Optimally sparse image representation by the easy path wavelet transform (Q2893485)

During the last few years an increasing effort to develop efficient representations of large high-dimensional data, especially for signals, has been done. In the 1D case, wavelets are particular efficient to represent piecewise smooth signals with point singularities. However, in the 2D case tensor product wavelet bases are no longer optimal for the representation of piecewise smooth functions with discontinuities along curves. Very recently, more sophisticated methods have been developed to design approximation schemes for efficient representations of 2D data, in particular for images, where correlations along curves are considered in order to recover the geometry of the given data. As examples for non-adaptive highly redundant function frames with strong anisotropic directional selectivity, curvelets, shearlets and directionlets, among others, have been introduced.NEWLINENEWLINEIn the paper under review, the authors deduce an optimal \(N\)-term approximation \(f_{N}\) for a piecewise Hölder continuous function \(f\) of order \(\alpha\), with \(0<\alpha\leq 1\) with singularities along curves using a locally adaptive Easy Path Wavelet Transform (EPWT) introduced in [\textit{G. Plonka}, Multiscale Model. Simul. 7, No. 3, 1474--1496 (2009; Zbl 1175.65158)]. Indeed, \(||f-f_{N}||_{2} ^{2} \leq C N^{-\alpha}\). The restriction concerning \(\alpha\) is due to the application of the adaptive Haar wavelet basis used for the EPWT. In order to achieve optimal approximation results, some specific side conditions for the path vectors that are used in the EPWT algorithm are required. A strategy for the construction of the path vectors satisfying such side conditions is presented. Finally, the authors deal with the error analysis and the above asymptotically optimal \(N\)-term error estimates for piecewise Hölder continuous functions is deduced.NEWLINENEWLINENotice that, for \(\alpha>1,\) the problem requires a complete different approach since the adaptive multiresolution analysis structure obtained for Haar wavelets cannot be achieved with other wavelet bases because the transfer from 1D wavelet functions (along paths) to suitable bivariate functions is no longer obvious.