Meyer, Yves Approximation by wavelets and nonlinear approximation. (Approximation par ondelettes et approximation non-linéaire.) (French) Zbl 1131.94301 Charpentier, Éric (ed.) et al., Leçons de mathématiques d’aujourd’hui. Paris: Cassini (ISBN 2-84225-007-9/pbk). Le Sel et le Fer 4, 193-222 (2000). The paper under review forms an essay which presents some ideas of a filter bank approach to image preprocessing for readers who need not to be specialists in the field of non-linear approximation protocols. Starting off with the problem of visual perception in cognitive neuroscience [D. Marr, E. Hildreth, “Theory of edge detection”, Proc. R. Soc. Lond. B 207, 187–217 (1980); D. Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information (W. H. Freeman, New York) (1982); G. J. Mitchison, G. Westheimer, “The perception of depth in simple figures”, Vision Res. 24, 1063–1073 (1984)], the conception of the essay inevitably leads to the problem of image reconstruction performed by the human retina which forms an image preprocessing outpost of the brain [G. Westheimer, “Sensitivity for vertical retinal image differences”, Nature 307, 632–634 (1984); J. E. Dowling, Retina: An Approachable Part of the Brain (The Belknap Press of Harvard University Press, Cambridge, Massachusetts, London) (1987); D. H. Hubel, Eye, Brain, and Vision (Scientific American Library, New York) (1988)]. The goal of the paper is to provide an introduction to some of the function spaces of approximation theory such as Besov and BMO (bounded mean oscillation) spaces [R. A. Adams, Sobolev spaces (Academic Press, New York, San Francisco, London) (1975; Zbl 0314.46030), 2nd ed. (2003; Zbl 1098.46001); E. M. Stein, Singular integrals and differentiability Properties of Functions (Princeton University Press, Princeton, New Jersey) (1970; Zbl 0207.13501)] by adopting the point of view of image processing. Vision or visual perception is fundamentally an information-processing task which is based on the concepts of rational approximation and filter bank [P. P. Vaidyanathan, Multirate systems and filter banks (Prentice Hall PTR, Englewood, New Jersey) (1993; Zbl 0784.93096); A. Mertins, Signal Analysis: Wavelets, Filter Banks, Time-Frequency Transforms and Applications (John Wiley & Sons, Chichester, New York, Weinheim) (1999; Zbl 0934.94001); E. Binz, W. Schempp, “Information technology: the Lie groups defining the filter banks of the compact disc”, J. Comput. Appl. Math. 144, 85–103 (2002; Zbl 1016.94517)]. Visual information processing admits a dual procedure, the internal representation of information or visualization [D. Marr, “Visual information processing: The structure and creation of visual representations”, Phil. Trans. R. Soc. Lond. B 290, 199–218 (1980); C. Blakemore (ed.) Vision: Coding and Efficiency (Cambridge University Press, Cambridge, New York, Port Chester) (1990); N. Dodgson, M. S. Floater, M. A. Sabin (eds.) Advances in Multiresolution for Geometric Modelling (Springer-Verlag, Berlin, Heidelberg, New York) (2005; Zbl 1047.65001)] which challenges the performances of the neurofunctional and morphological imaging protocols of the human retina and the visual cortex cerebri. The somewhat unbalanced discussion of visual information compression and the biased exposition of non-linear approximation protocols in the cognitive neuroscience part of the present paper unfortunately excludes the dual cross-sectional visualization modalities such as laser scanning tomography, optical coherence tomography (OCT), multislice helical computer tomography (X-ray CT), clinical magnetic resonance imaging (MRI), positron emission tomography (PET) and neurofunctional combined PET/CT imaging which nevertheless are of overwhelming importance in the field of cognitive neuroscience [R. O. W. Burke, K. Rohrschneider, H. E. Völcker, “Posterior segment laser scanning tomography: Contour line modulation in optic disc analysis”, Proc. SPIE 1357, 228–235 (1990); M. R. Hee, J. A. Izatt, E. A. Swanson, D. Huang, J. S. Schuman, C. P. Lin, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography of the human retina”, Arch. Ophthalmol. 113, 325–332 (1995); M. I. Posner, M. E. Raichle, Images of Mind (Scientific American Library, New York) (1994); C. T. W. Moonen, P. A. Bandettini (eds.), Functional MRI (Springer-Verlag, Berlin, Heidelberg, New York) (1999)].For the entire collection see [Zbl 1023.00004]. Reviewer: Walter Schempp (Siegen) MSC: 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 92C55 Biomedical imaging and signal processing 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 41A20 Approximation by rational functions 41A10 Approximation by polynomials Citations:Zbl 0314.46030; Zbl 1098.46001; Zbl 0784.93096; Zbl 0207.13501; Zbl 0934.94001; Zbl 1016.94517; Zbl 1047.65001 × Cite Format Result Cite Review PDF