodynamics Stability Analysis

SPH (smoothed particle hydrodynamics) is a gridless Lagrangian technique which is appealing as a possible alternative to numerical techniques currently used to analyze large deformation events. Recent tests of the standard SPH method using the cubic B-spline kernel indicated the possibility of an instability in the tensile regime, even though no such difficulties were observed in compression. A von Neumann stability analysis of the SPH algorithm has been carried out which identifies the criterion for stability or instability in terms of the stress state and the second derivative of the kernel function. The analysis explains the observation that the method is unstable in tension while apparently stable in compression but shows that it is possible to construct kernel functions which are stable in tension and unstable in compression. The instability is shown to result from an effective stress with a negative modulus (imaginary sound speed) being produced by the interaction between the constitutive relation and the kernel function and is not caused by the numerical time integration algorithm. That is, changes in the effective stress act to amplify, rather than reduce, perturbations in the strain. The analysis and the stability criterion provide insight into possible methods for removing the instability.