Image recovery techniques for x-ray computed tomography in limited data environments Page: 4 of 32
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advantage that are noniterative---hence they are computationally efficient---but they
suffer from a lack of flexibility in that they can't incorporate prior information about
the solution, nor can they use other than the squared error criterion to fit the data.
Because the image recovery problem associated with most tomography applications
is ill-posed, and this is particularly true for limited data situations, all possible prior
information about the unknown object must be utilized in order to produce high-
quality images. Furthermore, although least-squares (i.e. minimizing a sum of
squares error function that measures the mismatch between the data and the model)
is appropriate if the data has a Gaussian distribution with known constant variance,
using it when the data has a non-Gaussian distribution can seriously degrade the
quality of reconstructed images. An example is counting problems (e.g., emission
tomography) where the data consists of particle or photon counts that typically have
a Poisson distribution. Another example is outlier-corrupted data in which the
distribution function is not known exactly, but it is known that the data pixels are
subject to outliers that occur infrequently, but greatly distort those pixels where they
occur. These outliers result from a variety of problems including a few bad detectors
in a CCD array, improper assumptions about the model, or "hits" from extraneous
radiation or particles. Outliers have highly deleterious effects on the reconstructed
image when squared error is used.
The problems our techniques have been designed to handle can be described as
follows. First there is a linear equation that models the relationship of the unknown
to the data
9= Ax (1)
where the vector x represents the unknown image we wish to reconstruct, A is a
matrix that represents the effects of geometry, absorption, etc. that lead to the
expected data y if x were the true image. The actual observed data is y, and the
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Aufderheide, M B; Goodman, D M; Jackson, J A & Johansson, E M. Image recovery techniques for x-ray computed tomography in limited data environments, report, March 1, 1999; California. (https://digital.library.unt.edu/ark:/67531/metadc624723/m1/4/: accessed May 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.