Mathematics and Physics of Emerging Biomedical Imaging (1996) / Chapter Skim
Currently Skimming:
14 A CROSS-CUTTING LOOK AT THE MATHEMATICS OF EMERGING BIOMEDICAL IMAGING
14 A CROSS-CUTTING LOOK AT THE MATHEMATICS OF EMERGING BIOMEDICAL IMAGING
Pages 199-230
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 199... ...
14.1.1 Transmission Computed Tomography Transmission computed tomography (CT) is the original and simplest case of CT.
|
From page 200... ...
where H is the Hilbert transform. In fact the numerical implementation of equation 14.4 leads to the filtered backprojection algorithm, which is the standard algorithm in commercial CT scanners.
|
From page 201... ...
A variety of inversion formulas have been derived for these cases. If scatter is to be included, a transport model is more appropriate.
|
From page 202... ...
More on inverse problems can be found in section 14.3. 14.~.2 Emission Computed Tomography In emission computed tomography one determines the distribution f of radiating sources in the interior of an object by measuring the radiation outside the object in a tomographic fashion.
|
From page 203... ...
Thus, in addition to models using integral equations, stochastic models have been set up for emission tomography. These models are completely discrete.
|
From page 204... ...
lesions from images produced by two different algorithms concluded that maximum likelihood methods were superior to the filtered backprojection algorithm in certain clinical applications. The same type of study is needed to determine whether or not Gibbs priors will improve the maximum likelihood reconstruction Estopped short of convergence to avoid noise amplification)
|
From page 205... ...
Simple projections and linear integral equations wiB not suffice, and more sophisticated nonlinear models have to be used. In the following we consider an object Q C Ret with refractive index A
|
From page 206... ...
Numerical methods based on equation 14.20 and a similar equation for the Rytov approximation have become known as diffraction tomography. Unfortunately, the assumptions underlying the Born and Rytov approximations are not satisfied in medical imaging.
|
From page 207... ...
14.~.4 Optical Tomography With optical tomography one uses near-infrared lasers for the illumination of the body (see Chapter 11~. The process is described by the transport equation ~ (x, 8, t)
|
From page 208... ...
The numerical methods that have become known are of the Newton type, either applied directly to the transport equation or to the so-called diffusion approximation (e.g., Arridge et al., listed in section 14.~. The diffusion approximation is an approximation to the transport equation by a parabolic differential equation.
|
From page 209... ...
Unfortunately, the stability properties are very bad. Numerical methods based on Newton's method, linearization, simple backprojection, and layer stripping have been tried.
|
From page 210... ...
From here we can proceed in two ways. We can use equation 14.32 to determine the three-dimensional Fourier transform Mo of Mo and compute
|
From page 211... ...
This requires Mo to be known on a Cartesian grid, which can be achieved by a proper choice of the gradients or by interpolation. Alternatively, one can invoke the central slice theorem to obtain the three-dimensional Radon transform RMo Of Mo by a series of one-dimensional Fourier transforms.
|
From page 212... ...
Inverting R is a problem of vector tomography. Mathematically, this problem belongs to the recently developed field of integral geometry of tensor fields; see, for example, the paper by Sharafutdinov listed in section 14.~.
|
From page 213... ...
The appropriate mathematical model relating the electric field ~7 inside the body Q to the magnetic field B is the Biot-Savart law (compare section 10.2)
|
From page 214... ...
magnetic dipoles or by invoking a strong regularization scheme for a continuous dipole distribution. 14.~.10 Electrical Source Imaging With electrical source imaging (EST)
|
From page 215... ...
Considerable effort should be spent producing increasingly realistic models that allow for effective numerical simulation and validation in terms of real data. Numerical methods for solving inverse problems are usually iterative in nature solving the forward problem over anct over again.
|
From page 216... ...
Integral equations of the first kind with smooth kernels, such as 14.44, tend to be severely ill posed; see the next section.
|
From page 217... ...
In particular in the one-dimensional case, the Gelfand-Levitan procedure not only yields uniqueness and stability, but also provides efficient numerical methods (background can be found in the work by Burridge :Listed in section 14.~. 14.4 Ill-Poseciness and Regularization All problems in medical imaging are ill posed, even to very different degrees.
|
From page 218... ...
The only problems in medical imaging that have been solved successfully so far belong to the class of modestly ill-posed problems, for which s(~) is of the order [c with 0 < c < 1 (Holder stability)
|
From page 219... ...
It tells us precisely which features off cannot be recoverer! in a stable way, namely those that resemble functions fk that have small singular values (Jk A good example is the limited angle problem in transmission CT where the SVD has been computed analytically.
|
From page 220... ...
in section 14.~) , "finite series expansion methods" (see Censor's work in the suggested reading list)
|
From page 221... ...
Assume that the reconstruction region is the circle with radius p, and that we want to reconstruct reliably functions that are essentially "-band-limited, i.e., whose Fourier transform is negligible in some sense outside the circle of radius Lo. According to the Shannon sampling theorem, this restriction on the Fourier transform corresponds to a spatial resolution of 2:r/w.
|
From page 222... ...
Many methods have been developed for dealing with this situation: the "ridding method, a method based on the principle of the stationary phase, methods that estimate the phase of f based on small samples, localized polynomial approximation, autoregressive moving averages (ARMAs) , projection on convex subsets (POCS)
|
From page 223... ...
Equation 14.46 is replaced by the linear stochastic model Af =g+n, (14.59) where n is a family of random variables independent of f that is normally distributed with mean value O and covariance E
|
From page 224... ...
Various choices for F can be found in the paper by Geman and McClure in the suggested reading list. 14.7 Research Opportunities Investigation of the trade-offs of stability versus resolution for the inverse problem for the Helmholtz equation, as applied to ultrasound ~ .
|
From page 225... ...
· Development of faster methods for computing maximum likelihood estimates with priors, more efficient iterative methods and methods that exploit symmetries of the scanning geometries through efficient numerical algorithms such as the FFT. · Investigation of preconditioning for nonlinear iterations such as expectation maximization (EM)
|
From page 226... ...
· Reconstruction of a function from irregularly spaced samples of its Fourier transform, which has applications to CT and MRI. · Removal of artifacts caused by opaque objects, such as hip joints in CT.
|
From page 227... ...
SUGGESTED READING 227 7. Burridge, R., The Gelfand-I,evitan, the Marchenko, and the GopinathSondhi integral equations of inverse scattering theory, regarded in the context of inverse impuIse-response problems, Wave Motion 2 (1980)
|
From page 228... ...
28. Natterer, F., Determination of tissue attenuation in emission tomography of optically dense media, Inverse Problems 9 (1993)
|
From page 229... ...
SUGGESTED READING 229 32. Quinto, E.T., Cheney, M., and Kuchment, P., eds., Tomography, Impedance Imaging, and Integral Geometry, Proceedings of the AMSSIAM Summer Seminar, June 7-18, 1993, Mount Holyoke College, Massachusetts, American Mathematical Society, Providence, Ad., 1993.
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.