92
3: Hua-mei Chen
a
b
c
Fig. 3.2a-c. Joint histograms of a pair of Landsat TM band 3 and band 7 images. a Images
are completely registered. b Images are shifted by one pixel vertically. c Images are shifted
by three pixels vertically
joint entropy. Figure 3.2a-c show the joint histograms of a pair of Landsat TM
band 3 and band 7 images. Figure 3.2a is obtained when the two images are
registered. Figure 3.2b is obtained when the first image is shifted vertically
by one pixel. Figure 3.2c shows the joint histogram when the first image is
shifted vertically by three pixels. From Fig. 3.2, we can observe that when
two images are registered, the joint histogram is very sharp whereas it gets
more dispersed as the vertical displacement is increased. Therefore, it seems
reasonable to devise a similarity measure that meaningfully represents the
sharpness or degree of dispersion of the joint histogram. Joint entropy is one
such metric that measures the uncertainty of the joint histogram. Thus, under
the assumption of constant entropies H{F) and H{R), (3.2) may be used to
interpret the mutual information criterion in that it tries to minimize the
uncertainty of the joint histogram of the two images to be registered.
When the assumption of constant entropies H{F) and H{R) is not satisfied,
we may employ the notion of conditional entropy to interpret the MI criterion.
We rewrite MI in terms of conditional entropy as,
I{F, R) = H{F) - H{FIR)
= H{R) - H{RIF) ,
where the conditional entropies H{FIR)and H{RIF) are defined as:
H{FIR) = - L Pp,R (f, r) 10gPFI R (fIr) ,
f,r
H{RIF) = - L Pp,R (f, r) 10gPRIF (rlf) .
f,r
(3.10)
(3.11)
(3.12)
Conditional entropy H{FIR) represents the entropy (uncertainty) of F when R
is known. Similarly, H{RIF) is the entropy (uncertainty) of R given F. Therefore, mutual information can be realized as the reduction in the uncertainty
3: Hua-mei Chen
a
b
c
Fig. 3.2a-c. Joint histograms of a pair of Landsat TM band 3 and band 7 images. a Images
are completely registered. b Images are shifted by one pixel vertically. c Images are shifted
by three pixels vertically
joint entropy. Figure 3.2a-c show the joint histograms of a pair of Landsat TM
band 3 and band 7 images. Figure 3.2a is obtained when the two images are
registered. Figure 3.2b is obtained when the first image is shifted vertically
by one pixel. Figure 3.2c shows the joint histogram when the first image is
shifted vertically by three pixels. From Fig. 3.2, we can observe that when
two images are registered, the joint histogram is very sharp whereas it gets
more dispersed as the vertical displacement is increased. Therefore, it seems
reasonable to devise a similarity measure that meaningfully represents the
sharpness or degree of dispersion of the joint histogram. Joint entropy is one
such metric that measures the uncertainty of the joint histogram. Thus, under
the assumption of constant entropies H{F) and H{R), (3.2) may be used to
interpret the mutual information criterion in that it tries to minimize the
uncertainty of the joint histogram of the two images to be registered.
When the assumption of constant entropies H{F) and H{R) is not satisfied,
we may employ the notion of conditional entropy to interpret the MI criterion.
We rewrite MI in terms of conditional entropy as,
I{F, R) = H{F) - H{FIR)
= H{R) - H{RIF) ,
where the conditional entropies H{FIR)and H{RIF) are defined as:
H{FIR) = - L Pp,R (f, r) 10gPFI R (fIr) ,
f,r
H{RIF) = - L Pp,R (f, r) 10gPRIF (rlf) .
f,r
(3.10)
(3.11)
(3.12)
Conditional entropy H{FIR) represents the entropy (uncertainty) of F when R
is known. Similarly, H{RIF) is the entropy (uncertainty) of R given F. Therefore, mutual information can be realized as the reduction in the uncertainty