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We construct some smooth functions defined over a sphere that interpolate large sets of scattered data, using some modified Shepard methods, the least squares thin-plate spline and the inverse multiquadric functions. We illustrate the benefits of our methods in numerical examples for some test functions and some real data applications. [ABSTRACT FROM AUTHOR]

Standard interpolatory subdivision schemes and their underlying interpolating refinable functions are of interest in CAGD, numerical PDEs, and approximation theory. Generalizing these notions, we introduce and study n s -step interpolatory M -subdivision schemes and their interpolating M -refinable functions with n s ∈ N ∪ { ∞ } and a dilation factor M ∈ N \ { 1 } . We completely characterize C m -convergence and smoothness of n s -step interpolatory subdivision schemes and their interpolating M -refinable functions in terms of their masks. Inspired by n s -step interpolatory stationary subdivision schemes, we further introduce the notion of r-mask quasi-stationary subdivision schemes, and then we characterize their C m -convergence and smoothness properties using only their masks. Moreover, combining n s -step interpolatory subdivision schemes with r-mask quasi-stationary subdivision schemes, we can obtain r n s -step interpolatory subdivision schemes. Examples and construction procedures of convergent n s -step interpolatory M -subdivision schemes are provided to illustrate our results with dilation factors M = 2 , 3 , 4 . In addition, for the dyadic dilation M = 2 and r = 2 , 3 , using r masks with only two-ring stencils, we provide examples of C r -convergent r-step interpolatory r-mask quasi-stationary dyadic subdivision schemes. [ABSTRACT FROM AUTHOR]

This article explores various techniques for generating fractal-like Bézier curves in both 2D and 3D environments. It delves into methods such as subdivision schemes, Iterated Function System (IFS) theory, perturbation of Bézier curves, and perturbation of Bézier basis functions. The article outlines conditions on subdivision matrices necessary for convergence and demonstrates their use in creating an IFS with an attractor aligned to the convergent point of the subdivision scheme based on specified initial data. Additionally, it discusses conditions for obtaining a one-sided approximation of a given Bézier curve through perturbation. The article also addresses considerations for perturbed Bézier basis functions to construct fractal-like Bézier curves that remain within the convex hull polygon/polyhedron defined by control points. These methods find applications in various fields, including computer graphics, art, and design. [ABSTRACT FROM AUTHOR]

Composition detection and semantic line detection are important research topics in computer vision and play an important auxiliary role in the analysis of image esthetics. However, at present, few researchers have considered the internal relationship between these two related tasks for comprehensive research. In order to solve this problem, we propose a synchronous detection network of composition class and semantic lines based on cross-attention, which can realize the mutual supervision and guidance between composition class detection and semantic line detection, to improve the accuracy of each other’s detection. First, the pre-trained composition detection model and the pre-trained semantic line detection model as two teacher models to provide data labels of composition and semantic line information for the student model. Then, we train a student model with the help of the teacher model. The student model adopts the multi-task learning architecture by combining soft and hard parameter sharing, as we propose. At the same time, we develop a cross-attention module to ensure that both tasks get the help and supervision they need from each other. Experimental results show that our method can draw semantic lines while detecting composition classes, which increases the interpretability of composition class detection. Our composition detection accuracy reaches 92.57%, and for benchmark semantic lines, the accuracy of our AUC_A metric can reach 92.00%. [ABSTRACT FROM AUTHOR]

Linear approximation methods suffer from Gibbs oscillations when approximating functions with jumps. Essentially non oscillatory subcell-resolution (ENO-SR) is a local technique avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper jump or corner (jump in the first order derivative) discontinuity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. This new procedure can be applied as subdivision scheme to design curves and surfaces both in point-value and in cell-average contexts. Using the proposed algorithm, we are able to construct approximations with high precision, with high piecewise regularity, and without smearing nor oscillations in the presence of discontinuities. These are desired properties in real applications as computer aided design or car design, among others. [ABSTRACT FROM AUTHOR]

A novel approach to zipper fractal interpolation theory for functions of several variables is presented. Multivariate zipper fractal functions are constructed and then perturbed through free choices of base functions, scaling functions, and a binary matrix called signature to obtain their zipper α-fractal versions. In particular, we propose a multivariate Bernstein zipper fractal function and study its coordinate-wise monotonicity which depends on the values of signature. We derive bounds for the graph of a multivariate zipper fractal function by imposing conditions on the scaling factors and the Hölder exponent of the associated germ function and base function. The box dimension result for multivariate Bernstein zipper fractal function is derived. Finally, we study some constrained approximation properties for multivariate zipper Bernstein fractal functions. [ABSTRACT FROM AUTHOR]

In this paper, a kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator is studied by combining the known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate Bernoulli polynomials. Some error bounds and convergence rates of the combined operators are studied. A selection of numerical examples is presented to compare the performances of the obtained scheme. Furthermore, our method can be applied to time-dependent differential equations. Its advantage is that the algorithm is very simple and easy to implement. [ABSTRACT FROM AUTHOR]

Digitized handwritten documents have been used for various purposes, including age detection, a crucial area of research in fields like forensic investigation and medical diagnosis. Automated age recognition is deemed to be a difficult task, due to the great degree of similarity and overlap across people's handwriting. Consequently, the efficiency of the classification system is determined by extracting pertinent features from handwritten documents. This research proposes a set of age-related features suggested by a graphologist to classify handwritten documents into two age groups: youth adult and mature adult. The extracted features are: slant irregularity (SI), pen pressure irregularity (PPI), text lines irregularity (TLI) and the percentage of black and white pixels (PWB). Support Vector Machines (SVM) and Neural Network (NN) classifiers have been used to train, validate and test the proposed approach using two different datasets: the FSHS and the Khatt datasets. When applied to the FSHS dtaset using SVM and NN approaches, the proposed method resulted in a classification accuracy of 71% and 63.5%, respectively. Meanwhile, when applied to the Khatt dataset, our method outperformed state-of-the-art methods with a classification accuracy of 65.2% and 67% utilising SVM and NN classifiers, respectively. These are the best rates available right now in this field. [ABSTRACT FROM AUTHOR]

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed. [ABSTRACT FROM AUTHOR]

Facial expression generation has always been an intriguing task for scientists and researchers all over the globe. In this context, we present our novel approach for generating videos of the six basic facial expressions. Starting from a single neutral facial image and a label indicating the desired facial expression, we aim to synthesize a video of the given identity performing the specified facial expression. Our approach, referred to as FEV-GAN (Facial Expression Video GAN), is based on Spatio-temporal Convolutional GANs, that are known to model both content and motion in the same network. Previous methods based on such a network have shown a good ability to generate coherent videos with smooth temporal evolution. However, they still suffer from low image quality and low identity preservation capability. In this work, we address this problem by using a generator composed of two image encoders. The first one is pre-trained for facial identity feature extraction and the second for spatial feature extraction. We have qualitatively and quantitatively evaluated our model on two international facial expression benchmark databases: MUG and Oulu-CASIA NIR&VIS. The experimental results analysis demonstrates the effectiveness of our approach in generating videos of the six basic facial expressions while preserving the input identity. The analysis also proves that the use of both identity and spatial features enhances the decoder ability to better preserve the identity and generate high-quality videos. The code and the pre-trained model will soon be made publicly available.

An inverse central ordering of the nodes is proposed for the Newton interpolation formula. This ordering may improve the stability for certain distributions of nodes. For equidistant nodes, an upper bound of the conditioning is provided. This bound is close to the bound of the conditioning in the Lagrange interpolation formula, whose conditioning is the lowest. This ordering is related to a pivoting strategy of a matrix elimination procedure called Neville elimination. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]

The Floater--Hormann family of rational interpolants do not have spuriouspoles or unattainable points, are efficient to calculate, and have arbitrarilyhigh approximation orders. One concern when using them is that theamplification of rounding errors increases with approximation order, and canmake balancing the interpolation error and rounding error difficult. Thisarticle proposes to modify the Floater--Hormann interpolants by includingadditional local polynomial interpolants at the ends of the interval. Thisappears to improve the conditioning of the interpolants and allow higherapproximation orders to be used in practice.

This paper introduces a novel technique to approximate a given continuous function f defined on a real compact interval by a new class of zipper α -fractal functions which contain a scaling vector and a binary vector or signature. For specific choices of scaling and signature vectors, the corresponding zipper fractal functions simultaneously interpolate and approximate f. When signature is zero, the proposed zipper fractal functions coincide with existing α -fractal functions. Hence, the zipper approximation proposed in this manuscript generalizes the existing fractal and classical approximations. Zipper fractal analogue of some elementary results in the classical approximation theory are obtained. Using convex optimization technique, we investigate the existence of optimal zipper fractal function for a given continuous function. The parameter identification problems for zipper α -fractal approximants are investigated. We derive sufficient conditions on the parameters of zipper α -fractal functions so that these functions preserve the positivity, monotonicity and convexity of the original function f. Also, we have studied the constructions of k-times continuously differentiable zipper α -fractal functions and one sided zipper fractal approximants for f. Numerical illustrations are provided to support the proposed theoretical results on zipper α -fractal functions. [ABSTRACT FROM AUTHOR]

The paper introduces new sufficient conditions of strict positive definiteness for kernels on d-dimensional spheres which are not radially symmetric but possess specific coefficient structures. The results use the series expansion of the kernel in spherical harmonics. The kernels either have a convolutional form or are axially symmetric with respect to one axis. The given results on convolutional kernels generalise the result derived by Chen et al. (Proc Am Math Soc 131:2733–2740, 2003) for radial kernels. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to provide an efficient scheme for constructing kernel-based reduced interpolation models. In the existing literature such problems are mainly addressed via the well-established knot insertion or knot removal schemes. Such iterative strategies are usually quite demanding from a computational point of view and our goal is to study an efficient implementation for data removal approaches, namely efficient reduced basis algorithm (ERBA). Focusing on kernel-based interpolation, the algorithm makes use of two iterative rules for removing data. The former, called ERBA-r, is based on classical residual evaluations. The latter, namely ERBA-p, is independent of the function values and relies on error bounds defined by the power function. In both cases, inspired by the so-called extended Rippa’s algorithm, our ERBA takes advantage of a fast implementation. [ABSTRACT FROM AUTHOR]

Classical rational interpolation usually enjoys better approximation properties than polynomial interpolation because it avoids wild oscillations and exhibits exponential convergence for approximating analytic functions. We develop a rational interpolation operator, which not only preserves the advantage of classical rational interpolation, but also has a finite Lebesgue constant. In particular, it is convergent for approximating any continuous function, and the convergence rate of the interpolants approximating a function is obtained using the modulus of continuity. [ABSTRACT FROM AUTHOR]

In this paper we analyze a greedy procedure to approximate a linear functional defined in a reproducing kernel Hilbert space by nodal values. This procedure computes a quadrature rule which can be applied to general functionals. For a large class of functionals, that includes integration functionals and other interesting cases, but does not include differentiation, we prove convergence results for the approximation by means of quasi-uniform and greedy points which generalize in various ways several known results. A perturbation analysis of the weights and node computation is also discussed. Beyond the theoretical investigations, we demonstrate numerically that our algorithm is effective in treating various integration densities, and that it is even very competitive when compared to existing methods for Uncertainty Quantification. [ABSTRACT FROM AUTHOR]

The mapped bases or Fake Nodes Approach (FNA), introduced in De Marchi et al. (J Comput Appl Math 364:112347, 2020c), allows to change the set of nodes without the need of resampling the function. Such scheme has been successfully applied for mitigating the Runge's phenomenon, using the S-Runge map, or the Gibbs phenomenon, with the S-Gibbs map. However, the original S-Gibbs suffers of a subtle instability when the interpolant is constructed at equidistant nodes, due to the Runge'sphenomenon. Here, we propose a novel approach, termed Gibbs–Runge-Avoiding Stable Polynomial Approximation (GRASPA), where both Runge's and Gibbs phenomena are mitigated simultaneously. After providing a theoretical analysis of the Lebesgue constant associated with the mapped nodes, we test the new approach by performing various numerical experiments which confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

• New model for scene text VQA that jointly reasons about textual and visual modalities. • Attending on multi-modal features is better than attending separately to each modality. • Grid features from an object detection backbone proves to work well in this task. • The proposed model is up to 5 times faster than current state of the art models. • Combined with a standard VQA model reaches state of the art on mixed datasets. This paper presents a new model for the task of scene text visual question answering. In this task questions about a given image can only be answered by reading and understanding scene text. Current state of the art models for this task make use of a dual attention mechanism in which one attention module attends to visual features while the other attends to textual features. A possible issue with this is that it makes difficult for the model to reason jointly about both modalities. To fix this problem we propose a new model that is based on an single attention mechanism that attends to multi-modal features conditioned to the question. The output weights of this attention module over a grid of multi-modal spatial features are interpreted as the probability that a certain spatial location of the image contains the answer text to the given question. Our experiments demonstrate competitive performance in two standard datasets with a model that is × 5 faster than previous methods at inference time. Furthermore, we also provide a novel analysis of the ST-VQA dataset based on a human performance study. Supplementary material, code, and data is made available through this link. [ABSTRACT FROM AUTHOR]

The data driven extrapolation requires the definition of a functional model depending on the available data and has the application scope of providing reliable predictions on the unknown dynamics. Since data might be scattered, we drive our attention towards kernel models that have the advantage of being meshfree. Precisely, the proposed numerical method makes use of the so-called Variably Scaled Kernels (VSKs), which are introduced to implement a feature augmentation-like strategy based on discrete data. Due to the possible uncertainty on the data and since we are interested in modelling the behaviour of the target functions, we seek for a regularized solution by ridge regression. Focusing on polyharmonic splines, we investigate their implementation in the VSK setting and we provide error bounds in Beppo–Levi spaces. The performances of the method are then tested on functions showing exponential or rational decay. Comparisons with Support Vector Regression (SVR) are also carried out and highlight that the proposed approach is effective, particularly since it does not require to train complex architecture constructions. [ABSTRACT FROM AUTHOR]

• A mental stress state diagnostic for timely intervention from clinicians. • Deep neural network with hierarchical learning capabilities using IoT based biomarkers. • Learns using both wrist-based and chest-based sensor bio-signal features. • Model-level fusion of high-level representations for stress state classification. Affective state recognition at an early stage can help in mood stabilization, stress and depression management for mental well-being. Pro-active and remote mental healthcare warrants the use of various biomarkers to detect the affective mental state of the individual by evaluating the daily activities. With the easy accessibility of IoT-based sensors for healthcare, observable and quantifiable characteristics of our body, physiological changes in the body can be measured and tracked using various wearable devices. This work puts forward a model for mental stress state detection using sensor-based bio-signals. A multi-level deep neural network with hierarchical learning capabilities of convolution neural network is proposed. Multivariate time-series data consisting of both wrist-based and chest-based sensor bio-signals is trained using a hierarchy of networks to generate high-level features for each bio-signal feature. A model-level fusion strategy is proposed to combine the high-level features into one unified representation and classify the stress states into three categories as baseline, stress and amusement. The model is evaluated on the WESAD benchmark dataset for mental health and compares favourably to state-of-the-art approaches giving a superlative performance accuracy of 87.7%. [ABSTRACT FROM AUTHOR]

It is well understood that boundary conditions (BCs) may cause global radial basis function (RBF) methods to become unstable for hyperbolic conservation laws (CLs). Here we investigate this phenomenon and identify the strong enforcement of BCs as the mechanism triggering such stability issues. Based on this observation we propose a technique to weakly enforce BCs in RBF methods. In the case of hyperbolic CLs, this is achieved by carefully building RBF methods from the weak form of the CL, rather than the typically enforced strong form. Furthermore, we demonstrate that global RBF methods may violate conservation, yielding physically unreasonable solutions when the approximation does not take into account these considerations. Numerical experiments validate our theoretical results. [ABSTRACT FROM AUTHOR]

In this article we present a new adaptive algorithm for solving 2D interpolation problems of large scattered data sets through the radial basis function partition of unity method. Unlike other time-consuming schemes this adaptive method is able to efficiently deal with scattered data points with highly varying density in the domain. This target is obtained by decomposing the underlying domain in subdomains of variable size so as to guarantee a suitable number of points within each of them. The localization of such points is done by means of an efficient search procedure that depends on a partition of the domain in square cells. For each subdomain the adaptive process identifies a predefined neighborhood consisting of one or more levels of neighboring cells, which allows us to quickly find all the subdomain points. The algorithm is further devised for an optimal selection of the local shape parameters associated with radial basis function interpolants via leave-one-out cross validation and maximum likelihood estimation techniques. Numerical experiments show good performance of this adaptive algorithm on some test examples with different data distributions. The efficacy of our interpolation scheme is also pointed out by solving real world applications. [ABSTRACT FROM AUTHOR]

This document summarises the current theoretical and experimental status of the di-Higgs boson production searches, and of the direct and indirect constraints on the Higgs boson self-coupling, with the wish to serve as a useful guide for the next years. The document discusses the theoretical status, including state-of-the-art predictions for di-Higgs cross sections, developments on the effective field theory approach, and studies on specific new physics scenarios that can show up in the di-Higgs final state. The status of di-Higgs searches and the direct and indirect constraints on the Higgs self-coupling at the LHC are presented, with an overview of the relevant experimental techniques, and covering all the variety of relevant signatures. Finally, the capabilities of future colliders in determining the Higgs self-coupling are addressed, comparing the projected precision that can be obtained in such facilities. The work has started as the proceedings of the Di-Higgs workshop at Colliders, held at Fermilab from the 4th to the 9th of September 2018, but it went beyond the topics discussed at that workshop and included further developments. FERMILAB-CONF-19-468-E-T, LHCHXSWG-2019-005

In 2012 the ATLAS and CMS collaborations discovered at the LHC the Higgs boson decaying to vector bosons. This discovery has provided a strong indication that the mechanism of Electroweak Symmetry Breaking (EWSB) is similar to the one predicted by Brout-Englert-Higgs (BEH) nearly 50 years before. Since then, one of the priorities of the LHC program, as well as of the majority of the future collider proposals, is to measure directly the parameters of the EWSB potential. The goal is to identify if it has indeed the straightforward quartic shape predicted by BEH or it is more complex, as the result of an unexplored physics nature. The answer to this major scientific question will have a considerable impact on our understanding of vacuum properties and the history of the universe through the EWSB during the Big Bang. The only direct way to probe these couplings is through the measure of the production of multiple Higgs bosons, two being the simplest case. In this paper, we present a comprehensive review of the current searches and the state of the art insights on the topic. In particular, we explain why this ambitious project is even more challenging than the discovery of the Higgs boson itself. Finally, we sketch the plans of the HEP community for how to access the parameters of the BEH mechanism. This review is adapted to a curious reader familiar with particle physics in general or a scientist who wants to have a landscape overview of the topic.

Imaging methods based on the absorption or scattering of atmospheric muons, collectively named under the neologism “muography”, exploit the abundant natural flux of muons produced from cosmic-ray interactions in the atmosphere. Recent years have seen a steep rise in the development of muography methods in a variety of innovative multidisciplinary approaches to study the interior of natural or human-made structures, establishing synergies between usually disconnected academic disciplines such as particle physics, geology, and archaeology. Muography also bears promise of immediate societal impact through geotechnical investigations, nuclear waste surveys, homeland security, and natural hazard monitoring. Our aim is to provide an introduction to this vibrant research area, starting from the physical principles at the basis of the methods and describing the main detector technologies and imaging tools, including their combination with conventional techniques from other disciplines, where appropriate. Then, we discuss critically some outstanding issues that affect a broad variety of applications, and the current state of the art in addressing them. Finally, we review several recent developments in the application of muography methods to specific use cases, without any pretence of exhaustiveness.

Demonstration of the light radiation pressure on a microscopic level by A. Ashkin led to the invention of optical tweezers (OT). Applied in the studies of living systems, OT have become a preferable instrument for the noninvasive study of microobjects, allowing manipulation and measurement of the mechanical properties of molecules, organelles, and cells. In the present paper, we overview OT applications in hemorheological research, placing emphasis on red blood cells but also discussing OT applications for the investigation of the biomechanics of leukocytes and platelets. Blood properties have always served as a primary parameter in medical diagnostics due to the interconnection with the physiological state of an organism. Despite blood testing being a well-established procedure of conventional medicine, there are still many complex processes that must be unraveled to improve our understanding and contribute to future medicine. OT are advancing single-cell research, promising new insights into individual cell characteristics compared to the traditional approaches. We review the fundamental and practical findings revealed in blood research through the optical manipulation, stretching, guiding, immobilization, and inter-/intracellular force measurements of single blood cells.

The complexity of collider data analyses has dramatically increased from early colliders to the CERN LHC. Reconstruction of the collision products in the particle detectors has reached a point that requires dedicated publications documenting the techniques, and periodic retuning of the algorithms themselves. Analysis methods evolved to account for the increased complexity of the combination of particles required in each collision event (final states) and for the need of squeezing every last bit of sensitivity from the data; physicists often seek to fully reconstruct the final state, a process that is mostly relatively easy at lepton colliders but sometimes exceedingly difficult at hadron colliders to the point of requiring sometimes using advanced statistical techniques such as machine learning. The need for keeping the publications documenting results to a reasonable size implies a greater level of compression or even omission of information with respect to publications from twenty years ago. The need for compression should however not prevent sharing a reasonable amount of information that is essential to understanding a given analysis. Infrastructures like Rivet or HepData have been developed to host additional material, but physicists in the experimental Collaborations often still send an insufficient amount of material to these databases. In this manuscript I advocate for an increase in the information shared by the Collaborations, and try to define a minimum standard for acceptable level of information when reporting the results of statistical procedures in High Energy Physics publications.

Scattered data interpolation can be used to approximate a multivariate function by a linear combination of positive definite radial basis functions (RBFs). In practice, the approximation error stagnates (due to numerical instability) even if the function is smooth and the number of data centers is increased. A smaller approximation error can be obtained using multilevel interpolation on a sequence of nested subsets of the initial set of centers. For the construction of these nested subsets, we compare two thinning algorithms from the literature, a greedy algorithm based on nearest neighbor computations and a Poisson point process. The main novelty of our approach lies in the use of H -matrices both for the solution of linear systems and for the evaluation of residual errors at each level. For the solution of linear systems, we use GMRes combined with a domain decomposition preconditioner. Using H -matrices allows us to solve larger problems more efficiently compared with multilevel interpolation based on dense matrices. Numerical experiments with up to 50,000 scattered centers in two and three spatial dimensions demonstrate that the computational time required for the construction of the multilevel interpolant using H -matrices is of almost linear complexity with respect to the number of centers. [ABSTRACT FROM AUTHOR]

• Enhance face verification performance for demographics effected by decision bias. • Operates on comparison score-level to enable using existing face recognition models. • Based on the notation of individual fairness to treat similar face groups similarly. • Effectiveness proved on three databases and two face embeddings. • Unsupervised enhancement of the overall and intra-class recognition performance. Current face recognition systems achieve high progress on several benchmark tests. Despite this progress, recent works showed that these systems are strongly biased against demographic sub-groups. Consequently, an easily integrable solution is needed to reduce the discriminatory effect of these biased systems. Previous work mainly focused on learning less biased face representations, which comes at the cost of a strongly degraded overall recognition performance. In this work, we propose a novel unsupervised fair score normalization approach that is specifically designed to reduce the effect of bias in face recognition and subsequently lead to a significant overall performance boost. Our hypothesis is built on the notation of individual fairness by designing a normalization approach that leads to treating "similar" individuals "similarly". Experiments were conducted on three publicly available datasets captured under controlled and in-the-wild circumstances. Results demonstrate that our solution reduces demographic biases, e.g. by up to 82.7% in the case when gender is considered. Moreover, it mitigates the bias more consistently than existing works. In contrast to previous works, our fair normalization approach enhances the overall performance by up to 53.2% at false match rate of 10 − 3 and up to 82.9% at a false match rate of 10 − 5. Additionally, it is easily integrable into existing recognition systems and not limited to face biometrics. [ABSTRACT FROM AUTHOR]

The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders. [ABSTRACT FROM AUTHOR]

In this paper, the properties of a new family of nonlinear dyadic subdivision schemes are presented and studied depending on the conditions imposed to the mean used to rewrite the linear scheme upon which the new scheme is based. The convergence, stability, and order of approximation of the schemes of the family are analyzed in general. Also, the elimination of the Gibbs oscillations close to discontinuities is proved in particular cases. It is proved that these schemes converge towards limit functions that are Hölder continuous with exponent larger than 1. The results are illustrated with several examples. [ABSTRACT FROM AUTHOR]

Given discrete function values sampled at uniform centers, the iterated quasi-interpolation approach for approximating the m th derivative consists of two steps. The first step adopts m successive applications of the operator DQ (the quasi-interpolation operator Q first, and then the differentiation operator D) to get approximated values of the m th derivative at uniform centers. Then, by one further application of the quasi-interpolation operator Q to corresponding approximated derivative values gives the final approximation of the m th derivative. The most salient feature of the approach is that it approximates all derivatives with the same convergence rate. In addition, it is valid for a general multivariate function, compared with the existing iterated interpolation approaches that are only valid for periodic functions, so far. Numerical examples of approximating high-order derivatives using both the iterated and direct approach based on B-spline quasi-interpolation and multiquadric quasi-interpolation are presented at the end of the paper, which demonstrate that the iterated quasi-interpolation approach provides higher approximation orders than the corresponding direct approach. [ABSTRACT FROM AUTHOR]

In this paper, a kind of univariate multiquadric quasi-interpolants with the derivatives of approximated function is proposed by combining a univariate multiquadric quasi-interpolant with Lidstone interpolation polynomials proposed in Lidstone (Proc Edinb Math Soc 2:16–19, 1929), Costabile and Dell' Accio (App Numer Math 52:339–361, 2005) and Catinas (J Appl Funct Anal 4:425–439, 2006). For practical purposes, another kind of approximation operators without any derivative of the approximated function is given using divided differences to approximate the derivatives. Some error bounds and the convergence rates of new operators are derived, which demonstrates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter c and a non-negative integer n. Finally, we make extensive comparison with the other existing methods and give some numerical examples. Moreover, the associated algorithm is easily implemented. [ABSTRACT FROM AUTHOR]

As an extension of Lagrange interpolation, we introduce a class of interpolation formulas and study their existence and uniqueness. In the sequel, we consider some particular cases and construct the corresponding weighted quadrature rules. Numerical examples are finally given and compared. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of some new non-linear subdivision and multiresolution schemes using the Lehmer means. Our main objective is to attain adaption close to discontinuities. We present theoretical, numerical results and applications for different schemes. The main theoretical result is related to the four-point interpolatory scheme, that we write as a perturbation of a linear scheme. Our aim is to establish a one-step contraction property that allows to prove the stability of the new scheme. Indeed with a one-step contraction property for the scheme of differences, it is possible to prove the stability of the 2D algorithm constructed using a tensor product approach. In this article, we also consider the associated three points cell-average scheme, that we will use to present some results for image compression, and a non-interpolatory scheme, that we will use to introduce an application to subdivision curves in 2D. These applications show that the use of the Lehmer mean is suitable for the design of subdivision schemes for the generation of curves and for image processing. [ABSTRACT FROM AUTHOR]

We propose algorithms to take point sets for kernel-based interpolation of functions in reproducing kernel Hilbert spaces (RKHSs) by convex optimization. We consider the case of kernels with the Mercer expansion and propose an algorithm by deriving a second-order cone programming (SOCP) problem that yields n points at one sitting for a given integer n. In addition, by modifying the SOCP problem slightly, we propose another sequential algorithm that adds an arbitrary number of new points in each step. Numerical experiments show that in several cases the proposed algorithms compete with the P-greedy algorithm, which is known to provide nearly optimal points. [ABSTRACT FROM AUTHOR]

In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps. For the new basis we provide error bounds and numerical results that support our claims. The method is also effectively tested for approximating satellite images. [ABSTRACT FROM AUTHOR]

We study worst-case optimal approximation of positive linear functionals in reproducing kernel Hilbert spaces induced by increasingly flat Gaussian kernels. This provides a new perspective and some generalisations to the problem of interpolation with increasingly flat radial basis functions. When the evaluation points are fixed and unisolvent, we show that the worst-case optimal method converges to a polynomial method. In an additional one-dimensional extension, we allow also the points to be selected optimally and show that in this case convergence is to the unique Gaussian quadrature–type method that achieves the maximal polynomial degree of exactness. The proofs are based on an explicit characterisation of the reproducing kernel Hilbert space of the Gaussian kernel in terms of exponentially damped polynomials. [ABSTRACT FROM AUTHOR]

A geometric construction of a Bézier curve is presented by a unifiable way from the mentioned literature with some modification. A closed-form solution to the inverse problem in cubic Bézier-spline interpolation will be obtained. Calculations in the given examples are performed by a Maple procedure using this solution. [ABSTRACT FROM AUTHOR]