The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008
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2. THEORY
Before lens calibration, resolution should be examined. It is
conditioned by necessity of test and calibration model choice.
Because of the fact, that we cannot describe different distortions
by single scalar value, we have to use point spread function PSF.
This function is a real function of two spatial variables . It
determinates intensity distribution in output plane of the system
that is depended on point activation on input.
Unfortunately, though function plays important role in image
forming process, there is great problems with its measurement.
For lens with a diameter function PSF can be presented as a
function of radius r (Castleman, 1996):
-V7F(/„/ ) ) = |0[/>SF(x,>.)] (4)
Because phase shift O is also dependent on fx, f y , and can be
expressed by formula (5), where function f(fx, fy) determinate
changes in signal phase during its transfer. Such a function is
called Phase transfer function PTF.
<tij x .f r ) = arctm
WrJr)
KCfxfr)
Arg{s[PSF(x.y) J}
(5)
where:
PSF(r) =
F(fx-f,) = |f PST(x,y)sm[2*(/ r x+/,>))**
-« (6)
(I) fl PSF(x,y)a42x(f x x + f r y)}bc<fy
- (7)
where:
Ji(x)- is a first degree of Bessel function
ri - scale parameter
r - radial radius
On the basis of formula (4) and (5) we can state, that MTF and
PTF are module and argument of the same Fourier transform of
PSF function.
Whereas, line spread function LSF, describing intensity
variations across line in output plane, and related with point
broadening is easy to measure.
3[PSF(x,y)] - MTF(J x .f t )tx^JPTF(f x .f, )] (8)
Line broadening function can be obtain, registering of
luminance signal by displacing detector (narrow slit) along
dispersion spot.
If direction of this shift determines x axis, and slit is parallel to
y axis, it corresponds to relation between PSF and LSF (2).
With those broadening functions correlates subsequent function,
this is an answer to edge spread function ESF.
Because PSF presents point answer of system, equation (8) is an
expression for spatial transfer function Sp(fx,fV) of the image
processing system, that is named Optical Transfer Function -
OTF. From formulas (9) and (10) arise, that normalization of
PSF function means normalization of OTF function to unity
when spatial frequency values equal zero (Wozniak,1996).
■+*'
OTF(f x .f y ) = 0[.PSF(.v,.v)]
LSF(x) = f PSF(x.y)dy
— 30
(2)
MTF(f x ,f t ) = |07F(/ x ,/ r )|
The way of receiving differs from LSF determination. The main
difference is a detector form - now it is a half-plane, not a slit.
In can be described by formula (3).
ESF(x) = | dx [ PSF(x, y)dy
-® (3)
Broadening functions can be used not only in quality
assessment of image processing systems, but also can serve in
determination of resolution or modulation transfer function -
MTF. MTF function is an answer of optical system versus
spatial frequency.lt is contrast dependency on spatial frequency
in relation to contrast for low frequencies. MTF can be express
by relation of signal modulation factor in output to signal
modulation factor on input (depended on spatial frequencies fx,
fy) and present as Fourier transform module of function PSF.
The possibility of OTF normalization is very important, because
it enables conversion of absolute signal values into relative
signal values, that are contrast and image quality measure.
“Wave” deforms passing optical system (lens) or being
registered by photosensitive element. Wave is flattened, so
contrast is worse. The denser waves, the bigger flatness and the
higher contrast decrease.
Lens construction, quality of glass, type of photographic film
emulsion, type and technology of CCD matrix production -all
these elements and many other, influence on image sharpness.
3. METHOD DESCRIPTION
By my own researches, I reached a conclusion, that Siemens
star test is best in resolution of fish eye lens. I have planned test
with 192 Siemens stars, in order that all frame was fulfilled with
stars, as it is visible in the figure 1.
