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Description  |
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BACKGROUND OF THE INVENTION
An inherent advantage in using optical methods in information processing is
that these systems can perform data processing operations in parallel. In
contrast to the advantage provided by parallel processing, the space
invariance of most conventional imaging techniques limits the utilization
of optical data processing techniques in certain instances. For example,
the features of both parallel processing and space variance are necessary
for general map transformations using optical schemes.
Although prior art elements such as corrector plates, axicons, conical
lenses and ring lenses have been utilized to provide local image
modifications, they are generally inadequate for purposes of information
processing.
Alternate techniques for providing general types of coordinate
transformations as well as local image modifications include image
processing with digital techniques, fiber optic devices and scanning
systems, both optical and electronic. These techniques, however, are
indirect and relatively complex.
Therefore a need exists for a direct, relatively noncomplex optical
technique for providing a space variant system which, in turn, allows
general types of coordinate transformation as well as local image
modifications such as translation, stretching and rotation to be achieved.
SUMMARY OF THE PRESENT INVENTION
The present invention provides method and apparatus for providing
geometrical transformations using optical techniques wherein a
space-variant, optical coherent system is provided to influence the light
from each point of the object. The space-variant system introduces a
specific deflection and focusing power to the light from each point of the
object distribution independently allowing any light distribution on a
surface of general shape to be displayed onto another arbitrarily shaped
surface. In particular, geometrical image modifications such as coordinate
transformations and local translation, inversion, reflection, stretching,
which require space-variant systems are provided by introducing phase
filters having a predetermined phase function into optical coherent
systems in such a manner that the local phase variations influence light
from local object areas. In one embodiment, the object distribution is
multiplied by the phase function so that its spectrum at the frequency
plane constitutes the desired transformation. In a second embodiment, the
aforementioned concept is applied to produce a transformation in an image
plane. The phase filters, in a preferred embodiment, comprise computer
generated holograms. The space-variant, optical coherent system provided
in accordance with the teachings of the present invention may replace
conventional optical components utilized in lenses and corrector plates
for field flattening, or utilized as spatial filters for optical data
processing, coding and decoding optical data or imaging on curved
surfaces.
It is an object of the present invention to provide a space-variant,
optical coherent system for performing general types of map
transformations.
It is a further object of the present invention to provide a space-variant,
optical coherent system for performing geometrical image modifications
such as coordinate transformations and local translation, inversion,
reflection, stretching and rotation.
It is still a further object of the present invention to provide a
space-variant, optical coherent system for performing general types of map
transformations wherein the local phase variations of phase filters
intorduced into the system influence light from local object areas.
It is an object of the present invention to provide a space-variant,
optical coherent system for parallel processing of optical information.
DESCRIPTION OF THE DRAWINGS
For a better understanding of the invention as well as other objects and
further features thereof, reference is made to the following description
which is to be read in conjunction with the following drawing wherein.
FIG. 1 illustrates a conventional imaging system;
FIG. 2 is a first embodiment of an optically coherent space-variant system
in accordance with the teachings of the present invention;
FIG. 3 is a second embodiment of an optically coherent space-variant system
in accordance with the teachings of the present invention;
FIG. 4 illustrates examples of conformal mapping utilizing the embodiment
set forth in FIG. 2;
FIG. 5 illustrates examples of image modifications using the embodiment set
forth in FIG. 3;
FIGS. 6(a) and 6(b) illustrate computer-generated holograms of the grid
type which may be utilized as phase filters to realize map transformations
in accordance with the teaching of the present invention;
FIGS. 7(a) and 7(b) show the transformed object distributions produced by
the phase filters of FIGS. 6(a) and 6(b) respectively;
FIGS. 8(a)-8(g) illustrate one-dimensional image modifications using a
variable one-dimensional phase filter of the grid type;
FIGS. 9(a) and 9(b) illustrate computer generated hologram phase filters
for introducing local one-dimensional image modifications;
FIGS. 10(a)-10(d) illustrate computer generated hologram phase filters for
performing two-dimensional transformations;
FIGS. 11(a)-11(d) illustrate the image transforms corresponding to the
phase filters shown in FIGS. 10(a)-10(d);
FIG. 12 illustrates a computer generated hologram crossed grid phase filter
which transforms a square area conformally into a 90.degree. portion of an
annulus area; and
FIG. 13 is the image plane display made by using the phase filter of FIG.
12.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
The primary purpose of the present invention is to produce a system for
realizing general geometrical transformations using optical techniques. A
space-variant system is provided to influence the light from each point in
an object. Such a system will introduce a specific deflection and focusing
power to the light from each point of the object distribution
independently. This allows any light distribution on a surface of general
shape to be displayed on another arbitrarily shaped surface and
incorporate image modification such as local stretch rotation and
translation.
FIG. 1 illustrates a conventional imaging system. Object 10, located at
object plane O, is spaced apart from an objective lens 12, lens 12 being
separated from object 10 by a distance equal to the focal length f.sub.L
of lens 12. The objective lens 12 focuses and directs the light incident
thereon onto a second lens 14, separated from lens 12 by a distance equal
to the back focal length of lens 12 and the front focal length of lens 14.
Lens 14 directs the light incident thereon to image plane 16 via a phase
plate 18 located at plane P. Image plane 16 is separated from lens 14 by a
distance equal to the back focal length of lens 14. The distance between
phase plate 18 and image plane 16 is arbitrary.
In operation, the object 10, such as a transparency, is exposed to a source
of coherent illumination 20, such as a laser, the coherent light 22
emitted therefrom being directed through transparency 10 to lens 12. The
light amplitude distribution projected by lens 12 to frequency plane F
will be a Fourier transform of the light amplitude distribution
transmitted through the transparency 10. Lens 14, in essence, performs a
second or inverse Fourier transform on the light distribution thereon,
reconstructing the original light distribution transmitted by transparency
10, onto image plane 16 via phase plate 18. The use of a single phase
plate 18 (either reflective or refractive) will not make the system
completely space-variant. The influence of the irregular phase plate 18 is
strongly dependent on its location along the optical axis; the further
phase plate 18 is from either object 10 or image plane 16, the stronger is
its optical influence. Increasing this distance, however, eliminates the
possibility of separately influencing different object areas. Diffractive
systems allow multiplexing on optical images but refractive and reflective
systems do not. That is, the full aperture of the diffractive system may
be shared by a large number of image forming elements. Selectivity may be
achieved by incorporating volume effects. Thus, a thick hologram placed
somewhere between object 10 and image plane 16 may provide space
variation. However, volume holograms are difficult to produce because it
would be necessary to multiplex as many subholograms as there are number
of points in the object plane. Alternately, a phase filter may be placed
in the object (or an intermediate image) plane, i.e. the only plane where
the information about individual object elements is laterally separated.
However, any phase variation introduced in the object plane will have no
influence on the location of the image points in image plane 16.
In accordance with the teachings of the present invention, the prior art
system shown in FIG. 1 is modified as shown in FIG. 2 (and FIG. 3 as will
be set forth hereinbelow) to produce a space-variant system. In this
sytem, a phase filter with the proper phase variation is placed in the
object plane so that the transformation of the object distribution is
formed around the frequency plane F.
Object or transparency 30 is coherently illuminated from laser source 32.
Lens 34, separated from the object plane by a distance equal to its front
focal length f.sub.L, displays the Fourier transform of the complex
transmission of transparency 30 in the frequency plane F. Frequency plane
F is separated from lens 34 by a distance equal to the back focal length
f.sub.L of lens 34. A phase filter 36 having a predetermined phase
variation is positioned at the object plane so that the transformation of
the object distribution is formed around plane F. The local phase
variation of phase filter 36 is represented in the figure as a combination
of prismatic (first) and focusing (second order term) effects, the prism
42 bending the light, the optical elements 44 focusing the light incident
thereon. The transformed object distribution is displayed in the frequency
plane F of the optical system. As can be seen, the two dimensional optical
information in transparency 30 (one dimension into the plane of the
figure) is transformed into a curved surface 38 at frequency plane F.
The techniques utilized to fabricate a phase filter with the proper phase
variation will be set forth hereinafter.
Lens 34 as set forth hereinabove, produces in its back focal plane F a
frequency spectrum (Fourier transform) of the amplitude transmission
function in its front focal plane, i.e. the object plane. The concept of
providing Fourier transforms of amplitude distributions by optical
techniques is well known and will not be set forth herein.
Referring now to FIG. 3 a space-variant, coherent optical system for
producing transformations in an image plane is illustrated. FIG. 3 is, in
essence, the system shown in FIG. 1 with the addition of two phase
filters. One of the phase filters is placed in the object plane so that
light from different parts of the object will be laterally separated and
directed to the frequency plane F. The other phase filter is place in the
frequency plane F in order to modify the information from various portions
of the object field in a predetermined manner. Transparency 50 is placed
in contact with (or imaged onto) phase filter 52 located in the object
plane. A lens 54 is interposed between the object plane and a second phase
filter 56 positioned at the frequency plane F. A lens 58 images the light
amplitude distribution incident thereon onto image plane 59. Lens 54 is
separated from the transparency 50 by a distance equal to its front focal
length and phase filter 56 is separated from lens 54 by a distance equal
to the rear focal length of lens 54.
A source of coherent light 60, such as that produced by a laser, is
provided to expose object 50.
In operation, the coherent light emitted by source 60 is incident on
transparency 50, phase filter 52 laterally separating the light from
different parts of the transparency and directing the light to the
frequency plane F. Lens 54 forms a Fourier transform of the light
amplitude distribution transmitted through phase filter 52 at the plane of
phase filter 56 (frequency plane F). Phase filter 56, selected to have a
predetermined phase variation as will be described hereinafter, modifies
information from various portions of the object field in a predetermined
manner, the modified information being directed to lens 58 which projects
an image on image plane I. Lens 54, as set forth hereinabove, acts
essentially as a spectrum analyzer producing in its back focal plane
(frequency plane) the spectrum of the transmission function in the front
focal plane (object plane). The points of light in the frequency plane F
are thereafter used as the object in the front focal plane of lens 58, the
back focal plane (image plane 59) of lens 58 containing the spectrum of
the light in the frequency plane F. Lens 58 therefore produces a transform
of a transform that is in effect an inverse transform which gives the
original transmission function as modified by phase filters 52 and 56. Any
object in the object plane of lens 54 will be imaged by lenses 54 and 58
into the image plane I of the optical system. Phase filters 52 and 56 are
formed in a telecentric arrangement with lenses 54 and 58 forming an image
at image plane I of the modified object 50.
The following describes the procedure to be utilized for designing specific
phase filters to be utilized in the FIG. 2 and FIG. 3 embodiments to allow
the optical geometrical transformation of information from the object
plane into either the frequency or image planes.
The change in direction of a light wave caused by a phase filter is given
by the eikonal equation. In the simplified, specific case of a
transformation of an amplitude distribution, a(x, y), in the object plane
onto another flat surface, u, v, in the frequency plane F, of FIG. 2, the
normal to the wavefront at the point (x, y) represents the ray direction,
i.e., (l/k) (.differential..phi./.differential.x); and (l/k)
(.differential..phi./.differential.y), where .phi.(x, y) represents the
phase function introduced in the object plane O and k =
.differential..pi./.lambda.. Thus, in the paraxial region, light from (x,
y) will hit plane F in
##EQU1##
The relationship between the complex amplitude distribution in the object
or image planes, and the frequency plane, F, in FIGS. 2 and 3 is given by
the Fourier transform
##EQU2##
For large k, the method of stationary phase can be applied to find an
approximate solution to eq. (2). The main contribution to the integral
arises from the neighborhood of the saddle-points which are obtained when
the derivatives .differential./.differential.x and
.differential./.differential.y of the exponent are equal to zero, i.e.,
##EQU3##
which, is identical to eq. (1). The solution to eq. (2) can now be
calculated by approximating the exponent around the saddle-points and
adding the contributions to obtain the distribution, a. Since it is
desired to achieve specific, relative locations among the image points,
the solution looked for will be the phase function .phi.(x, y) which will
give the transformation
ti u = u(x,y), v=v(x,y) (4)
The answer is given by introducing into eq. (3) the transformation desired
as represented by eq. (4). The transformation described so far is obtained
between an object and a frequency (Fraunhofer diffraction) plane.
There are several ways to physically obtain the necessary phase variation
.phi.(x, y). For example, refractive or reflective elements can be formed
by varying the thickness or the shapes of their surfaces. If blazed
grating techniques are used to produce the components (then frequently
called kinoforms) the total phase variation required is limited to 2.pi..
This makes it possible to use photographic procedures for their formation.
A preferred technique for shaping wavefronts is to use computer-generated
binary holograms, because they may be made in a binary fashion and may be
utilized in line as well as off-axis configurations.
To illustrate the aforementioned technique in a situation wherein
presentation of geometric similarity is desired (geometrical
transformation of the object onto the frequency plane F), u = x and v = y
is introduced in equation 3. This results in
##EQU4##
The phase variation of equation (5) for a first transformation may be
obtained by replacing the phase filter 52 with a conventional lens having
a focal lens f.sub.L or, in holographic terms, a corresponding Fresnel
zone-plate, in the object plane.
The phase variation of phase filter 52 (FIG. 3) is determined by equation
(5) whereas the phase variation of phase filters 36 (FIG. 2) and 56 (FIG.
3) will be determined by the desired transformation. As set forth
hereinabove, computer-generated holograms may be utilized to provide the
desired wavefronts.
In order to provide a conformed transformation from the given x, y - plane
(z - plane) into any given domain of the u, v plane (w - plane) and the
function w = f(z) is known the following procedure is utilized to
optically realize the transformation.
A specific example will show how an area in the z - plane is mapped onto a
portion of the interior of the circle, r = p, in the w - plane, the
following equation being utilized:
w = pe.sup.-z*/q (6)
(p and q are lengths). The square area 0<x/q <.pi.; 0<y/q <.pi. in the
z-plane then corresponds to the half annulus pe.sup.-.pi.
<r<p,o<.phi.<.pi. in the w-plane. Introduction of eq. (6) in eq. (3)
results in
##EQU5##
which can be written
##EQU6##
Solving equation (8) results in
##EQU7##
As will be described hereinafter, the equation (9) solution was
incorporated in a computer-generated hologram. FIG. 4 shows some examples
of this transformation (image modifications). The light distributions
(object transparenices) that illuminated the hologram are on the left of
the figure, the corresponding reconstruction in the frequence plane F
(transformations) are on the right. Two diffraction orders are included in
the displays and, of course, the second order is twice the size of the
first. The computer generated hologram used had 80 fringes.
Examples of images modifications using the embodiment of FIG. 3 are
illustrated in FIGS. 5(b)-(g), FIG. 5(a) being the object. The resultant
transforms of the object are shown in FIGS. 5(b)-(g). The computer
generated holograms utilized contained approximately 300 fringes. The
transformed distribution appears in an image plane of the original object.
To achieve the transformation, a one-dimensional grid structure with a
linear increase in spatial frequency (other phase variation could also
have been chosen) was placed in the object plane. In the frequency plane,
F, the first diffraction orders are then extended and u .varies. x in a
manner as set forth hereinabove. A second one-dimensional grating
(hologram) was placed over one of the first diffraction orders in F. The
rest of this plane was blocked with a mask. In the image plane I, several
diffraction orders appear due to the grid in F. The zeroth order is a copy
of the object distribution. The first order will be a modified image
because of the phase variation introduced by the synthetic hologram in F.
Grids with an error-function frequency variation were used in FIGS.
5(b)-(d). A local stretch in the middle of the object is applied so that
the wheels of the car (object) remain round. In FIGS. 5e and g, a lateral
magnification and in FIG. 5f an exponentially increasing magnification is
introduced. This imaging scheme may be utilized to perform two-dimensional
transforms if desired.
As set forth hereinabove, two optical schemes for performing geometrical
transformations have been disclosed. In the first technique (one
transformation), a light distribution in a non-image plane (frequency
plane P) is achieved that corresponds to the configuration of the desired
transformation. In the second technique (two transformations), a modified
imaging system is provided by incorporation of phase filters.
The first technique introduces a phase function in the object plane so that
the lateral distribution of its frequency spectrum conforms with new
relative locations among the object points. The frequency spectrum is
displayed in the focal plane of a lens, as indicated in FIG. 2. The shape
of the phase function, .phi.(x,y), is given by Eqs. (3) with the
transformation wanted inserted according to Eqs. (4). The transformation
is independent of the object configuration as long as a pure amplitude
object distribution with relatively coarse structures is introduced or
imaged onto the object plane of FIG. 2. The irradiance of the transformed
distribution will vary with the amount of local magnification.
The second technique modifies an imaging system in such a way that each
point of the object can be individually influenced. A nonimage plane
distribution is needed in which light from each object point is laterally
separated. A phase filter is placed in the object plane so that different
prismatic and focusing effects influence the image location of the various
object points. A preferred solution is illustrated in FIG. 3. The
telecentric lens system, comprising lenses 54 and 58, images transparency
50 onto image plane I. Placing a lens in the object plane will result in
an extended illuminated area in the frequency plane F with a light
distribution similar to the one in the object plane. A convenient size of
this area is obtained by proper choice of the focal length of the added
lens. The filter performing the transformation wanted is placed in F. In
constructing this filter, the phase variation in the light illuminating it
should be considered. For convenience, it is practical to introduce a
quadratic phase variation, e.g., a conventional lens, in the object plane
which produces a spherical wave incident on the plane F. The amplitude in
the image plane is
##EQU8##
where .phi.(u,v) is the phase filter introduced in the frequency plane F
to obtain the final image transformation in the image plane I of FIG. 3.
The phase variation of the incident light on F, now due to a lens in the
object plane is described by exp(i.pi.(u.sup.2
+v.sup.2)/(.lambda./f.sub.L)) Equation (10) is solved by approximately the
saddle points and adding the contributions to obtain the distribution.
After introduction of the spherical phase factor, the result is
##EQU9##
which describe the achieved transformation.
Means for influencing the phase variation of the light in certain planes of
optical systems, i.e. phase filters, are needed to realize map
transformations. Several kinds of elements may be used. They can be either
refractive, reflective, diffractive, or a combination of these.
The most convenient way to manufacture the phase filter is to shape a
surface proportionally to .phi.(x,y). A refractive element, may be
obtained not only by surface relief but also by using internal reflective
index variations. Although some common wavefront shapes can be achieved
with available optical components, general shapes seem, however, difficult
to produce, especially using grinding techniques.
The blazed grating type filter is a hybrid between a refractive and a
diffractive element. The maximum phase retardation in the filter required
to form the .phi.(x,y)-shaped wavefront is reduced to 2.pi.. This amount
of variation is easily obtained by utilizing the changes of optical
thickness that occur in processed photographic emulsions. However, the
precision needed in the lateral variation of the optical path is still
high if in-line filters like kinoforms are produced. On the other hand, it
is a powerful and easily applicable technique for improving diffraction
efficiency in off-axis holograms.
Any grid structure may be considered as a synthetic interferogram (image
hologram). Thus, illumination with a plane wave will result in a
diffracted wave. Its phase variation is determined by the location of the
grid lines, which constitute niveau curves 2.pi. apart. This makes the
filter simple to manufacture, but it also has disadvantages. More waves
than the one wanted are formed, and spatial-filtering procedures are, in
general, complicated because of the inline character of the filter.
However, limited portions of grid structures often show off-axis effects.
Holograms possess attractive features that make them the preferred form of
phase filters. For example, the shape of the wavefront created is
determined by the lateral geometrical properties of the filter, and the
filters can produce any angular separation among the reconstructed
wavefronts. Furthermore, they can be made in a binary fashion, and
amplitude variations are easily incorporated. This is important when
extensive local-area changes are involved. Computer generated holograms
have been successfully utilized as phase filters. Computer-generated
holograms are binary i.e. they consist of many transparent areas on an
opaque background and have a real nonnegative amplitude transmittance and
are able to influence the phase of a light wave.
Several different techniques are available for forming computer generated
holograms. For practical production of phase filters, the binary technique
using the "detour phase" concept is convenient and has been utilized,
except for FIG. 4, to form the transformations set forth herein. In this
type of application, in which the phase varies in a continuous fashion,
the holograms are formed with continuous fringes and are identical in
appearance to hard-clipped interferograms. A technique for realizing the
detour plane hologram is set forth in the article Binary Fraunhofer
Holograms, Generated by Computer, A. W. Lohmann and D. P. Parris, Applied
Optics, Volume 6, No. 10, Page 1739 (1967) the teachings of which are
incorporated herein by reference. The specific problems here allow
considerable simplification of the statements that characterize the
filters so they may seem slightly different than the "detour phase"
hologram. Therefore, descriptions of some specifics of the approach will
be set forth.
The location of the fringes in an interferogram (hologram) formed between a
wave exp (i.phi.(x,y)) and a plane wave inclined at an angle .theta. to
the interferogram plane is
##EQU10##
where n are integers. This recording can now, in the conventional sense,
be used as a filter (hologram). In the first diffraction order,
sin.theta. = .lambda.V.sub.o
holds, where V.sub.o is the spatial frequency that corresponds to a
deflection .theta.. Illumination with a plane wave results in a
reconstructed wave
exp{iW} = exp{i.phi.(x,y) + i2.pi.xV.sub.o } (13)
If v(x,y) now indicates the frequency variation over the filter then its
components in the x and y direction are related to .phi. by
##EQU11##
Clearly, by proper variation of the spatial frequency and the orientation
of the grid structure in the filter any shape of the wavefront is
possible. In the computer-generated filters successfully utilized, the
frequency v(x,y) is the decisive parameter. Prismatic as well as lens
effects can be introduced. Here, lateral modifications of object-light
distributions are exemplified. Thus, a local variation of light deflection
over the grid structure is required. According to Eqs. (14), the angular
components of the deflection are
##EQU12##
Other types of computer-generated holograms have also been successfully
utilized. For example, the transformation shown in FIG. 4 was produced by
an image-plane binary hologram, the realization of which is disclosed in
the article Binary Synthetic Hologram, Wai-Hon Lee, Applied Optics, Volume
13, page 1677 (1974), the teachings of which are incorporated herein by
reference.
The optical geometrical transforms described hereinabove were successfully
implemented as will be described hereinafter.
To simplify the plotting procedure of the computer-generated phase filters,
they were constructed in the form of two orthogonal grating structures.
This is possible only if the x and y dependences of .phi. are separable.
To demonstrate preservation of geometrical similarity, a phase filter
.phi.(x) 32 kx.sup.2 /(2f.sub.L), .phi.(y) = ky.sup.2 /(2f.sub.L), (16)
has to be placed in the object plane in the FIG. 3 embodiment. Then, Eqs.
(15) give the necessary frequency variations in the filter
V(x) = V.sub.o + x/(.lambda.f.sub.L); V(y) = V.sub.o + y/(.lambda.f.sub.L).
(17)
this grid structure was plotted in a large format with 300 lines per
direction and with a frequency range 1 to 2. FIG. 6(a) shows one grating;
FIG. 6(b) shows the crossed-grating version, the grid structures having a
linear change of spatial frequency. This figure is illustrated for
clearness with 30 fringes and was photographically reduced to about
4.5.times.4.5 mm.sup.2. The spatial frequencies then ranged from 40 to 80
lines/mm. With this choice of range, no overlap between the lst and 2nd
diffraction orders occur. However, a larger range could have been used;
the 2nd order was weak, owing to the choice of opening ratios in the
grating. This filter was placed in contact with an amplitude object in the
front focal plane of a lens (f.sub.L= 200mm) as indicated in FIG. 2.
Illumination with collimated laser light gave the light distribution in
the back focal plane (F in FIG. 2) that is reproduced in FIG. 7(a). The
object here was a 90.degree. sector of a circular grating. The displayed
distribution appears as the cross terms due to the crossed gratings used
as a phase filter. It is interesting to note that, in this case, the
transforms in one diagonal of the figure are equivalent to introducing a
positive and a negative lens, respectively, in the object plane. Each
transform in the other diagonal is caused by a combination of a divergent
wave in the orthogonal direction.
Another example is shown in FIG. 7(b), which shows the display of
transformed object distributions in a Fraunhofer diffraction plane.
Crossed grids were placed in contact with the object, the grids being
linear in frequency in FIG. 7(a) and in period in FIG. 7(b). Here same
object is used, but the filter is now linear in period l/v instead of in v
as in FIG. 7(a). This filter also had 300 lines in a period range 1 to 2.
Its visual appearance was almost identical to the one used to obtain FIG.
7(a). However, the transform is strikingly different. The frequency
variation in the x-direction of this filter is
v=v.sub.0 /(1-px), (18)
where v.sub.0 and p are constant factors. The corresponding transformation
in the u direction in the frequency plane F, is found by introducing Eqs.
(18) and (15) in Eqs. (3),
u=.lambda.f.sub.L v.sub.0 px/(1-px). (19)
By this transformation, the circular grating has almost the appearance of a
one-dimensional grid structure, indicating that any type of geometrical
transformation is possible with this technique.
In order to produce one dimensional image modifications, the optical
arrangement shown in FIG. 2, with the corresponding modification of one of
the crossed gratings shown in FIG. 6(b), can be utilized to introduce the
changes in the display.
An alternate technique is to utilize the embodiment shown in FIG. 3 with a
one-dimensional grating placed in the object plane O. The frequency of
this grating varies linearly with the distance across it, as shown in FIG.
6(a). 300 lines occupied 4.5.times.4.5 mm.sup.2 and the frequency range
was about 40-80 lines/mm. In the back-focal plane, F, of the lens 54
(f.sub.L =200 mm), the different orders in the spectrum were elongated in
one dimension. The phase filter that gives rise to the image modifications
is now placed over one of the extended diffraction orders in F. The
relation between the F plane and image plane I in the distorted direction
is given by Eqs. (11) and (15),
x = u + .lambda.f.sub.L {V.sub.u -V.sub.o } (20)
The term within the braces is the change introduced between image and
object. A simple transform is a pure scale change. This is achieved by
placing in F a grid structure the frequency of which is linear in u. In
case V.sub.u =V.sub.o +U/(.lambda.f.sub.L), then, according to Eq. (20)
the image is magnified to double size in the x direction. On the other
hand, V.sub.u =V.sub.o -u/(.lambda.f.sub.L) reduces the image to a line in
the y direction. Thus, an increase of spatial frequency magnifies the
image and a decrease reduces it. This is shown in FIGS. 8(b) and 8(c). The
object used is shown in FIG. 8(a). The other examples in FIG. 8 show that
the technique is well suited to introducing local stretching in the image.
In FIG. 8(d), an exponentially increasing magnification toward the right
is shown. This was achieved by use of the filter in FIG. 9(a). The filter
shown in FIG. 9(a) shows an exponentially increasing spatial frequency and
is used to produce FIGS. 8(b) and 8(d). Stretches affecting only the
central portion of the image are shown in FIGS. 8(e)-8(g). If
.differential.v/.differential.u is described by a gaussian, then the
change of location of the points in the image will be proportional to a
normal distribution function. Three filters with different frequency
ranges but the same value of standard deviation were used to form the
images in FIGS. 8(e)-8(g). The filter corresponding to FIG. 8(f) is shown
in FIG. 9(b), the spatial frequency increasing according to a normal
distribution function.
The simplest technique for converting a conventional imaging system (see
FIG. 1) into one that is able to perform a general two-dimensional
geometrical transformation of any amplitude object distribution is
indicated in FIG. 3 as set forth hereinabove. A lens placed in the object
plane makes the lateral amplitude distribution in the frequency plane of
the system proportional to the object distribution. The transformed image
in the final image plane I [x=x(u,v), y=y(u,v)] is obtained by placing a
phase filter .phi.(u,v) in the frequency plane. The proper value for
.phi.(u,v) is obtained by solving Eqs. (11) after introducing the value of
x(u,v), y(u,v) (desired image configuration).
The phase filter that performs the mapping can be realized by constructing
a regular type computer generated hologram which provided the
transformations shown in FIG. 5.
Simple illustrations of the relations between frequency variations in the
filter and distribution changes in the display are set forth hereinafter.
FIG. 10 shows the filters
##EQU13##
for 0<u<1, 0<v<1. Each filter has 300 lines and v.sub.o =80 lines/mm. The
corresponding image-plane displays are contained in FIG. 12 for an object
with coarse grid structure (1 mm bar separation). In the central 0th
order, an image of the object appears. Both first diffraction orders are
shown. Observe the difference between these orders; in one
x=u-.lambda.f.sub.L V.sub.u, y=v-.lambda.f.sub.L V.sub.v and in the other
x=u+.lambda.f.sub.L V.sub.u, y=V+.lambda.f.sub.L V.sub.v. From this change
of sign, if follows that an area contraction in one of the distorted
images is related to an expansion in the other, a rotation in a positive
sense in one with a negative rotation, in the other, etc. One line in
FIGS. 11(a)-11(c) is not transformed, namely, the vertical bar to the very
left in the object and in FIG. 11(d) one point, the lower-left hand
corner.
Two crossed one-dimensional grid structures may be used as phase filters as
set forth hereinabove. This implies that the variables are separable. Even
in cases where this is not possible, the same type of approach may be
applied. In some situations, this possibility had advantages. For example,
different transforms of the same field can be displayed simultaneously,
plotter requirements may be simplified, amplitude-uniformity
considerations can be incorporated and it is helpful in some special
space-variant spatial-filtering problems.
An example in which the transformed image is formed with a crossed-grid
structure is illustrated in FIGS. 12 and 13. A filter of the shape shown
in FIG. 12 was made by double exposure. The intended transformation
(exponential transformation) was to map a square area conformally onto a
90.degree. sector of an annular area. The image plane display is shown in
FIG. 13, the crossed order to the upper right is the 90.degree. sector
transformation of the square grid structure (object) at the lower left.
The two crossed grid structures may be placed in different planes, e.g.,
one in the object plane and the other in F in the embodiment of FIG. 3.
The present invention provides convenient and powerful optical techniques
for establishing a space-variant system; they are practical and efficient
in most situations. General types of coordinate transformations as well as
local image modifications such as translation, stretching and rotation may
be achieved. The optical techniques can be used to solve problems in field
flattening, imaging on curved surfaces, distortion correction, or in
replacing complicated conventional devices such as zoom lenses and image
rotators. Coding and decoding of information as well as general
cryptography are among their many natural applications.
Although othe prior art techniques are available to provide modified
images, such as using corrector plates, axicons, conical lenses, ring
lenses; fun house mirrors and shower glass type plates for distorting
images; and image processing with digital techniques, fiber optics
devices, scanning systems, both optical and electronic; the prior art
techniques are mainly indirect and/or complex in contradistinction to the
simplified powerful and efficient technique of the present invention.
While the invention has been described with reference to its preferred
embodiment, it will be understood by those skilled in the art that various
changes may be made and equivalents may be substituted therefor without
departing from the true spirit and scope of the invention. In addition,
many modifications may be made to adapt a particular situation or material
to the teaching of the invention without departing from its essential
teachings.
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