Biomedical Engineering Reference
In-Depth Information
is characterized by very little data in the Fourier domain, and direct inversion
approaches produce severe artifacts. Difficulties in reconstructing volumes from
such incomplete tomographic datasets are often aggravated by noise in the
measurements and misalignments among projections.
Under-constrained problems are typically solved using one or both of two
different strategies. The first strategy is to choose from among feasible solu-
tions (those that match the data) by imposing some additional criterion, such
as finding the solution that minimizes an energy function. This additional crite-
rion should be designed to capture some desirable property, such as minimum
entropy. The second strategy is to parameterize the solution in a way that re-
duces the number of degrees of freedom. Normally, the model should contain
few enough parameters so that the resulting parameter estimation problem is
overconstrained. In such situations solutions are allowed to differ from the data
in a way that accounts for noise in the measurements.
In this section we consider a special class of underconstrained tomographic
problems that permits the use of a simplifying model. The class of problems we
consider are those in which the imaging process is targeted toward tissues or
organs that have been set apart from the other anatomy by some contrast agent.
This agent could be an opaque dye, as in the case of transmission tomography,
or an emissive metabolite, as in nuclear medicine. We assume that this agent
produces relatively homogeneous patches that are bounded by areas of high
contrast. This assumption is reasonable, for instance, in subtractive angiogra-
phy or CT studies of the colon. The proposed approach, therefore, seeks to find
the boundaries of different regions in a volume by estimating sets of closed
surface models and their associated density parameters directly from the in-
complete sinogram data [74]. Thus, the reconstruction problem is converted to
a segmentation problem. Of course, we can never expect real tissues to exhibit
truly homogeneous densities. However, we assert that when inhomogeneities
are somewhat uncorrelated and of low contrast the proposed model is adequate
to obtain acceptable reconstructions.
8.6.1 Related Work
Several areas of distinct areas of research in medical imaging, computer vision,
and inverse problems impact this work. Numerous tomographic reconstruc-
tion methods are described in the literature [75, 76], and the method of choice
depends on the quality of projection data. Filtered back projection (FBP), the
