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Unlike wavelet, shearlets have the capability to detect directional discontinuities together with their directions. To achieve this, the considered scaling matrices have to be not only expanding, but also anisotropic. Shearlets allow for the definition of a directional multiple multiresolution analysis, where perfect reconstruction of the filterbank can be easily ensured by choosing an appropriate interpolatory multiple subdivision scheme. A drawback of shearlets is their relative large determinant that leads to a substantial complexity. The aim of the paper is to find scaling matrices in $\Z^{d \times d}$ which share the properties of shearlet matrices, i.e. anisotropic expanding matrices with the so-called slope resolution property, but with a smaller determinant. The proposed matrices provide a directional multiple multiresolution analysis whose behaviour is illustrated by some numerical tests on images.

Given a diagonal anisotropic expanding matrix $D\in\ZZ^{\di\times\di},$ we show that it is always possible to detect a unimodular matrix $A\in\ZZ^{\di\times\di}$ that satisfies the pseudo-commuting properties \[DA=A^nD.\] In particular when $\di=2$ the more general relation \[DA^m=A^nD\] holds.

We present an adaptive method to extract shape-preserving information from a univariate data sample. The behavior of the signal is obtained by interpolating at adaptively selected few data points by a linear combination of multiquadrics with variable scaling parameters. On the theoretical side, we give a sufficient condition for existence of the scaled multiquadric interpolant. On the practical side, we give various examples to show the applicability of the method

In this note we shall present some problems about fitting scattered data in 2d from regular functions, and that, in our opinion, have to be studied more in depth.

The aim of the paper is to construct a multiresolution analysis of L2(IRd) based on generalized kernels which are fundamental solutions of differential operators of the form $\boldsymbol {\prod }_{\ell = 1}^{m}(-{\Delta }+\kappa _{\ell }^{2}\,I)$ . We study its properties and provide a set of pre-wavelets associated with it, as well as the filters which are indispensable to perform decomposition and reconstruction of a given signal, being very useful in applied problems thanks to the presence of the tension parameters κl.

Objectives: To evaluate patients’ participation during physical therapy sessions as assessed with the Pittsburgh rehabilitation participation scale (PRPS) as a possible predictor of functional gain after rehabilitation training. Methods: All patients aged 65 years or older consecutively admitted to a Department of Rehabilitation and Aged Care (DRAC) were evaluated on admission regarding their health, nutritional, functional and cognitive status. Functional status was assessed with the functional independence measure (FIM) on admission and at discharge. Participation during rehabilitation sessions was measured with the PRPS. Functional gain was evaluated using the Montebello rehabilitation factor score (MRFS efficacy), and patients stratified in two groups according to their level of functional gain and their sociodemographic, clinical and functional characteristics were compared. Predictors of poor functional gain were evaluated using a multivariable logistic regression model adjusted for confounding factors. Result: A total of 556 subjects were included in this study. Patients with poor functional gain at discharge demonstrated lower participation during physical therapy sessions were significantly older, more cognitively and functionally impaired on admission, more depressed, more comorbid, and more frequently admitted for cardiac disease or immobility syndrome than their counterparts. There was a significant linear association between PRPS scores and MRFS efficacy. In a multivariable logistic regression model, participation was independently associated with functional gain at discharge (odds ratio 1.51, 95 % confidence interval 1.19–1.91). Conclusion: This study showed that participation during physical therapy affects the extent of functional gain at discharge in a large population of older patients with multiple diseases receiving in-hospital rehabilitation

We have presented in Bozzini et al. (2011) a procedure in spaces of m-harmonic splines in Rd that starts from a simple generator φ0 and recursively defines generators φ1, φ2,.,φm-1 with corresponding quasi-interpolation operators reproducing polynomials of degrees 3, 5,.,2m-1 respectively. In this paper we study the properties of generators φj, and we prove that these new generators are positive definite functions, and are scaling functions whenever φ0 has those properties. Moreover φ0 and φj generate the same multiresolution analysis. We show that it is possible to define a convergent subdivision scheme, and to provide in this way a fast computation of the quasi-interpolant.

We generate various new radial kernels by taking derivatives of known kernels with respect to scale. This is different from the well-known scale mixtures used before. In addition, we provide a simple recipe that explicitly constructs new kernels from the negative Laplacian of known kernels. Surprisingly, these two methods for generating new kernels can be proven to coincide for certain standard classes of radial kernels. The resulting radial kernels are positive definite, and a few illustrations are provided.

We present a numerical method for the detection of faults and gradient faults of two dimensional functions when uniformly scattered and noisy data are given. The method recognizes the presence of a discontinuity curve by a suitable statistic. Moreover, we discuss the recovery of such curves and provide a computational bound for the pointwise error and convergence properties.

Within kernel-based interpolation and its many applications, the handling of the scaling or the shape parameter is a well-documented but unsolved problem. We consider native spaces whose kernels allow us to change the kernel scale of a d-variate interpolation problem locally, depending on the requirements of the application. The trick is to define a scale function c on the domain ? ? Rd to transform an interpolation problem from data locations xj in Rd to data locations (xj, c(xj)) and to use a fixed-scale kernel on Rd+1 for interpolation there. The (d+1)-variate solution is then evaluated at (x, c(x)) for x ? Rd to give a d-variate interpolant with a varying scale. A large number of examples show how this can be done in practice to get results that are better than the fixed-scale technique, with respect to both condition number and error. The background theory coincides with fixed-scale interpolation on the submanifold of Rd+1 given by the points (x, c(x)) of the graph of the scale function c.

In this paper, we deal with two different problems. First, we provide the convergence rates of multiresolution approximations, with respect to the supremum norm, for the class of elliptic splines defined in Ref. 10, and in particular for polyharmonic splines. Secondly, we consider the problem of recovering a function from a sample of noisy data. To this end, we define a linear and smooth estimator obtained from a multiresolution process based on polyharmonic splines. We discuss its asymptotic properties and we prove that it converges to the unknown function almost surely.

When the data are unevenly distributed and the behaviour of a function changes abruptly, the approximant can present undue oscillations. We present an algorithm to identify a domain decomposition, such that on each subdomain the behaviour of the function is sufficiently homogeneous in order to calculate separate approximants and to blend them together. © 2013 IMACS. Published by Elsevier B.V. All rights reserved.

The aim of the paper is to provide a method for approximating non regular surfaces from a set of scattered data in a faithful way. The method we propose is effective and particularly well-suited for recovering geophysical surfaces with faults or drainage patterns. Some real examples will be presented. © Springer-Verlag 2014.

Background:There are no studies that have identified the ability to recognize and manage delirium among Italian health providers caring for patients at risk. Therefore, the Italian Association of Psychogeriatrics (AIP) conducted a multicenter survey among doctors, nurses, psychologists and physiotherapists to assess their competence regarding the theme of delirium and its management in the everyday clinical practice.Methods:The survey period was 1st June 2013 to 30th November 2013. The invitation to participate was sent via email, with publication on the AIP website. The survey included 14 questions and two case vignettes.Results:A total of 648/1,500 responses were collected. Most responders were doctors (n= 322/800), followed by nurses (n= 225/500), psychologists (n= 51/100), and physiotherapists (n= 30/100). Generally, doctors and psychologists correctly defined delirium, while nurses and physiotherapists did not. The most frequently used diagnostic tools were the Confusion Assessment Method (CAM) and the Diagnostic and Statistical Manual of Mental Disorders-IV. Delirium intensity was rarely assessed. Hypoactive delirium was generally managed with non-pharmacological approaches, while hyperactive delirium with a combination of non-pharmacological and pharmacological approaches. However, possible causes of delirium were under-assessed by half of doctors and by the majority of other professionals. Nurses, psychologists and physiotherapists did not answer the case vignettes, while doctors identified the correct answer in most cases.Conclusions:This is the first Italian survey among health providers caring for patients at risk of delirium. This is also the first survey including doctors, nurses, psychologists and physiotherapists. The results emphasize the importance of training to improve knowledge of this relevant unmet medical need.

In this work, we study a method to recover surfaces with some discontinuity curves from a set of noisy data. By function with singularities, we mean that the function or its partial derivatives are discontinuous across planar curves. The method we present is well suited for correlated data on trajectories (i.e., the data are collected on given tracks as in the case for measurements gathered from ships or aircraft). We propose a method based on wavelet decomposition and splines as well. (C) 2000 Elsevier Science Ltd. All rights reserved

This paper simultaneously generalizes two standard classes of radial kernels, the polyharmonic kernels related to the differential operator (???Δ) m and the Whittle---Matern kernels related to the differential operator (???Δ?+?I) m . This is done by allowing general differential operators of the form $\prod_{j=1}^m(-\Delta+\kappa_j^2I)$ with nonzero ? j and calculating their associated kernels. It turns out that they can be explicity given by starting from scaled Whittle---Matern kernels and taking divided differences with respect to their scale. They are positive definite radial kernels which are reproducing kernels in Hilbert spaces norm-equivalent to $W_2^m(\ensuremath{\mathbb{R}}^d)$ . On the side, we prove that generalized inverse multiquadric kernels of the form $\prod_{j=1}^m(r^2+\kappa_j^2)^{-1}$ are positive definite, and we provide their Fourier transforms. Surprisingly, these Fourier transforms lead to kernels of Whittle---Matern form with a variable scale ?(r) between ? 1,...,? m . We also consider the case where some of the ? j vanish. This leads to conditionally positive definite kernels that are linear combinations of the above variable-scale Whittle---Matern kernels and polyharmonic kernels. Some numerical examples are added for illustration.

In this paper our concern is the recovery of a highly regular function by a discrete set of data with arbitrary distribution. We consider the case of a nonstationary multiquadric interpolant that presents numerical breakdown. Therefore we propose a global least squares multiquadric approximant with a center set of maximal size and obtained by a new thinning technique. The new thinning scheme removes the local bad conditions in order to obtain a well conditioned matrix. The choice of working on local subsets of the data set $X$ provides an effective solution. Some numerical examples to validate the goodness of our proposal are given.

The aim of the paper is to provide a computationally effective way to construct stable bases on general non-degenerate lattices. In particular, we define new stable bases on hexagonal lattices and we give some numerical examples which show their usefulness in applications.

In this paper, we propose a method belonging to the class of special meshless kernel techniques for solving Poisson problems. The method is based on a simple new construction of an approximate particular solution of the nonhomogeneous equation in the space of polyharmonic splines. High order Lagrangian polyharmonic splines are used as a basis to approximate the nonhomogeneous term and a closed-form particular solution is given. The coefficients can be computed by convolution products of known vectors. This can be done in all dimensions, without numerical integration nor solution of systems. Numerical experiments are presented in two dimensions and show the quality of the approximations for different test examples

In this paper we discuss the problem of detecting the faults and or the gradient faults of a function when scattered and noisy data are given.

We propose an adaptive local procedure, which uses the modified Shepard's method with local polyharmonic interpolants. The aim is to reconstruct, in a faithful way, a function known by a large and highly irregularly distributed sample. Such a problem is generally related to the recovering of geophysical surfaces, where the sample is measured according to the behaviour of the surface. The adaptive local procedure is used to calculate, by an efficient algorithm, an interpolating polyharmonic function, when a very large sample is assigned. When we consider a sample of size $N

In this paper we present a technique for recovering functions of three variables when scattered data on tracks are given. The aim is to obtain a good quality of reproduction with a contained computational cost. In order to do this we exploit the data structure which allows to solve lower dimension problems

In this paper we adress the construction of polyharmonic prewavelets in low dimensions, extending the approach of Riemenschneider and Scen [10] to the non-othonormal setting. Such prewavelets are derived from the polyharmonic Lagrangean spline and its dual. We provide formula for an efficient construction of the functions, filters, and the associated multiresolution analysis.

The aim of thepaper is to provide, in any dimensions ,a multiresolution analysis associated witha new family of polyharmonic B-splines, that unlike the elementary B-splines, havea more ``rotational invariant" behavior.

This paper applies difference operators to conditionally positive definite kernels in order to generate kernel $$B$$ -splines that have fast decay towards infinity. Interpolation by these new kernels provides better condition of the linear system, while the kernel $$B$$ -spline inherits the approximation orders from its native kernel. We proceed in two different ways: either the kernel $$B$$ -spline is constructed adaptively on the data knot set $$X$$ , or we use a fixed difference scheme and shift its associated kernel $$B$$ -spline around. In the latter case, the kernel $$B$$ -spline so obtained is strictly positive in general. Furthermore, special kernel $$B$$ -splines obtained by hexagonal second finite differences of multiquadrics are studied in more detail. We give suggestions in order to get a consistent improvement of the condition of the interpolation matrix in applications.

Many papers discuss the problem of recovering a function when a set of data is known on a domain. Most of these papers assume that the size of the data set is large. At our knowledge, little or practically nothing is said about the case in which the sample has a size that is moderate. In this paper we tackle this problem. We indicate a way for its solution and hence we give a concrete example. Some numerical experiments are shown

Under very mild additional assumptions, translates of conditionally positive definite radial basis functions allow unique interpolation to scattered multivariate data, because the interpolation matrices have a symmetric and positive definite dominant part. In many applications, the data density varies locally according to the signal behaviour, and then the translates should get different scalings that match the local data density. Furthermore, if there is a local anisotropy in the data, the radial basis functions should possibly be distorted into functions with ellipsoids as level sets. In such cases, the symmetry and the definiteness of the matrices are no longer guaranteed. However, this brief note is the first paper to provide sufficient conditions for the unique solvability of such interpolation processes. The basic technique is a simple matrix perturbation argument combined with the Ball-Narcowich-Ward stability results. © Springer-Verlag 2004.

In this paper we study a method for the approximation by a monotone smooth function of a discrete set of observations. By this method it is possible to obtain a completely automatic program. We show that the solution is monotone with probability tending to one and by the test examples that it is sufficient in the practical case