1. May 1, 2009AMSC 663/6641 Image Reconstruction from Non-Uniformly Sampled Spectral Data Alfredo Nava-Tudela AMSC 664, Spring 2009 Final Presentation Advisor: John J. Benedetto

2. May 1, 2009AMSC 663/6642 Signals and their spectral decomposition A signal can be decomposed in harmonics that reveal the frequency or spectral content contained in that signal

3. May 1, 2009AMSC 663/6643 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

4. May 1, 2009AMSC 663/6644 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

5. May 1, 2009AMSC 663/6645 The Algorithm Stage one:

6. May 1, 2009AMSC 663/6646 The Algorithm Stage two:

7. May 1, 2009AMSC 663/6647 The Algorithm Stage three: Image Reconstructed

8. May 1, 2009AMSC 663/6648 CG Experiments - Using the DFTxSinc input data

9. May 1, 2009AMSC 663/6649 CG Experiments - Using the DFTxSinc input data

10. May 1, 2009AMSC 663/66410 CG Experiments - Using the DFTxSinc input data

11. May 1, 2009AMSC 663/66411 CG Experiments - Using the DFTxSinc input data

12. May 1, 2009AMSC 663/66412 CG Experiments - Using the DFTxSinc input data

13. May 1, 2009AMSC 663/66413 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.

14. May 1, 2009AMSC 663/66414 CG Experiments - Runtime vs Precision

15. May 1, 2009AMSC 663/66415 CG Experiments - Number of iterations vs Precision

16. May 1, 2009AMSC 663/66416 CG Experiments - Convergence and time results

17. May 1, 2009AMSC 663/66417 CG Experiments - Convergence and time results The time is given in seconds

18. May 1, 2009AMSC 663/66418 CG Experiments - Convergence and time results

19. May 1, 2009AMSC 663/66419 CG Experiments - Convergence and time results The time is given in seconds

20. May 1, 2009AMSC 663/66420 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.

21. May 1, 2009AMSC 663/66421 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

22. May 1, 2009AMSC 663/66422 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

23. May 1, 2009AMSC 663/66423 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

24. May 1, 2009AMSC 663/66424 CG Experiments - Convergence N=16

25. May 1, 2009AMSC 663/66425 CG Experiments - Convergence N=32

26. May 1, 2009AMSC 663/66426 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 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.

27. May 1, 2009AMSC 663/66427 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.