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Using Catastrophe Theory to Derive Trees from Images

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Abstract

In order to investigate the deep structure of Gaussian scale space images, one needs to understand the behaviour of critical points under the influence of blurring. We show how the mathematical framework of catastrophe theory can be used to describe the different types of annihilations and the creation of pairs of critical points and how this knowledge can be exploited in a scale space hierarchy tree for the purpose of a topology based segmentation. A key role is played by scale space saddles and iso-intensity manifolds through them. We discuss the role of non-generic catastrophes and their influence on the tree and the segmentation. Furthermore it is discussed, based on the structure of iso-intensity manifolds, why creations of pairs of critical points don’t influence the tree. We clarify the theory with an artificial image and a simulated MR image.

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  1. Department of Innovation, IT University of Copenhagen, Rued Langgaardsvej 7, DK-2300, Copenhagen, Denmark

    Arjan Kuijper

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Arjan Kuijper received his M.Sc. degree in applied mathematics in 1995 with a thesis on the comparision of two image restoration techniques, from the University of Twente, The Netherlands. During the period 1996–1997 he worked at ELTRA Parkeergroep, Ede, The Netherlands. In the period 1997-2002 he has been a Ph.D. student and associate researcher at the Institute of Information and Computing Sciences of Utrecht University. In 2002 he received his Ph.D. degree with a thesis on “Deep Structure of Gaussian Scale Space Images” and worked as postdoc at Utrecht University on the project “Co-registration of 3D Images” on a grant of the Netherlands Ministry of Economic Affairs within the framework of the Innovation Oriented Research Programme. Since Januari 1st 2003 he has been working as an assistant research professor at the IT University of Copenhagen in Denmark funded by the IST Programme “Deep Structure, Singularities, and Computer Vision (DSSCV)” of the European Union. His interest subtends all mathematical aspects of image analysis, notably multiscale representations (scale spaces), catastrophe and singularity theory, medial axes and symmetry sets, and applications to medical imaging.

Luc M.J. Florack received his M.Sc. degree in theoretical physics invv 1989, and his Ph.D. degree in 1993 with a thesis on image structure, both from Utrecht University, The Netherlands. During the period 1994–1995 he was an ERCIM/HCM research fellow at INRIA Sophia-Antipolis, France, and INESC Aveiro, Portugal. In 1996 he was an assistant research professor at DIKU, Copenhagen, Denmark, on a grant from the Danish Research Council. From 1997 until June 2001 he was an assistant research professor at Utrecht University at the Department of Mathematics and Computer Science. Since June 1st 2001 he is with Eindhoven University of Technology, Department of Biomedical Engineering, currenlty employed as an associate professor. His interest subtends all structural aspects of signals, images and movies, notably multiscale representations, and their applications to imaging and vision.

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Kuijper, A. Using Catastrophe Theory to Derive Trees from Images. J Math Imaging Vis 23, 219–238 (2005). https://doi.org/10.1007/s10851-005-0481-9

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