Statistics of Medical Imaging

378 Statistics of Medical Imaging

a b c

d e f

g h i

j k l

FIGURE 12.1

Twelve simulated images with different variances.

detection performance for images with moderate and high SNR. Probabilities

of over- and under-detection of the number of image regions, P

ov

and P

ud

(by

±1), are also summarized in Table 12.3, which shows that for images with

SNR > 14.2 db, both P

ov

(+1) and P

ud

(−1) are almost zero.

Table 12.3 also shows that P

ov

decreases and P

ud

increases as σ

2

0

increases.

This result can be conceptually explained as follows. When σ

2

0

increases, the

intervals (x

′

k

, x

′′

k

) and (x

′

l

, x

′′

l

), which ar e intensity ranges of pixels in the k-

th and l-th image regions, may par tially overlap. This overlapping will cause

some image regions to merge into one region. As a result, the over-detection

probability will decrease and the under-detection probability will increase.

This obser vation can also be quantitatively illus trated as follows. For the

images s hown in Figures 12.1c, 12.1f, and 12.1i, their pdfs, h(z) o f Eq. (12.16),

are shown in Figures 12.2 and 12.3, respectively, where J = 4096, K

0

= 4,

K

1

= 5 and 3 (over and under), σ

2

0

= 10, 40, 70. These curves can be roughly

approximated by Gaussian pdfs, that is, Z ∼ N (µ, σ

2

), where the mean µ and

the standard deviation σ are determined by the location of the peak of h(z)

and

1

√

2πh(µ)

, respectively. Let Z

′

=

Z−µ

σ

; Z

′

w

ill have a standard Gaussian

distribution, that is, Z

′

∼ N(0, 1). For Z = ∆, the corresponding value for