1 / 27

Image Reconstruction from Non-Uniformly Sampled Spectral Data

Image Reconstruction from Non-Uniformly Sampled Spectral Data. Alfredo Nava-Tudela AMSC 664, Spring 2009 Final Presentation Advisor: John J. Benedetto. Signals and their spectral decomposition.

elyse
Download Presentation

Image Reconstruction from Non-Uniformly Sampled Spectral Data

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Image Reconstruction from Non-Uniformly Sampled Spectral Data Alfredo Nava-Tudela AMSC 664, Spring 2009 Final Presentation Advisor: John J. Benedetto AMSC 663/664

  2. Signals and their spectral decomposition • A signal can be decomposed in harmonics that reveal the frequency or spectral content contained in that signal AMSC 663/664

  3. Signals and their spectral decomposition • Often times, we have spectral information and we need to convert back to spatial information, for example Magnetic Resonance Imaging AMSC 663/664

  4. Problem Statement • We are particularly interested in the reconstruction of images given spectral information • More specifically, we are interested in image reconstruction given non-uniformly sampled spectral data • Given a two dimensional spectral data set, reconstruct an image in the spatial domain that matches as closely as possible that data set in the spectral domain AMSC 663/664

  5. The Algorithm • Stage one: AMSC 663/664

  6. The Algorithm • Stage two: AMSC 663/664

  7. The Algorithm • Stage three: Image Reconstructed AMSC 663/664

  8. CG Experiments - Using the DFTxSinc input data AMSC 663/664

  9. CG Experiments - Using the DFTxSinc input data AMSC 663/664

  10. CG Experiments - Using the DFTxSinc input data AMSC 663/664

  11. CG Experiments - Using the DFTxSinc input data AMSC 663/664

  12. CG Experiments - Using the DFTxSinc input data AMSC 663/664

  13. CG Experiments - Time of one iteration vs image size If N = 16 = 2^4, then ln(16)/ln(2) = 4. Time is given in seconds. AMSC 663/664

  14. CG Experiments - Runtime vs Precision AMSC 663/664

  15. CG Experiments - Number of iterations vs Precision AMSC 663/664

  16. CG Experiments - Convergence and time results AMSC 663/664

  17. CG Experiments - Convergence and time results The time is given in seconds AMSC 663/664

  18. CG Experiments - Convergence and time results AMSC 663/664

  19. CG Experiments - Convergence and time results The time is given in seconds AMSC 663/664

  20. CG Experiments - Memory usage • One iteration of the CG method issues 2 calls to the function A_times() and 2 calls to the function A_star_times(). • Both functions, by implementation, use the same amount of memory. • The CG method also has bookkeeping variables that require memory. AMSC 663/664

  21. CG Experiments - Memory usage A call to either A_times() or A_star_times() uses the following memory: Name Size Class Attributes KL N^2x2 double M 1x1 double N_square 1x1 double S Mx2 double a Mx1 double complex f N^2x1 double m 1x1 double n 1x1 double sum 1x1 double complex Which gives a sub-total for each call of: 3xN^2 + 4xM + 6 words AMSC 663/664

  22. CG Experiments - Memory usage A call of the CG code, without the previous taken into account, uses the following memory: Name Size Class Attributes KL N^2x2 double S Mx2 double alpha 1x1 double complex beta 1x1 double d N^2x1 double complex delta_0 1x1 double delta_new 1x1 double delta_old 1x1 double f_hat Mx1 double complex iteration 1x1 double q N^2x1 double complex r N^2x1 double complex tol 1x1 double x N^2x1 double y N^2x1 double complex Which gives a sub-total of: 11xN^2 + 4xM + 8 words AMSC 663/664

  23. CG Experiments - Memory usage Combined, we obtain the following grand total of: 14xN^2 + 8xM + 14 words needed to run our code. The direct method that saves the matrices A and its adjoint A* would need O(N^2 x M) words of memory. Clearly the CG method is the way to go memory wise! Direct Method CG Method We assume M = N^2, best case scenario AMSC 663/664

  24. CG Experiments - Convergence N=16 AMSC 663/664

  25. CG Experiments - Convergence N=32 AMSC 663/664

  26. References • Richard F. Bass and Karlheinz Groechenig “Random Sampling of Multivariate Trigonometric Polynomials” • Zhou Wang, Alan C. Bovik, Hamid R. Sheikh, and Eero P. Simoncelli “Image Quality Assessment: From Error Measurements to Structural Similarity”, IEEE Transactions on Image Processing, Vol. 13, No. 1, January 2004 • Conjugate Gradient Method: http://en.wikipedia.org/wiki/Conjugate_gradient_method • Jonathan Richard Shewchuk, “An Introduction to the Conjugate Gradient Method Without the Agonizing Pain”. August 4, 1994. • Adi Ben-Israel and Thomas N. E. Greville. Generalized Inverses. Springer-Verlag, 2003. AMSC 663/664

  27. References • John J. Benedetto and Paulo J. S. G. Ferreira. Moderm Sampling Theory: Mathematics and Applications. Birkhauser, 2001. • J. W. Cooley and J. W. Tukey. An algorithm for the machine computation of complex Fourier series. Math. Comp., 19:297-301, 1965. • E. H. Moore. On reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society, 26:85-100, 1920. • Diane P. O’Leary. Scientific computing with case studies. Book in preparation for publication, 2008. • Roger Penrose. On best approximate solution to linear matrix equations. Proceedings of the Cambridge Philosophical Society, 52:17-19, 1956. AMSC 663/664

More Related