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Adaptive Image Approximation using the EPWT
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In this talk we present the Easy Path Wavelet Transform (EPWT), a locally adaptive wavelet transform that makes use of strong correlations of adjacent pixels. First, a path through all pixels (i.e. a permutation of all pixels) is calculated, so that pixels with similar values are adjacent on the path. Then a one-dimensional wavelet transform (e.g. the biorthogonal Cohen-Daubechies-Feauveau 9/7-transform) is applied to the newly ordered pixels.

Unfortunately, like most adaptive wavelet transforms, the EPWT suffers from its adaptivity costs as the path vectors also need to be stored. One idea to minimize these costs is the so-called \textit{relaxed EPWT}.

In addition, we propose another algorithm that uses the EPWT for an efficient
representation of edges and texture and that exploits the advantages of the usual tensor product wavelet transform for the
representation of smooth images.
This hybrid algorithm works as follows. After the given image has been smoothed by a diffusion procedure, a biorthogonal tensor product wavelet transform is applied to the smoothed image. Further, the EPWT is used to construct a sparse representation of the (shrunken) difference image.

Finally we show some numerical results.
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This talk is based on joint work with Gerlind Plonka (Georg-August-Universität Göttingen), Armin Iske (Universität Hamburg)
and Daniela Ro\c{s}ca (Technical University of Cluj-Napoca, Romania).