Tomographic Imaging with Diffracting Sources

Tomographic imaging with diffracting sources can be modelled as inverse problems for partial differential equations. Linearized versions lead to problems similar to those in tomographic imaging with non-diffracting sources, except that the straight lines are replaced by more complex shapes. In this chapter, we single out three non-ionizing imaging methods: (i) electrical impedance tomography; (ii) ultrasound imaging, and (iii) microwave imaging. These three techniques form an important alternative to straight ray tomography (CT) and MRI. In ultrasound and microwave imaging modalities, the interaction of a field and an object is modelled with the Helmholtz equation while in EIT, the mathematical model reduces to the conductivity equation. One general reconstruction algorithm used in ultrasound and microwave imaging is the diffraction tomography.

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 electric, acoustic, or electromagnetic radiation, which are considered safe at low levels. In addition, these modalities measure the electrical, acoustic, and electromagnetic properties of tissues and thus make available information that is not obtainable from X-ray tomography or MRI images. Thirdly, they are easily portable and relatively inexpensive.

In this chapter we first describe general algorithms used in electrical impedance tomography. Then we present the mathematical basis of diffraction tomography.