ScaleSpaceViz: α-Scale spaces in practice

Kernels of the so-called α-scale space have the undesirable property of having no closed-form representation in the spatial domain, despite their simple closed-form expression in the Fourier domain. This obstructs spatial convolution or recursive implementation. For this reason an approximation of the 2D α-kernel in the spatial domain is presented using the well-known Gaussian kernel and the Poisson kernel. Experiments show good results, with maximum relative errors of less than 2.4%. The approximation has been successfully implemented in a program for visualizing α-scale spaces. Some examples of practical applications with scale space feature points using the proposed approximation are given. The text was submitted by the authors in English. Frans Kanters received his MSc degree in Electrical Engineering in 2002 from the Eindhoven University of Technology in the Netherlands. Currently he is working on his PhD at the Biomedical Imaging and Informatics group at the Eindhoven University of Technology. His PhD work is part of the “Deep Structure, Singularities, and Computer Vision (DSSCV)” project sponsored by the European Union. His research interests include scale space theory, image reconstruction, image processing algorithms, and hardware implementations thereof. Luc Florack received his MSc degree in theoretical physics in 1989 and his PhD degree cum laude in 1993 with a thesis on image structure, both from Utrecht University, the Netherlands. During the period from 1994 to 1995, he was an ERCIM/HCM research fellow at INRIA Sophia-Antipolis, France, and IN-ESC Aveiro, Portugal. In 1996 he was an assistant research professor at DIKU, Copenhagen, Denmark, on a grant from the Danish Research Council. From 1997 to June 2001, he was an assistant research professor at Utrecht University in the Department of Mathematics and Computer Science. Since June 1, 2001, he has been working as an assistant professor and, then, as an associate professor at Eindhoven University of Technology, Department of Biomedical Engineering. His interest includes all multiscale structural aspects of signals, images, and movies and their applications to imaging and vision. Remco Duits received his MSc degree (cum laude) in Mathematics in 2001 from the Eindhoven University of Technology, the Netherlands. Today he is a PhD student at the Department of Biomedical Engineering at the Eindhoven University of Technology on the subject of multiscale perceptual organization. His interest subtends functional analysis, group theory, partial differential equations, multiscale representations and their applications to biomedical imaging and vision, perceptual grouping. Currently, he is finishing his thesis “Perceptual Organization in Image Analysis (A Mathematical Approach Based on Scale, Orientation and Curvature).” During his PhD work, several of his submissions at conferences were chosen as selected or best papers—in particular, at the PRIA 2004 conference on pattern recognition and image analysis in St. Petersburg, where he received a best paper award (second place) for his work on invertible orientation scores. Bram Platel received his Masters Degree cum laude in biomedical engineering from the Eindhoven University of Technology in 2002. His research interests include image matching, scale space theory, catastrophe theory, and image-describing graph constructions. Currently he is working on his PhD in the Biomedical Imaging and Informatics group at the Eindhoven University of Technology. Bart M. ter Haar Romany is full professor in Biomedical Image Analysis at the Department of Biomedical Engineering at Eindhoven University of Technology. He has been in this position since 2001. He received a MSc in Applied Physics from Delft University of Technology in 1978, and a PhD on neuromuscular nonlinearities from Utrecht University in 1983. After being the principal physicist of the Utrecht University Hospital Radiology Department, in 1989 he joined the department of Medical Imaging at Utrecht University as an associate professor. His interests are mathematical aspects of visual perception, in particular linear and non-linear scale-space theory, computer vision applications, and all aspects of medical imaging. He is author of numerous papers and book chapters on these issues; he edited a book on non-linear diffusion theory and is author of an interactive tutorial book on scale-space theory in computer vision. He has initiated a number of international collaborations on these subjects. He is an active teacher in international courses, a senior member of IEEE, and IEEE Chapter Tutorial Speaker. He is chairman of the Dutch Biophysical Society.