Chapter 4
3D Moment Invariants to Translation, Rotation, and Scaling
Jan Flusser,
Tomáš Suk,
Barbara Zitová,
Jan Flusser
Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic
Search for more papers by this authorTomáš Suk
Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic
Search for more papers by this authorBarbara Zitová
Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic
Search for more papers by this authorBook Author(s):Jan Flusser,
Tomáš Suk,
Barbara Zitová,
Jan Flusser
Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic
Search for more papers by this authorTomáš Suk
Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic
Search for more papers by this authorBarbara Zitová
Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic
Search for more papers by this authorSummary
Current medical imaging devices, such as MRI, f-MRI, CT, SPECT, and PET, provide us with full 3D images of the patient's body. Numerous 3D image processing algorithms are just extensions of the 2D versions by an additional dimension. In order to avoid numerical problems with geometric and complex moments, some authors proposed 3D invariants from orthogonal moments, such as Zernike moments and Gaussian-Hermite moments. The design of rotation moment invariants in 3D is much more difficult than in 2D. This chapter presents three alternative approaches in the sequel. The first one uses geometric moments, and the invariance is achieved by means of tensor algebra. The second one is based on spherical harmonics and can be viewed as an analogue to 2D complex moment invariants. The last one is a normalization approach. The chapter presents a survey of all existing reflection and rotation symmetries in 3D.
References
- P. Shilane, P. Min, M. Kazhdan, and T. Funkhouser, “The Princeton Shape Benchmark,” 2004. http://shape.cs.princeton.edu/benchmark/ .
- F. A. Sadjadi and E. L. Hall, “Three dimensional moment invariants,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 2, no. 2, pp. 127–136, 1980.
-
X. Guo, “Three-dimensional moment invariants under rigid transformation,” in Proceedings of the Fifth International Conference on Computer Analysis of Images and Patterns CAIP'93 (D. Chetverikov and W. Kropatsch, eds.), vol. 719 of Lecture Notes in Computer Science, (Berlin, Heidelberg, Germany), pp. 518–522, Springer, 1993.
10.1007/3-540-57233-3_67 Google Scholar
- D. Cyganski and J. A. Orr, “Object recognition and orientation determination by tensor methods,” in Advances in Computer Vision and Image Processing (T. S. Huang, ed.), (Greenwich, Connecticut, USA), pp. 101–144, JAI Press, 1988.
- D. Xu and H. Li, “3-D affine moment invariants generated by geometric primitives,” in Proceedings of the 18th International Conference on Pattern Recognition ICPR'06, pp. 544–547, IEEE, 2006.
- D. Xu and H. Li, “Geometric moment invariants,” Pattern Recognition, vol. 41, no. 1, pp. 240–249, 2008.
-
T. Suk and J. Flusser, “Tensor method for constructing 3D moment invariants,” in Computer Analysis of Images and Patterns CAIP'11 (P. Real, D. Diaz-Pernil, H. Molina-Abril, A. Berciano, and W. Kropatsch, eds.), vol. 6854–6855 of Lecture Notes in Computer Science, (Berlin, Heidelberg, Germany), pp. 212–219, Springer, 2011.
10.1007/978-3-642-23678-5_24 Google Scholar
- C.-H. Lo and H.-S. Don, “3-D moment forms: Their construction and application to object identification and positioning,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, no. 10, pp. 1053–1064, 1989.
- J. Flusser, J. Boldyš, and B. Zitová, “Moment forms invariant to rotation and blur in arbitrary number of dimensions,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, no. 2, pp. 234–246, 2003.
- J. M. Galvez and M. Canton, “Normalization and shape recognition of three-dimensional objects by 3D moments,” Pattern Recognition, vol. 26, no. 5, pp. 667–681, 1993.
- T. Suk, J. Flusser, and J. Boldyš, “3D rotation invariants by complex moments,” Pattern Recognition, vol. 48, no. 11, pp. 3516–3526, 2015.
- T. Suk and J. Flusser, “Recognition of symmetric 3D bodies,” Symmetry, vol. 6, no. 3, pp. 722–757, 2014.
- N. Canterakis, “3D Zernike moments and Zernike affine invariants for 3D image analysis and recognition,” in Proceedings of the 11th Scandinavian Conference on Image Analysis SCIA'99 (B. K. Ersbøll and P. Johansen, eds.), DSAGM, 1999.
-
B. Yang, J. Flusser, and T. Suk, “3D rotation invariants of Gaussian–Hermite moments,” Pattern Recognition Letters, vol. 54, no. 1, pp. 18–26, 2015.
10.1016/j.patrec.2014.11.014 Google Scholar
- M. Kazhdan, T. Funkhouser, and S. Rusinkiewicz, “Rotation invariant spherical harmonic representation of 3D shape descriptors,” in Proceedings of the 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, SGP '03, (Aire-la-Ville, Switzerland), pp. 156–164, Eurographics Association, 2003.
- R. Kakarala, “The bispectrum as a source of phase-sensitive invariants for Fourier descriptors: A group-theoretic approach,” Journal of Mathematical Imaging and Vision, vol. 44, no. 3, pp. 341–353, 2012.
- J. Fehr and H. Burkhardt, “3D rotation invariant local binary patterns,” in 19th International Conference on Pattern Recognition ICPR'08, pp. 1–4, IEEE, 2008.
- T. Ojala, M. Pietikäinen, and T. Mäenpää, “Multiresolution gray-scale and rotation invariant texture classification with local binary patterns,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 24, no. 7, pp. 971–987, 2002.
- J.-F. Mangin, F. Poupon, E. Duchesnay, D. Rivière, A. Cachia, D. L. Collins, A. C. Evans, and J. Régis, “Brain morphometry using 3D moment invariants,” Medical Image Analysis, vol. 8, no. 3, pp. 187–196, 2004.
- M. Kazhdan, “An approximate and efficient method for optimal rotation alignment of 3D models,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 29, no. 7, pp. 1221–1229, 2007.
- M. Trummer, H. Suesse, and J. Denzler, “Coarse registration of 3D surface triangulations based on moment invariants with applications to object alignment and identification,” in Proceedings of the 12th International Conference on Computer Vision, ICCV'09, pp. 1273–1279, IEEE, 2009.
- R. D. Millán, M. Hernandez, D. Gallardo, J. R. Cebral, C. Putman, L. Dempere-Marco, and A. F. Frangi, “Characterization of cerebral aneurysms using 3D moment invariants,” in Proceedings of the conference SPIE Medical Imaging 2005: Image Processing, vol. 5747, pp. 743–754, 2005.
- R. D. Millán, L. Dempere-Marco, J. M. Pozo, J. R. Cebral, and A. F. Frangi, “Morphological characterization of intracranial aneurysms using 3-D moment invariants,” IEEE Transactions on Medical Imaging, vol. 26, no. 9, pp. 1270–1282, 2007.
-
K. Shoemake, “Animating rotation with quaternion curves,” SIGGRAPH Computer Graphics, vol. 19, no. 3, pp. 245–254, 1985.
10.1145/325165.325242 Google Scholar
- B. Chen, H. Shu, H. Zhang, G. Chen,C. Toumoulin, J. Dillenseger, and L. Luo, “Quaternion Zernike moments and their invariants for color image analysis and object recognition,” Signal Processing, vol. 92, no. 2, pp. 308–318, 2012.
-
F. Bouyachcher, B. Bouali, and M. Nasri, “Quaternion affine moment invariants for color texture recognition,” International Journal of Computer Applications, vol. 81, no. 16, pp. 1–6, 2013.
10.5120/14204-2405 Google Scholar
-
D. Rizo-Rodríguez, H. Méndez-Vázquez, and E. García-Reyes, “Illumination invariant face image representation using quaternions,” in Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications (I. Bloch and R. M. Cesar, Jr., eds.), vol. 6419 of Lecture Notes in Computer Science, pp. 434–441, Berlin, Heidelberg, Germany: Springer, 2010.
10.1007/978-3-642-16687-7_58 Google Scholar
-
J. Angulo, “Structure tensor of colour quaternion image representations for invariant feature extraction,” in Computational Color Imaging, Proceedings of the Second International Workshop CCIW'09 (A. Trémeau, R. Schettini, and S. Tominaga, eds.), vol. 5646 of Lecture Notes in Computer Science (Berlin, Heidelberg, Germany), pp. 91–100, Springer, 2009.
10.1007/978-3-642-03265-3_10 Google Scholar
- M. Pedone, E. Bayro-Corrochano, J. Flusser, and J. Heikkilä, “Quaternion Wiener deconvolution for noise robust color image registration,” IEEE Signal Processing Letters, vol. 22, no. 9, pp. 1278–1282, 2015.
- Wikipedia, “Einstein notation.”
- E. P. L. Van Gool, T. Moons and A. Oosterlinck, “Vision and Lie's approach to invariance,” Image and Vision Computing, vol. 13, no. 4, pp. 259–277, 1995.
- DIP, “3D rotation moment invariants,” 2011. http://zoi.utia.cas.cz/3DRotationInvariants .
- W. E. Byerly, An Elementary Treatise on Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics with Applications to Problems in Mathematical Physics. New York, USA: Dover Publications, 1893.
- E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics. Cambridge, U.K.: Cambridge University Press, 1931.
- N. M. Ferrers, An Elementary Treatise on Spherical Harmonics and Subjects Connected with Them. London, U.K.: Macmillan, 1877.
- Y.-F. Yen, “3D rotation matching using spherical harmonic transformation of k-space navigator,” in Proceedings of the ISMRM 12th Scientific Meeting, p. 2154, International Society for Magnetic Resonance in Medicine, 2004.
- J. P. Elliot and P. G. Dawber, Symmetry in Physics. London, U.K.: MacMillan, 1979.
- A. G. Mamistvalov, “ -dimensional moment invariants and conceptual mathematical theory of recognition -dimensional solids,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, no. 8, pp. 819–831, 1998.
- J. Boldyš and J. Flusser, “Extension of moment features' invariance to blur,” Journal of Mathematical Imaging and Vision, vol. 32, no. 3, pp. 227–238, 2008.
- G. Cybenko, A. Bhasin, and K. D. Cohen, “Pattern recognition of 3D CAD objects: Towards an electronic yellow pages of mechanical parts,” International Journal of Smart Engineering System Design, vol. 1, no. 1, pp. 1–13, 1997.
- G. Burel and H. Hénocq, “Determination of the orientation of 3D objects using spherical harmonics,” Graphical Models and Image Processing, vol. 57, no. 5, pp. 400–408, 1995.
-
H. Weyl, Symmetry. Princeton, USA: Princeton University Press, 1952.
10.1515/9781400874347 Google Scholar
- S. Mamone, G. Pileio, and M. H. Levitt, “Orientational sampling schemes based on four dimensional polytopes,” Symmetry, vol. 2, no. 3, pp. 1423–1449, 2010.
- M. Langbein and H. Hagen, “A generalization of moment invariants on 2D vector fields to tensor fields of arbitrary order and dimension,” in Proceedings of 5th International Symposium Advances in Visual Computing, ISVC'09, Part II, vol. 5876 of Lecture Notes in Computer Science, pp. 1151–1160, Springer, 2009.
- R. Bujack, Orientation Invariant Pattern Detection in Vector Fields with Clifford Algebra and Moment Invariants. PhD thesis, Fakultät für Mathematik und Informatik, Universität Leipzig, Leipzig, Germany, 2014.
- R. Bujack, J. Kasten, I. Hotz, G. Scheuermann, and E. Hitzer, “Moment invariants for 3D flow fields via normalization,” in Pacific Visualization Symposium, PacificVis'15, pp. 9–16, IEEE, 2015.
- D. Xu and H. Li, “3-D surface moment invariants,” in Proceedings of the 18th International Conference on Pattern Recognition ICPR'06, pp. 173–176, IEEE, 2006.
- S. A. Sheynin and A. V. Tuzikov, “Explicit formulae for polyhedra moments,” Pattern Recognition Letters, vol. 22, no. 10, pp. 1103–1109, 2001.
