Journal of Mathematical Imaging and Vision

, Volume 58, Issue 3, pp 349–381 | Cite as

Spherical Tensor Algebra: A Toolkit for 3D Image Processing

  • Henrik Skibbe
  • Marco Reisert


With the advent of novel 3D image acquisition techniques, their efficient and reliable analysis becomes more and more important. In particular in 3D, the amount of data is enormous and requires for an automated processing. The tasks are manifold, starting from simple image enhancement, image reconstruction, image description and object/feature detection to high-level contextual feature extraction. One important property that most of these tasks have in common is their covariance to rotations. Spherical Tensor Algebra (STA) offers a general framework to fulfill these demands. STA transfers theories from mathematical physics and harmonic analysis into the domain of image analysis and pattern recognition. The main objects of interest are orientation fields. The interpretations of the fields are manifold. Depending on the application, they can represent local image descriptors, features, orientation scores or filter responses. STA deals with the processing of such fields in the domain of the irreducible representations of the rotation group. Two operations are fundamental: the extraction/projection of the features by convolution-like procedures and the nonlinear covariant combination by spherical products. In this paper, we propose an open-source toolbox that implements, in addition to fundamental STA operators, advanced functions for feature detection and image enhancement and makes them accessible to the 3D image processing community. The core features are implemented in C (CPU and GPU) with APIs in C++ and MATLAB. As examples, we show applications for medical and biological images.


Biomedical 3D image processing Rotational invariance Spherical tensors 3D feature detection Bi-spectrum 



This research was supported by the program for Brain Mapping by Integrated Neurotechnologies for Disease Studies (Brain/MINDS) from Japan Agency for Medical Research and development, AMED. This study was supported by Deutsche Forschungsgemeinschaft (German Research Council) via grants DFG RE 3286/2-1 and DFG KI 1089/3-2. We would like to thank Professor Kei Ito from the Department of Computational Biology, The University of Tokyo, for providing us the image of neurite structures in a drosophila fly brain; Fig. 22.

Supplementary material


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Ishii-Lab, Department of Systems Science, Graduate School of InformaticsKyoto UniversityKyotoJapan
  2. 2.Medical Physics, Faculty of MedicineUniversity FreiburgFreiburg im BreisgauGermany

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