List of Presentations

Talks

1. 27-06-2024, Mathematical Methods for Curves and Surfaces 2024, Oslo

   • Authors: Michael S. Floater and Georg Muntingh*

   • Title: Injectivity of mean value mappings between quads

   • 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. 13-06-2024, Dagstuhl seminar on Geometric Modelling: Challenges for Additive Manufacturing, Design and Analysis, Dagstuhl

   • Authors: Sverre Briseid, Georg Muntingh*

   • Title: New developments in machine learning for geometry reconstruction

   • Abstract: Recent developments in machine learning from sensor data have enabled new possibilities in geometry reconstruction. We will discuss some of these new methods, such as neural implict representations and autoregressive models, and investigate some preliminary results. 

3. 29-09-2022, SESAR2020 Pj05.35 Open Day Malmö

   • Title: Fast-forward for remote tower

4. 27-09-2021, SIAM Conference on Geometric and Physical Modeling 

   • Title: Medical image segmentation using spline representations

   • Abstract: Automatic image segmentation is an important tool in medical imaging, saving a great amount of time and costs when compared to manual segmentation performed by expert radiologists. One popular approach to image segmentation is to use deep learning, where convolutional neural networks (CNNs) are used to determine how the image should be segmented. In previous work, the idea of using implicit curves to define the segmentation boundaries in 2D via CNNs was introduced. By making use of tensor-product splines for the implicit function, this naturally equipped the method with a smoothness prior, which is well suited to medical data given the smooth nature of organs. In this work we aim to expand upon the approach introduced above by performing the segmentation on the entire 3D volume. Allowing the model to infer the segmentation boundary in 3D has several advantages, such as avoiding artifacts seen in single slices and building in the smoothness in all three spatial dimensions. Nevertheless, neural networks that perform convolutions in 3D often suffer from issues with memory consumption, simply due to the huge amounts of data that need to be concurrently stored in memory. Here, we exploit a recent approach that tackles this issue known as Dense V-net. The approach is implemented in PyTorch and is evaluated against state-of-the-art methods on various medical datasets, such as CT and MRI, and on segmentation classes of various organs. 

5. 11-06-2021, Deep Learning seminar, Rikshospitalet, Oslo

   • Title: ANALYST: LR splines & deep learning

   • Abstract: ANALYST is a research project funded by the Norwegian Research Council that aims to combine the novel LR-spline representation with modern AI techniques. The project specifically targets applications related to geospatial and medical imaging data. In this talk we will provide an introduction to the project concept and mention some of the results achieved in the project. We will then go in more depth into a recent publication that was made in collaboration between IVS and SINTEF. In that paper, the idea was to combine tensor-product splines and deep learning to provide compact representations of smooth segmentation boundaries. This paper has led to several new avenues for future research, so we will conclude the talk by describing some of these future directions and the challenges involved.

6. 02-06-2021, Confer Conference, Oslo

   • Title: Machines drawing hearts

   • Abstract: Machine learning -- in the form of convolutional neural networks -- has come a long way, from recognizing handwritten digits in the 90s to generating convincing deepfakes and providing highly accurate medical diagnoses based on medical images. In these examples the inputs and outputs are 1D, 2D, or 3D images, or numbers. But how can a computer generate something more complex, like a shape? In this talk we will investigate one way of doing this, using our method involving implicit spline representations. We will demonstrate how this works on a congenital heart disease data-set, where it is able to reconstruct heart and connecting vessels from CT images with state-of-the-art accuracy.

   • Video: https://www.youtube.com/watch?v=xn0SDt8Z0yY 

7. 20-03-2019, Validation Exercise and Open Day PJ05, Växjö.

   • Title: Tracking demo

8. 04-09-2018, Benefits of Industry 4.0 for Manufacturing Industries, Raufoss.

   • Title: Data analytics for production processes

9. 21-08-2018, BIT Circus, Helsinki.

   • Title: Symbols and exact regularity of symmetric pseudo-splines of any arity

10. 13-12-2017, NORTEM-Oslo Christmas Seminar, Oslo. Annett Thøgersen, Torunn Kjelstad, Ingvild J. T. Jensen, Georg Muntingh, Marit Stange, Alexander Ulyashin, Augustinas Galeckas, Ola Nilsen, Edouard Monakhov, and Spyros Diplas

    • Title: Plasmonic properties of aluminium nanowires in amorphous silicon and inverted silicon nanowires

11. 25-10-2017, Manufuture 2017, Tallinn (PDF, video).

    • Title: CAxMan: Computer-Aided Technologies for Additive Manufacturing

12. 30-05-2017, Dagstuhl Seminar on Geometry Modelling, Interoperability and New Challenges, Schloss Dagstuhl.

    • Title: B-spline-like bases for C2 cubics on the Powell-Sabin 12-split

    • Abstract: For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, Cohen, Lyche and Riesenfeld recently discovered so-called S-bases. These are simplex spline bases with B-spline-like properties on a single macrotriangle, which are tied together across macrotriangles in a Bézier-like manner. In this talk 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. Although the cubic bases can only be used to define smooth surfaces over specific triangulations, we envision applications for local constructions, such as hybrid meshes and extra-ordinary points, with the potential to be used in isogeometric analysis.

13. 25-06-2016, 9th International Conference on Mathematical Methods for Curves and Surfaces (MMCS9), Tønsberg.

    • Title: Symbols and exact regularity of symmetric pseudo-splines of any arity

    • Abstract: 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 a positive (discrete-time) Fourier transform, making it possible to compute the exact Hölder regularity algebraically as the spectral radius of a matrix. We apply this method to compute the regularity explicitly in some special cases, such as the symmetric binary and ternary pseudo-spline schemes.

14. 12-10-2015, SIAM Conference on Geometric and Physical Modeling (GD/SPM15), Salt Lake City. (PDF)

    • Title: Stable Simplex Spline Bases for C3 Quintics on the Powell-Sabin 12-Split

    • Abstract: See below.

15. 01-06-2015, 9th International Conference on Geometric Modeling and Processing (GMP 2015), Lugano. (PDF)

    • Title: A Hermite Subdivision Scheme for C2 Quintics on the Powell-Sabin 12-Split

    • Abstract: See below.

16. 20-04-2015, Multivariate Splines and Algebraic Geometry, Oberwolfach. (PDF)

    • Title: Simplex Spline Bases on the Powell-Sabin 12-Split: Part II

    • 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, and domain points with an intuitive control net. For one of these bases we derive C0, C1, C2 and C3 conditions on the control points of two splines on adjacent macrotriangles.

17. 30-05-2014, Dagstuhl Seminar on Geometry Modeling, Schloss Dagstuhl. (PDF)

    • Title: A Hermite subdivision scheme for C2 quintic macro-elements on the Powell-Sabin 12-split

    • Abstract: See below.

18. 30-10-2013, Geometry Seminar, Oslo.

    • Title: A Hermite subdivision scheme for C2 quintic macro-elements on the Powell-Sabin 12-split

    • Abstract: See below.

19. 30-09-2013, Multivariate Approximation and Interpolation with Applications, Erice, Italy.  

    • Title: A Hermite subdivision scheme for smooth macro-elements on the Powell-Sabin 12-split (PDF)

    • Abstract: In order to construct a C1-smooth 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 was shown by Dyn and Lyche that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme. In this talk we introduce a macro-element on the 12-split for the space of quintic splines that are locally C3 and globally C2-smooth. As in the quadratic case, these quintic splines can be evaluated quickly by means of a Hermite subdivision scheme, involving the values and first, second, and third derivatives.

20. 25-02-2013, Geometry Seminar, Oslo.

    • Title: Exact regularity of subdivision schemes of higher arity

    • Abstract: See below.

21. 22-02-2013, New Trends in Applied Geometry, Bad Herrenalb.

    • Title: Exact regularity of subdivision schemes of higher arity

    • Abstract: In a paper of 1992, Rioul introduced a technique to compute the exact regularity of a wide class of binary subdivision schemes. In this talk we will look at the extension of this technique to subdivision schemes of higher arity. We use this technique to compute the exact regularity of various subdivision schemes encountered in the literature, including pseudo-splines of higher arity defined by Conti and Hormann.

22. 18-12-2012, Seminario de Matemática Aplicada, Alcalá de Henares.

    • A Classification of Generalized Principal Lattices in Space (PDF)

23. 23-08-2012, BIT Circus, Copenhagen.

    • Title: A Classification of Generalized Principal Lattices in Space (PDF)

24. 30-05-2012, Sigma Xi, Nordic chapter, Annual Meeting and Research Conference, Oslo.

    • Title: A chi-distribution model of hail storm damage (PDF)

25. 23-02-2012, Geometry Seminar, Oslo.

    • Title: A Classification of Generalized Principal Lattices in Space (PDF)

26. 16-02-2012, New Trends in Applied Geometry, Gazzada.

    • Title: A Classification of Generalized Principal Lattices in Space (PDF)

27. 03-10-2011, Geometry Seminar, Oslo.

    • Title: Divided Differences of Multivariate Implicit Functions (PDF)

    • Abstract: See below.

28. 26-09-2011, International Conference on Multivariate Approximation, Hagen.

    • Title: Divided Differences of Multivariate Implicit Functions (PDF)

    • Abstract: Under general conditions, the equation g(x1, ..., xq, y) = 0 implicitly defines y locally as a function of x1, ..., xq. In this talk we present a formula that expresses 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 indicate how to arrive at a simpler formula for derivatives. 

29. 29-03-2011, PhD Defense, Oslo.

    • Title: Topics in Polynomial Interpolation Theory (PDF, Sage Worksheet)

30. 29-03-2011, PhD Defense, Oslo.

    • Title: Kergin Interpolation, Divided Differences, and Simplex Splines (PDF)

31. 17-03-2011, CMA Seminar, Oslo.

    • Title: Generalized Principal Lattices in Space (PDF)

    • Abstract: See below.

32. 11-02-2011, International Conference and Workshop on New Materials and Devices for Photovoltaic Applications, Madurai (joint talk with Annett Thøgersen).

    • Title: Light Trapping: Plasmonic Nanoparticles (PDF)

    • Abstract: Thin Si wafers are attractive for solar cell applications, because they can reduce the material consumption and thereby decrease the cost of solar electricity. However, when a Si wafer decreases in thickness, much of the light passes through the wafer and will not be absorbed. In order to capture the transmitted light, light trapping structures at the front or back side can be incorporated in the solar cell structure.

    • One way to capture more light is to deposit metallic nanoparticles on the solar cell. Metallic nanoparticles can scatter light into the solar cell by the localized surface plasmon resonance, which is a collective oscillation of the conduction electrons. This strong absorption band is not present in the spectrum of the corresponding bulk metal. The valence electron oscillation results in a local electromagnetic field at the nanocrystal surface and in wavelength selective photon absorption and scattering. By dispersing metal nanocrystals on top of an optically thin solar cell, the localized surface plasmon resonance will scatter the light further into the solar cell, thereby increasing the absorption and efficiency. The material properties, structure, particle size and shape is crucial for this effect to take place. These properties have been studied using TEM.

33. 17-01-2011, Geometry Seminar, Oslo.

    • Title: Generalized Principal Lattices in Space (PDF)

    • Abstract: In multivariate polynomial interpolation theory, the properties of polynomial interpolants depend very much on the configuration of the interpolation points in space. An important class is made up by the generalized principal lattices, which form a corner stone in the classification of the meshes with simple Lagrange formula and can be viewed as a generalization of the triangular meshes. While generalized principal lattices are defined by an abstract combinatorial definition, all generalized principal lattices in the projective plane arise from a real cubic curve in the dual projective plane. As all such curves are of arithmetic genus 1, one can ask the question: Which space curves of arithmetic genus 1 and degree 4 give rise to generalized principal lattices in dual projective space? In this talk we show how generalized principal lattices arise naturally from the notion of a triangular mesh and try to give an answer to this question.

34. 23-06-2009, Geometry Seminar, Oslo.

    • Title: Divided Differences of Implicit Functions (PDF)

    • Abstract: Under general conditions, the equation g(x,y) = 0 implicitly defines y locally as a function of x. In this talk we express divided differences of y in terms of bivariate divided differences of g, generalizing a recent result on divided differences of inverse functions by Michael Floater and Tom Lyche. As in the inverse case, the formula is a sum of terms over a combinatorial structure whose elements can be viewed either as partitions of a convex polygon or as planar trees. Using this connection, we prove as a corollary a recent formula of Tom Wilde for higher order derivatives of y in terms of partial derivatives of g.

35. 11-06-2008, CMA Seminar, Oslo.

    • Title: Dicritical Singularities and the Poincaré Problem (PDF)

    • Abstract: In the early nineties, an article by Lins-Neto revived a question that later became known as the Poincaré Problem: Given a differential equation y' = P(x,y)/Q(x,y), can we bound the degree of any algebraic solution in terms of the degrees of the polynomials P and Q? In this CMA-talk we'll address this problem from a geometric angle, allowing us to express most of our statements as pictures. As it turns out, a principal obstruction to solving the Poincaré Problem is the presence of so-called dicritical singularities. Such dicritical singularities can be resolved by blowing them up, which is just as spectacular as it sounds. I will end by hinting at two directions we are trying to take these results: How can we create foliations with prescribed singularities and how can we use dicritical singularity types to solve the Poincaré Problem?

36. 01-10-2007, Algebra Seminar, Oslo.

    • Title: Unifying Mathematical Systems with SAGE (PDF,ODP,HTML-archive)

    • Abstract: In a world where computers increasingly play a supporting role in mathematics, mathematicians have lots of mathematical systems to choose from. Although many of these systems have overlapping goals, they tend to reinvent the wheel and do not cooperate. In this talk we will discuss a project that builds a layer on top of these systems to connect them, and we'll put some emphasis on the algebra systems that are included.

37. 15-06-2007, Algebra Seminar, Oslo.

    • Title: The First Order Ordinary Differential Equations with the Painlevé property (PDF)

    • Abstract: See below.

38. 14-09-2006, Tsukuba.

    • Title: The First Order Ordinary Differential Equations with the Painlevé property (PDF)

    • Abstract: See below.

39. 23-03-2006, Master Thesis Defense, Groningen.

    • Title: The First Order Ordinary Differential Equations with the Painlevé property (PDF)

    • Abstract: Differential equations with the Painlevé Property play a special role in physics. They have been studied extensively, and have connections to many branches of mathematics, including function theory, dynamical systems and differential algebraic geometry. In some sense, one can say that they form a class broad enough to capture a wide range of phenomena, but on the other hand narrow enough to be able to say sensible things about them. Assuming only some mathematics taught in high school, we shall start with elucidating the concept of a differential equation, followed by an explanation of what it means for a differential equation to have the Painlevé property. After this, we will consider in a conceptual way first order ordinary differential equations with the Painlevé property from two points of view: the classical function theoretic approach, and a modern differential algebraic geometric approach. Historical details will be given, and the talk ends with a classification theorem.