19-03-2012, 01:34 PM
Optimally Sparse Image Representation by the Easy Path Wavelet Transform
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Introduction
During the last few years, there has been an increasing interest in efficient representations
of large high-dimensional data, especially for signals. In the one-dimensional case, wavelets
are particularly efficient to represent piecewise smooth signals with point singularities. In
higher dimensions, however, tensor product wavelet bases are no longer optimal for the
representation of piecewise smooth functions with discontinuities along curves.
Just very recently, more sophisticated methods were developed to design approximation
schemes for efficient representations of two-dimensional data, in particular for images,
where correlations along curves are essentially taken into account to capture the geometry
of the given data. Curvelets [2, 3], shearlets [12, 13] and directionlets [24] are examples
for non-adaptive highly redundant function frames with strong anisotropic directional
selectivity.
Approximation Properties of the EPWT Algorithm
Recall that for given integer J > 0, the function F2J is assumed to be the piecewise
constant approximation of the image F satisfying (2.2). In this section, we shall prove the
optimal N-term approximation to F by a suitably chosen EPWT, where the path vectors
are required to satisfy the region condition and the diameter condition of Subsection 2.3.
Let us first prove suitable estimates for the scaling and the wavelet coefficients.