UC SPARK - University of Canterbury - New Zealand

Professor Rick Beatson

Mathematics and Statistics

Fields of Research

  • Approximation theory and numerical analysis
  • Fast algorithms for computing with radial basis functions and splines
  • Fitting spline and radial basis function surfaces to large data sets
  • Applications to geophysical, laser scanner and medical imaging data sets
  • Kriging and cross validation
  • Neural networks
  • Radial basis functions
  • Visualisation

Researcher Summary

My recent research has focused on the Approximation Theory and Numerical Analysis of fast computation and smoothing with radial basis functions. This work is motivated by applications to geophysics, image processing, computer graphics and custom manufacture.

Subject Area: Disciplines

Research Groups

Key Methodologies

  • Fast algorithms

Prizes and Awards

  • Erskine Grant (14/07/01-27/08/01). Algorithms for Approximation IV, Huddersfield; SIGGRAPH, Los Angeles. ( 1997 - 2002)

Research/Scholarly/Creative Works

  • Beatson RK. and zu Castell W. (2017) Dimension hopping and families of strictly positive definite zonal basis functions on spheres. Journal of Approximation Theory 221: 22-37. http://dx.doi.org/10.1016/j.jat.2017.04.001. (Journal Articles)
  • Beatson RK. and zu Castell W. (2016) Invited Speaker. Title: Dimension Hopping Operators and Families of Strictly Positive Definite Functions on Spheres. Schloss Rauischholtzhausen, Germany: International Conference on Multivariate Approximation, 31 Mar-5 Apr 2016 (Conference Contributions - Other)
  • Beatson RK. and zu Castell W. (2016) One-Step Recurrences for Stationary Random Fields on the Sphere. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 12 (2016), 043: 19pp. http://www.emis.de/journals/SIGMA/2016/043/. (Journal Articles)
  • Beatson RK. (2015) Colloquium talk at the University of Bayreuth. Hosted by Prof. Holger Wendland. Bayreuth, Germany. (Oral Presentations)
  • Beatson RK. (2015) Colloquium talk at the University of Passau. Hosted by Prof. Tomas Sauer. Passau, Germany. (Oral Presentations)
  • Beatson RK. (2015) Seminar, Physics department, University of Otago. Dunedin, New Zealand. (Oral Presentations)
  • Beatson RK. (2015) Kernel based methods for fitting scattered data with applications. Christchurch, New Zealand: New Zealand Mathematics Colloquium 2015, 1-3 Dec 2015 (Conference Contributions - Other)
  • Beatson RK. (2015) New Zealand Mathematical Society Lecturer. Title: Kernel based methods for fitting scattered data with applications. Christchurch, New Zealand: New Zealand Mathematics Colloquium 2015, 1-3 Dec 2015 (Conference Contributions - Other)
  • Beatson RK., Ong WE. and Rychkov I. (2014) Faster fast evaluation of thin plate splines in two dimensions. Journal of Computational and Applied Mathematics 261(1): 201-212. http://dx.doi.org/10.1016/j.cam.2013.11.005. (Journal Articles)
  • Beatson RK., zu Castell W. and Xu Y. (2014) A Pólya criteria for (strict) positive-definiteness on the sphere. IMA Journal of Numerical Analysis 34(2): 550-568. http://dx.doi.org/10.1093/imanum/drt008. (Journal Articles)
  • Naik N., Beatson R. and Eriksson J. (2014) Radial-basis-function level-set-based regularized Gauss-Newton-filter reconstruction scheme for dynamic shape tomography. Applied Optics 53(29): 6872-6884. http://dx.doi.org/10.1364/AO.53.006872. (Journal Articles)
  • Beatson RK. (2013) Invited speaker. Title: Radial basis functions applications and theory. Glasgow: 25th Biennial Numerical Analysis Conference, 25-28 Jun 2013 (Conference Contributions - Other)
  • Beatson RK. (2013) Invited Speaker. Title: Radial basis functions on the sphere. Justus -Leibig University, Giessen, Germany: Radial basis function day, 30-30 Jul 2013 (Conference Contributions - Other)
  • MacMahan J., Vennell R., Beatson R., Brown J. and Reniers A. (2012) Divergence-free spatial velocity flow field interpolator for improving measurements from ADCP-equipped small unmanned underwater vehicles. Journal of Atmospheric and Oceanic Technology 29(3): 478-484. http://dx.doi.org/10.1175/JTECH-D-11-00084.1. (Journal Articles)
  • Beatson RK., Levesley J. and Mouat CT. (2011) Better bases for radial basis function interpolation problems. Journal of Computational and Applied Mathematics 236(4): 434-446. http://dx.doi.org/10.1016/j.cam.2011.06.030. (Journal Articles)
  • Beatson RK., zu Castell W. and Schrodl SJ. (2011) Kernel-based methods for vector-valued data with correlated components. SIAM Journal on Scientific Computing 33(4): 1975-1995. http://dx.doi.org/10.1137/090758076. (Journal Articles)
  • Ong WE., Beatson R. and Price CJ. (2011) Reconstruction with blobby shapes. University of New South Wales, Sydney, Australia: 15th Computational Techniques and Applications Conference (CTAC2010), 28 Nov-1 Dec 2010. In ANZIAM journal 52: C596-C611. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3885. (Conference Contributions - Published)
  • De Las Heras H., Beatson R., Castell WZ., Tischenko O., Xu Y. and Hoeschen C. (2009) CT with Dual Optimal Reading: Compatibility of the two data sets and interpolation issues. In Ifmbe Proceedings 25(2): 579-582. http://dx.doi.org/10.1007/978-3-642-03879-2-162. (Conference Contributions - Published)
  • de las Heras H., Beatson R., Castell WZ., Tischenko O., Xu Y. and Hoeschen C. (2009) CT with dual optimal reading: compatibility of the two data sets and interpolation issues. Munich, Germany: World Congress 2009 - Medical Physics & Biomedical Engineering (WC2009), 7-12 Sep 2009. In International Federation for Medical and Biological Engineering Proceedings 25(2): 579-582. http://dx.doi.org/10.1007/978-3-642-03879-2_162. (Conference Contributions - Published)
  • Vennell R. and Beatson R. (2009) A divergence-free spatial interpolator for large sparse velocity data sets. Journal of Geophysical Research 114 C10024 http://dx.doi.org/10.1029/2008JC004973. (Journal Articles)
  • Beatson R. (2008) Computational issues in RBF fitting. Tonsberg, Norway: Minisymposium: Radial Basis Functions; Seventh International Conference on Mathematical Methods for Curves and Surfaces, 26 Jun-1 Jul 2008 (Conference Contributions - Other)
  • Naik N., Beatson R., Eriksson J. and Van Houten E. (2008) An implicit radial basis function based reconstruction approach to electromagnetic shape tomography. Inverse Problems 25(2): 25004. http://dx.doi.org/10.1088/0266-5611/25/2/025004. (Journal Articles)
  • Beatson RK. and Bui HQ. (2007) Mollification formulas and implicit smoothing. Advances in Computational Mathematics 27(2): 125-149. (Journal Articles)
  • Beatson RK. and Langton MK. (2007) Integral Interpolation. In Iske A; Levesley J (Ed.), Algorithms for Approximation: Proceedings of the 5th International Conference, Chester, July 2005: 199-218. Heidelberg: Springer-Verlag. (Chapters)
  • Beatson RK., Davydov O. and Levesley J. (2006) Preconditioning radial basis function computations using mean value coordinates. Avignon, France: Sixth International Conference on Curves and Surfaces, 29 Jun-5 Jul 2006 (Conference Contributions - Other)
  • Beatson RK., Powell MJD. and Tan AM. (2006) Fast evaluation of polyharmonic splines in three dimensions. IMA Journal of Numerical Analysis 27(3): 427-450. http://dx.doi.org/10.1093/imanum/drl027. (Journal Articles)
  • Vennell R. and Beatson RK. (2006) Moving vessel acoustic Doppler current profiler measurement of tidal stream function using radial basis functions. Journal of Geophysical Research 111(C9) http://dx.doi.org/10.1029/2005JC003321. (Journal Articles)
  • Beatson RK. (2005) Invited speaker. Topic: RBFs. Goettingen, Germany: Radial basis functions and beyond, 25-26 Nov 2005 (Conference Contributions - Other)
  • Beatson RK., Bui HQ. and Levesley J. (2005) Embeddings of Beppo-Levi spaces in Holder-Zygmund spaces and a new method for radial basis function interpolation error estimates. Journal of Approximation Theory 137: 166-178. (Journal Articles)
  • Beatson RK. (2004) Methods for surface reconstruction with an emphasis on polyharmonic spline methods. Tromso, Norway: 6th International Conference on Curves and Surfaces, 1 Jul 2004 (Conference Contributions - Other)
  • Beatson RK., Bui HQ. and Levesley J. (2004) Embeddings of Beppo-Levi spaces in Holder-Zygmund spaces and a new method for radial basis function interpolation error estimates, Research report UCDMS 2004/6Embeddings of Beppo-Levi spaces in Holder-Zygmund spaces and a new method for radial basis function interpolation error estimates, Research report UCDMS 2004/6Embeddings of Beppo-Levi spaces in Holder-Zygmund spaces and a new method for radial basis function interpolation error estimates, Research report UCDMS 2004/6Embeddings of Beppo-Levi spaces in Holder-Zygmund spaces and a new method for radial basis function interpolation error estimates, Research report UCDMS 2004/6Embeddings of Beppo-Levi spaces in Holder-Zygmund spaces and a new method for radial basis function interpolation error estimates, Research report UCDMS 2004/6Embeddings of Beppo-Levi spaces in Holder-Zygmund spaces and a new method for radial basis function interpolation error estimates, Research report UCDMS 2004/6Embeddings of Beppo-Levi spaces in Holder-Zygmund spaces and a new met.Commissioned by Mathematics and Statistics. 12. http://www.math.canterbury.ac.nz/. (Reports)
  • Beatson RK., Cherrie JB., McLennan TJ., Mitchell TJ., Carr JC., Fright WR. and McCallum BC. (2004) Surface reconstruction via smoothest restricted range approximation. Seattle, WA, USA: Geometric Design and Computing, 9 Nov 2003 (Conference Contributions - Other)
  • Beatson RK., Levesley J. and Light WA. (2004) Fast evaluation on the sphere. Witten-Blommerholz, Germany: IDOMAT 2004, 15 Feb 2004 (Conference Contributions - Other)
  • Beatson RK. (2003) An introduction to radial basis function theory and applications. ICIAM, 18 Dec 2003. (Oral Presentations)
  • Beatson RK. (2003) Fast computation with radial basis functions for applications to image reconstruction and geophysics. ANODE Conference, 04 May 2001. (Oral Presentations)
  • Beatson RK. (2003) Image reconstruction via smoothest restricted range approximation. SIAM conference on Geometric Design. (Oral Presentations)
  • Beatson RK. (2003) Smoothing of radial basis functions with applications to image reconstruction and geophysics. Will Light Memorial Conference, 21 Sep 2003. (Oral Presentations)
  • Beatson RK. and Bui HQ. (2003) Mollification formulas and implicit smoothing, UCDMS 2003/19.Commissioned by Department of Mathematics and Statistics, University of Canterbury. 11. http://www.math.canterbury.ac.nz/. (Reports)
  • Carr JC., Beatson RK., McCallum BC., Fright WR., McLennan TJ. and Mitchell TJ. (2003) Smooth surface reconstruction from noisy range data. Melbourne, Australia: Graphite 2003, 11 Feb 2003 (Conference Contributions - Other)
  • Cowan EJ., Beatson RK., Ross HJ., Fright WR., McLennan TJ., Evans TR., Carr JC., Lane RG., Bright DV. and Gillman AJ. (2003) Practical implicit geological modelling. Bendigo, Australia: Fifth International Mining Geology Conference, 17-19 Nov 2003. In Proceedings of the 5th International Mining Geology Conference, Simon Dominy ed. Australian Institute of Mining and Metalurgy: 17-19. (Conference Contributions - Published)
  • Cowan EJ., Beatson RK., Ross HJ., Fright WR., McLennan TJ., Evans TR., Carr JC., Lane RG., Bright DV. and Gillman AJ. (2003) Practical implicit geological modelling. In 5TH INTERNATIONAL MINING GEOLOGY CONFERENCE 2003(8): 89-99. (Conference Contributions - Published)
  • Beatson RK. (2002) Mollification formulas and Implicit Smoothing. (Oral Presentations)
  • Beatson RK. and Levesley J. (2002) Good point/bad point iterations for solving the thin-plate spline interpolation equations. St. Louis, MO, USA: Approximation Theory X: Wavelets, Splines and Applications. Vanderbilt University Press., 1 Jan 2002 (Conference Contributions - Other)
  • Billings SD., Beatson RK. and Newsam GN. (2002) Interpolation of geophysical data with continuous global surfaces. Geophysics 67: 1810-1822. (Journal Articles)
  • Billings SD., Newsam GN. and Beatson RK. (2002) Smooth fitting of geophysical data using continuous global surfaces. Geophysics 67: 1823-1834. (Journal Articles)
  • Cherrie JB., Beatson RK. and Newsam GN. (2002) Fast evaluation of radial basis functions: Methods for generalised multiquadrics in R^n. SIAM Journal on Scientific Computation 23: 1549-1571. (Journal Articles)
  • Cowan EJ., Beatson RK., Fright WR., McLennan TJ. and Mitchell TJ. (2002) Rapid geological modelling (9 page extended abstract). Kalgoorlie, Australia: Applied Structural Geology for Mineral Exploration and Mining, 23-25 Sep 2002 (Conference Contributions - Other)
  • Mouat CT. and Beatson RK. (2002) On the boundary over distance preconditioner for radial basis function interpolation. Chester, United Kingdom.: Algorithms for Approximation IV. University of Huddersfield Press., 1 Jan 2002 (Conference Contributions - Other)
  • Mouat CT. and Beatson RK. (2002) RBF Collocation, UCDMS 2002/3.30pp. (Reports)
  • Beatson RK. (2001) Radial basis function methods for 3D applications: Fast evaluation of 3D polyharmonic splines. 3rd International Meeting on Approximation Theory, 01 May 2000. (Oral Presentations)
  • Beatson RK., Cherrie JB. and Ragozin D. (2001) Fast evaluation of radial basis functions: Methods for four-dimensional polyharmonic splines. SIAM Journal on Mathematical Analysis 32: 1272-1310. (Journal Articles)
  • Carr JC., Beatson RK., Cher̀rie JB., Mitchell TJ., Fright WR., McCallum BC. and Evans TR. (2001) Reconstruction and representation of 3D objects with radial basis functions. In Proceedings of the ACM SIGGRAPH Conference on Computer Graphics: 67-76. (Conference Contributions - Published)
  • Carr JC., Beatson RK., Cherrie JB., Mitchell TJ., Fright WR., McCallum BC. and Evans TR. (2001) Reconstruction and representation of 3D objects with radial basis functions. Los Angeles, CA, USA: Computer Graphics, SIGGRAPH 2001 Proceedings, 1 Jan 2001 (Conference Contributions - Other)
  • Mouat CT. and Beatson RK. (2001) Some properties of the boundary over distance preconditioner for radial basis function interpolation, UCDMS 2001/6.20pp. (Reports)
  • Beatson RK. and Chacko E. (2000) Fast evaluation of radial basis functions: A multivariate momentary evaluation scheme. Saint-Malo Conference on Approximation Theory, 1-1 Jul 1999. In Curve and Surface Fitting: Saint-Malo 1999 Nashville: Vanderbilt University Press: 37-46. (Conference Contributions - Published)
  • Beatson RK. and Mouat CT. (2000) Fast Kriging. Trends in Approximation Theory, 01 Aug 2002. (Oral Presentations)
  • Beatson RK., Cherrie JB. and Ragozin D. (2000) Polyharmonic splines in R^d. Tools for fast evaluation.. St Malo, France: Curve and Surface Fitting:Saint-Malo, 1999. Vanderbilt University Press., 1 Jan 2000 (Conference Contributions - Other)
  • Beatson RK., Light WA. and Billings S. (2000) Fast solution of the radial basis function interpolation equations: domain decomposition methods. SIAM Journal on Scientific Computing 22: 1717-1740. (Journal Articles)
  • Beatson RK., Cherrie JB. and Mouat CT. (1999) Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration. Advances in Computational Mathematics 11(2-3): 253-270. (Journal Articles)
  • Beatson RK., Cherrie JB. and Mouat CT. (1999) Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration. Advances in Computational Mathematics 11: 253-270. (Journal Articles)
  • Beatson RK. (1998) Radial basis function methods for large data sets. CAGD and Wavelet Methods for Reconstructing Functions, 01 Aug 2001. (Oral Presentations)
  • Beatson RK. and Newsam GN. (1998) Fast evaluation of radial basis functions: Moment based methods. SIAM Journal on Scientific Computing 19: 1428-1449. (Journal Articles)
  • Beatson RK. and Greengard LL. (1997) A short course on fast multipole methods. In Ainsworth M; Levesley J; Light WA; Marletta M (Ed.), Wavelets, Multilevel Methods and Elliptic PDEs: 1-37. Oxford: Oxford University Press. (Chapters)
  • Beatson RK. and Light WA. (1997) Fast evaluation of radial basis functions: Methods for 2--dimensional polyharmonic splines. IMA Journal of Numerical Analysis 17: 343-372. (Journal Articles)
  • Carr JC., Fright WR. and Beatson RK. (1997) Surface interpolation with radial basis functions for medical imaging.. IEEE Transactions on Medical Imaging 16: 77-97. (Journal Articles)
  • Beatson RK., Goodsell G. and Powell MJD. (1995) On multigrid techniques for thin-plate spline interpolation in two dimensions. Park City, UT, USA: 1995 AMS-SIAM Summer Seminar in Applied Mathematics, 17 Jul-11 Aug 1995. In The Mathematics of Numerical Analysis, J.M. Renegar, M. Shub and S. Smale eds American Mathematical Society: 77-97. (Conference Contributions - Published)
  • BEATSON RK. and POWELL MJD. (1994) AN ITERATIVE METHOD FOR THIN-PLATE SPLINE INTERPOLATION THAT EMPLOYS APPROXIMATIONS TO LAGRANGE FUNCTIONS. UNIV DUNDEE, DUNDEE, SCOTLAND: 15th Dundee Conference on Numerical Analysis, 29 Jun-2 Jul 1993. In Pitman Research Notes in Mathematics Series 303: 17-39. (Conference Contributions - Published)
  • Beatson RK. and Light WA. (1993) Quasi-interpolation by thin-plate splines on a square. Constructive Approximation 9(4): 407-433. http://dx.doi.org/10.1007/BF01204649. (Journal Articles)
  • BEATSON RK. and CHACKO E. (1992) WHICH CUBIC SPLINE SHOULD ONE USE. SIAM Journal on Scientific Computing 13(4): 1009-1024. http://dx.doi.org/10.1137/0913059. (Journal Articles)
  • Beatson RK. and Light WA. (1992) Quasi-interpolation in the absence of polynomial reproduction. Oberwolfach, Germany: Numerical Methods of Approximation Theory, 24-30 Nov 1992. In Numerical Methods of Approximation Theory vol 9, ISNM 105, D. Braess and L.L. Schumaker eds, Birkhauser Verla (Conference Contributions - Published)
  • Beatson RK. and Powell MJD. (1992) Univariate interpolation on a regular finite grid by a multiquadric plus a linear polynomial. IMA Journal of Numerical Analysis 12(1): 107-133. http://dx.doi.org/10.1093/imanum/12.1.107. (Journal Articles)
  • Beatson RK. and Powell MJD. (1992) Univariate multiquadric approximation: Quasi-interpolation to scattered data. Constructive Approximation 8(3): 275-288. http://dx.doi.org/10.1007/BF01279020. (Journal Articles)
  • BEATSON RK. and WOLKOWICZ H. (1989) POST-PROCESSING PIECEWISE CUBICS FOR MONOTONICITY. SIAM Journal on Numerical Analysis 26(2): 480-502. http://dx.doi.org/10.1137/0726028. (Journal Articles)
  • Beatson RK. (1986) ON THE CONVERGENCE OF SOME CUBIC SPLINE INTERPOLATION SCHEMES.. SIAM Journal on Numerical Analysis 23(4): 903-912. (Journal Articles)
  • Beatson RK. and Ziegler Z. (1985) MONOTONICITY PRESERVING SURFACE INTERPOLATION.. SIAM Journal on Numerical Analysis 22(2): 401-411. (Journal Articles)
  • Beatson RK. and Chacko E. (1983) A quantitative comparison of end conditions for cubic spline interpolation. College Station, TX, USA: International Symposium on Approximation Theory, 10-14 Jan 1983. In Approximation Theory IV, Academic Press 1989, C.K. Chui, L.L. Schumaker and J.D. Ward editors: 77-79. (Conference Contributions - Published)
  • BEATSON RK. (1982) MONOTONE AND CONVEX APPROXIMATION BY SPLINES - ERROR-ESTIMATES AND A CURVE FITTING ALGORITHM. SIAM Journal on Numerical Analysis 19(6): 1278-1285. http://dx.doi.org/10.1137/0719092. (Journal Articles)
  • BEATSON RK. (1982) RESTRICTED RANGE APPROXIMATION BY SPLINES AND VARIATIONAL-INEQUALITIES. SIAM Journal on Numerical Analysis 19(2): 372-380. http://dx.doi.org/10.1137/0719023. (Journal Articles)
  • Beatson RK., Chui CK. and Hasson M. (1982) Degree of best inverse approximation by polynomials. Illinois Journal of Mathematics 26(1): 173-180. (Journal Articles)
  • BEATSON RK. (1980) ADDITION. Journal of Approximation Theory 29(4): 336-338. (Journal Articles)
  • Beatson RK. (1980) Joint approximation of a function and its derivatives. Austin, Texas: Symposium on Approximation Theory, 8-12 Jan 1980. In Approximation Theory III, E.W. Cheney ed, Academic Press, 1980: 199-206. (Conference Contributions - Published)
  • Beatson RK. and Chui CK. (1980) Best multipoint local approximation. Oberwolfach, Germany: Functional Analysis and Approximation, 9-16 Aug 1980. In Functional Analysis and Approximation, P.L. Butzer, B. Sz.-Nagy and E. Gorlich eds, Birkauser Verlag ISNM 60: 283-296. (Conference Contributions - Published)
  • Beatson RK. (1978) The degree of monotone approximation. Pacific Journal of Mathematics 74(1): 5-14. (Journal Articles)