A unique shape is mathematically prescribed for a "lens" which refracts cimated light passing through it in such a way that any linear scale modulating the incident light is distorted into a logarithmic scale at the final image plane.

BACKGROUND OF THE INVENTION 
Replotting data from a linear scale onto a logarithmic scale in order to 
test the data for exponential dependence has always been time consuming 
and often tedious. A need exists for display systems which can optically 
convert a linear representation to a logarithmic one. 
This conversion is also needed for more sophisticated technology. With the 
advent of coherent optical correlation technology, character recognition 
(for reading machines) and terrain pattern recognition (for aerial 
reconnaissance, etc.) programs having received great emphasis in optical 
data processing. By means of lasers, halographic quality lenses and the 
fabrication of matched filters in the so-called Fourier Transform plane, 
certain patterns hidden among a confusion of shapes or background noise 
can be "recognized." This recognition consists of a strong optical signal 
in an output plane which contains the cross-correlation between the input 
image and matched filter. The location of this strong optical signal in 
the output plane is indicative of the location of the recognizable pattern 
in the input image. This recognition ability persists despite variation in 
image intensity, certain obscurations, and translation of the input image. 
However, if there is a magnification or scaling factor change between the 
input image and that which contained the reference pattern from which the 
matched filter was made, no strong, localized optical signal will result 
in the output plane. In other words, image recognition cannot take place 
after pattern magnification. This makes it impossible for coherent optical 
correlation systems to rapidly indicate recognizable content in an aerial 
reconnaissance photo unless the altitude (or, at any rate, the scaling of 
photo content) is identical for reference photos (from which matched 
fibers are made) and the photos to be examined. 
The log-log scale refractor of the present invention is, in part, designed 
to solve the aforementioned problem of replotting data for which a 
logarithmic rendition is desired. This may be done quite simply by 
photographing the linear plot with polaroid transparency film, placing the 
transparency in a well-collimated light beam, allowing the modulated beam 
to pass through the log-scale lens (or simply log-scale lens, if desired), 
and viewing the result in a screen appropriately placed. 
The log-log scale refractor is also disposed to eliminate the restriction 
that scaling factor (or magnification) be the same for reference and 
examined patterns in order to obtain "recognition" in coherent optical 
correlation systems. Such is possible with the log-log scale refractor. 
Provided the image content at the zero coordinate of the log-log scale 
refractor plane is the same for both reference pattern and examined 
pattern, a magnification of the examined pattern simply converts to 
translation of the log-log scale image. The optical correlation is fully 
capable of pattern recognition when mere translation is involved. The key 
to making this possible is adapting the system to recognize the log-log 
scale rendition of patterns, rather than linear scale renditions.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
As seen in FIG. 1, a refractor 10 includes a pair of surfaces 12 and 14. 
Surface 14 is provided with a predetermined curvature and surface 12 is 
planar. 
FIGS. 1 and 2 show a typical configuration depicting curvature as a 
function of one of the planar coordinates (y.sub.1 for the lens plane, 
y.sub.2 for the image plane), and it also illustrates the path of an 
incident ray, normal to the refractor. Parameters A.sub.1 and A.sub.2 are 
arbitrary equal distances measured in terms of y.sub.1 and y.sub.2 units, 
(inches, meters, etc), respectively; a is the displacement between origins 
in the refractor and image planes while L is the separation between the 
planes (both a and L are measured in the same units as y.sub.1); .theta. 
is the angle through which the LLS refractor bends the incident ray, while 
.phi..sub.i and .phi..sub.o are the angles of ray incidence and exit with 
respect to the curved surface normal N(angles measured clockwise are 
negative, counterclockwise are positive); and n is the refractive index of 
the LLS refractor material. For Y.sub.2 =log Y.sub.1, then 
##EQU1## 
where 
EQU tan .theta.=-(y.sub.1 -a-(A.sub.1 /A.sub.2) log y.sub.1)/L. (2) 
For cases where tan.sup.2 .theta.&lt;&lt;1, Eq. (1) simplifies to 
EQU tan .phi..sub.1 =tan .phi./(n-1). (3) 
For curvature along two orthogonal directions, it is convenient to express 
the distance Z from the planar side of the refractor to the curved surface 
(thickness) in terms of the lens plane coordinates x and y. The following 
is derived from the Eqs. (2) and (3) by extension to two planar 
coordinates and employing a notational change. The applicable formula here 
is a small angle approximation (tan.sup.2 .phi.&lt;&lt;1) and is given by 
EQU Z(x,y)=[1/2(y.sup.2 +x.sup.2)-(A.sub.1 /A.sub.2) (0.43429)(y lny+x 
lnx-y-x)-a(y+x)]/L (n-1)+Z(0,0), (4) 
where Z(0,0) is the lens thickness at x=y=0, and, again, n is the 
refractive index of lens material. 
Mathematics has been presented assuming log =log base 10. The natural 
logarithm (denoted in) relationship between incident and final image may, 
however, be desired; for this case, the factor 0.43429 in Eq. (4) should 
be replaced by unity. 
The LLS refractor is designed only for appropriate refraction of 
well-collimated light, parallel to its axis. It is not a lens in the 
general sense that an image is formed of an object located at a finite 
distance from it. Where collimated light contains slight divergence, best 
results will be attained if the collimating lens also focuses the incident 
image on or slightly beyond the LLS refractor plane. 
The log-log scale refractor is fashioned from a material which is 
transparent to the wavelength of electromagnetic radiation of interest. 
Normally, this radiation will be light (4000-6000 Angstroms). The 
refractor is aspheric and possesses the property of refracting collimated 
light (parallel to its axis) in such a way that the coordinates of a ray 
intersecting an image plane a distance L from the lens are logarithmically 
related to the coordinates in the plane of the lens. The two sets of 
coordinate are two-dimensional and posses origins in their respective 
planes. 
The effect of the LLS refractor is to modify the coordinates of an image 
modulating the incident light beam so that, if a reference point on the 
incident image remains at the origin throughout the system, incident image 
magnification is converted to mere displacement in the image plane. 
The description of a log-log scale refractor is simultaneously the 
description of a log-scale refractor. Where coordinate transformation in 
only one dimension is required, the lens will possess the prescribed 
curvature in only one dimension. 
The LLS refractor thickness (as well as thickness variation) has been 
implicitly assumed as very small compared to L. If the thickness is large, 
L should be taken as the distance from the plane defined by Z=Z.sub.avg 
(Where Z.sub.avg is the average value of Z) to the image plane. The 
thickness variation should always be kept small compared to L. 
It was emphasized that there must be a common point at the origin of the 
log-log scale refractor plane for both reference pattern and examined 
pattern in order that magnification convert to translation. In other 
words, if the coordinates x, y of an examined pattern are both magnified 
and translated (e.g., ax+b, cy+d), simple translation of the log-log scale 
image is not the entire result. If connected to a coherent optical 
processor with a matched filter for the image, the correlation signal 
would decrease. This could very well be the basis of a missile homing 
system based on optical correlation. Via feedback circuits, the 
correlation signal keeps the missile on an optically recognized target, 
despite the image magnification which occurs with the approach. If 
wandering from an optical reference point occurs, the drop of correlation 
signal is sensed and automatic course corrections are made.