Page 290 - Biomedical Engineering and Design Handbook Volume 2, Applications

268 DIAGNOSTIC EQUIPMENT DESIGN

discussed in Sec. 10.3. It also permits the observation that from a mathematical view, a perfect repre-
sentation of the body is theoretically possible, provided the requisite type and number of projected
images are available.
An important feature in the formulation of the tomographic reconstruction process is the assump-
tion of linearity attached to the various operations involved. This leads to the concept of a spatially
invariant point-spread function that is a measure of the performance of a given operation. In practice
the transformation associated with an operation involves the convolution of the input with the point-
spread function in the spatial domain to provide the output. This is recognized as a cumbersome
mathematical process and leads to an alternative representation that describes the input in terms of
sinusoidal functions. The associated transformation is now conducted in the frequency domain and
with the transfer function described by the Fourier integral. In discussing these principles, the func-
tions in the spatial domain and the frequency domain are considered to be continuous in their respec-
tive independent variables. However, for practical applications the relevant processes involve
discrete and finite data sampling. This has a significant effect on the accuracy of the reconstruction,
and in this respect certain conditions are imposed on the type and amount of data in order to improve
the result.
In Sec. 10.3 the mathematical principles of computed tomography are developed by considering
the formation of images by the relative attenuation of a series of parallel ray paths. The funda-
mental quantity in this process is the line integral ∫ AB f(x, y) ds of the object function f(x, y), which
is the radon transform, or projection, of the object along the ray path AB. This leads to the concept
that a parallel projection of an object, taken at an angle q in physical space, is equivalent to a slice
of the two-dimensional Fourier transform of the object function f(x, y), inclined at an angle q in fre-
quency space. According to a process known as the filtered back-projection, the object can be
reconstructed by taking the Fourier transform of the radon transform, applying a weighting to rep-
resent the effect of discrete steps in projection angle q, and back-projecting the result through the
reconstruction volume.
A natural progression is to consider the advantages that divergent ray-path scanning provides and
the increased complexity that this introduces. It should be appreciated that the cardinal measure for
developments in computed tomography is, according to the mathematical view, increased computa-
tional efficiency with enhanced modeling accuracy, and from the systems view is faster data capture
with improved image contrast and higher spatial resolution. These two aspects are inextricably
linked, though it is reasonable to conclude that the extensive and continuous development in recon-
struction algorithms has tended to be the catalyst for advancement.
The question of absolute accuracy will depend on considerations of what is practically achiev-
able and directly relates to the type and extent of measured data and the configuration of the associ-
ated system. The components of a basic laboratory system will comprise a source of x-rays, a
specimen stage, one or more imaging detector/camera devices, and a data acquisition system. The
type of x-ray source employed will depend on the permissible exposure period and on the degree of
resolution required. Large-area sources, measured in square millimeters, with high x-ray flux will be
used where the object is liable to movement and the length of exposure is important, as is usually the
case in medical imaging. However, small-area sources, measured in square micrometers, with rela-
tively low x-ray flux will be used where resolution is important. This is usually the case for biolog-
ical research where the application is generally referred to as microtomography, as discussed in
Sec. 10.4. This activity is properly classed as microscopy, where the physical layout of the related
system can provide high geometric magnification at the detector input plane of about 100 times and
high spatial resolution of about 1 mm or better.
With provision for projected high magnification, the beam-line area is relatively extensive and
can accommodate a range of ancillary equipment. This attribute makes the laboratory microtomog-
raphy facility a very versatile instrument. It can operate in a variety of modes and support scanning
strategies that are tailored to particular subject shapes, sizes, and material content. With large-area
detectors, magnified images can be recorded with divergent cone-beam projection and provide infor-
mation of the internal structure with a combination of specimen rotation and translation motion. With
small-area detectors, images can be recorded with a collimated pencil-beam and transverse raster
scanning of the specimen in the manner of a conventional light microscope. These detectors can be