Copyright © 2013 Henrik Skibbe and Marco Reisert. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
With the advent of novel biomedical 3D image acquisition techniques, the efficient and reliable analysis of volumetric images has become more and more important. The amount of data is enormous and demands an automated processing. The applications are manifold, ranging from image enhancement, image reconstruction, and image description to object/feature detection and high-level contextual feature extraction. In most scenarios, it is expected that geometric transformations alter the output in a mathematically well-defined manner. In this paper we emphasis on 3D translations and rotations. Many algorithms rely on intensity or low-order tensorial-like descriptions to fulfill this demand. This paper proposes a general mathematical framework based on mathematical concepts and theories transferred from mathematical physics and harmonic analysis into the domain of image analysis and pattern recognition. Based on two basic operations, spherical tensor differentiation and spherical tensor multiplication, we show how to design a variety of 3D image processing methods in an efficient way. The framework has already been applied to several biomedical applications ranging from feature and object detection tasks to image enhancement and image restoration techniques. In this paper, the proposed methods are applied on a variety of different 3D data modalities stemming from medical and biological sciences.