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Image Reconstruction from Non-Uniformly Sampled Spectral
Data
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
OutlineOutline
Background/Problem Statement Algorithm Database and Validation Test Results Future Work
Background/Problem Statement Algorithm Database and Validation Test Results Future Work
BackgroundBackground
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
Problem StatementProblem 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
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
Problem StatementProblem 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
The AlgorithmThe Algorithm
Stage one: Stage one:
The AlgorithmThe Algorithm
Stage two: Stage two:
The AlgorithmThe Algorithm
Stage three: Stage three:
The AlgorithmThe 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
ValidationValidation
We select from a standard set of image processing images a subset
We select from a standard set of image processing images a subset
ValidationValidation
We convert the images to grayscale, in case they are in color
We convert the images to grayscale, in case they are in color
ValidationValidation
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
Test Results: BaboonTest Results: Baboon
Test Results: BaboonTest Results: Baboon
Test Results: BaboonTest Results: Baboon
Test Results: LenaTest Results: Lena
Test Results: LenaTest Results: Lena
Test Results: LenaTest Results: Lena
Test Results: PeppersTest Results: Peppers
Test Results: PeppersTest Results: Peppers
Test Results: PeppersTest Results: Peppers
Future WorkFuture 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
ReferencesReferences
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.
