Advertisement

Tight Graph Framelets for Sparse Diffusion MRI q-Space Representation

  • Pew-Thian YapEmail author
  • Bin Dong
  • Yong Zhang
  • Dinggang Shen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9902)

Abstract

In diffusion MRI, the outcome of estimation problems can often be improved by taking into account the correlation of diffusion-weighted images scanned with neighboring wavevectors in q-space. For this purpose, we propose in this paper to employ tight wavelet frames constructed on non-flat domains for multi-scale sparse representation of diffusion signals. This representation is well suited for signals sampled regularly or irregularly, such as on a grid or on multiple shells, in q-space. Using spectral graph theory, the frames are constructed based on quasi-affine systems (i.e., generalized dilations and shifts of a finite collection of wavelet functions) defined on graphs, which can be seen as a discrete representation of manifolds. The associated wavelet analysis and synthesis transforms can be computed efficiently and accurately without the need for explicit eigen-decomposition of the graph Laplacian, allowing scalability to very large problems. We demonstrate the effectiveness of this representation, generated using what we call tight graph framelets, in two specific applications: denoising and super-resolution in q-space using \(\ell _{0}\) regularization. The associated optimization problem involves only thresholding and solving a trivial inverse problem in an iterative manner. The effectiveness of graph framelets is confirmed via evaluation using synthetic data with noncentral chi noise and real data with repeated scans.

References

  1. 1.
    Dong, B.: Sparse representation on graphs by tight wavelet frames and applications. Applied and Computational Harmonic Analysis (2015)Google Scholar
  2. 2.
    Hammond, D.K., Vandergheynst, P., Gribonval, R.: Wavelets on graphs via spectral graph theory. Appl. Comput. Harmonic Anal. 30(2), 129–150 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ron, A., Shen, Z.: Affine systems in \(L_{2}(\mathbb{R}^{d})\): the analysis of the analysis operator. J. Funct. Anal. 148(2), 408–447 (1997)Google Scholar
  4. 4.
    Michailovich, O., Rathi, Y.: On approximation of orientation distributions by means of spherical ridgelets. IEEE Trans. Image Process. 19(2), 461–476 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chan, R.H., Chan, T.F., Shen, L., Shen, Z.: Wavelet algorithms for high-resolution image reconstruction. SIAM J. Sci. Comput. 24(4), 1408–1432 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Zhang, Y., Dong, B., Lu, Z.: \(\ell _{0}\) minimization for wavelet frame based image restoration. Math. Comput. 82, 995–1015 (2013)Google Scholar
  7. 7.
    Tuch, D.S.: Q-ball imaging. Magn. Reson. Med. 52, 1358–1372 (2004)CrossRefGoogle Scholar
  8. 8.
    Belkin, M., Niyogi, P.: Towards a theoretical foundation for laplacian-based manifold methods. In: Auer, P., Meir, R. (eds.) COLT 2005. LNCS (LNAI), vol. 3559, pp. 486–500. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Lu, Z., Zhang, Y.: Sparse approximation via penalty decomposition methods. SIAM J. Optim. 23(4), 2448–2478 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lu, Z.: Iterative hard thresholding methods for \(l_0\) regularized convex cone programming. Math. Prog. Ser. A B 147(1–2), 125–154 (2014)Google Scholar
  11. 11.
    Yap, P.-T., Zhang, Y., Shen, D.: Diffusion compartmentalization using response function groups with cardinality penalization. In: Navab, N., Hornegger, J., Wells, W.M., Frangi, A.F. (eds.) MICCAI 2015. LNCS, vol. 9349, pp. 183–190. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-24553-9_23CrossRefGoogle Scholar
  12. 12.
    Donoho, D.L.: De-noising by soft-thresholding. IEEE Trans. Inf. Theor. 41(3), 613–627 (1995)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Koay, C.G., Özarslan, E., Basser, P.J.: A signal transformational framework for breaking the noise floor and its applications in MRI. J. Magn. Reson. 197, 108–119 (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Authors and Affiliations

  • Pew-Thian Yap
    • 1
    Email author
  • Bin Dong
    • 2
  • Yong Zhang
    • 3
  • Dinggang Shen
    • 1
  1. 1.Department of Radiology and BRICUniversity of North CarolinaChapel HillUSA
  2. 2.Beijing International Center for Mathematical ResearchPeking UniversityBeijingChina
  3. 3.Department of Psychiatry and Behavioral SciencesStanford UniversityStanfordUSA

Personalised recommendations