Meshfree Thinning of 3D Point Clouds

An efficient data reduction scheme for the simplification of a surface given by a large set X of 3D point-samples is proposed. The data reduction relies on a recursive point removal algorithm, termed thinning, which outputs a data hierarchy of point-samples for multiresolution surface approximation. The thinning algorithm works with a point removal criterion, which measures the significances of the points in their local neighbourhoods, and which removes a least significant point at each step. For any point x in the current point set Y⊂X, its significance reflects the approximation quality of a local surface reconstructed from neighbouring points in Y. The local surface reconstruction is done over an estimated tangent plane at x by using radial basis functions. The approximation quality of the surface reconstruction around x is measured by using its maximal deviation from the given point-samples X in a local neighbourhood of x. The resulting thinning algorithm is meshfree, i.e., its performance is solely based upon the geometry of the input 3D surface point-samples, and so it does not require any further topological information, such as point connectivities. Computational details of the thinning algorithm and the required data structures for efficient implementation are explained and its complexity is discussed. Two examples are presented for illustration. This paper is dedicated to Arieh Iserles on the occasion of his 60th anniversary.     

Editorial: Math. Log. Quart. 1/2007

Dear Readers, A happy and successful 2007 to all of you. I wish to draw your attention again to an enlargement of the Editorial Board of our journal. We are pleased to welcome three new members: Ramon Jansana (Barcelona), Klaus Meer (Odense), and Andrzej Roslanowski (Omaha). I would like to thank the outgoing member, Peter Koepke (Bonn), for the work he has done in supporting MLQ. Also, it is more than justified that the name of our Editorial Assistant, who always faithfully compiles the final layout, can now be seen at the cover. All readers are once more encouraged to contact the Editorial Office or a member of the Editorial Board with any suggestions concerning the contents or the editorial work of our journal. In particular, proposals for issues devoted to special subjects or conferences are highly welcome. In 2007 there will be two such issues: one will be dedicated to “Trends in Constructive Mathematics” and a further one will deal with “Computability and Complexity in Analysis”. It should finally be mentioned that our publishing house is celebrating its 200th anniversary in 2007. During its whole history, Wiley has grown and evolved and has always been successful in serving the needs of its customers. So we can be sure to have a good background for the further development of our journal. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Neuere Untersuchungen über die Struktur thixotroper Gele

Zusammenfassung Die thixotrope Erstarrung beruht auf der Ausbildung dreidimensionaler Gerüste von Feststoffpartikeln. Dies wird unter anderem auch durch die in der vorliegenden Arbeit beschriebene anomale Gefrierpunktserniedrigung erwiesen. Die Architektur des Gerüsts wird bestimmt durch die Gestalt und die Größe der Kristalle, die Ladungsdichte und die Verteilung der Ladungen auf ihrer Oberfläche, die Wertigkeit der Gegenionen und die Möglichkeit zur Ausbildung von Solvatbrücken.Vorgetragen auf der Rheologen-Tagung vom 15. bis 17. Mai 1962 in Berlin-Dahlem.