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A simple class of fractal transforms for hyperspectral images. (English) Zbl 1410.94019

Summary: A complete metric space of function-valued mappings appropriate for the representation of hyperspectral images is introduced. A class of fractal transforms is then formulated on this space. Under certain conditions, the fractal transform \(T\) can be contractive, implying the existence of a unique fixed point. We then formulate a simple class of block transforms for the fractal coding of digital hyperspectral images, and illustrate with an example.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
28A80 Fractals
68U10 Computing methodologies for image processing
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[1] Alexander, S. K.; Vrscay, E. R.; Tsurumi, S., A simple model for the affine self-similarity of images, (Image Analysis and Recognition, ICIAR, 2008, Lecture Notes in Computer Science, vol. 5112, (2008), Springer-Verlag Berlin, Heidelberg), 192-203
[2] AVIRIS hyperspectral image, Yellowstone calibrated scene 0, available from Information Processing Group website, Jet Propulsion Laboratory, California Institute of Technology, <http://compression.jpl.nasa.gov/hyperspectral/>.
[3] Barnsley, M. F., Fractals everywhere, (1988), Academic Press New York · Zbl 0691.58001
[4] Barnsley, M. F.; Hurd, L., Fractal image compression, (1993), A.K Peters Wellesley, Mass · Zbl 0796.68186
[5] Brunet, D.; Vrscay, E. R.; Wang, Z., Structural similarity-based affine approximation and the self similarity of images revisited, (Image Analysis and Recognition, ICIAR 2011, Lecture Notes in Computer Science, vol. 6754, (2011), Springer-Verlag Berlin, Heidelberg), 264-275
[6] Buades, A.; Coll, B.; Morel, J. M.; Buades, A.; Coll, B.; Morel, J. M., An updated version of this paper by the same authors: image denoising methods, a new nonlocal principle, Multiscale Model. Simul., SIAM Rev., 52, 113-147, (2010) · Zbl 1182.62184
[7] Chang, C., Hyperspectral data exploitation, theory and applications, (2007), John Wiley and Sons Hoboken, NJ
[8] Dabov, K.; Foi, A.; Katkovnik, V.; Egiazarian, K., Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Trans. Image Process., 16, 2080-2095, (2007)
[9] Ebrahimi, M.; Vrscay, E. R., Solving the inverse problem of image zooming using self-examples, (Image Analysis and Recognition, ICIAR 2007, Lecture Notes in Computer Science, vol. 4633, (2007), Springer-Verlag Berlin, Heidelberg), 117-130
[10] Elad, M.; Datsenko, D., Example-based regularization deployed to super-resolution reconstruction of a single image, Comput. J., 50, 1-16, (2007)
[11] Etemoglu, C.; Cuperman, V., Structured vector quantization using linear transforms, IEEE Trans. Signal. Process., 51, 1625-1631, (2003) · Zbl 1369.94442
[12] (Fisher, Y., Fractal Image Compression: Theory and Application, (1995), Springer-Verlag New York)
[13] Hyperspectral images of natural scenes 2004, Scene 2, Available from the personal page of David Foster, Manchester University. <http://personalpages.manchester.ac.uk/staff/david.foster/default.htm>.
[14] Freeman, W. T.; Jones, T. R.; Pasztor, E. C., Example-based super-resolution, IEEE Comput. Graphics Appl., 22, 56-65, (2002)
[15] Ghazel, M.; Freeman, G.; Vrscay, E. R., Fractal image denoising, IEEE Trans. Image Process., 12, 12, 1560-1578, (2003)
[16] Jacquin, A., Image coding based on a fractal theory of iterated contractive image transformations, IEEE Trans. Image Process., 1, 18-30, (1992)
[17] Kunze, H.; La Torre, D.; Mendivil, F.; Vrscay, E. R., Fractal-based methods of analysis, (2012), Springer New York · Zbl 1237.28002
[18] Lu, N., Fractal imaging, (1997), Academic Press New York
[19] Mayer, G.; Vrscay, E. R., Iterated Fourier transform systems: A method for frequency extrapolation, (Image Analysis and Recognition, ICIAR 2007, Lecture Notes in Computer Science, vol. 4633, (2007), Springer-Verlag Berlin, Heidelberg), 728-739
[20] Michailovich, O.; La Torre, D.; Vrscay, E. R., Function-valued mappings and total variation and compressed sensing for diffusion magnetic resonance imaging, (Image Analysis and Recognition, ICIAR 2012, Lecture Notes in Computer Science, vol. 7325, Part II, (2012), Springer-Verlag Berlin, Heidelberg), 286-295
[21] (Motta, G.; Rizzo, F.; Storer, J. A., Hyperspectral Data Compression, (2006), Springer Science + Business Media Inc. New York) · Zbl 1088.68050
[22] Vrscay, E. R.; Otero, D.; La Torre, D., Hyperspectral images as function-valued mappings, their self-similarity and a class of fractal transforms, (Image Analysis and Recognition, ICIAR 2013, Lecture Notes in Computer Science, vol. 7950, (2013), Springer-Verlag Berlin, Heidelberg), 225-234
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