Fourier Analysis of CCD Sampled Imaging Systems
William Shamblin and Charles Bennett
Modulation Contrast Method
The modulation contrast method is the standard form of analysis for
determining the quality of an imaging system. High contrast edges are scanned,
but due to system degradation the scanned image has a much smoother transition
along the edge than did the orginal object. The modulation transfer values
obtained in this way are defined to be the contrast or the image relative to
that of the object.
Edge Gradient Method
The edge gradient method has been used in the past to analyze
photographs taken with imaging systems. This method originally used
photographic grains measured with a microdensitometer to determine the
quality of the imaging system on which the photograph was taken. High
resolution film contains photographic grains much smaller than any CCD
array detectors. Our experiment was to determine if current CCD detectors
are small enough to allow this kind of analysis.
The edge gradient technique involves fitting a function to our
digitized data to obtain an estimate of the edge spread function E(x).
This allows us to get an estimate of the line spread function g(x). With
knowledge of the line spread function we can calculate the MTF, since the
MTF is defined to be the normed modulus of the Fourier transform of the
line spread function.
To accomplish this we expand the line spread function in a Fourier series.
We used Gaussian-Hermite polynomials as basis functions becuse they
efficiently model the particular data feature that we are measuring. The edge
spread function is the convolution of the line spread function and the step
function. In our case this gives the equation
Using this equation as a fitting function gives an estimate of the Fourier
coefficients. This gives an estimate of E(x) and hence the line spread
function g(x). Now to find the MTF we simply take the Fourier transform of
the line spread function. The MTF is now a function of spatial frequency nu,
The Fourier coefficients, and the spread parameter b.
The dilemma in implementing this technique is the lack of data
points along the transition region from a black square to a white square.
Our CCD array has 4096 elements in a 35mm package, so that each detector
is approx. 7 microns wide. The f/4 optics utilized in the scanner produce
a diffraction limited blur of about 5 microns. This gives approx. 2 or three
data points along an edge.
We needed a way to generate many more points along an edge transition
region. A novel resolution target was designed to increase the density of
The reolution target that we developed is similar to a checker
board. It whas 81 squares across and down, and each square is just a little
smaller than 50 pixels wide. This allows us to generate many data points
in the following way. We can start in the middle of a black or a white
square and jump 100 pixels at a time, and end up a little closer to an edge
each succesive jump. If this is done 8 times we would have jumped over 8
different squares. Incrementing the stating pixel and performing 8 jumps
each increment would allow us to build up information over an average edge.
Applying this technique to the entire digitized resolution target
we can develop a surface map of the entire image plane of the optical
device that we are analyzing. This kind of data is practically impossible
to generate with the modulation contrast method.
The picture below contains edge data along with a nonlinear fit, this result
was obtained using only the leading Fourier coefficient ( the zeroth order).
The fit was generated using code developed in Mathematica running on a Sun
The picture below shows higher order fits along with the resulting MTF
determinations. The MTF values that we obtained by the edge gradient method
are plotted against the values obtained usiong the modulation contrast method.
Agreement between the two seems to diminish for fits above the second order.
One possible explanation for this could be an unaccounted for MTF stemming
from the fabrication of the resolution target, and we are still investigating
other basis functions to see if they more closely model the data feature we
need them to model and hence model the MTF more accurately.
Below is the surface map of the image plane of the Dalsa-pr37
scanner at Oak Ridge National Labs. Each individual square on the map
represents an average over 8 squares of the resolution target. In order to
get the rows we skipped four lines in the vertical direction of the
resolution target, and performed our technique on that horizontal line.
We have demonstrated that the current CCD detectors of width 7
microns are wide enough to permit an MTF determination of a high quality
imaging lens. An inovative resolution target has been designed and tested
which allows many data points to be generated along a transition region.
Using Fourier techniques and nonlinear fitting routines we were able to
generate an expression for the MTF, and compare it to the standard modulation
contrast method. The fitting procedure based on Gaussian-Hermite polynomials
has been shown to produce some variabilty in final modulation transfer
If you would like to see the Mathematica code for this experiment it can
be found here Source Code. Or if you would like the Mathematica notebook and postscript graphics, NoteBook.