Physics and Engineering of Radiation Detection

436 Chapter 7. Position Sensitive Detection and Imaging

modulation transfer function orMTF. Fig.7.1.13 shows the shape of a typical
MTF.

0.5

1.0

0

0 5 10

M

od

ul

at

ion

T

rans

fer

R

at

io

Spatial Frequency (cycles/mm)

MTF

Figure 7.1.13: Typical dependence of

modulation transfer ratio on spatial

frequency. The parameterized function

is called the modulation transfer func-

tion.

SinceMTFcharacterizes how well the contrast in the object gets transferred
to the image, therefore it is inherently related to the various spread functions we
discussed earlier. In fact, MTFcan be defined as the modulus of the Fourier
transform of the line spread function, that is

MTF(ν)=|F{LSF(x)}|

=

1

√

2 π

∫∞

−∞

LSF(x)eı^2 πνxdx, (7.1.16)

whereν is the spatial frequency,xis the spatial distance, and{represents the
Fourier transform. In case of digital sampling, the above integral should be changed
to a summation. The problem with the above formalism is that the line spread
function is not easy to determine directly. Instead, as mentioned earlier, the edge
spread function is normally determined. Since line spread function can be obtained
by differentiating the edge spread function therefore theMTFformula given above
can be used.
A big advantage of usingMTFto characterize the contrast resolution is that if
the system consists of different components having their ownMTFs, then the overall
modulation transfer functionMTFtotalcan be obtained by simply multiplying them
together. That is

MTFtotal=

∏

i

MTFi (7.1.17)

whereMTFirepresents the modulation transfer function ofith component of the
system.
Most modern imaging systems, such as CCD cameras, are based on regularly
spaced pixel detectors. The maximum MTF of such a pixel is well known and is
given by the sinc function (see example below)

MTF(ν)pix =sinc(dν)

=

sin(dνπ)

dνπ

, (7.1.18)