Laura Bacchelli Montefusco

Radial basis functions for the multivariate interpolation of large scattered data sets

An efficient method for the multivariate interpolation of very large scattered data sets is presented. It is based on the local use of radial basis functions and represents a further improvement of the well known Shepards method. Although the latter is simple and well suited for multivariate interpolation, it does not share the good reproduction quality of other methods widely used for bivariate interpolation. On the other band, radial basis functions, which have proven to be highly useful for multivariate scattered data interpolation, have a severe drawback. They are unable to interpolate large sets in an efficient and numerically stable way and maintain a good level of reproduction quality at the same time. Both problems have been circumvented using radial basis functions to evaluate the nodal function of the modified Shepards method. This approach exploits the flexibility of the method and improves its reproduction quality. The proposed algorithm has been implemented and numerical results confirm its efficiency.

Multiwavelet analysis and signal processing

In this paper we present some results and applications concerning the recent theory of multiscaling functions and multiwavelets. In particular, we present the theory in compact notation with the use of some types of recursive block matrices. This allows a flexible schematization of the construction of semi-orthogonal multiwavelets. As in the scalar case, an efficient algorithm for the computation of the coefficients of a multiwavelet transform can be obtained, in which r input sequences are involved. This is a crucial point: the choice of a good prefilter which can provide a good approximation of the true initial coefficient sequences, when applied to the input data, is critical in the context of multiwavelet analysis. We explore this problem with concrete examples, showing the strong dependence of the prefilter on the chosen multiwavelet basis. Finally, an application of the multiwavelet-based algorithm to signal compression is shown. The goal is both to compare the results obtained with different multiwavelet bases, and to test the effectiveness of multiwavelets in this kind of problem with respect to scalar wavelets.

Image compression through embedded multiwavelet transform coding

In this paper, multiwavelets are considered in the context of image compression and two orthonormal multiwavelet bases are experimented, each used in connection with its proper prefilter. For evaluating the effectiveness of multiwavelet transform for coding images at low bit-rates, an efficient embedded coding of multiwavelet coefficients has been realized. The performance of this multiwavelet-based coder is compared with the results obtained for scalar wavelets.

A Fast Compressed Sensing Approach to 3D MR Image Reconstruction

The problem of high-resolution image volume reconstruction from reduced frequency acquisition sequences has drawn significant attention from the scientific community because of its practical importance in medical diagnosis. To address this issue, several reconstruction strategies have been recently proposed, which aim to recover the missing information either by exploiting the spatio-temporal correlations of the image series, or by imposing suitable constraints on the reconstructed image volume. The main contribution of this paper is to combine both these strategies in a compressed sensing framework by exploiting the gradient sparsity of the image volume. The resulting constrained 3D minimization problem is then solved using a penalized forward-backward splitting approach that leads to a convergent iterative two-step procedure. In the first step, the updating rule accords with the sequential nature of the data acquisitions, in the second step a truly 3D filtering strategy exploits the spatio-temporal correlations of the image sequences. The resulting NFCS-3D algorithm is very general and suitable for several kinds of medical image reconstruction problems. Moreover, it is fast, stable and yields very good reconstructions, even in the case of highly undersampled image sequences. The results of several numerical experiments highlight the optimal performance of the proposed algorithm and confirm that it is competitive with state of the art algorithms.

Shape preserving surface reconstruction using locally anisotropic radial basis function interpolants

In this paper we deal with the problem of reconstructing surfaces from unorganized sets of points, while capturing the significant geometry details of the modelled surface, such as edges, flat regions, and corners. This is obtained by exploiting the good approximation capabilities of the radial basis functions (RBF), the local nature of the method proposed in [1], and introducing information on shape features and data anisotropies detected from the given surface points. The result is a shape-preserving reconstruction, given by a weighted combination of locally aniso tropic interpolants. For each local interpolant the anisotropy is obtained by replacing the Euclidean norm with a suitable metric which takes into account the local distribution of the points. Thus hyperellipsoid basis functions, named anisotropic RBFs, are defined. Results from the application of the method to the reconstruction of object surfaces in @?^3 are presented, confirming the effectiveness of the approach.

Nonlinear Filtering for Sparse Signal Recovery From Incomplete Measurements

The problem of recovering sparse signals and sparse gradient signals from a small collection of linear measurements is one that arises naturally in many scientific fields. The recently developed Compressed Sensing Framework states that such problems can be solved by searching for the signal of minimum L 1-norm, or minimum Total Variation, that satisfies the given acquisition constraints. While L 1 optimization algorithms, based on Linear Programming techniques, are highly effective at generating excellent signal reconstructions, their complexity is still too high and renders them impractical for many real applications. In this paper, we propose a novel approach to solve the L 1 optimization problems, based on the use of suitable nonlinear filters widely applied for signal and image denoising. The corresponding algorithm has two main advantages: low computational cost and reconstruction capabilities similar to those of Linear Programming optimization methods. We illustrate the effectiveness of the proposed approach with many numerical examples and comparisons.

Fast Sparse Image Reconstruction Using Adaptive Nonlinear Filtering

Compressed sensing is a new paradigm for signal recovery and sampling. It states that a relatively small number of linear measurements of a sparse signal can contain most of its salient information and that the signal can be exactly reconstructed from these highly incomplete observations. The major challenge in practical applications of compressed sensing consists in providing efficient, stable and fast recovery algorithms which, in a few seconds, evaluate a good approximation of a compressible image from highly incomplete and noisy samples. In this paper, we propose to approach the compressed sensing image recovery problem using adaptive nonlinear filtering strategies in an iterative framework, and we prove the convergence of the resulting two-steps iterative scheme. The results of several numerical experiments confirm that the corresponding algorithm possesses the required properties of efficiency, stability and low computational cost and that its performance is competitive with those of the state of the art algorithms.

Fast surface reconstruction and hole filling using positive definite radial basis functions

Abstract Surface reconstruction from large unorganized data sets is very challenging, especially if the data present undesired holes. This is usually the case when the data come from laser scanner 3D acquisitions or if they represent damaged objects to be restored. An attractive field of research focuses on situations in which these holes are too geometrically and topologically complex to fill using triangulation algorithms. In this work a local approach to surface reconstruction from point-clouds based on positive definite Radial Basis Functions (RBF) is presented that progressively fills the holes by expanding the neighbouring information. The method is based on the algorithm introduced in [7] which has been successfully tested for the smooth multivariate interpolation of large scattered data sets. The local nature of the algorithm allows for real time handling of large amounts of data, since the computation is limited to suitable small areas, thus avoiding the critical efficiency problem involved in RBF multivariate interpolation. Several tests on simulated and real data sets demonstrate the efficiency and the quality of the reconstructions obtained using the proposed algorithm.

The regularizing properties of anisotropic Radial Basis Functions

In the present work we consider the problem of interpolating scattered data using radial basis functions (RBF). In general, it is well known that this leads to a discrete linear inverse problem that needs to be regularized in order to provide a meaningful solution. The work focuses on a metric-regularization approach, based on a new class of RBF, called anisotropic RBF. The work provides theoretical justifications for the regularization approach and it considers a suitable proposal for the metric, supporting it by numerical examples.

Recursive properties of Toeplitz and Hurwitz matrices

Abstract Banded Toeplitz and Hurwitz matrices are shown to be particular cases of a more general class of biinfinite matrices, called recursive matrices. The main features of Toeplitz and Hurwitz matrices can thereby be seen to be immediate consequences of a fundamental theorem about recursive matrices, called the product rule. Moreover, some properties of products of Toeplitz and Hurwitz matrices can be proved by similar arguments. Some applications related to the general theory of compactly supported wavelets are presented.