|Magnetic resonance spectroscopic imaging (MRSI) is a completely non-invasive
method for obtaining quantitative information regarding biochemical parameters .
It shows great promise for use in basic physiological research and for clinical imaging of
metabolic function [2, 3]. The information obtained with MRSI can be used to assess
regional metabolic abnormalities in various pathologies. However, MRSI requires a great
deal of time to gather the data necessary to achieve satisfactory resolution. When prior
information about the image is available, it may be possible to reconstruct the image
from a subset of k-space samples. Therefore, we desire to choose the best possible combination
of a small number of k-space samples to guarantee the quality of the reconstructed
image by using the available prior knowledge. In this thesis, we assume prior knowledge
only of the region of support (ROS) of the spatial-domain image.
Sequential forward selection (SFS) is appealing as an optimization method because
the previously selected sample can be observed while the next sample is selected. We
develop an efficient computational strategy for this algorithm that allows SFS to be applied to this problem when the image region has a known region of support (ROS).
The combined algorithm efficiently selects a reduced set of k-space samples from which
the ROS can be reconstructed with minimal noise amplification. Furthermore, if there
is no noise, the minimum density can be reached with this algorithm.
Hexagonal sampling gives a 13.4% sampling density reduction compared to rectangular
sampling for images with a circular ROS. However, nonuniform sampling patterns
are more efficient than hexagonal sampling for the same ROS. To reduce selection time
and achieve higher resolution, we develop a sequential backward selection (SBS) algorithm
from samples on a hexagonal grid. Simulation results show that more efficiency
and reduced selection time can be achieved with the proposed method in comparison
with SBS on a rectangular grid.
We develop two efficient algorithms for optimizing the dithering pattern so that an
image can be reconstructed as reliably as possible from a periodic nonuniform set of
samples, which can be obtained from a dithered rectangular-grid array.
One algorithm is SBS of sample arrays. Taking into account the ROS of the image,
we sequentially eliminate the least informative array recursively until the minimal
number of arrays remain. In this scheme, we provide an efficient update formula for the
criterion based on convolution kernels rather than large, non-sparse matrices, an efficient
update formula for the convolution kernels based on the deleted array, and an efficient reconstruction
method based on convolutions. The proposed method dramatically reduces
storage and computational complexity.
The other algorithm is SFS of sample arrays. Based on the ROS image, we sequentially
select the array that minimizes the noise amplification recursively until the desired number of arrays are selected. To avoid the singularity of the criterion when extending
the selection procedure to more samples than unknowns, we propose a modified criterion
for the case when the number of unknowns is more than the number of selected samples
and the complementary case. We also propose an efficient method to update the criterion
based only on the deleted array in the previous step to greatly reduce computational
time and avoid the inversion of a huge matrix. This method has great practical potential
because it can finish the selection process within half a minute for practical sizes.
The proposed schemes in this dissertation efficiently optimize the MRSI observation
in different ways. In general, they will reduce observation time and overcome the
problems in various available optimization methods.