Statistics of Medical Imaging

Performance Evaluation of Image Analysis Methods 397

for images with moderate quality, the pro bability of over-detection of the

number of image re gions is negligible, and the probability of under-detection

of the numbe r of image regions is very small. These results are similar to those

in the iFNM model-ba sed image analysis method.

2) For e stimation performance, the cFNM model-based image analysis

metho d used the EM algorithm (with Gibbs distribution or MRF as a pri-

ori). Its performance is essentially the same as that of the iFNM model-based

image analysis method given in Section 12.2.2.

3) For classification performance, the cFNM model-based image analysis

metho d uses the MAP criterion. A strategy has been developed to assess the

validity of using MAP with the underlying imaging principles. That is, the

results obtained using MAP are judged if they are toward the physical ground

truth that is to be imaged. To elaborate this strategy, MR images are used as

an example.

In statistical physics, let Ω = {ω} denote the possible configurations of a

nuclear spin system. If the system is in thermal equilibrium with its surround-

ings, then the probability, or Boltzmann’s factor, of ω is given by

p(ω) =

1

P

ω

exp(−γε(ω))

exp(−γε(ω)), (12.52)

where ε(ω) is the energy function of ω and γ =

1

κT

, κ is the Boltzmann’s

constant, and T is the absolute temperature [53–56]. Eq. (12.52) indicates

that the nuclear spin system tends to be in the low energy state, or with the

higher probability to be in the low energy state.

In MR imaging, pixel intensities represent thermal equilibrium macroscopic

magnetization (TEMM), that is, the vector sum o f nuclear spin moments

in unit volume. Chapter 9 shows that pixel intensities of MR images are

characterized by an MRF X or a corr esponding Gibbs dis tribution P (x).

P (x) =

1

P

x

exp(−β

−1

P

c

V

c

(x))

exp(−β

−1

X

c

V

c

(x)), (12.53)

which, in the functional fo rm, is identical to the Boltzmann’s factor

Eq. (12.52).

Eq. (12.52) and Eq. (12.53) characterize the s ame nuclear spin system in the

microscopic and the macroscopic sta tes, respectively. Thus, given an image x,

maximizing the posterior probability P (y|x) = P (x|y)P (y), that is, seeking

a configuration

ˆ

y for the underlying MRF or for the corresponding Gibbs

distribution, is actually to seek a thermal equilibrium of the nuclear spin

system. This thermal equilibrium is well defined and is the physical ground

truth to be ima ged.

Given the interplay between the microscopic and macroscopic states of a

process and of the ana lytical modeling of the propagations from one to the

other, we can account for the geo metric locality of the pixels/spins as well as

for their probabilistic dynamics suited to the underlying struc ture of an image

and its perturbations.