Mathematical Methods for Spectral Image Reconstruction

Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 3)

Abstract

We present a method for recovery of damaged parts of old paintings (frescoes), caused by degradation of the pigments contained in the paint layer. The original visible colour information in the damaged parts can be faithfully recovered from measurements of absorption spectra in the invisible region (IR and UV) and from the full spectral data of the well preserved parts of the image. We test algorithms recently designed for low-rank matrix recovery from few observations of their entries. In particular, we address the singular value thresholding (SVT) algorithm by Cai, Candès and Shen, and the iteratively re-weighted least squares minimization (IRLS) by Daubechies, DeVore, Fornasier and Güntürk, suitably adapted to work for low-rank matrices. In addition to these two algorithms, which are iterative in nature, we propose a third non-iterative method (which we call block completion, BC), which can be applied in the situation when the missing elements of a low-rank matrix constitute a block (submatrix); this is always true in our application. We shortly introduce the SVT and IRLS algorithms and present a simple analysis of the BC method. We eventually demonstrate the performance of these three methods on a sample fresco.

Keywords

Image recovery UV and IR absorption spectra Matrix completion 

References

  1. 1.
    Cai JF, Candès EJ, Shen Z (2008) A singular value thresholding algorithm for matrix completion. October. arXiv:0810.3286Google Scholar
  2. 2.
    Candès EJ, Recht B (2009) Exact matrix completion via convex optimization. Foundations of Computational Mathematics. arXiv:0805.4471v1Google Scholar
  3. 3.
    Candès EJ, Tao T (in press) The power of matrix completion: near-optimal convex relaxation. IEEE Inf TheoryGoogle Scholar
  4. 4.
    Chistov AL, Grigoriev Dyu (1984) Complexity of quantier elimination in the theory of algebraically closed elds. In: Proceedings of the 11th symposium on mathematical foundations of computer science. Lecture notes in computer science, vol 176. Springer, Berlin, p 1731Google Scholar
  5. 5.
    Daubechies I, DeVore R, Fornassier M, Güntürk CS (2009) Iteratively re-weighted least squares minimization for sparse recovery. Commun Pure Appl Math 35Google Scholar
  6. 6.
    Fornasier M, Haskovec J, Vybiral J (in preparation) The block completion method for matrix reconstructionGoogle Scholar
  7. 7.
    Fornasier M, Rauhut H, Ward R (in preparation) Iteratively re-weighted least squares for low-rank matrix completionGoogle Scholar
  8. 8.
    Levinson R, Berdahl P, Akbari H (2005) Solar spectral optical properties of pigments–Part II: survey of common colorants. Sol Energ Mat Sol C 89:351–389CrossRefGoogle Scholar
  9. 9.
    Recht B, Fazel M, Parrilo P (2007, submitted) Guaranteed minimum rank solutions of matrix equations via nuclear norm minimization. SIAM Rev. arXiv: 0706.4138Google Scholar
  10. 10.
    Wright J, Ma Y, Ganesh A, Rao S (2009) Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization. J ACM (to appear)Google Scholar
  11. 11.
    Wyszecki G, Stiles WS (1982) Color science: concepts and methods, quantitative data and formulae. Wiley, New YorkGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Wolfgang Baatz
    • 1
  • Massimo Fornasier
    • 2
  • Jan Haskovec
    • 2
  1. 1.Akademie der Bildenden KünsteWienAustria
  2. 2.Johann Radon Institute for Computational and Applied MathematicsLinzAustria

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