Statistics of Medical Imaging

5

Statistics of X-Ray CT Imaging

5.1 Introduction

As shown in Chapter 2, X-ray CT images are reconstructed from projections ,

while the projections are formulated from photon measurements. In X-ray CT

imaging, photon measurements and projections are called CT imaging data,

abbreviated as CT data.

∗

Similar to any type of realistic data, ea ch type of CT data consists of its

signal and noise components. In photon measurements, the instrumental and

environmental noise form the noise component of photon measurements, which

is rando m. The emitted and detected photons (numbers) form the signal com-

ponent of the photon measurements, which is also random, due to the intrinsic

variations in the emission and detec tion processes.

The parallel projection is a translation-rotation mode (Sections 2.3 a nd 2.4).

Within one vie w , the photon measur ements and the projections are ac quired

and formulated sequentially in time from one projection location to another.

The divergent projection is a rotation mode (Sections 2.3 and 2.4). Within

one view, the photon measurements and the projections ar e acquired and

formulated simultaneously for all projections. Over all views , both parallel and

divergent projectio ns are collected sequentially in time from one view location

to another. Thus, projections are spatio-temporal in nature. Because the time

interval of CT data collection, particularly in the dievergent projection, is

very short, the time argument in CT data is excluded in the process of image

reconstruction.

This chapter describes the statistics of both signal a nd noise components of

each type of CT da ta, and is focused on their second-order statistics. Based

on the physical principles of X- ray CT des cribed in Chapter 2 and according

to CT data ac quisition procedures , the statistical description of X-ray CT

imaging is progress es in the fo llowing order : photon measurement (emission

→ detection → emission and detection) =⇒ projection.

This chapter also provides signal proc essing paradigms for the convolution

image reconstruction method for the parallel and divergent projections. Then,

∗

Imaging i s often referred to as a process or an operation from the imaging data (e.g., the

acquired m easurements) to the reconstructed pi ctures.

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126 Statistics of Medical Imaging

it gives a statistical interpretation to CT image reconstruction. C T image

reconstruction can be viewed as a transform from a set of ra ndom variables

(projections) to another set of random variables (pixel intensities). These ne w

random variables form a spatial random process, also known as a random

field. Statistics of CT data in the imaging domain propagate to the statistics

in the image domain through image reconstruction.

Discussions in this chapter are confined to the monochromatic X-ray, the

basic parallel and divergent projections, and the convolution image recon-

struction method [2, 2–4, 7–9, 18, 37, 53].

5.2 Statistics of Photon Measurements

This section describes statistics of the photon measurements in terms of their

signal and noise components.

5.2.1 Statistics of Signal Component

Physical principles of photon emission, attenuation, and detection are de-

scribed in Sec tion 2.2; so me parameters in these processes are given in Section

2.3. In this section, statistics of the signal component of photon measurements

are analy zed in the following order: photo n emission → photon de tec tion →

photon emission and detection, based on their physical principles.

5.2.1.1 Photon Emission

In the process of photon emission, let t

i

and (t

i

, t

i+1

] (i = 0, 1, 2, ···) de note

the time instant and the time interval, respectively; n(t

i

, t

i+1

] repre sents the

number of photons emitted in the interval (t

i

, t

i+1

]. n(t

i

, t

i+1

] is a random

variable. P (n(t

i

, t

i+1

] = m) denotes the proba bility that m photons are emit-

ted in the time interval (t

i

, t

i+1

].

Generally, similar to other particle emissions in physics, photon emission

in X-ray C T is consider ed to follow the Poisson law. Specifically, the photon

emission in X-ray CT satisfies the following four conditions:

1) P (n(t

i

, t

i+1

] = m) depends on m a nd the interval τ

i

= t

i+1

− t

i

only; it

does not depend on the time instant t

i

.

2) For the nonoverlapping intervals (t

i

, t

i+1

], random variables n(t

i

, t

i+1

]

(i = 0, 1, 2, ···) are independent

3) There ar e only finite numbers of photons emitted in a finite interval,

P (n(t

i

, t

i+1

] = ∞) = 0. Also, P (n(t

i

, t

i+1

] = 0) 6= 1.

4) The probability that more than one pho ton is emitted in an interval