Biomedical Engineering Reference
In-Depth Information
most widely used approach, works well in the case of the fully constrained
reconstruction where one is given
enough high-quality projections over 180
◦
angular range
. Statistical, iterative approaches such as maximum likelihood
(ML) and maximum
a posteriori
(MAP) estimation have been proven to work
well with
noisy
projection data, but do not systematically address the under-
constrained reconstruction problem and generally rely on complete datasets.
An exception is [77], which proposes an iterative algebraic approach that in-
cludes some assumptions about the homogeneity of the solution to compute a
full grayscale reconstruction. Also, some hybrid approaches [78, 79] are specif-
ically developed to deal with
limited-angle
tomography by extrapolating the
missing sinogram data.
Other tomographics reconstruction techniques have been proposed, for ex-
ample those that utilize discrete tomography strategies [73, 80-82], and de-
formable models [83-87]. The literature also describes many examples of level
sets as curve and surface models for image segmentation [6, 7, 41, 88]. The au-
thors have examined their usefulness for 3D segmentation of TEM
reconstruc-
tions
[37]. Several authors have proposed solving
inverse problems
using level
sets [89-95], but are mostly limited to solving 2D problems.
We make several important contributions to this previous body of work; first
we give a formal derivation of the motion of deformable surface models as the
first variation of an error term that relates the projected model to the noisy
tomographic data. This formulation does not assume any specific surface repre-
sentation, and therefore applies to a wide range of tomographic, surface-fitting
problems. Second we present a level set implementation of this formulation that
computes incremental changes in the radon transform of the projected model
only along the wave front, which makes it practical on large datasets. Third
we examine the specific problem of initializing the deformable surface in the
absence of complete sinogram data, and demonstrate, using real and synthetic
data, the effectiveness of direct surface estimation for a specific class of tomo-
graphic problems which are underconstrained.
8.6.2 Mathematical Formulation
As an introduction, we begin with the derivation of surface estimation problem
in two dimensions. The goal is to simultaneously estimate the interface between
two materials and their densities,
β
0
and
β
1
. Thus we have a background with
