6 Tomographic Imaging with Diffracting Sources

Diffraction tomography is an important alternative to straight ray tomog-raphy. For some applications, the harm caused by the use of x-rays, an ionizing radiation, could outweigh any benefits that might be gained from the tomogram. This is one reason for the interest in imaging with acoustic or electromagnetic radiation, which are considered safe at low levels. In addition, these modalities measure the acoustic and electromagnetic refrac-tive index and thus make available information that isn't obtainable from x-ray tomography. As mentioned in Chapter 4, the accuracy of tomography using acoustic or electromagnetic energy and straight ray assumptions suffers from the effects of refraction and/or diffraction. These cause each projection to not represent integrals along straight lines but, in some cases where geometrical laws of propagation apply, paths determined by the refractive index of the object. When the geometrical laws of propagation don't apply, one can't even use the concept of line integrals-as will be clear from the discussions in this chapter. There are two approaches to correcting these errors. One approach is to use an initial estimate of the refractive index to estimate the path each ray follows. This approach is known as algebraic reconstruction and, for weakly refracting objects, will converge to the correct refractive index distribution after a few iterations. We will discuss algebraic techniques in Chapter 7. When the sizes of inhomogeneities in the object become comparable to or smaller than a wavelength, it is not possible to use ray theory (geometric propagation) based concepts; instead one must resort directly to wave propagation and diffraction based phenomena. In this chapter, we will show that if the interaction of an object and a field is modeled with the wave equation, then a tomographic reconstruction approach based on the Fourier Diffraction Theorem is possible for weakly diffracting objects. The Fourier Diffraction Theorem is very similar to the Fourier Slice Theorem of conventional tomography: In conventional (or straight ray) tomography, the Fourier Slice Theorem says that the Fourier transform of a projection gives the values of the Fourier transform of the object along a straight line. When diffraction effects are included, the Fourier Diffraction Theorem says that a " projection " yields the Fourier transform of the object over a semicircular arc. This result is fundamental to diffraction tomography. In this chapter the basics of diffraction tomography are presented for application with acoustic, microwave, and optical energy. For each case we …