Statistics of Medical Imaging

10

Statistical Imag e Analysis – I

10.1 Introduction

In most cases, images generated by X-ray CT and MR are all-inc lusive. That

is, these two imaging modalities cannot directly produce the images of the

selected tissue types or organ sy stems. For example, when the human abdomen

is imaged by X-ray CT, the liver, kidney, stomach, pancre as, gallbladder,

adrenal glands, spleen, etc., are all shown in the resultant image ; when a cross

section of the human brain is imaged by MRI, the scalp, bone, gray matter ,

white matter, and cer ebrospinal fluid, etc., are all included in the resultant

image. In order to obtain an image of the selected targets of interest, an image

processing or image analysis method is often required.

Generally, imaging refers to an operation or a process from the data to the

picture. X-ray CT re presents an operation from photon measurements to a

2-D display of the spa tial distribution o f the r elative linear attenuation co-

efficient (RLAC); MRI represents a process from free induction decay (FID)

signal measurements to a 2-D display of the spatial distribution of the ther-

mal e quilibrium macroscopic magnetization (TEMM). Image processing refers

to an operation or a process from the picture to the picture. The commonly

used image processing approaches may include but are not limited to trans-

form, enhancement, and restoration. Im age analysis r efers to an operation or

a process from the picture to the “data.” Here, data may include some im-

age primitives such as edges or regions as well as some quantities and labels

related to these primitives. This chapter and the following chapters focus on

image analysis.

Various image analysis methods have b een developed, and some of them are

applied to X-ray CT and MR images. The graph a pproach [1, 3, 5], the classical

snakes and active contour appro aches [5, 6, 8, 9], the Level set methods [10–

13], and Active Shape model (ASM) a nd Active Appearance model (AAM)

approaches [] are edge-based approaches. Fuzzy connected object delineation

[15–18, 20–22] and Markov random field (MRF) [24, 26, 28, 30, 32 , 34, 36 ,

38] are the region-based approaches. This and the next chapter describe two

statistical image analysis methods for X-ray CT and MR images based on the

stochastic models I and II given in Chapter 9 , respectively.

In analyzing the so-called all-inclusive images as illustrated by the e xam-

299

300 Statistics of Medical Imaging

ples given in the beginning of this section, the first step is to determine how

many imag e reg ions are presented in the image. After this number is de-

tected, the second step is to estimate region parameters, for example, the

mean and variance, etc. After these parameters are estimated, the third step

is to cla ssify e ach pixel to the corresponding imag e regions. By implementing

these three steps, an all-inclusive image is partitioned into the separated im-

age regions; e ach of them represents a tissue type or an organ system. The

above detection-estimation-classification approach forms an unsupervised im-

age analysis technique; it is a model-based, data driven approach.

10.2 Detection of Number of Image R egions

Property 9.1 shows that an image whose pixel intensities ar e s tatistically inde-

pendent can be modeled by an independent Finite Normal Mixture (iFNM).

Let the image be denoted by IMG(J, K); J and K are the numbers of pixels

and image regions, respectively. iFNM pdf is given by

f(x) =

P

K

k=1

π

k

g(x|θ

k

), (10.1)

where x denotes the pixel intensity, θ

k

= (µ

k

, σ

2

k

) (k = 1, ···, K) is the

parameter vector of the k-th image region R

k

(µ

k

– the mean, σ

2

k

– the

variance), g(x|θ

k

) is a Gaussian pdf of pixel intensities in the k-th image

region given by

g(x|θ

k

) =

1

p

2πσ

2

k

exp(−

(x − µ

k

)

2

2σ

2

k

), (

10.2)

and π

k

(k = 1, ···, K) represents the probability of the occurrence of the kth

image region in the image and is characterized by a multinomial distribution

0 < π

k

< 1 and

P

K

k=1

π

k

= 1. (10.3)

The 3K-dimens ional model para meter vector of iFNM (Eq. (10.1)) is defined

as

r = (π

1

, µ

1

, σ

2

1

, ······, π

K

, µ

K

, σ

2

K

)

T

. (10.4)

In this section and Appendix, for the purpose of derivation, pdf f(x) of

Eq. (10.1) is also written as f (x|r) or f(x, r).

Based on Eq. (10.1), detecting the number o f image regions is a ctually the

selecting the o rder of iFNM model. Traditionally, this is implemented by two

types of approaches: (1) hypothesis test [40, 41], and (2) power spectrum

analysis [42, 43]. The hypothesis test is a general standard method. It first

establishes a null hypothesis H

0

: the or de r is K

0

and an alternative hypothesis

H

1

: the order is not K

0

, then creates test statistics and derives its distribution;