Common approximation tools return low-order approximations in the vicinities of singularities. Mo... more Common approximation tools return low-order approximations in the vicinities of singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form f = g + r+ where g, r ∈ C M +1 (R n ) and the function r+ is defined by Given scattered (or uniform) data points X ⊂ R n , we investigate approximation by quasi-interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities.
Given gridded cell-average data of a smooth multivariate function, we present a constructive expl... more Given gridded cell-average data of a smooth multivariate function, we present a constructive explicit procedure for generating a high-order global approximation of the function. One contribution is the derivation of high order approximations to point-values of the function directly from the cell-average data. The second contribution is the development of univariate B-spline based high order quasi-interpolation operators using cell-average data. Multivariate spline quasiinterpolation approximation operators are obtained by tensor products of the univariate operators.
Journal of Computational and Applied Mathematics, 1975
A direct generalization of a convergence acceleration algorithm due to the author to infinite int... more A direct generalization of a convergence acceleration algorithm due to the author to infinite integrals is presented, and is used to obtain an efficient method for numerical inversion of the Laplace transform. Some numerical examples are given and compared with results of methods due to Salzer and Longman.
We consider a problem of great practical interest: the repairing and recovery of a low-dimensiona... more We consider a problem of great practical interest: the repairing and recovery of a low-dimensional manifold embedded in high-dimensional space from noisy scattered data. Suppose that we observe a point cloud sampled from the low-dimensional manifold, with noise, and let us assume that there are holes in the data. Can we recover missing information inside the holes? While in low-dimension the problem was extensively studied, manifold repairing in high dimension is still an open problem. We introduce a new approach, called Repairing Manifold Locally Optimal Projection (R-MLOP), that expands the MLOP method [14], to cope with manifold repairing in low and high-dimensional cases. The proposed method can deal with multiple holes in a manifold. We prove the validity of the proposed method, and demonstrate the effectiveness of our approach by considering different manifold topologies, for single and multiple holes repairing, in low and high dimensions.
This work suggests a new variational approach to the task of computer aided segmentation and rest... more This work suggests a new variational approach to the task of computer aided segmentation and restoration of incomplete characters, residing in a highly noisy document image. We model character strokes as the movement of a pen with a varying radius. Following this model, in order to fit the digital image, a cubic spline representation is being utilized to perform gradient descent steps, while maintaining interpolation at some initial (manually sampled) points. The proposed algorithm was used in the process of restoring approximately 1000 ancient Hebrew characters (dating to ca. 8 th-7 th century BCE), some of which are presented herein and show that the algorithm yields plausible results when applied on deteriorated documents.
The generalization of Shanks’ e-transformation to double series is discussed and a class of nonli... more The generalization of Shanks’ e-transformation to double series is discussed and a class of nonlinear transformations, the [ A / S ] R {[A/S]_R} transformations, for accelerating the convergence of infinite double series is presented. It is constructed so as to sum exactly infinite double series whose terms satisfy certain finite linear double difference equations; in that sense it is a generalization of Shanks’ e-transformation or its equivalent Wynn’s ε \varepsilon -algorithm. A generalization of the [ A / S ] R {[A/S]_R} transformation to N-dimensional series is also presented and their application to power series is discussed and exemplified. Some transformations for accelerating the convergence of infinite double integrals are also obtained, generalizing the confluent ε \varepsilon -algorithm of Wynn and the G-transformation of Gray, Atchison, and McWilliams for infinite 1-D integrals.
In recent years, the moving least-square (MLS) method has been extensively studied for approximat... more In recent years, the moving least-square (MLS) method has been extensively studied for approximation and recon- struction of surfaces. The MLS method involves local weighted least-squares polynomial approximations, using a fast decaying weight function. The local approximating polynomial may be used for approximating the under- lying function or its derivatives. In this paper we consider locally supported weight functions, and
In this paper we introduce positive mean value coordinates (PMVC) for mesh deformation. Following... more In this paper we introduce positive mean value coordinates (PMVC) for mesh deformation. Following the obser- vations of Joshi et al. (JMD 07) we show the advantage of having positive coordinates. The control points of the deformation are the vertices of a "cage" enclosing the deformed mesh. To define positive mean value coordinates for a given vertex, the visible portion of the cage is integrated over a sphere. Unlike MVC (JSW05), PMVC are computed numerically. We show how the PMVC integral can be efficiently computed with graphics hardware. While the properties of PMVC are similar to those of Harmonic coordinates (JMD 07), the setup time of the PMVC is only of a few seconds for typical meshes with 30K vertices. This speed-up renders the new coordinates practical and easy to use.
Journal of Computational and Applied Mathematics, 1992
Dyn, N., D. Levin and S. Rippa, Boundary correction for piecewise linear interpolation defined ov... more Dyn, N., D. Levin and S. Rippa, Boundary correction for piecewise linear interpolation defined over data-dependent triangulations, Journal of Computational and Applied Mathematics 39 (1992) 179-192. Given a set of data points in R' and corresponding data values it is clear that the quality of a Piecewise Linear Interpolating Surface (PLIS) over triangles depends on the specific triangulation of the data points. While conventional triangulation methods depend only on the distribution of the data points in R2, the authors (1990) suggested to construct data-dependent triangulations which depend on the data values as well. Numerical examples indicate that this method improves the PLIS only away from the boundary of the triangulation. In this paper we present and test a numerical scheme for boundary correction to be used as a complementary step to the creation of data-dependent triangulations. This scheme adds more points on the boundary edges of "bad" triangles and estimates corresponding function values. Numerical tests show the success of the scheme in improving the PL!S also near the boundary of the triangulation.
Approximation properties of the dilations of the integer translates of a smooth function, with so... more Approximation properties of the dilations of the integer translates of a smooth function, with some derivatives vanishing at infinity, are studied. The results apply to fundamental solutions of homogeneous elliptic operators and to "shifted" fundamental solutions of the iterated Laplacian. Following the approach from spline theory, the question of polynomial reproduction by quasiinterpolation is addressed first. The analysis makes an essential use of the structure of the generalized Fourier transform of the basis function. In contrast with spline theory, polynomial reproduction is not sufficient for the derivation of exact order of convergence by dilated quasi-interpolants. These convergence orders are established by a careful and quite involved examination of the decay rates of the basis function. Furthermore, it is shown that the same approximation orders are obtained with quasi-interpolants defined on a bounded domain.
We consider the possibility of using locally supported quasi-interpolation operators for the appr... more We consider the possibility of using locally supported quasi-interpolation operators for the approximation of univariate non-smooth functions. In such a case one usually expects the rate of approximation to be lower than that of smooth functions. It is shown in this paper that prior knowledge of the type of 'singularity' of the function can be used to regain the full approximation power of the quasi-interpolation method. The singularity types may include jumps in the derivatives at unknown locations, or even singularities of the form (x − s) α , with unknown s and α. The new approximation strategy includes singularity detection and high-order evaluation of the singularity parameters, such as the above s and α. Using the acquired singularity structure, a correction of the primary quasi-interpolation approximation is computed, yielding the final high-order approximation. The procedure is local, and the method is also applicable to a non-uniform data-point distribution. The paper includes some examples illustrating the high performance of the suggested method, supported by an analysis proving the approximation rates in some of the interesting cases.
We investigate non-stationary orthogonal wavelets based on a non-stationary interpolatory subdivi... more We investigate non-stationary orthogonal wavelets based on a non-stationary interpolatory subdivision scheme reproducing a given set of exponentials with real-valued parameters. The construction is analogous to the construction of Daubechies wavelets using the subdivision scheme of Deslauriers-Dubuc. The main result is the existence and smoothness of these Daubechies type wavelets.
Using refinement subdivision techniques, we construct smooth multiwavelet bases for L 2 (R) and L... more Using refinement subdivision techniques, we construct smooth multiwavelet bases for L 2 (R) and L 2 ([0, 1]) which are in an appropriate sense dual to Alpert orthonormal multiwavelets. Our new multiwavelets allow one to easily give smooth reconstructions of a function purely from knowledge of its local moments. At the heart of our construction is the concept of moment-interpolating (MI) refinement schemes, which interpolate sequences from coarse scales to finer scales while preserving the underlying local moments on dyadic intervals. We show that MI schemes have smooth refinement limits. Our proof technique exhibits an intimate intertwining relation between MI schemes and Hermite schemes. This intertwining relation is then used to infer knowledge about moment-interpolating schemes from knowledge about Hermite schemes. Our MI multiwavelets make Riesz bases for L 2 and unconditional bases of a variety of smoothness spaces, so they can efficiently represent smooth functions. We here derive an algorithm which rapidly develops a piecewise polynomial fit to data by recursive dyadic partitioning and then rapidly produces a smooth reconstruction with matching local moments on pieces of the partition. This avoids the blocking effect suffered by piecewise polynomial fitting.
Subdivision schemes are special multi-resolution analysis (MRA) methods that have become prevalen... more Subdivision schemes are special multi-resolution analysis (MRA) methods that have become prevalent in computer-aided geometric design. This paper draws useful analogies between the mathematics of subdivision schemes and the hierarchical structures of music compositions. Based on these analogies, we propose new methods for music synthesis and analysis through MRA, which provide a different perspective on music composition, representation and analysis. We demonstrate that the structure and recursive nature of the recently proposed subdivision models [S. Hed and D. Levin, Subdivision models for varying-resolution and generalized perturbations, Int. J. Comput. Math. 88(17) (2011), pp. 3709-3749; S. Hed and D. Levin, A 'subdivision regression' model for data analysis, 2012, in preparation] are well suited to the synthesis and analysis of monophonic and polyphonic musical patterns, doubtless due in large part to the strongly hierarchical nature of traditional musical structures. The analysis methods demonstrated enable the compression and decompression (reconstruction) of selected musical pieces and derive useful features of the pieces, laying groundwork for music classification.
Subdivision schemes generate self-similar curves and surfaces. Therefore there is a close connect... more Subdivision schemes generate self-similar curves and surfaces. Therefore there is a close connection between curves and surfaces generated by subdivision algorithms and self-similar fractals generated by Iterated Function Systems (IFS). We demonstrate that this connection between subdivision schemes and fractals is even deeper by showing that curves and surfaces generated by subdivision are also attractors, fixed points of IFS's. To illustrate this fractal nature of subdivision, we derive the associated IFS for many different subdivision curves and surfaces without extraordinary vertices, including B-splines, piecewise Bezier, interpolatory four-point subdivision, bicubic subdivision, three-direction quartic box-spline subdivision and Kobbelt's √3-subdivision surfaces. Conversely, we shall show how to build subdivision schemes to generate traditional fractals such as the Sierpinski gasket and the Koch curve, and we demonstrate as well how to control the shape of these fracta...
The problem of interpolation by a convex curve to the vertices of a convex polygon is considered.... more The problem of interpolation by a convex curve to the vertices of a convex polygon is considered. A natural 1-parameter family of C a algebraic curves solving this problem is presented. This is extended to a solution of a general Hermite-type problem, in which the curve also interpolates to one or two prescribed tangents at any desired vertices of the polygon. The construction of these curves is a generalization of well known methods for generating conic sections. Several properties of this family of algebraic curves are discussed. In addition, the method is generalized to convex C a interpolation of strictly convex data sets in II~ 3 by algebraic surfaces.
Proceedings of the National Academy of Sciences of the United States of America, Jan 11, 2016
The relationship between the expansion of literacy in Judah and composition of biblical texts has... more The relationship between the expansion of literacy in Judah and composition of biblical texts has attracted scholarly attention for over a century. Information on this issue can be deduced from Hebrew inscriptions from the final phase of the first Temple period. We report our investigation of 16 inscriptions from the Judahite desert fortress of Arad, datedca 600 BCE-the eve of Nebuchadnezzar's destruction of Jerusalem. The inquiry is based on new methods for image processing and document analysis, as well as machine learning algorithms. These techniques enable identification of the minimal number of authors in a given group of inscriptions. Our algorithmic analysis, complemented by the textual information, reveals a minimum of six authors within the examined inscriptions. The results indicate that in this remote fort literacy had spread throughout the military hierarchy, down to the quartermaster and probably even below that rank. This implies that an educational infrastructure ...
Advances in Computational Mathematics - Adv. Comput. Math., 1999
Non‐uniform binary linear subdivision schemes, with finite masks, over uniform grids, are studied... more Non‐uniform binary linear subdivision schemes, with finite masks, over uniform grids, are studied. A Laurent polynomial representation is suggested and the basic operations required for smoothness analysis are presented. As an example it is shown that the interpolatory 4‐point scheme is C 1 with an almost arbitrary non‐uniform choice of the free parameter.
Journal of Computational and Applied Mathematics, 1999
A general method for near-best approximations to functionals on R d , using scattered-data inform... more A general method for near-best approximations to functionals on R d , using scattered-data information, is applied for producing stable multidimensional integration rules. The rules are constructed to be exact for polynomials of degree 6m and, for a quasi-uniform distribution of the integration points, it is shown that the approximation order is O(h m+1) where h is an average distance between the data points.
This paper presents a new subdivision scheme that operates over an infinite triangulation, which ... more This paper presents a new subdivision scheme that operates over an infinite triangulation, which is regular except for a single extraordinary vertex. The scheme is based on the quartic three-directional Box-spline scheme, and is guaranteed to generate C 2 limit functions whenever the valency n of the extraordinary vertex is in the range 4 ≤ n ≤ 20. The new scheme differs from the commonly used subdivision schemes by the fact that it applies special subdivision rules near edges of the original triangulation, which emanate from the extraordinary vertex, and not only in the vicinity of the extraordinary vertex.
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Papers by David Levin