Abstract
Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling, continuous surface deformation, shape morphing, reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.
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Acknowledgements
This work was funded by the Swiss National Science Foundation under Grant 200020-162343. We are grateful to Zsuzsanna Püspöki for help with the figures and to Irina Radu for help with the video. We also appreciate the interesting discussions on the subject that we had with Masih Nilchian and Emrah Bostan. We thank Mike McCann for proof-reading the manuscript.
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D. Schmitter received his master degree in bioengineering and biomedical technologies from the École Polytechnique Fédérale de Lausanne (EPFL), Switzerland, in 2013. He was with the Advanced Clinical Imaging Technology Group, Siemens, at the Center for Biomedical Imaging, Switzerland, where he was one of the main contributors working on brain-imaging software and related imageprocessing algorithms. Currently, he is a Ph.D. student at the Biomedical Imaging Group, EPFL, where he is working on spline-based shape representation and segmentation problems. He has developed several segmentation and tracking methods in the field of biomedical imaging.
P. García-Amorena was ranked in the top ten of the Spanish regional university entrance exam of Catalunya in 2009 and obtained double-bachelor degrees in mathematics and industrial engineering at Barcelona Tech (UPC), Spain. He received an Introduction to Research Grant from the Spanish National Research Council (CSIC) to work on 3D image reconstruction from deformation scans. In 2014 he won a La Caixa award for top undergraduate Spanish students. In 2016 he obtained his master degree in computational science and engineering at EPFL, Switzerland. His research interests include numerical methods and mathematical foundations for shape modeling, computer vision, and image processing.
M. Unser is the professor and director of EPFL’s Biomedical Imaging Group, Lausanne, Switzerland. His primary area of investigation is biomedical image processing. He is internationally recognized for his research contributions to sampling theory, wavelets, the use of splines for image processing, stochastic processes, and computational bioimaging. He has published over 250 journal papers on those topics. He is the author with P. Tafti of the book An Introduction to Sparse Stochastic Processes, Cambridge University Press, 2014. From 1985 to 1997, he was with the Biomedical Engineering and Instrumentation Program, National Institutes of Health, Bethesda, USA, conducting research on bioimaging. Dr. Unser has held the position of associate Editor-in- Chief (2003–2005) for the IEEE Transactions on Medical Imaging. He is currently a member of the editorial boards of SIAM J. Imaging Sciences, IEEE J. Selected Topics in Signal Processing, and Foundations and Trends in Signal Processing. He is the founding chair of the technical committee on Bio Imaging and Signal Processing (BISP) of the IEEE Signal Processing Society. Prof. Unser is a Fellow of the IEEE (1999), an EURASIP Fellow (2009), and a member of the Swiss Academy of Engineering Sciences. He is the recipient of several international prizes including three IEEE-SPS Best Paper Awards and two Technical Achievement Awards from the IEEE (2008 SPS and EMBS 2010).
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Schmitter, D., García-Amorena, P. & Unser, M. Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction. Comp. Visual Media 3, 199–215 (2017). https://doi.org/10.1007/s41095-017-0086-4
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DOI: https://doi.org/10.1007/s41095-017-0086-4