List of Publications

Journal/Conference Publications and Book Chapters

1. Michael S. Floater and Georg Muntingh. On the injectivity of mean value mappings between quadrilaterals. Submitted. (arXiv)

   • Abstract: 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.

2. Heidi E.I. Dahl, Georg Muntingh. Chapter 8: "Industriell datavitenskap" in Industri 4.0: Digitalisering av prosesser i produksjon (2023). Fagbokforlaget. https://www.fagbokforlaget.no/Industri-4.0/I9788245039931 

3. Oliver J. D. Barrowclough, Sverre Briseid, Tor Dokken, Konstantinos Gavriil and Georg Muntingh,
   Data-driven geometric modelling methods for digital twinning: manufacturing, geospatial and medical applications. ECCOMAS Congress 2022 - 8th European Congress on Computational Methods in Applied Sciences and Engineering. (online)

   • Abstract: In recent years there has been an explosion of interest in digital twinning in many disciplines, including the manufacturing, geospatial, and medical domains. A core topic of importance in modelling digital twins, is reconstruction of geometric models from raw data. Despite the diversity of requirements in the vast space of digital twin applications, methods for geometric reconstruction can often be transferred between disciplines with only minor modifications. In this paper we present some recent results related to how advances in machine learning over the last decade can be used for data-driven geometric reconstruction in the medical, geospatial and manufacturing domains.

4. J.G. Alcazar, G. Muntingh. Affine equivalences of rational surfaces of translation, and applications to rational minimal surfaces. Journal of Computational and Applied Mathematics (2022). (arXiv, journal, code) 

   • Abstract: We provide an algorithm for determining whether two rational surfaces of translation are affinely equivalent. In turn, this also provides an algorithm for determining whether two rational minimal surfaces are affinely equivalent. This algorithm is applied to determine the symmetries of rational minimal surfaces, in particular the higher-order Enneper surfaces. Finally certain parity-like conditions in the Weierstrass form of minimal surfaces are used to construct minimal surfaces with prescribed symmetries. 

5. O.J.D. Barrowclough, G. Muntingh, V. Nainamalai, I. Stangeby. Binary segmentation of medical images using implicit spline representations and deep learning. Computer Aided Geometric Design, Volume 85, February 2021. (arXiv, journal, code)

   • Abstract: 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 x 128 coefficient resolution performed optimally for 512 x 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. 

6. O.J.D. Barrowclough, S. Briseid, G. Muntingh, T. Viksand, Real-time processing of high resolution video and 3D model-based tracking in remote tower operations. SN Computer Science 1, Article number: 296 (2020). (arXiv, journal)

   • Abstract: During the past decade, a new approach to providing air traffic services to airports from a remote location has been established, known as remote or digital tower. High quality video data is a core component in remote tower operations as it inherently contains a huge amount of information on which a controller can base decisions. The total resolution of a typical remote tower setup 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 describe the development of number of improvements and discuss how they can be implemented efficiently on a single workstation by decoupling processes, implementing attention mechanisms and utilizing hardware for parallel computing.

7. K. Gavriil, G. Muntingh, O.J.D. Barrowclough, Void filling of Digital Elevation Models with Deep Generative Models, IEEE Geoscience and Remote Sensing Letters, Volume 16, Issue 10 (October 2019), pp. 1645–1649. (arXiv, journal, code)

   • Abstract: 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 a superhuman performance in various tasks, such as classification, localization, and segmentation, whereas unsupervised learning techniques based on increasingly advanced generative models have been applied to generate high-resolution synthetic images indistinguishable from real images. In this letter, 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 be successfully transferred to the setting of digital elevation models for the purpose of generating semantically plausible data for filling voids. Training, testing, and experimentation are done on GeoTIFF data from various regions in Norway, made openly available by the Norwegian Mapping Authority.

8. T. Lyche, G. Muntingh. B-spline-like bases for C2 cubics on the Powell-Sabin 12-split. The SMAI journal of computational mathematics, Volume S5 (2019) , pp. 129–159. (arXiv, journal, code)

   • Abstract: 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. IIn 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.

9. A. Raffo, O.J.D. Barrowclough, G. Muntingh. Reverse engineering of CAD models via clustering and approximate implicitization. CAGD, Volume 80, June 2020. (arXiv, journal, code)

   • Abstract: 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.

10. J.G. Alcázar, H.E.I. Dahl, G. Muntingh. Symmetries of canal surfaces and Dupin cyclides.  Computer Aided Geometric Design, Volume 59, January 2018, pp. 68–85. (arXiv, journal, code)

    • Abstract: We develop a characterization for the existence of symmetries of canal surfaces defined by a rational spine curve and rational radius function. This characterization leads to a method for constructing rational canal surfaces with prescribed symmetries, and it 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.

11. G. Muntingh. Symbols and exact regularity of symmetric pseudo-splines of any arity. BIT Numerical Mathematics, September 2017, Volume 57, Issue 3, pp. 867–900. (arXiv, code, journal, Springer ShareIt version)

    • Abstract: 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 symmetricm-ary pseudo-spline subdivision schemes. We show that their masks have positive Fourier transform, making it possible to compute the exact Hölder 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.

12. J.G. Alcázar, C. Hermoso, G. Muntingh. Similarity detection of rational space curves. Journal of Symbolic Computation, Volume 85, March–April 2018, pp. 4–24. (arXiv, journal)

    • Abstract: 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 Gröbner 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.

13. J.G. Alcázar, C. Hermoso, G. Muntingh. Detecting similarities of rational space curves. ISSAC '16: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, 2016, Waterloo, ON, Canada. (ACM Digital Library, PDF)

14. T. Lyche and G. Muntingh. Stable simplex spline bases for C3 quintics on the Powell-Sabin 12-split. Constructive Approximation (2017) 45: 1. (arXiv, code, journal) 

    • Abstract: For the space of C3 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∞ 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 h2 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 C0, C1, C2 and C3 conditions on the control points of two splines on adjacent macrotriangles.

15. J.G. Alcázar, C. Hermoso, G. Muntingh. Symmetry detection of rational space curves from their curvature and torsion. Computer Aided Geometric Design. Volume 33, February 2015, Pages 51–65.(arXiv, code, journal)

    • Abstract:  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. We present timings from an implementation in Sage to support this claim, and a complexity analysis of the algorithm completes the study.

16. T. Lyche and G. Muntingh. A Hermite interpolatory subdivision scheme for C2-quintics on the Powell-Sabin 12-split. Computer Aided Geometric Design. Volume 31, Issues 7 – 8, October 2014, Pages 464–474. (arXiv, code, journal)

    • Abstract: In order to construct a C1-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 C3 and globally C2. 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.

17. J.G. Alcázar, C. Hermoso, G. Muntingh. Detecting similarity of rational plane curves. Journal of Computational and Applied Mathematics, Volume 269 (2014), Pages 1–13. (arXiv, code, journal)

    • Abstract: A novel and deterministic algorithm is presented to detect whether two given planar rational 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.

18. J.G. Alcázar, C. Hermoso, G. Muntingh. Detecting symmetries of rational plane and space curves. Computer Aided Geometric Design, Volume 31, Issues 3–4, March–May 2014, Pages 199–209. (arXiv, code, journal)

    • Abstract: This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univariate polynomials. These algorithms have been implemented and tested in the Sage system.

19. A. Thøgersen, G. Muntingh. Solar induced growth of silver nanocrystals. Journal of Applied Physics, Volume 113, Issue 14 (2013) (arXiv, pre-print PDF, journal)

    • Abstract: 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.

20. G. Muntingh. Implicit Divided Differences, Little Schröder Numbers, and Catalan Numbers.  Journal of Integer Sequences, Vol. 15 (2012), Article 12.6.5 (arXiv, journal)

    • Abstract: 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 an in this formula. The main result describes six equivalent characterizations of the sequence {an}.

21. G. Muntingh. Divided Differences of Multivariate Implicit Functions. BIT Numerical Mathematics, Volume 52, Issue 3 (2012), Pages 703–723 (arXiv, journal)

    • Abstract: Under general conditions, the equation g(x1, ..., xq, y) = 0 implicitly defines y locally as a function of x1, ..., xq. 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 planar trees. Through this connection we prove as a corollary a formula for derivatives of y in terms of derivatives of g.

22. A. Thøgersen, J. Bonsak, C. Huseby Fosli, G. Muntingh. Size Distributions of Chemically Synthesized Ag Nanocrystals. Journal of Applied Physics, Volume 110, Issue 4 (2011) (arXiv page, journal)

    • Abstract: Silver nanocrystals made by a chemical reduction of silver salts (AgNO3) by sodium borohydride (NaBH4) were studied using Transmission Electron Microscopy (TEM) and light scattering simulations. For various AgNO3/NaBH4 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, SiO2, SiN, and Al2O3 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.

23. G. Muntingh, M. Floater. Divided Differences of Implicit Functions. Mathematics of Computation, Volume 80, Number 276, October 2011, Pages 2185–2195 (arXiv, journal)

    • Abstract: 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.

24. G. Muntingh, M. van der Put. Order One Equations with the Painlevé Property. Indagationes Mathematicae, N.S., 18 (1), 2007, Pages 83–95 (arXiv, journal)

    • Abstract: Differential equations with the Painlevé 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.

Popular Scientific

1. G. Muntingh and K. Olson Lye. Hva er algoritmer? Og styrer de livene våre? (October 28, 2022)
   https://ung.forskning.no/data-it/hva-er-algoritmer-og-styrer-de-livene-vare/2098859 

2. G. Muntingh and B. van Lieshout. Verslag — Fysica en Feesten. Nederlands Tijdschrift voor Natuurkunde, 71 (10), 2005, Page 313.

3. G. Muntingh and B. van Lieshout. Verslag — EuroPhysicsFun. Nederlands Tijdschrift voor Natuurkunde 71 (10), 2005, Page 319.

Reports

1. T. Lyche and G. Muntingh. Simplex Spline Bases on the Powell-Sabin 12-Split: Part 1. In: Schenck H., Schumaker L., Sorokina T. (eds) Multivariate Splines and Algebraic Geometry. Oberwolfach Report 12 (2015), Pages 1166–1169. (Arxiv page, publisher) 

2. T. Lyche and G. Muntingh. Simplex Spline Bases on the Powell-Sabin 12-Split: Part 2. In: Schenck H., Schumaker L., Sorokina T. (eds) Multivariate Splines and Algebraic Geometry. Oberwolfach Report 12 (2015), Pages 1169–1172. (Arxiv page, publisher) 

3. P.C. Kettler,  G. Muntingh. A χ-distribution model of hail storm damage. Submitted. (Arxiv page)

   • Abstract: 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 χ-distribution with two degrees of freedom. We then extend the model 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.

4. M. Floater, G. Muntingh. Exact regularity of pseudo-splines. Submitted. (Arxiv page)

   • Abstract: In this paper we review and refine a technique of Rioul to determine the Hölder 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.

5. T. Dokken and G. Muntingh. Introduction to ShApes, Geometry, and Algebra. Appeared in SAGA — Advances in ShApes, Geometry, and Algebra (2014), Pages 1–8.  (Preprint)

   • Abstract:  We present a brief overview of the ShApes, Geometry and Algebra Initial Training Network, including its motivation and results. After this we provide a preview of this volume, where we briefly describe the flavor of each chapter.

6. G. Muntingh. Topics in Polynomial Interpolation Theory, PhD Thesis (PDF, DUO)

   • Summary: This thesis was carried out at the Centre of Mathematics for Applications at the University of Oslo. Two topics were studied with the goal to improve methods for finding smooth curves and surfaces passing through — or interpolate — certain known points. Such methods are used in a variety of contexts where shapes are constructed from discrete data, from typography to the computer-aided geometric design of cars, ships, and airplanes. The special case where the curves and surfaces are represented locally by polynomials is particularly popular in industrial applications. 

   • The numerical properties of such interpolants depend very much on how the data points are situated in space. The simplest configurations of data points are the rectangular grids, which are the points of intersection of some collection of horizontal and vertical lines. The triangular grids form another classical type of configurations; they are the points in the plane with nonnegative integral coordinates (i, j) with i+j bounded from above by some fixed integer. Because of their convenient properties, both configurations have been studied extensively in the literature.

   • The first part of the thesis sets forth explicit formulas for the coefficients of polynomial interpolants to implicit functions on rectangular grids. A closed formula for the higher-order derivatives of implicit functions appears as a limiting case of these formulas. The second part delves into certain generalizations of triangular grids — generalized principal lattices — that are well suited for interpolation. Applying the theory of real algebraic curves then allows the construction of many new examples of such configurations.

7. G. Muntingh. The Classification of the First Order Ordinary Differential Equations with the Painlevé Property, Master Thesis (PDF)

8. H. Kloosterman, D. Land, J. Massolt, G. Muntingh, F. van den Berg. Hoge molens vangen veel wind – wind- en geluidmetingen bij een hoge windturbine, Science Shop for Physics, University of Groningen, NWU-106, 2002. Also available in German under the title Hohe Mühlen fangen viel Wind (Dutch version, German version)

• Abstract: Dit project probeert een verklaring te geven voor het feit dat windturbines bij bepaalde weersomstandigheden meer geluid maken en daardoor op grotere afstand te horen zijn dan volgens de gebruikelijke theorie mogelijk is. Deze theorie voorspelt dat de windsnelheid logaritmisch toeneemt met de hoogte. Uit ons project blijkt dat dit verband tussen windsnelheid en hoogte niet geldt bij een stabiele atmosfeer. De windsnelheid neemt dan sneller toe bij het toenemen van de hoogte. Uitgaande van de windsnelheid op 10 meter hoogte zal de windsnelheid op ashoogte groter zijn dan volgens het logaritmische verband. Een windturbine zal daardoor meer geluid produceren.

Manuscripts in progress

1. G. Muntingh. A classification of generalized principal lattices in P3

2. T. Lyche and G. Muntingh. B-spline-like bases for C2 cubics on the Powell-Sabin 12-split

Miscellaneous

1. New sequences in The Online Encyclopedia of Integer Sequences

   • A162326

   • A172003

   • A172004