Polyharmonic B-splines Research Articles

On MRAs and Prewavelets Based on Elliptic Splines

We consider shift-invariant multiresolution spaces generated by q-elliptic splines in $${\mathbb {R}^{d}},d\ge 2,$$ which are tempered distributions characterized by a complex-valued elliptic homogeneous polynomial q of degree $$m>d$$ . To construct Riesz bases of $$L^{2} ({\mathbb {R}^{d}}),$$ a family of non-separable basic smooth functions are obtained by localizing a fundamental solution of the operator q(D), properly. The construction provides a generalization of some known elliptic scaling functions, the most famous being polyharmonic B-splines. Here, we prove that real-valued q leads to r-regular multiresolution analysis, with $$r=m-d-1.$$ In addition, we prove that there exist r-regular non-separable prewavelet systems associated with not necessarily regular multiresolution analyses. These prewavelets have $$m-1$$ vanishing moments and the approximation order of the prewavelet decomposition can be established.

Approximation based on elliptic splines

In this paper, we study the elliptic splines which include the well-known polyharmonic B-splines. We analyze their Fourier transforms, decay behaviors and polynomial reproducing properties. We also study the order of approximation in Sobolev spaces and consider their characterizations of Besov spaces by the scale projection operators, quasi-interpolation operators and wavelet operators.

Polyharmonic splines: An approximation method for noisy scattered data of extra-large size

The aim of this paper is to provide a fast method, with a good quality of reproduction, to recover functions from very large and irregularly scattered samples of noisy data, which may present outliers. To the given sample of size N, we associate a uniform grid and, around each grid point, we condense the local information given by the noisy data by a suitable estimator. The recovering is then performed by a stable interpolation based on isotropic polyharmonic B-splines. Due to the good approximation rate, we need only M@?N degrees of freedom to recover the phenomenon faithfully.

Shift-invariant spaces from rotation-covariant functions

We consider shift-invariant multiresolution spaces generated by rotation-covariant functions ρ in R 2 . To construct corresponding scaling and wavelet functions, ρ has to be localized with an appropriate multiplier, such that the localized version is an element of L 2 ( R 2 ) . We consider several classes of multipliers and show a new method to improve regularity and decay properties of the corresponding scaling functions and wavelets. The wavelets are complex-valued functions, which are approximately rotation-covariant and therefore behave as Wirtinger differential operators. Moreover, our class of multipliers gives a novel approach for the construction of polyharmonic B-splines with better polynomial reconstruction properties.

Isotropic polyharmonic B-splines: scaling functions and wavelets

In this paper, we use polyharmonic B-splines to build multidimensional wavelet bases. These functions are nonseparable, multidimensional basis functions that are localized versions of radial basis functions. We show that Rabut's elementary polyharmonic B-splines do not converge to a Gaussian as the order parameter increases, as opposed to their separable B-spline counterparts. Therefore, we introduce a more isotropic localization operator that guarantees this convergence, resulting into the isotropic polyharmonic B-splines. Next, we focus on the two-dimensional quincunx subsampling scheme. This configuration is of particular interest for image processing because it yields a finer scale progression than the standard dyadic approach. However, up until now, the design of appropriate filters for the quincunx scheme has mainly been done using the McClellan transform. In our approach, we start from the scaling functions, which are the polyharmonic B-splines and, as such, explicitly known, and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi-orthogonal wavelet transform; e.g., orthonormal, B-spline, and dual. The filters are automatically specified by the scaling relations satisfied by these functions. We prove that the isotropic polyharmonic B-spline wavelet converges to a combination of four Gabor atoms, which are well separated in the frequency domain. We also show that these wavelets are nearly isotropic and that they behave as an iterated Laplacian operator at low frequencies. We describe an efficient fast Fourier transform-based implementation of the discrete wavelet transform based on polyharmonic B-splines.

Decomposition and reconstruction of multidimensional signals using polyharmonic pre-wavelets

In this paper, we build a multidimensional wavelet decomposition based on polyharmonic B-splines. The pre-wavelets are polyharmonic splines and so not tensor products of univariate wavelets. Explicit construction of the filters specified by the classical dyadic scaling relations is given and the decay of the functions and the filters is shown. We then design the decomposition/recomposition algorithm by means of downsampling/upsampling and convolution products.

Construction of Real-Valued Wavelets by Symmetry

A construction of orthogonal wavelet bases in L2(Rd) from a multiresolution analysis is given when the scaling function is skew-symmetric about an integer point. These wavelets are real-valued when the scaling function is real-valued. As an application, orthogonal wavelets generated by polyharmonic B-splines are obtained. We also present an example to show that the center of a skew-symmetric scaling function may not be in Zd/2.

Using the refinement equation for the construction of pre-wavelets III: Elliptic splines

The purpose of this paper is to provide multiresolution analysis, stationary subdivision and pre-wavelet decomposition onL 2(R d ) based on a general class of functions which includes polyharmonic B-splines.