Statistics of Medical Imaging

Performance Evaluation of Image Analysis Methods 383

TABLE 12.6

Detection Results.

K 2 3 4 5 6 K

1

P

ov

(+1)

P

ud

(−1)

MDL(K) 3452 3355 3279 3116 3246 5 ≃ 0 5.1 × 10

−1

Images in Figures 12.4a (K

0

=

5).

correct: K

1

= Arg{min

1<K<7

I

K

} = 5 is equal to the correc t number of image

regions K

0

= 5. Probabilities of over- and under-detectio n of the number of

image regions ar e P

ov

(+1) ≃ 0 and P

ud

(−1) ≃ 51%, which are similar to

P

ov

(+1) ≃ 0 and P

ud

(−1) ≃ 62% of the image in Figure 12.1h (Table 12.3),

where SNR = 12.4 db (Table 12.2).

12.2.2 Estimat ion Performance

This subsection describes the estimation performance of the iFNM model-

based image analysis method. It first shows that both the EM and CM algo-

rithms produce asymptotically unbiased ML estimates o f model parameters in

the case of no-overlap, then derives Cramer-Rao bounds of variances of these

estimates, and fina lly shows some results.

12.2.2.1 Asymptotically Unbiased ML Estimates of FNM Model

Parameters

1) ML Estimates of iFNM Model Parameters

Let IMG(J, K) denote an image of J pixels (x

j

, j = 1, ···, J) and K

image regions (R

k

, k = 1, ···, K). The iFNM model-based image analysis

metho d (Chapter 10) uses the EM algorithm to estimate the model param-

eters. Eqs. (10.11)-(10.13 ) are the EM solution de rived by maximizing the

expectation of iFNM likelihood function (Appendix 10C of Chapter 10). By

maximizing the iFNM likelihood function its elf, the ML estimates of iFNM

model parameters can be obtained. The joint likelihood of J independent

pixel intensities x

j

(j = 1, ···, J) is given by Eq. (12.2). The ML estima te

ˆ

r = {(ˆπ

k

, ˆµ

k

, ˆσ

2

k

), k = 1, ···, K} of iFNM model parameters of Eq. (10.4) can

be obtained by

maximizing L = ln L(

ˆ

r) + λ(

K

X

k=1

π

k

− 1) (12.26)

over the parameter set {(π

k

, µ

k

, σ

2

k

), k = 1, ···, K} and a Lagrangian multi-

plier λ.