Multiscale sharpening and smoothing in Besov spaces with applications to image enhancement (Q5946554)

A common way of modeling an image for Fourier- and wavelet-based analysis tools is as an element in the Hilbert space \(L^2\) of square integrable functions. Based on this assumption, the developed imaging algorithms are located in the \(L^2\) space. A deeper photonic reason for the \(L^2\) framework of image processing areas is the fact that the irreducible unitary linear Schrödinger representation is square integrable modulo the center of the three-dimensional Heisenberg nilpotent Lie group \(N\). Indeed, harmonic analysis on \(N\) is based on the \(L^2\) energy norm and embraces the quantum aspects of image processing [\textit{W. Schempp}, Radar ambiguity functions, the Heisenberg group, and holomorphic theta series, Proc. Am. Math. Soc. 92, 103-110 (1984; Zbl 0525.43007); Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory, Pitman Research Notes in Mathematics Series, 147 (Longman Scientific and Technical, London) (1986; Zbl 0632.43001); and Magnetic resonance imaging. Mathematical foundations and applications (Wiley-Liss, New York) (1998; Zbl 0930.92015); and \textit{E. Binz} and \textit{W. Schempp}, Digital information processing: The Lie groups defining the filter banks of the compact disc, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 68, 269-292 (2002)]. Function spaces that contain more detail about smoothness of signals provide suitable signal characterizations. Spaces of Besov type are an example of those function spaces in which a wavelet basis forms an unconditional basis. As a consequence, an equivalence to the Besov norm can be defined by weighted sums of wavelet coefficients [\textit{M. Frazier, B. Jawerth} and \textit{G. Weiss}, Littlewood-Paley theory and the study of function spaces, Am. Math. Soc., Regional Conference Series in Mathematics 79 (1991; Zbl 0757.42006); \textit{Y. Meyer}, Wavelets and operators, Cambridge Studies in Advanced Mathematics 37 (Cambridge University Press, Cambridge) (1992; Zbl 0776.42019)]. Roughly speaking this means that shrinking wavelet coefficients in size shrinks the norm in the considered Besov space. Signs and phases of coefficients do not influence the norm. This property is exploited in deriving the wavelet denoising technique. The paper under review tries to bridge the theory of Besov type spaces with the engineering problems of image processing by defining a general concept of smoothing and sharpening with wavelet decompositions in Besov spaces. Multiscale sharpening leads to a switching from a Besov space with large degree of smoothness to one with a lower degree of smoothness. This method combined with the wavelet denoising technique demonstrates a possible application for enhancement of scanned documents.