Invertible orientation scores as an application of generalized wavelet theory

Inspired by the visual system of many mammals, we consider the construction of—and reconstruction from—an orientation score of an image, via a wavelet transform corresponding to the left-regular representation of the Euclidean motion group in (ℝ²) and oriented wavelet φ ∈ (ℝ²). Because this representation is reducible, the general wavelet reconstruction theorem does not apply. By means of reproducing kernel theory, we formulate a new and more general wavelet theory, which is applied to our specific case. As a result we can quantify the well-posedness of the reconstruction given the wavelet φ and deal with the question of which oriented wavelet φ is practically desirable in the sense that it both allows a stable reconstruction and a proper detection of local elongated structures. This enables image enhancement by means of left-invariant operators on orientation scores. The text was submitted by the authors in English. Remco Duits received his M.Sc. degree (cum laude) in Mathematics in 2001 from Eindhoven University of Technology, The Netherlands. He received his PhD degree (cum laude) at the Department of Biomedical Engineering at Eindhoven University of Technology on the subject of multiscale perceptual organization. His interests include functional analysis, group theory, partial differential equations, multiscale representations and their applications to biomedical imaging and vision, and perceptual grouping. His PhD thesis is titled Perceptual Organization in Image Analysis (A Mathematical Approach Based on Scale, Orientation and Curvature). Several of his submissions at conferences have been selected/best papers, in particular, at the PRIA 2004 conference on pattern recognition and image analysis in St. Petersburg, he received a best paper award (second prize) for his work on invertible orientation scores. Currently, he is working at Eindhoven University of Technology as an assistant professor at both the Department of Applied Mathematics and Computer Science and the Department of Biomedical Engineering. Maurice Duits received his MSc degree (cum laude) in Mathematics in 2004 from Eindhoven University of Technology, The Netherlands on the subject of reproducing kernels in frame and wavelet transforms. Now he is a PhD student at the Department of Mathematics at Katholieke Universiteit Leuven on the subject of random matrices. His interests include Riemann-Hilbert problems, random matrices, orthogonal polynomials and Toeplitz matrices. Markus van Almsick earned a master degree in physics at the Technical University of Munich in 1990. From 1988 until 1992, he worked for the University of Illinois at Urbana-Champaign as a research and teaching associate. He taught undergraduate chemistry as well as graduate courses in advanced quantum mechanics, for which he developed Mathematica course material. His research interest has been quantum logic and quantization procedures of space-time. Since 1990, he has been a freelance applications consultant for Wolfram Research, Inc., USA, and Wolfram Research Europe Ltd., United Kingdom, promoting Mathematica at universities and research institutions in the U.S., Europe, and Israel, as well as developing Mathematica packages and application material. In 1996, Mr. van Almsick joined the Max Planck Insitut fur Biophysik in Frankfurt am Main, Germany, where he addressed problems in nonequilibrium thermodynamics until the theoretical department closed in 1997. Then, until 2001 he worked in collaboration with QT Software GmbH, Munich, as a full-time Mathematica consultant on a wide variety of assignments, e.g., designing the geometry of slides for playgrounds, modeling human interaction via graph theory (“social networks”), lossless image compression, vibration control in electric engines, and the isomer enumeration of libraries containing chemical diamutamers. Since 2001, he has been a part-time employee of the Technische Universiteit Eindhoven, where he develops Math VisionTools, a biomedical image analysis toolkit based on Mathematica. 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. An erratum to this article is available at .