1. Image Reconstruction from Non- Uniformly Sampled Spectral Data Alfredo Nava-Tudela AMSC 663, Fall 2008 Midterm Progress Report Advisor: John J. Benedetto Alfredo Nava-Tudela AMSC 663, Fall 2008 Midterm Progress Report Advisor: John J. Benedetto

2. Outline  Background/Problem Statement  Algorithm  Database and Validation  Test Results  Future Work  Background/Problem Statement  Algorithm  Database and Validation  Test Results  Future Work

3. Background  Sometimes there is a need to reconstruct from spectral data an object in the spatial/time domain  For example, and image from an MRI machine  Sometimes there is a need to reconstruct from spectral data an object in the spatial/time domain  For example, and image from an MRI machine

4. Problem Statement  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. Problem Statement  In a real life application, the spectral data set is generated by some physical process  In our case, we generate artificial spectral data from a known high resolution image  We use a down-sampled version of that image to compare the goodness of our reconstruction  In a real life application, the spectral data set is generated by some physical process  In our case, we generate artificial spectral data from a known high resolution image  We use a down-sampled version of that image to compare the goodness of our reconstruction

6. The Algorithm  Stage one:

7. The Algorithm  Stage two:

8. The Algorithm  Stage three:

9. The Algorithm  This algorithm corresponds to the direct solution of the linear system of equations presented in the project proposal  This has the drawback of having to store a potentially very big matrix  This algorithm corresponds to the direct solution of the linear system of equations presented in the project proposal  This has the drawback of having to store a potentially very big matrix

10. Validation  We select from a standard set of image processing images a subset

11. Validation  We convert the images to grayscale, in case they are in color

12. Validation  These are the images that we feed to our algorithm  Select the desired resolution for the reconstruction: 16 by 16 and 32 by 32  Higher resolutions take longer  These are the images that we feed to our algorithm  Select the desired resolution for the reconstruction: 16 by 16 and 32 by 32  Higher resolutions take longer

13. Test Results: Baboon

16. Test Results: Lena

19. Test Results: Peppers

22. Future Work  Allow arbitrary size input images, currently only square images processed  Implement algorithm that doesn’t store matrices  Write C++ code, explore parallelization  Explore other ways to assess goodness of reconstruction  Explore different sampling geometries  Allow arbitrary size input images, currently only square images processed  Implement algorithm that doesn’t store matrices  Write C++ code, explore parallelization  Explore other ways to assess goodness of reconstruction  Explore different sampling geometries

23. References  Adi Ben-Israel and Thomas N. E. Greville. Generalized Inverses. Springer- Verlag, 2003.  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.  Adi Ben-Israel and Thomas N. E. Greville. Generalized Inverses. Springer- Verlag, 2003.  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.