Statistics of Medical Imaging

4

Non-Diffraction Computed Tomography

4.1 Introduction

In X-ray CT imaging, the measurements are photons, which demonstrate

wave-particle duality, that is, the properties of both waves and particle s. X-

rays have wavelengths roughly 10

−13

m to 10

−8

m, or frequencies of 10

16

Hz

to 10

21

Hz. In MR imaging, the measurements are free induction decay (FID)

signals, which are radio frequency (RF) signals with the wavelengths roughly

10

0

m to 10

2

m, or the frequencies 10

6

Hz to 10

8

Hz. X-ray and FID signal are

electromagne tic in nature. In this chapter, imaging sources in several medical

imaging techniques are characteriz ed a s electromagnetic (EM) waves.

When EM waves impinge on an object, or an object is immersed in the

EM field, several physical pheno mena occur on the object: its surface, inside,

and surrounding. These phenomena include, but are not limited to, absorp-

tion, diffra ction, non-diffrac tion, 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 a pplications, these imaging techniques are X- ray CT [2, 2–

4, 7–9, 18, 37, 53], MRI [12–16, 25, 25, 54–56], positron emission tomog-

raphy (PET) [21–24, 37, 53], single photon emission computed tomography

(SPECT) [22, 24, 25, 37, 53], ultra sonic (US) [26–29, 37, 53], etc. Although

these techniques were developed based on different physica l phenomena and

principles, a ccording to the nature o f source-medium interaction, they can

be classified into a category o f 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 index es 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 meas urements are

modeled with the wave equatio n and the tomog raphic reconstruction approach

is based on the Fourier diffra ction theorem [30, 37]. The former inc ludes X-ray

CT, MRI, emission CT, ultrasonic CT (e.g., refractive index CT and attenu-

∗

Its counterpart is called the reflection computed tomography, which is outside the scope

of this book.

107

108 Statistics of Medical Imaging

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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, c ertain 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 fr om a specific standpoint, a nd

show that they belong to a category of imaging—the non-diffraction computed

tomography. This insight may explain why X-r ay CT and MRI have very

similar statistical properties that are described in the remaining chapters of

this book.

4.2 Interaction between EM Wave and Object

When EM waves impinge on an o bject, due to the interaction between the

wave and the object, the following physical phenomena occur at the o bject,

which are shown in Figure 4.1. All these physical phenomena can be used for

imaging the shape of the object, its external surfa ce, and internal s tructure,

and have practical a pplications:

1. Absorption

2. Diffraction

Non-Diffraction Computed Tomography 109

3. Non-diffraction

4. Reflection

5. Refraction

6. Scattering

For the given E M wave and the fixed points on the s urface of the o bject, the

diffracted, reflected, and refracted waves are in specified directions, while the

scattered waves a re in all directions. Thus, when the detectors (or r eceivers)

are randomly placed around the object, the diffracted, reflected and refracted

waves are received with probability zero , and the scattered waves are received

with probability one. The invers e scattering pro ble m in the imaging ar ises

from scattered EM waves.

4.3 Inverse Scattering Problem

4.3.1 Relationship between Incident and Scattered Waves

The inverse scattering problem may be the simplest imaging principle. We

use it as an example. In Figure 4.2, S and V denote the surface and volume

of the object,

~

I is the unit vector in the direction of the incident wave, ~r is

the vector from the origin of the coordinate system to a point on the surface

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) wavefront

object

FIGURE 4.2

Incident and scattered waves.