V.G. Labunets. Hypercomplex models of multichannel images ... P. 69-83

We present a new theoretical approach to the processing of multidimensional and multicomponent images based on the theory of commutative hypercomplex algebras, which generalize the algebra of complex numbers. The main goal of the paper is to show that commutative hypercomplex numbers can be used in multichannel image processing in a natural and effective manner. We suppose that animal brain operates hypercomplex numbers when processing multichannel retinal images. In our approach, each multichannel pixel is regarded as a K–D hypercomplex number rather than a K–D vector, where K is the number of different optical channels. This creates an effective mathematical basis for various function–number transformations of multichannel images and invariant pattern recognition.

Keywords: multichannel images, hypercomplex algebra, image processing

Received May 12, 2020

Revised June 10, 2020

Accepted July 6, 2020

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 19-29-09022\19.)

Valerii Grigor’evich Labunets, Dr. Techn. Sci., Prof., Ural State Forest Engineering University, Yekaterinburg, 620100 Russia, e-mail: vlabunets05@yahoo.com

REFERENCES

1.    Cronin T. A retina with at least ten spectral types of photoreceptors in a mantis shrimp. Nature, 1989, vol. 339, pp. 137–140. doi: 10.1038/339137a0 

2.   Chang C. Hyperspectral data processing: Algorithm design and analysis. N Y: Wiley Press, 2013, 1164 p.

3.   Schowengerdt R. A. Remote sensing — Models and methods for image processing, N Y: Acad. Press, 1997, 558 p.

4.   Soifer V. A. Computer image processing. Part II: Methods and algorithms. Berlin: VDM, Verlag, 2010, 584 p.

5.   Luneburg R. K. The metric methods in binocular visual space. J. Opt. Soc. Amer., 1950, vol. 40, no. 1, pp. 627–642.

6.   Luneburg R. K. The metric methods in binocular visual // Studies and Essays. Courant Anniv., 1948, vol. 11, no. 1, pp. 215–239.

7.   Labunets V. Clifford algebra as unified language for image processing and pattern recognition. In: Computational Noncommutative Algebra and Applications, eds. J. Byrnes, G. Ostheimer. Dordrect; Boston; London: Kluwer Acad. Publ., 2003, pp. 197–225. doi: 10.1007/1-4020-2307-3_8 

8.   Labunets V. G., Rundblad E. V., Astola J. Is the Brain a “Clifford algebra quantum computer”? In Applied Geometrical Algebras in Computer Science and Engineering, eds. L. Dorst, C. Doran, J. Lasenby, N Y: Birkhauser, 2002, pp. 486–495. doi: 978-1-4612-0089-5_25 

9.   Labunets V., Labunets-Rundblad E. V. Algebra and geometry of color images. In Proc. of the First Int. Workshop on Spectral Tecniques and Logic Design for Future Digital Systems, eds. J. Astola, R. Stancovic, Tampere: Tampere University Publ., 2000, pp. 231–261.

10.   Doran C. J. L. Geometric algebra and its application to mathematical physics, Cambridge: Cambridge University Publ., 1994, 324 p.

11.   Greaves Ch. On algebraic triplets. Proc. Irisn Acad., 1847, vol. 3, pp. 51–108.

12.   Rundblad-Ostheimer E., Labunets V. Spatial-color Clifford algebras for invariant image recognition. In Geometric Computing with Clifford Algebras, ed. G. Sommer, Berlin: Springer, 2001, pp. 155–185. doi: 10.1007/978-3-662-04621-0_7 

13.   Rundblad-Ostheimer E., Nikitin I., Labunets V. Unified approach to Fourier-Clifford-Prometheus sequences, transforms and filter banks. In Computational Noncommutative Algebra and Applications, eds. J. Byrnes, G. Ostheimer. Dordrect, Boston, London: Kluwer Acad. Publ., 2003, pp. 389–400. doi: 10.1007/1-4020-2307-3_14 

14.   Rundblad-Ostheimer E., Maidan E. A., Novak P., Labunets V. G. Fast color Haar-Prometheus wavelet transforms for image processing. In Computational Noncommutative Algebra and Applications, ed. J. Byrnes, G. Ostheimer. Dordrect, Boston, London, Kluwer Acad. Publ., 2003, pp. 389–400. doi: 10.1007/1-4020-2307-3_15 

15.   Rundblad-Ostheimer E., Labunets V., Astola J. Is the visual cortex a “Fast Clifford algebra quantum computer”? In Clifford analysis and its applications / Mathematics, Physics and Chemistry. NATO Science Series, 2001, vol. 25, pp. 173–183. doi: 10.1007/978-94-010-0862-4_17 

16.   Labunets V. G., Maidan A., Rundblad-Ostheimer E., Astola J. Colour triplet-valued wavelets and splines. Proc. of the 2nd International Symposium on Image and Signal Processing and Analysis. In conjunction with 23rd International Conference on Information Technology Interfaces (IEEE Cat. No.01EX480), Pula, 2001, pp. 535–541. doi: 10.1109/ISPA.2001.938687 

Cite this article as: V.G. Labunets. Hypercomplex models of multichannel images, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 69–83; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2021, Vol. 313, Suppl. 1, pp. S155–S168.