Hyperspectral spectrometry of the kind proposed by Okamoto et al.[33] and demonstrated by Descour et al. Vose and Horton have devised another technique for CTIS reconstruction, using an almost explicit solution based on the circulating properties of the defining system matrix[14]. The amount of light detected is limited by the size of the local planar array, the aperture.
An intuitive approach would be to formulate the inverse problem from this perspective: The system matrix itself is an approximately shear-invariant representation of the basis. This preconditioning matrix will be used to compute an initializer, f 0, for each of the iterative routines mentioned above. As defined, regularization "imposes stability on an ill-posed problem in a way that yields accurate approximate solutions, often by incorporating prior information." The solutions consist of a parametric family of approximate inverse operators Rα : Y → X so that if yη =A xapprox + ηη, and if ηη→0 there exist parameters αη such that xαη = Rαη y → xapprox.
The approach in this work is direct construction, as the structure of the system matrix facilitates it, as will be discussed later. The Cholesky factorization is used to construct the approximate inverse preconditioners due to the easy parallel implementation compared to the SVD algorithm [ 12 , 19 ]. Exploiting the features of the CTIS system matrix proved to be more practical to construct a robust approximate inverse.
This approach is also used to check the sparsity of the resulting approximate inverse as governed by the overlap product term. It is used to construct the submatrices of the approximate inverse A such that A(H'H) ~ I + ε, where ε is very small. The results of using these constructors will be discussed after a brief review of algebraic reconstruction techniques.
3. Adjust the threshold for accepting submissions to reduce retained indexes. 4. The estimated inverse is then formed by shifting this dense inverse to all positions covered by the source image dimension. The third estimated inverse consists of eliminating the overlapping terms of this matrix to match the parsimony pattern of the system matrix. Initial estimates of the following types will be used to measure the effectiveness of this estimated inverse with the next one.
For some data, the first entry is reserved to give the dimensionality of the full expanded matrix as is the case for the base representation data. The structure of the routines facilitates the ordering of matrix operations and the construction of different reconstruction techniques as needed. However, the execution of the code was cumbersome due to the sheer size of the data generated.
The Zero-order image of the white light source data is provided below and will be compared to the reconstructions from each iterative reconstruction technique.
Dense Preconditioner Convergence
MART raw spectral reconstruction of the mean intensity number for the central pixel in 200 angstrom spectral bandwidth steps covering 4200 angstroms to 7000 angstroms. "Brightness" remains consistent except for the midbands, where dense preconditioning redistributes the overall intensity. The spectral crowding effect of progressively denser preconditioners was detrimental to additive ART methods.
Interestingly, the basic preconditioner built with the same sparse pattern as the system matrix produced a spectral profile very similar to the simple backprojection for the multiplicative ARTs. The quality of the resulting image is still much less than the forward backward routine used by Descour et al [9]. The amount of time to process 50 iterations was not significantly different for all routines examined.
Overall, the performance of the parallelized MART routine is significantly faster than what has been reported for a single CPU. implementation with the IDL programming language [17]. More complex reconstruction routines can make use of the parallelized linear algebra operations developed during this thesis research. During this project, it was hoped that a new algorithm could be developed along either an approximate inverse scheme or a conjugate gradient scheme with preconditioning similar to MINRES and other numerical techniques.
It is possible to parallelize this algorithm in the future and apply it to the CTIS problem. The rapid acquisition of the CTIS system avoids long-term exposure and rapid data collection that offers the promise of seeing rapid biochemical reactions in real time. Forward projection of the projected object image, in this case, the dense preconditioner for regular MART.
MART with dense preconditioned initialization, 50 iterations. 2000) "Implementation and Performance Evaluation of Iterative Reconstruction Algorithms in SPECT: A Simulation Study Using ESG4". Harvey AR, Fletcher-Holmes DW. 2004) "Birefringent Fourier transform imaging spectrometer.". 2000) "A Compact Visible/Near Infrared Hyperspectral Imager". Ford BK, Descour MR and Lynch RM. 2001) "Large-format computed tomography imaging spectrometer for fluorescence microscopy.".
Proc. 2004) “Experiments with the Nonlinear and Chaotic Behavior of the Multiplicative Algebraic Reconstruction Technique (MART) Algorithm for Computed Tomography.”. -Bakr Al-Mehdi, Mita Patel, Abu Haroon, et al., “Increased depth of cellular imaging in the intact lung using far-red and near-infrared fluorescent probes,” International Journal of Biomedical Imaging, vol. 2005) "Approximation of Mie scattering parameters in near-infrared tomography of normal breast tissue in vivo.".
