Statistics of Medical Imaging

7

Statistics of MR Imaging

7.1 Introduction

As shown in Chapter 3, MR images are reconstructed from k-space samples,

while k-space samples are formulated from analog-to-digital co nversion (ADC)

signals by applying adequate pulse s equences. An ADC s ignal is a discrete

version of a phase sensitive detection (PSD) signal (sampled at the proper

frequency), while a PSD signal is formed fro m a free induction decay (FID)

signal via quadrature P SD. FID signals are induced by transverse prec essing

macroscopic magnetization (TPMM), while TPMM originates from thermal

equilibrium macroscopic magnetization (TEMM).

Thus, in MR imaging, the term MR data means the macroscopic magnetiza-

tions (TEMM, TPMM), MR signals (FID, PSD, ADC), and k-space samples.

Among them, TEMM is spatially distributed, T PMM varies with both time

and the spatial location, FID, P SD and ADC are the temporal sig nals, a nd

k-space samples are in the spatial frequency domain. TEMM and FID are real

data; TPMM, PSD, ADC, and k-space samples a re complex data . Similar to

any type of realistic data, each type o f MR data consists of its signal and noise

components.

In some statistics studies on MR imaging, the randomness of the signal

components of MR data is ignored. Therefore, the ra ndomness of MR data

is mainly derived from their no ise components, particularly from the thermal

noise [1–3, 5–7, 9–11, 13, 14, 16, 17, 23, 27, 53–56]. MR noise studies are o ften

associated with the signal-to-noise ratio (SNR) evaluation. This is because

the SNR can provide an absolute scale for assessing imaging system perfor-

mance and lead to instrumentation design goals and constraints for system

optimization; also, it is one of the fundamental measures of image quality.

This chapter describes statistics of both signal and noise components of each

type of MR data, and focuses on their second-order statistics. Based on the

physical principles of MRI de scribed in Chapter 3 and according to MR data

acquisition procedures, the statistical description of MR imaging progresses in

the following natural and logical order: macroscopic magnetizations (TEMM

→ TPMM) =⇒ MR signals (FID → PSD → ADC) =⇒ k-spa ce samples.

When MR data travel in the space–time–(temporal and spatial)–frequency

domains, their statistics are evolving step by step.

167

168 Statistics of Medical Imaging

For the typical MR data acquisition protocols (the rectilinear and the r a-

dial k-spa ce sampling), this chapter pr ovides signal processing par adigms for

the basic image reconstr uc tion methods (Fourier transform (FT) and projec-

tion reconstruction (PR)). Then it gives a statistical interpretation o f MR

image reconstruction. That is, MR image reconstruction can be viewed as a

transform fr om a set of random variables (k-space samples ) to another set

of random variables (pixel intensities). These new random variables form a

random process, also known as a random field. Statistics of MR data in the

imaging domain propagate to the s tatistics in the image domain through im-

age reconstruction.

7.2 Statistics of Macroscopic Magnetizations

Several studies on magnetization are at the microscopic scale and have es-

tablished theoretical models for (a) the classical response of a single spin to

a magnetic field and (b) the correlation of two individua l spins in 1-D Ising

model [20, 27 , 55]. Macroscopic magnetization represents a vector sum of all

microscopic magnetic moments of spins in a unit volume of sample. As shown

in Chapter 3, because signal components of k-space samples (Eq. (3.110)

and MR signals (Eqs. (3.96), (3.101), (3.102)) represent the colle ctive be-

havior of a spin system, medical applications utilizing magnetic resonance for

imaging objects (tissues or organs) are based on macroscopic magnetization,

which is often calle d the bulk magnetization. Statistics of two types of bulk

magnetizations—TEMM and TPMM—are analyzed in this section.

7.2.1 Statistics of Therma l Equilibrium Magnetization

As shown in Section 3 .4.2, when a sample is placed in an external, static mag-

netic field

~

B

0

= B

0

~

k (wher e

~

k is the unit directional vector at the Z direction

of a Cartesian coordinate sy stem {U, V, Z}),

∗

the magnitude of TEMM of the

spin-

1

2

systems such as

1

H,

13

C,

19

F , and

31

P , etc., is given by

M

o

z

(r) =

1

2

(n

l

− n

h

)γ~ =

1

2

γ~ǫn, (7.1)

where r = (u, v, z) denotes a location, n

l

and n

h

are the numbers of spins

at the lower and higher energy states, n = n

l

+ n

h

, γ is the g yromagnetic

∗

Instead of {X, Y, Z} used in the previous chapters, starting from this chapter, {U, V, Z}

denotes a C artesian coordinate system.

~

i,

~

j ,and

~

k are the unit directional vectors at the

U, V , and Z directions.

~

i and

~

j define the transverse plane and

~

k specifies the longitudinal

direction.