ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002
explicitely h(z,y) — f f H(u, v)ei?r(vwtvvdudu or
x)= ] Bene? au
using the definition ofthe Fourier transform of (Castleman,
1979).
In case we have a sinus-type pattern s(x) = asin(2xux) =
a sin(27 $) the response of the system is a sine-wave with
contrast à = H(u)a. As the MTF usually falls off for large
frequencies, contrast of tiny details is diminished heavily.
In our special context we obtain the MTF for the Gaussian
shaped PSF
Gx (x) 0-6 e 20u' Xu
which again is a Gaussian, however, with the matrix P —
D /47? as parameter. Observe that we have
1
P-R 41202 g R!.
0 4 2
2
Eo,
2.3 Contrast Sensitivity Function
In order to evaluate the usefulness of the imaging system
with a certain PSF or MTF the so called contrast sensitiv-
ity function (CSF) is used. The contrast sensitivity func-
tion gives the minimum contrast at a periodic edge pattern
which can be perceived by a human. In our case we want
to apply this notion to edge detectors.
Assume we have a periodic pattern of edges characterized
by the wavelength A and the contrast c. Further assume the
image to be sampled with a pixel size of Az and the noise
has standard deviation o4. An ideal edge detector would
adapt to the wavelength of the pattern and perform an op-
timal test whether an edge exists or not. For simplicity we
assume that the pattern is parallel to one of the two coordi-
nate systems and that the edge detector uses the maximum
possible square of size À x A. The difference Ag between
the means / and ji? of the two neighboured areas can be
determined from the N/2 = (A/Az)?/2 pixels in the two
areas. It has standard deviation
/ / 2 2Ax
CAg = oi, + Ohio = V20y = v2. Won = TA 9n
Thus in case we perform the test with a significance num-
ber a and require a minimum probability o for detecting
the edge we can detect edges with a minimum height
2Ax
Aog = do(a, Bo)oag = do(@, Bo) ——0n-
The factor &o(a, Bo) depends on the significance level of
the test and the required probability of detecting an edge.
It is reasonable to fix it; in case we choose a small sig-
nificance number a = 0.001 and a minimum detectability
Bo = 0.8 we have & = 4.17 ~ 4. The minimum de-
tectable contrast in a reasonable manner depends on the
size of the window and the noise level: The larger the
noise standard deviation and the smaller the window the
larger the contrast of the edge needs to be in order to be
detectable.
As we finally want to relate the contrast sensitivity to the
frequency u — 1/A and obtain the contrast sensitivity func-
tion
CSF(u) - Aog(u) — 269 Ar uo;
It goes linear with the frequency, indicating higher fre-
quency edge patterns require higher contrast.
2.4 Resolving power
The resolving power RP usually is defined as that fre-
quency u where the contrast is too small due to the prop-
erties of the imaging system to be detectable. As periodic
patterns with small wave length will loose contrast heavily
they may not be perceivable any more.
The MTF has maximum value 1 and measures the ratio
in contrast MTF(u) = a(u)/a(u), whereas the CSF mea-
sures the minimum contrast being detectable. In order to
be able to compare the MTF with the CSF we need to nor-
malize the CSF. This easily can be done in case we intro-
duce the signal to noise ratio
SNR. = =
On
with k being the contrast. Then the relative contrast sensi-
tivity function reads as
ICSE(u) Ere 200 a UOn _ RAT
which immediately can be compared with the MTF.
One usually argues, that the resolving power is the fre-
quency where the relative contrast, measured by the MTF,
is identical to the minimum relative contrast being detecta-
ble (cf. fig. 3). Thus the resolving power RP=u is implic-
itly given by
MTF (ug) = rCSF (uo).
usable image contrast
MTF
CSF
I
I
I
|
Ug
resolving power
Figure 3: Relations between the modulation transfer func-
tion (MTF), the contrast sensitivity function (CSF) and the
resolving power (RP).
In the 1-dimensional case we can explicitely give ug
1 TT? 0?
= wea fLambertW a. =5 res SNR? J.
Ee (5 Ax? )
A - 207
