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Mean value coordinates can be used to map one polygon into another, with application to computer graphics and curve and surface modelling. In this paper we show that if the polygons are quadrilaterals, and if the target quadrilateral is convex, then the mapping is injective., Comment: Barycentric mapping, mean value coordinates, injectivity, Jacobian determinant

We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces, and therefore, in particular, the symmetries of a surface of translation or a minimal surface of the considered types. Additionally, we apply our results to designing surfaces of translation and minimal surfaces with symmetries, and to computing the symmetries of the higher-order Enneper surfaces., Comment: 24 pages

We propose a novel approach to image segmentation based on combining implicit spline representations with deep convolutional neural networks. This is done by predicting the control points of a bivariate spline function whose zero-set represents the segmentation boundary. We adapt several existing neural network architectures and design novel loss functions that are tailored towards providing implicit spline curve approximations. The method is evaluated on a congenital heart disease computed tomography medical imaging dataset. Experiments are carried out by measuring performance in various standard metrics for different networks and loss functions. We determine that splines of bidegree $(1,1)$ with $128\times128$ coefficient resolution performed optimally for $512\times 512$ resolution CT images. For our best network, we achieve an average volumetric test Dice score of almost 92%, which reaches the state of the art for this congenital heart disease dataset., Comment: 17 pages, 5 figures

High quality video data is a core component in emerging remote tower operations as it inherently contains a huge amount of information on which an air traffic controller can base decisions. Various digital technologies also have the potential to exploit this data to bring enhancements, including tracking ground movements by relating events in the video view to their positions in 3D space. The total resolution of remote tower setups with multiple cameras often exceeds 25 million RGB pixels and is captured at 30 frames per second or more. It is thus a challenge to efficiently process all the data in such a way as to provide relevant real-time enhancements to the controller. In this paper we discuss how a number of improvements can be implemented efficiently on a single workstation by decoupling processes and utilizing hardware for parallel computing. We also highlight how decoupling the processes in this way increases resilience of the software solution in the sense that failure of a single component does not impair the function of the other components.

For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a B\'ezier-like manner. In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for $C^0$-, $C^1$-, and $C^2$-smoothness are derived.

In recent years, advances in machine learning algorithms, cheap computational resources, and the availability of big data have spurred the deep learning revolution in various application domains. In particular, supervised learning techniques in image analysis have led to superhuman performance in various tasks, such as classification, localization, and segmentation, while unsupervised learning techniques based on increasingly advanced generative models have been applied to generate high-resolution synthetic images indistinguishable from real images. In this paper we consider a state-of-the-art machine learning model for image inpainting, namely a Wasserstein Generative Adversarial Network based on a fully convolutional architecture with a contextual attention mechanism. We show that this model can successfully be transferred to the setting of digital elevation models (DEMs) for the purpose of generating semantically plausible data for filling voids. Training, testing and experimentation is done on GeoTIFF data from various regions in Norway, made openly available by the Norwegian Mapping Authority., Comment: 5 pages; 4 figures; corrected names in references; clarifications regarding the two generators in the paper; added reference (Borji 2018) on GAN evaluation measures; extended future work discussion; changed (Fig. 4.f) to show a failure case

In applications like computer aided design, geometric models are often represented numerically as polynomial splines or NURBS, even when they originate from primitive geometry. For purposes such as redesign and isogeometric analysis, it is of interest to extract information about the underlying geometry through reverse engineering. In this work we develop a novel method to determine these primitive shapes by combining clustering analysis with approximate implicitization. The proposed method is automatic and can recover algebraic hypersurfaces of any degree in any dimension. In exact arithmetic, the algorithm returns exact results. All the required parameters, such as the implicit degree of the patches and the number of clusters of the model, are inferred using numerical approaches in order to obtain an algorithm that requires as little manual input as possible. The effectiveness, efficiency and robustness of the method are shown both in a theoretical analysis and in numerical examples implemented in Python.

We develop a characterization for the existence of symmetries of canal surfaces defined by a rational spine curve and rational radius function. In turn, this characterization inspires an algorithm for computing the symmetries of such canal surfaces. For Dupin cyclides in canonical form, we apply the characterization to derive an intrinsic description of their symmetries and symmetry groups, which gives rise to a method for computing the symmetries of a Dupin cyclide not necessarily in canonical form. As a final application, we discuss the construction of patches and blends of rational canal surfaces with a prescribed symmetry., Comment: 27 pages

Pseudo-splines form a family of subdivision schemes that provide a natural blend between interpolating schemes and approximating schemes, including the Dubuc-Deslauriers schemes and B-spline schemes. Using a generating function approach, we derive expressions for the symbols of the symmetric $m$-ary pseudo-spline subdivision schemes. We show that their masks have positive Fourier transform, making it possible to compute the exact H\"older regularity algebraically as a logarithm of the spectral radius of a matrix. We apply this method to compute the regularity explicitly in some special cases, including the symmetric binary, ternary, and quarternary pseudo-spline schemes., Comment: 31 pages

We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Gr\"obner basis for the special case of helical curves. Details on the implementation and experimentation carried out using the computer algebra system Maple 18 are provided.

For the space $\mathcal{S}$ of $C^3$ quintics on the Powell-Sabin 12-split of a triangle, we determine the simplex splines in $\mathcal{S}$ and the six symmetric simplex spline bases that reduce to a B-spline basis on each edge, have a positive partition of unity, a (barycentric) Marsden identity, and domain points with an intuitive control net. We provide a quasi-interpolant with approximation order 6 and a Lagrange interpolant at the domain points. The latter can be used to show that each basis is stable in the $L_\infty$ norm, which yields an $h^2$ bound for the distance between the B\'ezier ordinates and the values of the spline at the corresponding domain points. Finally, for one of these bases we provide $C^0$, $C^1$, and $C^2$ conditions on the control points of two splines on adjacent macrotriangles, and a conversion to the Hermite nodal basis., Comment: Oberwolfach report for the conference Multivariate Splines and Algebraic Geometry

For the space of $C^3$ quintics on the Powell-Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge, have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive domain mesh. The bases are stable in the $L_\infty$ norm with a condition number independent of the geometry, have a well-conditioned Lagrange interpolant at the domain points, and a quasi-interpolant with local approximation order 6. We show an $h^2$ bound for the distance between the control points and the values of a spline at the corresponding domain points. For one of these bases we derive $C^0$, $C^1$, $C^2$ and $C^3$ conditions on the control points of two splines on adjacent macrotriangles., Comment: 26 pages

We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem. To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage., Comment: 25 pages

In order to construct a $C^1$-quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme. In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally $C^3$ and globally $C^2$. For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements., Comment: 17 pages, 7 figures

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees., Comment: 22 pages

The effect of solar irradiation on plasmonic silver nanocrystals has been investigated using Transmission Electron Microscopy and size distribution analysis, in the context of solar cell applications for light harvesting. Starting from an initial collection of spherical nanocrystals on a carbon film whose sizes are log-normally distributed, solar irradiation causes the nanocrystals to grow, with one particle reaching a diameter of 638 nm after four hours of irradiation. In addition some of the larger particles lose their spherical shape. The average nanocrystal diameter was found to grow as predicted by the Ostwald ripening model, taking into account the range of area fractions of the samples. The size distribution stays approximately log-normal and does not reach one of the steady-state size distributions predicted by the Ostwald ripening model. This might be explained by the system being in a transient state., Comment: 5 pages, 3 figures

This paper addresses the pattern of damage, and investigates its properties, of a theoretical hail storm which gathers in intensity before subsiding, and which travels linearly across the landscape at constant velocity. We start by assuming a simpler model, that of a storm which does not move, restricted to having an uncorrelated binormal distribution of damage. This model, expressed in the natural polar coordinates, leads to a 1-dimensional pattern of damage as a function of the marginal radial distance conforming to the $\chi$-distribution with two degrees of freedom. The model is then extended to the traveling form, allowing further for a correlation of the variables, extending, as well, to the multidimensional case. In its full florescence the model produces hyperellipsoidal hypersurfaces of equal intensity for the correlated multinormal assumption. We provide closed-form solutions for the totality of damages upon these hypersurfaces as proxies for the insurance claims to follow. Finally the model is applied to extensive datasets of hail events, as detected by the NEXRAD network of weather radars., Comment: 15 pages, 3 figures, 2 tables

In this paper we review and refine a technique of Rioul to determine the H\"older regularity of a large class of symmetric subdivision schemes from the spectral radius of a single matrix. These schemes include those of Dubuc and Deslauriers, their dual versions, and more generally all the pseudo-spline and dual pseudo-spline schemes. We also derive various comparisons between their regularities using the Fourier transform. In particular we show that the regularity of the Dubuc-Deslauriers family increases with the size of the mask., Comment: 23 pages

Under general conditions, the equation $g(x,y) = 0$ implicitly defines $y$ locally as a function of $x$. In this short note we study the combinatorial structure underlying a recently discovered formula for the divided differences of $y$ expressed in terms of bivariate divided differences of $g$, by analyzing the number of terms $a_n$ in this formula. The main result describes six equivalent characterizations of the sequence $\{a_n\}$., Comment: 11 pages, 2 figures

Under general conditions, the equation $g(x^1, ..., x^q, y) = 0$ implicitly defines $y$ locally as a function of $x^1, ..., x^q$. In this article, we express divided differences of $y$ in terms of divided differences of $g$, generalizing a recent formula for the case where $y$ is univariate. The formula involves a sum over a combinatorial structure whose elements can be viewed either as polygonal partitions or as plane trees. Through this connection we prove as a corollary a formula for derivatives of $y$ in terms of derivatives of $g$., Comment: BIT Numerical Mathematics (2012)

Under general conditions, the equation $g(x,y) = 0$ implicitly defines $y$ locally as a function of $x$. In this article, we express divided differences of $y$ in terms of bivariate divided differences of $g$, generalizing a recent result on divided differences of inverse functions.

Silver nanocrystals made by a chemical reduction of silver salts (AgNO$_3$) by sodium borohydride (NaBH$_4$) were studied using Transmission Electron Microscopy (TEM) and light scattering simulations. For various AgNO$_3$/NaBH$_4$ molar ratios, the size distributions of the nanocrystals were found to be approximately log-normal. In addition, a linear relation was found between the mean nanocrystal size and the molar ratio. In order to relate the size distribution of Ag nanocrystals of the various molar ratios to the scattering properties of Ag nanocrystals in solar cell devices, light scattering simulations of Ag nanocrystals in Si, SiO$_2$, SiN, and Al$_2$O$_3$ matrices were carried out using Mie Plot. These light scattering spectra for the individual nanocrystal sizes were combined into light scattering spectra for the fitted size distributions. The evolution of these scattering spectra with respect to an increasing mean nanocrystal size was then studied. From these findings, it is possible to find the molar ratio for which the corresponding nanocrystal size distribution has maximum scattering at a particular wavelength in the desired matrix., Comment: 8 figures

Differential equations with the Painlev\'e property have been studied extensively due to their appearance in many branches of mathematics and their applicability in physics. Although a modern, differential algebraic treatment of the order one equations appeared before, the connection with the classical theory did not. Using techniques from algebraic geometry we provide the link between the classical and the modern treatment, and with the help of differential Galois theory a new classification is derived, both for characteristic 0 and p., Comment: 1 figure