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. 

Resolution Target 

	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 
our data.  
	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.

Surface Map 

	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.