Method of determining a modulation transfer function of a digital imaging system

The modulation transfer function of a digital imaging system is measured using a test object comprising a number of slits. The effect of geometrical distortions on the MTF measurement, is minimized by making the dimension of the test object as small as possible with respect to the image detector. The minimum slit spacing d enabling suitable MTF measurement is expressed in the number of pixels N over a dimension L of the image detector: d>8L/N.

The invention relates to a method of determining a modulation transfer 
function of a digital imaging system, comprising an image detector, where 
an image to be formed is composed of a number of N discrete pixels, a 
radiation intensity distribution spatially line-wise modulated by a test 
object being presented to an image detector entrance the imaging system 
into electric signals which are a measure of the radiation intensity 
distribution of the pixels, an arithmetic device determining from said 
signals a spectrum wherefrom the modulation transfer function is derived. 
A method of this kind is known from an article by R. A. Sones, G. T. 
Barnes, "A method to measure the MTF of digital X-ray systems"; Med. Phys. 
11(2), March/April, 1984. 
This article describes how the modulation transfer function, referred to 
hereinafter as MTF for the sake of brevity, is determined in a digital 
imaging system such as a radiography or fluorescopy system. The MTF of a 
system is an objective measure of the imaging quality of an imaging 
system. In the absence of geometrical distortion, an imaging system will 
reproduce a sinusoidal intensity distribution on a detector entrance face 
as a sinusoidal intensity distribution whose contrast has been reduced and 
whose phase has been shifted with respect to the original intensity 
distribution. By dividing the ratio of the contrast in the displayed 
intensity distribution by the contrast of the intensity distribution on 
the entrance face of the detector, the MTF of the detector can be measured 
for different spatial frequencies. For a spatial frequency zero, the MTF 
amounts to 1 and decreases to Zero as the frequency increases. 
A more effective method of measuring the MTF of an imaging system is based 
on the fact that the MTF can be described as the modulus of the 
one-dimensional Fourier-transform of the line spread function of the 
imaging system. The line spread function describes the image of a line 
displayed by the detector on the detector entrance face. For example in 
digital detectors whose detector entrance face is, for example subdivided 
into a matrix of separate detection sub-faces a problem is encountered in 
the determination of the MTF from the Fourier transform of the line spread 
function in that aliasing occurs because the sampling frequency is too 
low. Aliasing is the occurrence of components of the spectrum of the image 
displayed at frequencies which are lower than the frequencies with which 
the spectrum components are actually associated. This is because the 
spectrum of a sampled signal in the frequency domain is a periodic version 
of the actual spectrum. When these periodic spectra overlap, 
reconstruction of the original spectrum is not possible. By constructing 
the test object as a number of parallel slits, the problem imposed by 
aliasing is mitigated and the MTF can be determined by Fourier 
transformation of the detector signals which are proportional to the image 
to be displayed. 
The accuracy of the MTF measurements is limited by the occurrence of 
geometrical distortions of the image detector, for example cushion-shaped 
distortion. In order to minimize the effect of the distortions on the MTF 
measurements, it is necessary to use a test object which is as small as 
possible with respect to the detector entrance face. It is a drawback of 
the known test object that it is comparatively large and that effects of 
the distortion influence the accuracy of the MTF measurements. 
It is the object of the invention to provide a method where the effect of 
the geometrical distortion on the accuracy of the MTF measurement is 
comparatively small. 
To achieve this, a method in accordance with the invention is characterized 
in that a radiation-absorbing plate provided with substantially parallel 
equidistant slits is used as the test object, the slit spacing being at 
least 8 L/N. 
By minimizing the slit spacing, the dimension of the test object can be 
reduced, but the distance in frequency of peaks in the spectrum increases. 
The optimum slit spacing is a compromise between these two effects. 
Moreover, the peak width of the peaks occurring in the spectrum implies 
that the number of slits must be large enough for discrimination of the 
individual peaks. Calculations have shown that accurate MTF measurements 
can be performed in an X-ray imaging system when use is made of a test 
object having a number of slits equal to 5 and a slit spacing amounting 
to 5 mm. 
A version of a method of determining a modulation transfer function in 
accordance with the invention is characterized in that :rom an image of 
the test object formed by the image detector a geometrical distortion 
caused by the image detector is determined after which a 
distortion-dependent reliability value is derived for the measured 
modulation transfer function, which reliability value is found by 
calculating the transmission of a test object with a varying slit spacing. 
Because the geometrical distortion of the imaging system can be measured 
with the aid of the test object, the accuracy of the MTF measurement can 
be determined for the given distortion. 
A further version of a method of determining a modulation transfer function 
in accordance with the invention is characterized in that the envelope is 
determined by interconnecting discrete points by way of a Gaussian curve. 
The Gaussian curve provides an accurate mathematical description of the MTF 
.

FIG. 1 shows an X-ray source 1 which irradiates a test object 5 by means of 
an X-ray beam 3. The entrance screen 7 of an X-ray image intensifier tube 
9 receives a radiation intensity distribution which has been spatially 
line-wise modulated by the test object 5. The radiation intensity 
distribution is displayed on a television monitor 15 by means of a 
television camera tube 11, via an arithmetic device 13. 
FIG. 2a shows the spectrum of a test object consisting of line-shaped 
slits. An amplitude A is plotted as a function of the frequency f. FIG. 2b 
shows the MTF of the imaging system, for example a combination of an X-ray 
image intensifier tube and a television camera tube. When the slit-shaped 
test object is displayed by the imaging system, in the frequency 
domain/the spectrum is multiplied by the MTF of the imaging system. The 
sampling operation is represented in the frequency domain by the 
convolution of the pulse train modulated by the MTF with a sampling pulse 
train as shown in FIG. 2c. As a result of this convolution, the pulse 
train modulated with the MTF becomes periodic with the sampling frequency 
f.sub.s. Even when the condition: f.sub.s .gtoreq.2f.sub.m, where f.sub.m 
is the maximum frequency in the MTF, is not satisfied, the MTF can still 
be reconstructed from the correct values of the envelope of the spectrum 
as shown in FIG. 2d. 
The MTF is the discrete Fourier transform of the line spread function 
LSF(x) sampled by the discrete detector with a period .DELTA.x: 
EQU MTF(f)=DFT{LSF(x).comb(k.DELTA.x)} (1) 
Therein, comb(k.DELTA.x) is the pulse train having the period .DELTA.x, k 
is a natural number, and the operator DFT denotes the discrete Fourier 
transform. The spatial dimensions of the test object are represented as 
LSF(x) convoluted with a pulse train comb(x/d) which introduces spatial 
periodicity d, and multiplied by a rectangular function rect(x/c) Which 
introduces the finite dimensions c of the object (see FIG. 3). The 
intensity distribution I(x) measured by the imaging system is given by: 
EQU I(x)=[LSF(x)*comb(x/d)].rect(x/c).comb(k.DELTA.x) (2) 
Therein, the symbol * denotes the convolution operation. It follows 
therefrom that: 
EQU DFT{I(x)}=[MTF(f).comb(fd)]*sinc(fc).comb(f.sub.s) (3) 
Therein, sinc(x)=sin(x}/x and f.sub.s is the sampling frequency. 
It appears from the formula (3) that the MTF is the envelope of a number of 
successive sinc functions (comb(fd),sinc(fc)) which is repeated with the 
sampling frequency f.sub.s. When the slit spacing d is minimized, the 
dimension of the test object can remain small. As appears from the formula 
(3) and FIG. 4, the peaks of the MTF are spaced increasingly further apart 
in the case of a small slit spacing d. An optimum slit spacing d consists 
of a compromise between the dimension of the test object and the latter 
effect. It also follows from the formula (3) and FIG. 4 that the dimension 
c of the test object determines the width of the peaks in the spectrum and 
must be large enough so as to produce adequate separation of the 
individual peaks. 
The dimension of the pixels Ax in the entrance plane of an image detector 
can be described as .DELTA.x=L/N where L is a dimension of the image 
detector entrance face and N is the number of discrete pixels over 
dimension L. It follows therefrom that the number of pixels d.sub.n 
between two slits amounts to d.N/L. The number of pixels between two peaks 
in the spectrum is N/d.sub.n =L/D. The number of pixels in the frequency 
domain below half the sampling frequency amounts to N/2. The number of 
peaks n.sub.p then amounts to n.sub.p =N.d/2 L. For accurate determination 
of the MTF a minimum number of four peaks is required. It follows 
therefrom that Nd/2 L&gt;4 or d&gt;8 L/N. it follows from the state of the art 
that d&lt;c/2 must be satisfied in order to obtain a suitable resolution of 
the peaks in the spectrum, so that: 
EQU 8L/N&lt;d&lt;c/2. 
If this condition is satisfied, a suitable resolution and a minimum number 
of peaks are ensured. 
The effect of geometrical distortions can be determined by measuring the 
magnitude of the peaks of the MTF for a varying line spacing in the test 
object. FIG. 5 shows the calculated magnitude of the MTF value for a 
frequency equal to one quarter of the sampling frequency with respect to 
the MTF value for a spatial frequency 0 as a function of the geometrical 
distortion. Calculations were performed on a test object comprising 5 
slits whose spacing linearly increases between the successive slits. 
E=(dev/d.sub.av)*100% is given as a measure of the relative distortion. 
Therein, d.sub.av is the mean slit spacing and dev is the maximum 
deviation from d.sub.av. It appears from FIG. 5 that the variation of the 
MTF value is less than 1% for a distortion amounting to 6%, this is an 
acceptable result in most cases. Using data as shown in FIG. 5, an 
estimation of the accuracy of the MTF measurement can be given for a given 
geometrical distortion.