Search

In this paper, we propose a novel model to analyze serially correlated two-dimensional functional data observed sparsely and irregularly on a domain which may not be a rectangle. Our approach employs a mixed effects model that specifies the principal component functions as bivariate splines on triangles and the principal component scores as random effects which follow an auto-regressive model. We apply the thin-plate penalty for regularizing the bivariate function estimation and develop an effective EM algorithm along with Kalman filter and smoother for calculating the penalized likelihood estimates of the parameters. Our approach was applied on simulated datasets and on Texas monthly average temperature data from January year 1915 to December year 2014. Supplementary materials accompanying this paper appear online. [ABSTRACT FROM AUTHOR]

Screening mammography is the primary preventive strategy for early detection of breast cancer and an essential input to breast cancer risk prediction and application of prevention/risk management guidelines. Identifying regions of interest within mammogram images that are associated with 5- or 10-year breast cancer risk is therefore clinically meaningful. The problem is complicated by the irregular boundary issue posed by the semi-circular domain of the breast area within mammograms. Accommodating the irregular domain is especially crucial when identifying regions of interest, as the true signal comes only from the semi-circular domain of the breast region, and noise elsewhere. We address these challenges by introducing a proportional hazards model with imaging predictors characterized by bivariate splines over triangulation. The model sparsity is enforced with the group lasso penalty function. We apply the proposed method to the motivating Joanne Knight Breast Health Cohort to illustrate important risk patterns and show that the proposed method is able to achieve higher discriminatory performance. [ABSTRACT FROM AUTHOR]

Featured Application: The material in this article draws on, and has applications across a range of applications in civil and structural engineering, mechanical engineering, computer science, biochemistry, geometry and applied mathematics. In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas. [ABSTRACT FROM AUTHOR]

This article considers high-dimensional image-on-scalar regression, where the spatial heterogeneity of covariate effects on imaging responses is investigated via a flexible partially linear spatially varying coefficient model. To tackle the challenges of spatial smoothing over the imaging response's complex domain consisting of regions of interest, we approximate the spatially varying coefficient functions via bivariate spline functions over triangulation. We first study estimation when the active constant coefficients and varying coefficient functions are known in advance. We then further develop a unified approach for simultaneous sparse learning and model structure identification in the presence of ultrahigh-dimensional covariates. Our method can identify zero, nonzero constant, and spatially varying components correctly and efficiently. The estimators of constant coefficients and varying coefficient functions are consistent and asymptotically normal for constant coefficient estimators. The method is evaluated by Monte Carlo simulation studies and applied to a dataset provided by the Alzheimer's Disease Neuroimaging Initiative. for this article are available online. [ABSTRACT FROM AUTHOR]

Motivated by recent work involving the analysis of biomedical imaging data, we present a novel procedure for constructing simultaneous confidence corridors for the mean of imaging data. We propose to use flexible bivariate splines over triangulations to handle an irregular domain of the images that is common in brain imaging studies and in other biomedical imaging applications. The proposed spline estimators of the mean functions are shown to be consistent and asymptotically normal under some regularity conditions. We also provide a computationally efficient estimator of the covariance function and derive its uniform consistency. The procedure is also extended to the two‐sample case in which we focus on comparing the mean functions from two populations of imaging data. Through Monte Carlo simulation studies, we examine the finite sample performance of the proposed method. Finally, the proposed method is applied to analyze brain positron emission tomography data in two different studies. One data set used in preparation of this article was obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of partially linear spatially varying coefficient autoregressive models for data distributed over complex domains. We propose approximating the varying coefficient functions via bivariate splines over triangulation to deal with the complex boundary of the spatial domain. Under some regularity conditions, the estimated constant coefficients are asymptotically normally distributed, and the estimated varying coefficients are consistent and possess the optimal convergence rate. A penalized bivariate spline estimation method with a more flexible choice of triangulation is proposed. We further develop a fast algorithm to calculate the geodesic distance. The proposed method is much more computationally efficient than the local smoothing methods, and thus capable of handling large scales of spatial data. In addition, we propose a model selection approach to identify predictors with constant and varying effects. The performance of the proposed method is evaluated by simulation examples and the Sydney real estate dataset. [ABSTRACT FROM AUTHOR]

In many application areas, data are collected on a count or binary response with spatial covariate information. In this article, we introduce a new class of generalized geoadditive models (GGAMs) for spatial data distributed over complex domains. Through a link function, the proposed GGAM assumes that the mean of the discrete response variable depends on additive univariate functions of explanatory variables and a bivariate function to adjust for the spatial effect. We propose a two-stage approach for estimating and making inferences of the components in the GGAM. In the first stage, the univariate components and the geographical component in the model are approximated via univariate polynomial splines and bivariate penalized splines over triangulation, respectively. In the second stage, local polynomial smoothing is applied to the cleaned univariate data to average out the variation of the first-stage estimators. We investigate the consistency of the proposed estimators and the asymptotic normality of the univariate components. We also establish the simultaneous confidence band for each of the univariate components. The performance of the proposed method is evaluated by two simulation studies. We apply the proposed method to analyze the crash counts data in the Tampa-St. Petersburg urbanized area in Florida. for this article are available online. [ABSTRACT FROM AUTHOR]

In this study, we consider the estimation of partially linear models for spatial data distributed over complex domains. We use bivariate splines over triangulations to represent the nonparametric component on an irregular two-dimensional domain. The proposed method is formulated as a constrained minimization problem that does not require constructing finite elements or locally supported basis functions. Thus, it allows an easier implementation of piecewise polynomial representations of various degrees and various smoothness over an arbitrary triangulation. Moreover, the constrained minimization problem is converted to an unconstrained minimization using a QR decomposition of the smoothness constraints, enabling us to develop a fast and efficient penalized least squares algorithm for fitting the model. The estimators of the parameters are proved to be asymptotically normal under some regularity conditions. The estimator of the bivariate function is consistent, and its rate of convergence is also established. The proposed method enables us to construct confidence intervals and permits inferences for the parameters. The performance of the estimators is evaluated using two simulation examples and a real-data analysis. [ABSTRACT FROM AUTHOR]

In this paper, we focus on variable selection techniques for a class of semiparametric spatial regression models, which allow one to study effects of explanatory variables in the presence of spatial information. The spatial smoothing problemin the nonparametric part is tackled by means of bivariate splines over triangulation, which is able to deal efficiently with data distributed over irregularly shaped regions. In addition, we develop a unified procedure for variable selection to identify significant covariates under a double penalization framework, and we show that the penalized estimators enjoy the "oracle" property. The proposed method can simultaneously identify nonzero spatially distributed covariates and solve the problem of "leakage" across complex domains of the functional spatial component. To estimate the standard deviations of the proposed estimators for the coefficients, a sandwich formula is developed as well. In the end, Monte Carlo simulation examples and a real data example are provided to illustrate the proposed methodology. [ABSTRACT FROM AUTHOR]

In Tohǎneanu (2005), Tohǎneanu introduced a triangulation Δ T with the following property: the space of splines of smoothness r and degree 2 r defined on Δ T has a dimension not equal to the lower bound found by Schumaker in Schumaker (1979). In Minàč and Tohǎneanu (2013), it was shown that for d ≥ 2 r + 1 , the dimension of the space of splines on Δ T is equal to the lower bound. We show that this phenomenon can be explained by intrinsic supersmoothness of splines. Moreover, using this supersmoothness we obtain the dimension of the space of splines of smoothness r and degree ≤ 2 r on Δ T . [ABSTRACT FROM AUTHOR]

Spatially varying coefficient models are a classical tool to explore the spatial nonstationarity of a regression relationship for spatial data. In this paper, we study the estimation and inference in spatially varying coefficient models for data distributed over complex domains. We use bivariate splines over triangulations to represent the coefficient functions. The estimators of the coefficient functions are consistent, and rates of convergence of the proposed estimators are established. A penalized bivariate spline estimation method is also introduced, in which a roughness penalty is incorporated to balance the goodness of fit and smoothness. In addition, we propose hypothesis tests to examine if the coefficient function is really varying over space or admits a certain parametric form. The proposed method is much more computationally efficient than the well-known geographically weighted regression technique and thus usable for analyzing massive data sets. The performances of the estimators and the proposed tests are evaluated by simulation experiments. An environmental data example is used to illustrate the application of the proposed method. [ABSTRACT FROM AUTHOR]

Health care utilization is an outcome of interest in health services research. Two frequently studied forms of utilization are counts of emergency department (ED) visits and hospital admissions. These counts collectively convey a sense of disease exacerbation and cost escalation. Different types of event counts from the same patient form a vector of correlated outcomes. Traditional analysis typically model such outcomes one at a time, ignoring the natural correlations between different events, and thus failing to provide a full picture of patient care utilization. In this research, we propose a multivariate semiparametric modeling framework for the analysis of multiple health care events following the exponential family of distributions in a longitudinal setting. Bivariate nonparametric functions are incorporated to assess the concurrent nonlinear influences of independent variables as well as their interaction effects on the outcomes. The smooth functions are estimated using the thin plate regression splines. A maximum penalized likelihood method is used for parameter estimation. The performance of the proposed method was evaluated through simulation studies. To illustrate the method, we analyzed data from a clinical trial in which ED visits and hospital admissions were considered as bivariate outcomes. [ABSTRACT FROM AUTHOR]

We introduce bivariate C 1 piecewise quintic finite element spaces for curved domains enclosed by piecewise conics satisfying homogeneous boundary conditions, construct local bases for them using Bernstein–Bézier techniques, and demonstrate the effectiveness of these finite elements for the numerical solution of the Monge–Ampère equation over curved domains by Böhmer's method. [ABSTRACT FROM AUTHOR]

This article develops a method for simultaneous estimation of density functions for a collection of populations of protein backbone angle pairs using a data-driven, shared basis that is constructed by bivariate spline functions defined on a triangulation of the bivariate domain. The circular nature of angular data is taken into account by imposing appropriate smoothness constraints across boundaries of the triangles. Maximum penalized likelihood is used to fit the model and an alternating blockwise Newton-type algorithm is developed for computation. A simulation study shows that the collective estimation approach is statistically more efficient than estimating the densities individually. The proposed method was used to estimate neighbor-dependent distributions of protein backbone dihedral angles (i.e., Ramachandran distributions). The estimated distributions were applied to protein loop modeling, one of the most challenging open problems in protein structure prediction, by feeding them into an angular-sampling-based loop structure prediction framework. Our estimated distributions compared favorably to the Ramachandran distributions estimated by fitting a hierarchical Dirichlet process model; and in particular, our distributions showed significant improvements on the hard cases where existing methods do not work well. [ABSTRACT FROM AUTHOR]

We study how to use bivariate splines for scattered data interpolation with nonnegativity preservation of the given data values. That is, we propose a constrained minimal energy method to find a C 1 smooth interpolation of nonnegative data values over scattered locations using bivariate splines. We establish the existence and uniqueness of the minimizer under mild assumptions on the data locations and triangulations. Next we show some approximation properties of the minimizer. We then use the classic projected gradient algorithm to find the minimizer using a simplified nonnegative constraint. Some synthetic as well as a real life example demonstrate the performance, producing nonnegative interpolatory spline surfaces from nonnegative data values. Experimental results also show that the approximation of the nonnegative interpolatory spline surfaces is slightly better than the approximation of the classic minimal energy interpolatory splines. [ABSTRACT FROM AUTHOR]

We show that for many spaces of bivariate splines imposing certain additional individual smoothness conditions on a spline space does not change the space. Such spaces possess supersmoothness that is not reflected in their standard definitions. Smoothness is treated as vanishing of strongly supported smoothness functionals. We investigate the dependence of this phenomenon on the geometry of the underlying partition and the degree of splines. [ABSTRACT FROM AUTHOR]

This article presents and compares two approaches of principal component (PC) analysis for two-dimensional functional data on a possibly irregular domain. The first approach applies the singular value decomposition of the data matrix obtained from a fine discretization of the two-dimensional functions. When the functions are only observed at discrete points that are possibly sparse and may differ from function to function, this approach incorporates an initial smoothing step prior to the singular value decomposition. The second approach employs a mixed effects model that specifies the PC functions as bivariate splines on triangulations and the PC scores as random effects. We apply the thin-plate penalty for regularizing the function estimation and develop an effective expectation–maximization algorithm for calculating the penalized likelihood estimates of the parameters. The mixed effects model-based approach integrates scatterplot smoothing and functional PC analysis in a unified framework and is shown in a simulation study to be more efficient than the two-step approach that separately performs smoothing and PC analysis. The proposed methods are applied to analyze the temperature variation in Texas using 100 years of temperature data recorded by Texas weather stations. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Abstract: We present a method for isogeometric analysis on the triangulation of a domain bounded by NURBS curves. In this method, both the geometry and the physical field are represented by bivariate splines in Bernstein–Bézier form over the triangulation. We describe a set of procedures to construct a parametric domain and its triangulation from a given physical domain, construct -smooth basis functions over the domain, and establish a rational Triangular Bézier Spline (rTBS) based geometric mapping that -smoothly maps the parametric domain to the physical domain and exactly recovers the NURBS boundaries at the domain boundary. As a result, this approach can achieve automated meshing of objects with complex topologies and allow highly localized refinement. Isogeometric analysis of problems from linear elasticity and advection–diffusion analysis is demonstrated. [Copyright &y& Elsevier]

Abstract: Given a set of scattered data, we usually use a minimal energy method to find a Lagrange interpolation in a bivariate spline space over a triangulation of the scattered data locations. It is known that the approximation order of the minimal energy spline interpolation is only 2 in terms of the size of triangulation. To improve this order of approximation, we propose several new schemes in this paper. Mainly we follow the ideas of clamped cubic interpolatory splines and not-a-knot interpolatory splines in the univariate setting and extend them to the bivariate setting. In addition, instead of the energy functional of the second order, we propose to use higher order versions. We shall present some theoretical analysis as well as many numerical results to demonstrate that our new interpolation schemes indeed have a higher order of approximation than the classic minimal energy interpolatory spline. [Copyright &y& Elsevier]

Given a non-uniform criss-cross triangulation of a rectangular domain Ω, we consider the approximation of a function f and its partial derivatives, by general C quadratic spline quasi-interpolants and their derivatives. We give error bounds in terms of the smoothness of f and the characteristics of the triangulation. Then, the preceding theoretical results are compared with similar results in the literature. Finally, several examples are proposed for illustrating various applications of the quasi-interpolants studied in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we forecast ground level ozone concentrations over the USA, using past spatially distributed measurements and the functional linear regression model. We employ bivariate splines defined over triangulations of the relevant region of the USA to implement this functional data approach in which random surfaces represent ozone concentrations. We compare the least squares method with penalty to the principal components regression approach. Moderate sample sizes provide good quality forecasts in both cases with little computational effort. We also illustrate the variability of forecasts owing to the choice of smoothing penalty. Finally, we compare our predictions with the ones obtained using thin-plate splines. Predictions based on bivariate splines require less computational time than the ones based on thin-plate splines and are more accurate. We also quantify the variability in the predictions arising from the variability in the sample using the jackknife, and report that predictions based on bivariate splines are more robust than the ones based on thin-plate splines. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we consider the numerical integration on a polygonal domain Ω in ℝ2 of a function F( x, y) which is integrable except at a point $P_{0}=(x_{0},y_{0})\in{\stackrel{\circ}{\Omega}}$, where F becomes infinite of order two. We approximate either the finite-part or the two-dimensional Cauchy principal value of the integral by using a spline finite element method combined with a subdivision technique also of adaptive type. We prove the convergence of the obtained sequence of cubatures. Finally, to illustrate the behaviour of the proposed method, we present some numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: We study two kinds of quasi-interpolants (abbr. QI) in the space of piecewise cubics in the plane, or in a rectangular domain, endowed with the highly symmetric triangulation generated by a uniform 6-direction mesh. It has been proved recently that this space is generated by the integer translates of two multi-box splines. One kind of QIs is of differential type and the other of discrete type. As those QIs are exact on the space of cubic polynomials, their approximation order is 4 for sufficiently smooth functions. In addition, they exhibit nice superconvergent properties at some specific points. Moreover, the infinity norms of the discrete QIs being small, they give excellent approximations of a smooth function and of its first order partial derivatives. The approximation properties of the QIs are illustrated by numerical examples. [Copyright &y& Elsevier]

Abstract: We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods. [Copyright &y& Elsevier]

Abstract: In this paper, a cubature formula over polygons is proposed and analysed. It is based on an eight-node quadrilateral spline finite element [C.-J. Li, R.-H. Wang, A new 8-node quadrilateral spline finite element, J. Comp. Appl. Math. 195 (2006) 54–65] and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on rectangular partitions. The convergence of sequences of the above cubatures is proved for continuous integrand functions and error bounds are derived. Some numerical examples are given, by comparisons with other known cubatures. [Copyright &y& Elsevier]

Given a set of scattered data with derivative values. If the data is noisy or there is an extremely large number of data, we use an extension of the penalized least squares method of von Golitschek and Schumaker [Serdica, 18 (2002), pp.1001-1020] to fit the data. We show that the extension of the penalized least squares method produces a unique spline to fit the data. Also we give the error bound for the extension method. Some numerical examples are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Nöther’s theorem of algebraic curves plays an important role in classical algebraic geometry. As the zero set of a bivariate spline, the piecewise algebraic curve is a generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves is very important to construct the Lagrange interpolation sets for bivariate spline spaces. In this paper, using the characteristics of quasi-cross-cut partition, properties of bivariate splines and results in algebraic geometry, the Nöther-type theorem of piecewise algebraic curves on the quasi-cross-cut is presented. [ABSTRACT FROM AUTHOR]

Abstract: We develop a local Lagrange interpolation scheme for quartic splines on triangulations. Given an arbitrary triangulation , we decompose into pairs of neighboring triangles and add “diagonals” to some of these pairs. Only in exceptional cases, a few triangles are split. Based on this simple refinement of , we describe an algorithm for constructing Lagrange interpolation points such that the interpolation method is local, stable and has optimal approximation order. The complexity for computing the interpolating splines is linear in the number of triangles. For the local Lagrange interpolation methods known in the literature, about half of the triangles have to be split. [Copyright &y& Elsevier]

Abstract: Given a set of scattered data with derivatives values, we use a minimal energy method to find Hermite interpolation based on bivariate spline spaces over a triangulation of the scattered data locations. We show that the minimal energy method produces a unique Hermite spline interpolation of the given scattered data with derivative values. Also we show that the Hermite spline interpolation converges to a given sufficiently smooth function f if the data values are obtained from this f. That is, the surface of the Hermite spline interpolation resembles the given set of derivative values. Some numerical examples are presented to demonstrate our method. [Copyright &y& Elsevier]

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the Nöther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented. [ABSTRACT FROM AUTHOR]

Bivariate splines with various degrees are considered in this paper. A matrix form of the extended smoothness conditions for these splines is presented. Upon this form, the multivariate spline method for numerical solution of partial differential equations (PDEs) proposed by Awanou, Lai, and Wenston in [The multivariate spline method for scattered data fitting and numerical solutions of partial differential equations, in Wavelets and Splines, G. Chen and M. J. Lai, eds., Nashboro Press, Brentwood, TN, 2006, pp. 24-76] is generalized to obtain a new spline method. It is observed that, combined with prelocal refinement of triangulation and automatic degree raising over triangles of interest, the new spline method of bivariate splines of various degrees is able to solve linear PDEs very effectively and efficiently. [ABSTRACT FROM AUTHOR]

Shape-based solutions have recently received attention for certain ill-posed inverse problems. Their advantages include implicit imposition of relevant constraints and reduction in the number of unknowns, especially important for nonlinear ill-posed problems. We apply the shape-based approach to current-injection electrical impedance tomography (EIT) reconstructions. We employ a boundary element method (BEM) based solution for EIT. We introduce two shape models, one based on modified B-splines, and the other based on spherical harmonics, for BEM modeling of shapes. These methods allow us to parameterize the geometry of conductivity inhomogeneities in the interior of the volume. We assume the general shape of piecewise constant inhomogeneities is known but their conductivities and their exact location and shape is not. We also assume the internal conductivity profile is piecewise constant, meaning that each region has a constant conductivity. We propose and test three different regularization techniques to be used with either of the shape models. The performance of our methods is illustrated via both simulations in a digital torso model and phantom experiments when there is a single internal object. We observe that in the noisy environment, either simulated noise or real sources of noise in the experimental study, we get reasonable reconstructions. Since the signal-to-noise ratio (SNR) expected in modern EIT instruments is higher than that used in this study, these reconstruction methods may prove useful in practice. [ABSTRACT FROM AUTHOR]

Abstract: The convergences of three spline methods for scattered data interpolation and fitting using bivariate spline spaces are studied in this paper. That is, interpolatory splines, splines of least absolute deviation, and smoothing splines are shown to converge to the given data function under some conditions and hence, the surfaces from these three methods will resemble the given data values. [Copyright &y& Elsevier]

Abstract: We suggest a local hybrid approximation scheme based on polynomials and radial basis functions, and use it to improve the scattered data fitting algorithm of (Davydov, O., Zeilfelder, F., 2004. Scattered data fitting by direct extension of local polynomials to bivariate splines. Adv. Comp. Math. 21, 223–271). Similar to that algorithm, the new method has linear computational complexity and is therefore suitable for large real world data. Numerical examples suggest that it can produce high quality artifact-free approximations that are more accurate than those given by the original method where pure polynomial local approximations are used. [Copyright &y& Elsevier]

Abstract: The piecewise algebraic curve is a generalization of the classical algebraic curve. In this paper, we give the Nöther-type theorem of piecewise algebraic curves on cross-cut partitions. By using interpolation along a piecewise algebraic curve, we also present a method to construct Lagrange interpolation sets for bivariate spline spaces on cross-cut partitions. [Copyright &y& Elsevier]

We investigate spline quasi-interpolants defined by C1 bivariate quadratic B-splines on nonuniform type-2 triangulations and by discrete linear functionals based on a fixed number of triangular mesh-points either in the support or close to the support of such B-splines.We show they can approximate a real function and its partial derivatives up to an optimal order and we derive local and global upper bounds.We also present some numerical and graphical results. [Copyright &y& Elsevier]

The piecewise algebraic curve is a generalization of the classical algebraic curve. This paper describes the improvement of the Nöther-type theorem of piecewise algebraic curves on the star region. Moreover, the Nöther-type theorem of piecewise algebraic curves on the cross-cut partition is discussed. [ABSTRACT FROM AUTHOR]

The piecewise algebraic curve, determined by a bivariate spline function, is a generalization of the classical algebraic curve. In this paper, by using Bezout's theorem and No¨ther-type theorem of piecewise algebraic curves, the Cayley–Bacharach theorem and Hilbert function of C0 piecewise algebraic curves are presented. [Copyright &y& Elsevier]

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, a conjecture on triangulation is confirmed. The relation between the piecewise linear algebraic curve and four-color conjecture is also presented. By Morgan-Scott triangulation, we will show the instability of Bezout number of piecewise algebraic curves. By using the combinatorial optimization method, an upper bound of the Bezout number defined as the maximum finite number of intersection points of two piecewise algebraic curves is presented. [ABSTRACT FROM AUTHOR]

Many image processing systems, including those for printing applications, need sampling conversions for the representation of an image from one lattice to another. For example in the case of printing, classical halftoning requires new sample values on the halftone lattice. Although often considered as a straightforward procedure, resampling can cause so-called sampling moire due to aliasing. These artifacts are often very noticeable and as such undesirable, in particular for high-quality printing. In color printing, each color separation uses its own halftone lattice. Therefore, moire patterns will not only display an unexpected new frequency and orientation, but also influence the color appearance itself. These artifacts are frequently encountered in commercial (even high-quality) printing since the interpolation algorithms used in RIPs are simple (e.g., bilinear interpolation) and do not take into account the nature of the target lattice. Approaches such as simple low-pass filtering unacceptably blur the edges, while manual selective smoothing by an operator is very time-consuming.This paper proposes an optimal prefilter which is based on the least-squares linear resampling paradigm. Our approach requires proper discrete/continuous models, i.e., for both the source and the target lattices, and computes the associated reconstruction function which minimizes the error between the representations in the continuous domain. The reconstruction function jointly takes into account the Nyquist areas of every color separation using a novel hexagonal spline model resulting into an optimal prefilter before halftoning. Experimental results show that after prefiltering, the images are much less prone to moire while not suffering from noticeable edge blurring. [Copyright &y& Elsevier]

Resampling is a common operation in digital image processing systems. The standard procedure involves the (conceptual) reconstruction of a continuous image succeeded by sampling on the new lattice sites. When the reconstruction is done by classical interpolation functions, results might be sub-optimal because the information loss is not minimized. In the particular case of subsampling (i.e., resampling to a coarser lattice), aliasing artifacts might arise and produce disturbing moire patterns. This paper first introduces a spline model for different orders, both for orthogonal and hexagonal lattices. Next, an expression for a least-squares approximation is derived which can be applied to convolution-based resampling. Experimental results for a printing application demonstrate the feasibility of the proposed method and are compared against the standard approach. Our technique can be applied to general least-squares resampling between regular lattices. [Copyright &y& Elsevier]

Copyright of Computing is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)