composed of the pixel numbers per grey interval or
the frequency distritution according to statisti-
cal method after digitization, Thus we should use
the entropy to estimate the actual information
content Ha from the following equation:
$
He - MN(-2. P;1og, P) (bits) (3)
i!
where P; is a frequency of the grey level i in
the histogram.
In Information theory one explains the information
as an extent of eliminating uncertainty in his
mind after acceptance of a message. When all S
kinds of situation for a pixel happen with the
same opportunity it contains the maximum uncer-
tainty. Therefore, the clearance of that uncer-
tainty gives a maximal amount of information con-
tent as well.
Making a comparison between the actual entropy Ha
and the maximum Hmax we can derive the relative en-
tropy Hr and the redundancy R:
Hmax
R s: Hp. (5)
3. APPLICATION OF SAMPLING THEOREM
Each row of digital image virtually is a sample of
a continuous strip with the variable tone on an
aerial photograph. In the scanning digitizer it is
implemented to sample a frame of image in accor-
dance with the eaual interval, in other words the
digitizer records the signals from a continuous
imagery in the constant cycle length t, This pro-
cess means to convert a continual mersage f(t) into
the time sampling ones f (nat) in which 4t becomes
the sampling interval and n = 1,2,....
It is learned from Information theory that ramp-
ling à continual sirnal in the time 4cmain, g(t),
must obey the sampling theorem, This theorem indi-
cates if the freauency spectrum of 8 continual sig-
nal g(t) possesses a limited band width of W, there
will exicta the following relation:
OX k , sin T(2Wt-k)
OS EG Out e
Based on the theorem it is able to know the inten-
rity value at any moment for the signal with a
bandwidth W provided that there are given the va-
lues at t=k/2W. (k is an integer.) As to image
digitization it is necessary to observe the samp-
ling theorem during recording pixels in order to
recover all the changeable grey values between
scanned points of the original imagery afterwards.
Theoretically, the sampling cycle should be At =
1/2W, but one often makes At € 1/2W in practice.
When the sampling duration T is known the number
of sample point should not be less than 2WT. Hav-
ing taken the interval between the neighbouring
pixels,4X, corresponding to At into account we are
in a position to realize a complete recovery of
the original imagery from the sampled analog data
provided that the selected sampling interval en-
ables the highest frequency of the sequential im-
ages to have two pixel records at one cycle, or to
sample a frame of image in terms of a half of the
shortest cycle. This is an important theoretical
basis on which the resampling or interpolation of
data relies.
In Information theory sis x 1s called the samr=
ling function which corresronds to the output of
an ideal low-pass filter with the cut-off frequen-
Cy W while a Dirac function acts on that filter (
See Figure 5b). Figure 3a shows the sampling va-
jues of g(t) at t-k/2W (Ys0,41,42,...), that is
£(k/2W), and two curves of the sampling function
for k=0 and k=5 respectively. It is clear from the
graph that the curve of g(t) is just made of a su-
rerposition of the individual nampling functions,
In Figure 3c there is an illustration of recover-
ing the original signal from the sampled data.
During airphoto disitization the interval between
the adjacent pixels should not be far less than
the amount calculated by the sampling theorem,
Sin2rwt
900) SVT —y
\ 1
Mo
eh
T Jem hm Si EN E dal
- -2— 4 0 == NT 5
RRC Mk AA a A
(e)The continuous signal 9(t) and its
individual sampling functions.
IE AC ill ee = TM
Q) The function d(t) and its output
after passing through a (ow-pass filter.
15 Si QWt-5)*
269 (Qwt-5)"
Sampling cycle
tc) The sampling signal and its output
after passing through a. (ow-pass filter.
Fig. 3. The illustration of sampling theorem on time domain,
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B2. Vienna 1996
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