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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 Bézier-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 C0-, C1-, and C2-smoothness are derived.

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., 17 pages, 5 figures

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

In this chapter we consider LU factorizations of Hermitian and positive definite matrices. Recall that a matrix \({\boldsymbol {A}}\in {\mathbb {C}}^{n\times n}\) is Hermitian if A∗ = A, i.e., \(a_{ji}=\overline {a}_{ij}\) for all i, j. A real Hermitian matrix is symmetric. Since \(a_{ii}=\overline {a}_{ii}\) the diagonal elements of a Hermitian matrix must be real.

In this chapter we first review Gaussian elimination. Gaussian elimination leads to an LU factorization of the coefficient matrix or more generally to a PLU factorization, if row interchanges are introduced. Here P is a permutation matrix, L is lower triangular and U is upper triangular.

The factorization is easily verified. Here \( \mathit\mathbf{U}=\mathit\mathbf{V}=\frac{1}{\sqrt{2}} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right], \)which is clearly unitary. Since in addition the matrix in the middle is diagonal, it follows that this is both a spectral and a singular value decomposition.

Matrices arising from 2D and 3D problems sometimes have a Kronecker product structure. Identifying a Kronecker structure can be very rewarding since it simplifies the derivation of properties of such matrices.

Given a symmetric positive definite matrix \( \mathit\mathbf{A}\epsilon\mathbb{R}^{n \times n} \) with eigenvalues \( 0 < \lambda_{\mathit n} \leq \cdots \leq \lambda_{1} \).

In this chapter we present a fast method for solving Ax = b, where A is the Poisson matrix ( 10.8).

The QR algorithm is a method to find all eigenvalues and eigenvectors of a matrix. In this chapter we give a brief informal introduction to this important algorithm.

In Gaussian elimination and LU factorization we solve a linear system by transforming it to triangular form. These are not the only kind of transformations that can be used for such a task. Matrices with orthonormal columns, called unitary matrices can be used to reduce a square matrix to upper triangular form and more generally a rectangular matrix to upper triangular (also called upper trapezoidal) form. This lead to a decomposition of a rectangular matrix known as a QR decomposition and a reduced form which we refer to as a QR factorization. The QR decomposition and factorization will be used in later chapters to solve least squares- and eigenvalue problems.

No, we have seen that 4n3/3 operations are required in order to bring a matrix to upper triangular form using Householder transformations.

The conjugate gradient method was published by Hestenes and Stiefel in 1952, as a direct method for solving linear systems. Today its main use is as an iterative method for solving large sparse linear systems. On a test problem we show that it performs as well as the SOR method with optimal acceleration parameter, and we do not have to estimate any such parameter. However the conjugate gradient method is restricted to positive definite systems. We also consider a mathematical formulation of the preconditioned conjugate gradient method. It is used to speed up convergence of the conjugate gradient method. We only give one example of a possible preconditioner.

In this introductory chapter we give a compact introduction to linear algebra with emphasis on \({\mathbb {R}}^n\) and \({\mathbb {C}}^n\).

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.

In this chapter we consider three problems originating from: cubic spline interpolation, a two point boundary value problem, an eigenvalue problem for a two point boundary value problem.

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 and 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 and 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.

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.

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

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

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

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 (Dyn and Lyche, 1998). 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.

We provide an algorithm to check whether two rational space curves are related by a similarity, i.e., whether they are equal up to position, orientation and scale. 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 Grobner bases for the special case of helical curves.