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Anisotropic Curvature Motion for Structure Enhancing Smoothing of 3D MR Angiography Data

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Abstract

We propose a novel concept of shape prior for the processing of tubular structures in 3D images. It is based on the notion of an anisotropic area energy and the corresponding geometric gradient flow. The anisotropic area functional incorporates a locally adapted template as a shape prior for tubular vessel structures consisting of elongated, ellipsoidal shape models. The gradient flow for this functional leads to an anisotropic curvature motion model, where the evolution is driven locally in direction of the considered template. The problem is formulated in a level set framework, and a stable and robust method for the identification of the local prior is presented. The resulting algorithm is able to smooth the vessels, pushing solution toward elongated cylinders with round cross sections, while bridging gaps in the underlying raw data. The implementation includes a finite-element scheme for numerical accuracy and a narrow band strategy for computational efficiency.

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Authors and Affiliations

  1. Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Nussallee 15, 53115, Bonn, Germany

    Oliver Nemitz & Martin Rumpf

  2. 3450 Merrill Engineering Building, School of Computing, University of Utah, 50 S. Central Campus Drive, Salt Lake City, UT, 84112

    Tolga Tasdizen & Ross Whitaker

Authors
  1. Oliver Nemitz
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  2. Martin Rumpf
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  3. Tolga Tasdizen
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  4. Ross Whitaker
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Correspondence to Oliver Nemitz.

Additional information

Oliver Nemitz received his Diploma in mathematics from the university of Duisburg, Germany in 2003. Then he started to work on his Ph.D. thesis in Duisburg. Since 2005 he is continuing the work on his Ph.D. project at the Institute for Numerical Simulation at Bonn University. His Ph.D. subject is fast algorithms for image manipulation in 3d, using PDE’s, variational methods, and level set methods.

Martin Rumpf received his Ph.D. in mathematics from Bonn University in 1992. He held a postdoctoral research position at Freiburg University. Between 1996 and 2001, he was an associate professor at Bonn University and from 2001 until 2004 full professor at Duisburg University. Since 2004 he is now full professor for numerical mathematics and scientific computing at Bonn University. His research interests are in numerical methods for nonlinear partial differential equations, geometric evolution problems, calculus of variations, adaptive finite element methods, image and surface processing.

Tolga Tasdizen received his B.S. degree in Electrical Engineering from Bogazici University, Istanbul in 1995. He received the M.S. and Ph.D. degrees in Engineering from Brown University in 1997 and 2001. From 2001 to 2004 he was a postdoctoral research associate with the Scientific Computing and Imaging Institute at the University of Utah. Since 2004 he has been with the School of Computing at the University of Utah as a research assistant professor. He also holds an adjunct assistant professor position with the Department of Neurology and the Center for Alzheimer’s Care, Imaging and Research, and a research scientist position with the Scientific Computing and Imaging Institute at the University of Utah.

Ross Whitaker received his B.S. degree in Electrical Engineering and Computer Science from Princeton University in 1986, earning Summa Cum Laude. From 1986 to 1988 he worked for the Boston Consulting Group, entering the University of North Carolina at Chapel Hill in 1989. At UNC he received the Alumni Scholarship Award, and completed his Ph.D. in Computer Science in 1994. From 1994–1996 he worked at the European Computer-Industry Research Centre in Munich Germany as a research scientist in the User Interaction and Visualization Group. From 1996–2000 he was an Assistant Professor in the Department of Electrical Engineering at the University of Tennessee. He is now an Associate Professor at the University of Utah in the College of Computing and the Scientific Computing and Imaging Institute.

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Nemitz, O., Rumpf, M., Tasdizen, T. et al. Anisotropic Curvature Motion for Structure Enhancing Smoothing of 3D MR Angiography Data. J Math Imaging Vis 27, 217–229 (2007). https://doi.org/10.1007/s10851-006-0645-2

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