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Multiscale Approximation

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Mathematical Methods in Signal Processing and Digital Image Analysis

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

During the last two decades wavelet methods have developed into powerful tools for a wide range of applications in signal and image processing. The success of wavelet methods is based on their potential for resolving local properties and to analyze non-stationary structures. This is achieved by multiscale decompositions, e.g., a signal or image is mapped to a phase space parametrized by a time/space- and a scale/size/resolution parameter. In this respect, wavelet methods offer an alternative to classical Fourier- or Gabortransforms which create a phase space consisting of a time/space- frequency parametrization. Hence, wavelet methods are advantageous whenever local, non-stationary structures on different scales have to be analyzed.

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References

  1. R. A.Adams. Sobolev Spaces. Academic Press, New York, 1975.

    Google Scholar 

  2. K. Bredies and D. A. Lorenz. Iterated hard shrinkage for minimization problems with sparsity constraints. To appear in SIAM J. Sci. Comput., 2007.

    Google Scholar 

  3. K. Bredies, D. A.Lorenz, and P. Maass. A generalized conditional gradient method and its connection to an iterative shrinkage method. To appear in Comput. Optim. Appl., 2008.

    Google Scholar 

  4. J. C.Brown. Inverse problems in astrophysical spectrometry. Inverse Probl., 11:783–794, 1995.

    Article  MATH  Google Scholar 

  5. J. C.Brown. Overview of topical issue on inverse problems in astronomy. Inverse Probl., 11:635–638, 1995.

    Article  Google Scholar 

  6. E. J.Candès. Compressive sampling. Sanz-Solé, Marta (ed.) et al., Proceedings of the International Congress of Mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume III: Invited lectures. Zürich: European Mathematical Society (EMS), 2006.

    Google Scholar 

  7. E. J. Candès, J.K. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math., 59:1207–1223, 2005.

    Article  Google Scholar 

  8. C. Cabrelli, C. Heil, and U. Molter. Accuracy of lattice translates of several multidimensional refinable functions. J. Approx. Theory, 95:5–52, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Cohen. Numerical Analysis of Wavelet Methods. Studies in Mathematics and its Applications. 32. Amsterdam: North-Holland, xviii, 336 p., 2003.

    MATH  Google Scholar 

  10. A. Cohen, R. DeVore, P. Petrushev, and H. Xu. Nonlinear approximation and the space BV(ℝ2). Am. J. Math., 121:587–628, 1999.

    Article  MATH  Google Scholar 

  11. A. Cohen, R. DeVore, G. Kerkyacharian, and D. Picard. Maximal spaces with given rate of convergence for thresholding algorithms. Appl. Comput. Harmon. Anal., 11(2):167–191, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. R.Coifman and D. Donoho. Translation-invariant de-noising. Wavelets and Statistics, A. Antoniadis and G. Oppenheim, eds., SpringerVerlag, 21:125–150, 1995.

    Google Scholar 

  13. S. Dahlke, E. Novak, and W. Sickel. Optimal approximation of elliptic problems by linear and nonlinear mappings II. J. Complexity, 22:549–603, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  14. I. Daubechies. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics. 61. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics, xix, 357 p., 1992.

    MATH  Google Scholar 

  15. I. Daubechies and G. Teschke. Wavelet-based image decomposition by variational functionals. Wavelet Applications in Industrial Processing; Frederic Truchetet; Ed., 5266:94–105, 2004.

    Google Scholar 

  16. I. Daubechies and G. Teschke. Variational image restoration by means of wavelets: simultaneous decomposition, deblurring and denoising. Appl. Comput. Harmon. Anal., 19(1):1–16, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  17. I. Daubechies, M. Defrise, and C. DeMol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math., 57(11):1413–1457, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  18. I. Daubechies, G. Teschke, and L. Vese. Iteratively solving linear inverse problems with general convex constraints. Inverse Probl. Imaging, 1(1): 29–46, 2007.

    MATH  MathSciNet  Google Scholar 

  19. B. Han. Symmetry property and construction of wavelets with a general dilation matrix. Linear Algebra Appl., 353(1–3):207–225, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Han. Symmetric multivariate orthogonal refinable functions. Appl. Comput. Harmon. Anal., 17(3):277–292, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Hanke and J.G. Nagy. Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques. Inverse Probl., 12(2):157–173, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Hegland and R.S. Anderssen. Resolution enhancement of spectra using differentiation. Inverse Probl., 21(3):915–934, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  23. R.Q. Jia. Shift-invariant spaces and linear operator equations. Isr. J. Math., 103:259–288, 1998.

    Article  MATH  Google Scholar 

  24. Q.T. Jiang. Multivariate matrix refinable functions with arbitrary matrix dilation. Trans. Am. Math. Soc., 351:2407–2438, 1999.

    Article  MATH  Google Scholar 

  25. N. Kinsbury. Image processing with complex wavelets. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 357(1760):2 43-2560, 1999.

    Google Scholar 

  26. E. Klann, M. Kuhn, D.A. Lorenz, P. Maass, and H. Thiele. Shrinkage versus deconvolution. Inverse Probl., 23(5):2 31-2248, 2007.

    Google Scholar 

  27. K. Koch. Interpolating scaling vectors. Int. J. Wavelets Multiresolut. Inf. Process., 3(3):3 9-416, 2005.

    Google Scholar 

  28. K. Koch. Nonseparable orthonormal interpolating scaling vectors. Appl. Comput. Harmon. Anal., 22(2):1 8-216, 2007.

    Google Scholar 

  29. K. Koch. Multivariate symmetric interpolating scaling vectors with duals. Preprint Nr. 145, SPP 1114, to appear in J. Fourier Anal. Appl, 2006.

    Google Scholar 

  30. K. Koch. Interpolating Scaling Vectors and Multi-wavelets in ℝ d. PhD thesis, Philipps-University of Marburg, 2006.

    Google Scholar 

  31. M. Lindemann, M. Diaz, P. Maass, F.-M. Schleif, J. Decker, T. Elssner, M. Kuhn, and H. Thiele. Wavelet based feature extraction in the analysis of clinical proteomics mass spectra. In 54th Conference of the American Society for Mass Spectrometry, 2006.

    Google Scholar 

  32. A. Louis, P. Maass, and A. Rieder. Wavelets: Theory and Applications. Wiley, New York, 1997.

    Google Scholar 

  33. S. Mallat. Multiresolution approximation and wavelet orthonormal bases for L2(ℝ). Trans. Am. Math. Soc., 315:69–88, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  34. Y. Meyer. Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series, 22, 2002.

    Google Scholar 

  35. S. Osher and L. Vese. Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput., 19(1–3):5 3-572, 2003.

    Google Scholar 

  36. S. Osher and L. Vese. Image denoising and decomposition with total variation minimization and oscillatory functions. J. Math. Imaging Vis., 20(1–2):7 18, 2004.

    Google Scholar 

  37. S. Osher, A. Sole, and L. Vese. Image decomposition and restoration using total variation minimization and the H −1 norm. SIAM J. Multiscale Model. Simul., l(3):34-370, 2003.

    Google Scholar 

  38. R. Ramlau and G. Teschke. Tikhonov replacement functionals for iteratively solving nonlinear operator equations. Inverse Probl, 21:1571–1592, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  39. L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variations based noise removal algorithms. Physica D, 60:2 9-268, 1992.

    Google Scholar 

  40. I. W. Selesnick. Hilbert transform pairs of wavelet bases. IEEE Signal Process. Lett., 8(6):170–173, 2001.

    Article  Google Scholar 

  41. P. Wojtaszczyk. A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge 1997.

    MATH  Google Scholar 

  42. X.-G. Xia and Z. Zhang. On sampling theorem, wavelets and wavelet transforms. IEEE Trans. Signal Process., 41:3 24-3535, 1993.

    Google Scholar 

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Author information

Authors and Affiliations

  1. FB 12, Mathematics and Computer Sciences, Philipps-University of Marburg, Hans-Meerwein-Strasse, Lahnberge, D-35032, Marburg, Germany

    Stephan Dahlke, Karsten Koch & Manuel Werner

  2. Center for Industrial Mathematics (ZeTeM), FB 3, University of Bremen, Postfach 330440, D-28334, Bremen, Germany

    Peter Maass, Dirk Lorenz & Stefan Schiffler

  3. Zuse Institute Berlin (ZIB), Takustrasse 7, D-14195, Berlin-Dahlem, Germany

    Gerd Teschke

  4. Bruker Daltonics GmbH, Fahrenheitstrasse 4, D-28359, Bremen, Germany

    Herbert Thiele

  5. F. Hoffmann-La Roche AG, Grenzacherstrasse 124, CH-4070, Basel, Switzerland

    Stephan Müller & Andreas Stämpfli

Authors
  1. Stephan Dahlke
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  2. Peter Maass
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  3. Gerd Teschke
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  4. Karsten Koch
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  5. Dirk Lorenz
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  6. Stephan Müller
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  7. Stefan Schiffler
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  8. Andreas Stämpfli
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  9. Herbert Thiele
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  10. Manuel Werner
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Editor information

Editors and Affiliations

  1. Inst. Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany

    Rainer Dahlhaus

  2. Inst. Physik, LS Theoretische Physik, Universität Potsdam, Am Neuen Palais 19, 14469, Potsdam, Germany

    Jürgen Kurths

  3. FB 3 Mathematik/Informatik Zentrum Technomathematik, Universität Bremen, 28334, Bremen, Germany

    Peter Maass

  4. Zentrum Datenanalyse, Universität Freiburg, Eckerstr. 1, 79104, Freiburg, Germany

    Jens Timmer

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© 2008 Springer-Verlag Berlin Heidelberg

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Dahlke, S. et al. (2008). Multiscale Approximation. In: Dahlhaus, R., Kurths, J., Maass, P., Timmer, J. (eds) Mathematical Methods in Signal Processing and Digital Image Analysis. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75632-3_3

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