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Tree approximation and optimal image coding with shearlets

by P. Grohs

(Report number 2010-45)

Abstract
It is by now classical that various anisotropic frame decompositions such as curvelets or shearlets guarantee (almost) optimal >i>N -term approximation rates for functions which are C^2 apart from a C 2 discontinuity curve. However, if no structure is present in the set of retained indices, the cost of transmitting the location of the indices might dominate the cost of transmitting the actual coefficients. Therefore, as far as bit rate coding is concerned, simply storing the N largest (curvelet- or shearlet-) coefficients possibly leads to non-optimal codes. In the wavelet case this issue can be resolved by requiring that the set of indices which are kept possesses a tree structure which can be encoded more efficiently. In the present work we show how an analogous procedure can be carried out for curvelets or shearlets. The main result is that the N -term approximation rate can be essentially retained while imposing the additional constraint that the set of indices is a tree.

Keywords: Shearlets, Curvelets, Tree Approximation, Optimal Encoding, Kolmogorov Entropy

@Techreport{G10_443,
  author = {P. Grohs},
  title = {Tree approximation and optimal image coding with shearlets },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-45},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-45.pdf },
  year = {2010}
}

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