Adaptive Approximation Algorithms for Sparse Data Representation

(10/2008-10/2014)

DFG project within the priority program 1324 ( Mathematical methods for extracting quantifiable information from complex systems ) in cooperation with Armin Iske, University of Hamburg. We focus on the development and numerical analysis of novel adaptive approximation methods for high-dimensional signal data processing, where our joint research will provide efficient multiscale algorithms relying on modern tools from approximation theory, harmonic analysis, differential geometry, and algebraic topology. Special emphasis is placed on scattered data denoising by using wavelet transforms and on nonlinear dimensionality reduction by manifold learning.

Principal investigator: Gerlind Plonka-Hoch
Staff: Stefanie Tenorth, Dennis Heinen

Corresponding Publications

• Adaptive approximation algorithms for sparse data representation
  M. Guillemard, Dennis Heinen, A. Iske, S. Krause-Solberg, Gerlind Plonka
  Lecture Notes in Computational Science and Engineering 102, pp. 281-302, 2014, preprint as download.
• Optimal representation of piecewise Hölder smooth bivariate functions by the easy path wavelet transform
  Gerlind Plonka, Armin Iske, Stefanie Tenorth
  Journal of Approximation Theory 176, 2013, 42-67, revised preprint as download.
• Wavelet shrinkage on paths for denoising of scattered data
  Dennis Heinen, Gerlind Plonka
  Results in Mathematics 62(3), 2012, 337-354, open access, preprint as download.
• Compressive Video Sampling with Approximate Message Passing Decoding
  Jianwei Ma, Gerlind Plonka, M. Yousuff Hussaini
  IEEE Transactions on Circuits and Systems for Video Technology 22(9), 2012, 1354-1364, revised preprint as download.
• Optimally sparse image representation by the easy path wavelet transform
  Gerlind Plonka, Stefanie Tenorth, Armin Iske
  International Journal of Wavelets, Multiresolution and Information Processing 10(1), 2012, 1250007 (20 pages), revised preprint as download.
• Adaptive Waveletmethoden zur Approximation von Bildern
  Stefanie Tenorth
  Universität Göttingen, Juli 2011, Dissertation.
• Curvelet-wavelet regularized split Bregman iteration for compressed sensing
  Gerlind Plonka, Jianwei Ma
  International Journal of Wavelets, Multiresolution and Information Processing 9(1), 2011, 79-110, preprint as download.
• A hybrid method for image approximation using the easy path wavelet transform
  Gerlind Plonka, Stefanie Tenorth, Daniela Rosca
  IEEE Transactions on Image Processing 20(2), 2011, 372-381, preprint as download.
• Easy Path Wavelet Transform on triangulations of the sphere
  Gerlind Plonka, Daniela Rosca
  Mathematical Geosciences 42(7), 2010, 839-855, preprint as download.
• Bildapproximation mittels Wedgelets
  Dennis Heinen
  Universität Duisburg-Essen, Fachbereich Mathematik, März 2010, Diploma thesis, supervisor: Prof. Dr. Plonka-Hoch.
• A new sparse representation of seismic data using adaptive easy-path wavelet transform
  Jianwei Ma, Gerlind Plonka, Hervé Chauris
  IEEE Geoscience and Remote Sensing Letters 7(3), 2010, 540-544, preprint as download.
• Image approximation by a hybrid method based on the easy path wavelet transform
  Gerlind Plonka, Stefanie Tenorth, Daniela Rosca
  Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers, 2009, 442-446. (ISBN: 978-1-4244-5825-7).
• Sparse data representation on the sphere using the easy path wavelet transform
  Gerlind Plonka, Daniela Rosca
  Sampling Theory and Applications (SampTA'09), L. Fesquet and B. Torresani (eds.), 2009, 255–258, preprint as download.
• Nonlinear locally adaptive wavelet filter banks
  Gerlind Plonka, Stefanie Tenorth
  Sampling Theory and Applications (SampTA'09), L. Fesquet and B. Torresani (eds.), 2009, 381–384, preprint as download.
• The easy path wavelet transform: A new adaptive wavelet transform for sparse representation of two-dimensional data
  Gerlind Plonka
  Multiscale Modeling and Simulation 7(3), 2009, 1474-1496, revised preprint as download.

Corresponding Software

• Easy Path Wavelet Transform (EPWT)
  Wavelet shrinkage on paths for denoising of scattered data including an adaptive deterministic and an adaptive random path construction.
  Dennis Heinen, Gerlind Plonka, Stefanie Tenorth
  
• Region Based Easy Path Wavelet Transform (RBEPWT)
  A Region Based EPWT-like method for Sparse Image Representation
  Renato Budinich