Non-Diffraction Computed Tomography

In X-ray CT imaging, the measurements are photons, which demonstrate wave-particle duality, that is, the properties of both waves and particles. Xrays have wavelengths roughly 10−13 m to 10−8 m, or frequencies of 1016 Hz to 1021 Hz. In MR imaging, the measurements are free induction decay (FID) signals, which are radio frequency (RF) signals with the wavelengths roughly 100 m to 102 m, or the frequencies 106 Hz to 108 Hz. X-ray and FID signal are electromagnetic in nature. In this chapter, imaging sources in several medical imaging techniques are characterized as electromagnetic (EM) waves. When EM waves impinge on an object, or an object is immersed in the

EM field, several physical phenomena occur on the object: its surface, inside, and surrounding. These phenomena include, but are not limited to, absorption, diffraction, non-diffraction, reflection, refraction, scattering, etc. Many of these phenomena can be utilized for imaging the object: its shape or surface or the internal structure. In medical applications, these imaging techniques are X-ray CT [2, 2-

4, 7-9, 18, 37, 53], MRI [12-16, 25, 25, 54-56], positron emission tomography (PET) [21-24, 37, 53], single photon emission computed tomography (SPECT) [22, 24, 25, 37, 53], ultrasonic (US) [26-29, 37, 53], etc. Although these techniques were developed based on different physical phenomena and principles, according to the nature of source-medium interaction, they can be classified into a category of imaging, transmission computed tomography.∗

Transmission CT can be further divided into two groups: (1) a non-diffraction CT imaging, in which the interaction model and the external measurements are characterized by the straight line integrals of some indexes of the medium and the image reconstruction is based on Fourier Slice theorem [37, 53], and (2) a diffraction CT imaging, in which the interaction and measurements are modeled with the wave equation and the tomographic reconstruction approach is based on the Fourier diffraction theorem [30, 37]. The former includes X-ray CT, MRI, emission CT, ultrasonic CT (e.g., refractive index CT and attenu-

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FIGURE 4.1

Physical phenomena caused by interaction between EM wave and an object.

ation CT), etc. The latter includes acoustic, certain seismic, microwave, and optical imaging, etc. In this chapter, we first use the inverse scattering problem as an example

to demonstrate how the interactions between the incident EM wave and the object can be used to generate the image of the object. Then we revisit X-ray CT and MRI and briefly review emission CT from a specific standpoint, and show that they belong to a category of imaging-the non-diffraction computed tomography. This insight may explain why X-ray CT and MRI have very similar statistical properties that are described in the remaining chapters of this book.