f
For the final position estimation the least square
template matching method was implemented. The
template is a model of a reseaucross image. A
binary image of an ideal reseaucross image (4
pixels wide and 100 pixels long) was generated and
smoothed with a 2 x 2 averaging filter to
approximate a convolution with the 20 um wide
Point Spread Function (P.S.F.) of regular
photographic emulsions. The template and a
sample reseau cross image is shown in figure 3.
The matching of the template with the reseaucross
image involves four parameters. The model can be
written as follows:
g(x,y) = d, * t(X-X,,Y-Yo) + Bo
with:
g (x,y) = density at position x,y (the image)
t (x,y) = template
d, = radiometric scale factor
£o = radiometric offset
Xoyog = relative position template - image
The parameters can be split in two groups: the
radiometric (d, g,) and the geometric parameters
(x, y). The addition of a geometric parameter for
the rotation of the reseaucross relative to the
coordinate system defined by the pixels has been
tested. Although the estimated shift did not change
significantly due to this model improvement (in the
order of 0.1 um), the estimated noise reduced by a
factor 1.5. About the same reduction of the noise
estimate was obtained by reducing the width of the
cross-shaped matching window from 8 to 6 pixels.
This is due to the reduction of the number of
background pixels (see furtheron). The
approximate position being known within the one
pixel range, convergence did not pose any
problem.
Although the density noise depends on the
measured density itself (approximately linear for
densities greater than 1), all pixels have been
equally weighted. The estimated noise level should
be interpreted as an average for the pixels within
the matching window. The RMS-granularity of the
emulsion used for the experiment is 1%. The
effective pixelsize being 13 x 13 um? the emulsion
noise is about 3% for D(ensity) = 1. For an average
density the emulsion noise level can be expected to
be 4 - 5% for our pixelsize.
225
The reseaucrosses with the lowest noise estimates
were in the order of 6% of the average density level
(5 parameter solution, matching window 6 pixels
wide). This relatively high noise level originates
mainly from the unmodelled non-uniform exposure
of the reseaucross background. As mentioned
before the width of the matching window has an
influence on the noise estimate. This problem is
inherent to the way in which the reseaucrosses are
projected on to the emulsion. Pre-illumination of
the reseau grid can reduce, but not eliminate, the
errors resulting from the non-uniformity of the
background. Pre-illumination however solves the
problems of lack of contrast between reseaucrosses
and background that occurs in cases of extreme
under exposure of the background. However as the
legs of the crosses are very narrow (about 40 pm =
2 P.S.F.) the bias that might result from an
inadequate model for the background is assumed to
be negligible. Only with a very steep mean gradient
perpendicular to a leg might one expect a slight
bias, which could hardly exceed 1 um. So only the
noise estimates are effected. As the precision of the
position estimates for the crosses exceeds the
quality of the description of film distortion by a
large factor this presently is not relevant as the
noise estimate cannot be exploited in a
straightforward weighting scheme.
5.2 Targets
As for the reseaucrosses, the startpositions for the
targets should have a precision in the order of 2
pixels to guarantee convergence of the iterative
parameter matching procedure. Although an
optimized search procedure is expected to fulfil this
requirement, an algorithm for startposition
improvement has been implemented. This
algorithm is based on the principle of
autocorrelation and results in the symmetry point of
the digital image. The idea is to find the symmetry
point by finding the maximum correlation of the
image with the image mirrored with respect to the
(approximate) symmetry point. This is done
through minimizing the two-dimensional function
C (X9,y 9):
