ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
(PSF) or its amplitude spectrum, the modulation transfer 
function (MTF). Together with the contrast sensitivity func- 
tion (CSF), giving the least detectable contrast at an edge 
as a function of the spatial frequency of intensity changes, 
one can derive the resolving power. It is the maximum fre- 
quency of a periodic signal which can be detected with a 
given certainty. 
Now, the precise determination of the PSF is quite involv- 
ing, and usually derived from the intensity transition at 
edges, yielding the cumulative distribution of the PSF, in- 
terpreted as probability density function. Moreover, the 
classical CSF refers to a human observer. 
This paper assumes the PSF to be a Gaussian function. 
We will introduce a simple procedure for measuring the 
main characteristics of the PSF, namely its width. We give 
a definition for the CSF based on an ideal edge detector 
for straight edges between noisy homogeneous regions. It 
therefore allows to fully automatically determine the re- 
solving power of such an ideal edge detector. Experiments 
with synthetic and real data demonstrate the usefulness of 
the proposed approach. 
2 THEORETICAL BASIS 
As we are interested in simplifying the characteristic mea- 
sures of image quality we summarize the basic relations. 
2.1 Point and edge spread function 
The quality of an imaging system may be evaluated using 
the un-sharpness or blur at edges. The edge spread func- 
tion of a 1-dimensional signal is the response 5(x) of the 
system to an ideal edge s(x) of height 1 (cf. the first row 
in fig. 2). 
The quality of an imaging system usually is described by 
the point spread function (cf. the second row of fig. 2), 
being the response h(x) of the system to a delta func- 
tion d(x). As the imaging system is assumed to be linear 
and the ideal edge s(x) is the integral of the ó-function, 
the point spread function is the first derivative of the edge 
spread function: s(z) = h'(x). Observe, we may in- 
terprete the point spread function as a probability density 
function and the corresponding edge response function as 
its cumulative distribution function resp. distribution func- 
tion. 
In two dimensions the situation is a bit more involving. 
If we differentiate the 1-dimensional cross section of the 
response 8(u) to an ideal two dimensional edge s(u) we 
obtain a bell shaped function. It is the marginal distribu- 
tion of the point spread function along the edge direction. 
Fusing a large number of such marginal distributions of 
the PSF can only be done in the Fourier domain using to- 
mographic reconstruction techniques (cf. (Rosenfeld and 
Kak, 1982)). 
The situation becomes much easier in case we can approx- 
imate the 2-dimensional PSF by a Gaussian. Then the edge 
A - 206 
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| Ideal Signal Blurring Blurred signal | 
4H = TT i ln i 
| Edge-spread blurred Edge ! 
$ i Ideal edge function | 
E = | | 
"i i 
= / 
S| — / i 
2. i 
2) i 
Permet ete in rear dr eb en a 
Point-spread | 
function I 
Delta—Function J 
function i 
  
Spatial domain 
fr me fm ome Sn te mn fr tn ven emp mm, me. ma mi comet) 
TT oc TR yt fe om mm mt ed 
domain 
  
| 
| 
| 
| 
el LL | 
Di | 
Si ——— mL | 
A | | 
gt MTF | 
m | 
| 
  
  
Lu 
MM — rn en ie ie ee ee a ar mt CM" MUI M cda san 
Figure 2: Edge spread function, point spread function and 
modulation transfer function. 
spread function, i. e. the response to an arbitrary edge is 
an integrated Gaussian function. 
In detail we assume 
aun EU. 
(2) 
  
where the matrix Xi can be written as 
2 
AP 
Here the two parameters v and o» represent the width of 
the PSF in two orthogonal directions and R is the corre- 
sponding rotation matrix. In case we have two edges on 
the principle directions & and 7 of the PSF we obtain the 
two edge response functions 
51€) = zer) 52(n) = Let (2) 
with the error function erf(x) = JS G1 (t)dt. 
z-a( 
We refer to the individual values c as /ocal scale as it cor- 
responds to the notion of scale in a multi-scale analysis of 
an image. The matrix X is called scale matrix. 
2.2 Modulation Transfer Function (MTF) 
It is convenient to describe the characteristics of the imag- 
ing system by its response to periodic patterns, leading to 
the modulation transfer function H (u,v). It is the ampli- 
tude spectrum of the point spread function, 
hz, y) oe H(u,v),