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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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.

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

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

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

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

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

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: Jonathan Richard Shewchuk, “An Introduction to the Conjugate Gradient Method Without the Agonizing Pain”. August 4, Adi Ben-Israel and Thomas N. E. Greville. Generalized Inverses. Springer-Verlag, 2003.

May 1, 2009AMSC 663/66427 References John J. Benedetto and Paulo J. S. G. Ferreira. Moderm Sampling Theory: Mathematics and Applications. Birkhauser, J. W. Cooley and J. W. Tukey. An algorithm for the machine computation of complex Fourier series. Math. Comp., 19: , E. H. Moore. On reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society, 26:85-100, Diane P. O’Leary. Scientific computing with case studies. Book in preparation for publication, Roger Penrose. On best approximate solution to linear matrix equations. Proceedings of the Cambridge Philosophical Society, 52:17-19, 1956.