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Description  |
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BACKGROUND OF THE INVENTION
This invention relates to a method and apparatus for real time optical
image processing, i.e., for computing and processing images utilizing a
passive phase conjugate mirror.
There has been a growing interest in real-time optical image processing for
convolving and correlating objects with spatial information, performing
inversion of gray-scale objects, subtraction, and differentiation. In
recent years, photorefractive crystals have been used to perform such
real-time image processing operations. For example, real-time convolution
and correlation have been demonstrated by J. O. White and A. Yariv,
"Real-time image processing via four-wave mixing in a photorefractive
medium," Appl. Phys. Lett., Vol. 37, pp 5-7 (1980). Edge enhancement has
been reported by J. Feinberg, "Realtime edge enhancement using
photorefractive effect," Opt. Lett., Vol. 5, pp 330-332 (1980). Work
related to image subtraction has also been reported by J. P. Huignard, J.
P. Herrian, and F. Micherson, "Coherent selective erasure of superimposed
volume holograms in LiNbO.sub.3," Appl. Phys. Lett., Vol. 26, pp 225-258
(1975) and by Y. H. Ja, "Real-time image subtraction in four-wave mixing
with photorefractive Bi.sub.12 GeO.sub.20 crystal," Opt. Comm., Vol. 42,
pp 377-388 (1982). The latter employed two sequential exposures of the
hologram and was thus not strictly real-time. Differentiation, division
and inversion have also been reported by Y. H. Ja., "Real-time optical
image differentiation by degenerate four-wave mixing," Appl. Phys. B.,
Vol. 36 pp 21-24 (1985), "Real-time image division in four-wave mixing
with photorefractive Bi.sub.12 GeO.sub.20 crystal, " Opt. Comm., Vol. 44,
pp 24-28 (1982), and E. Ochoa, et al., "Real-time intensity inversion
using two waves and four-wave mixing in photorefractive Bi.sub.12
GeO.sub.20," Appl. Opt., Vol. 24, pp 1826-32 (1985), respectively. An
object of this invention is to provide simple, exact and high performance
optical systems for real-time image processing utilizing a passive
(self-pumped) phase conjugate mirror which reflects back through the
medium a coherent beam on the axis of the incident beam, as disclosed in
U.S. Pat. No. 4,529,273.
SUMMARY OF THE INVENTION
In accordance with this invention, a passive phase conjugate mirror (PPCM)
is employed to reflect two coherent beams derived from a laser by a beam
splitter back on their respective axis. Each beam is directed to the PPCM
through separate transparencies by a properly positioned mirror for each
beam. Since the beams pass through the beam splitter to be transmitted and
reflected, but in reverse order, a phase inversion occurs between them due
to the time reversibility of light. Upon being recombined
interferometrically at the beam splitter, the coherent reflected beams
produce an output beam incident on an image detector. The image of that
output beam is the "exclusive or" of the images of the two transparencies,
thus producing as an output image only the differences between the images
of the two transparencies. Image inversion may also be achieved by placing
a transparency of the image in one beam path and leaving the second beam
path clear. In a similar arrangement, a beam splitter is used to provide
two intersecting coherent beams, and a single transparency is placed at
some small distance from the intersection such that each beam reads a
slightly shifted image of the other. The phase locked conjugate images are
recombined at the beam splitter to provide an output beam proportional to
the first order differential. Second order image differentiation is
implemented in a similar way, but using a second beam splitter to provide
a third coherent beam intersecting with the other two at the same point.
The third beam is opposite the second beam with respect to the first so
that the beams read T(x,y), T(x+.DELTA.x,y) and T(x-.DELTA.,y). The
reflectivity of the second beam splitter positioned between the first beam
splitter and the intersection point is 50%, giving an output amplitude
proportional to the second order image differential at the output of the
first beam splitter.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates schematically an arrangement utilizing a passive phase
conjugate mirror (PPCM) to obtain the "exclusive-or" (difference) between
two image transparencies T.sub.1 and T.sub.2. With only one transparency
T.sub.1 or T.sub.2 in place, the arrangement performs an inversion
operation of the one image. FIG. 1a illustrates the arrangement for
performing an inversion operation achieved by setting .vertline.T.sub.2
.vertline..sup.2 =1, thus effectively removing transparency T.sub.2 in
FIG. 1.
FIG. 2a illustrates the image on a transparency T.sub.1 of a colon after
phase conjugation and FIG. 2b illustrates the image on a transparency
T.sub.2 of a semicolon after phase conjugation. FIG. 2c then illustrates
the "exclusive-or," i.e., image difference (T.sub.2 -T.sub.1) between
transparencies. FIG. 2d illustrates the image output of the second
detector D.sub.2 downstream from the recombining beam splitter in the path
of the incident beam. That image is proportional to the sum of the
intensities [.vertline.T.sub.1 .vertline..sup.2 +.vertline.T.sub.2
.vertline.].sup.2 when the square of the coefficient of reflection of the
beam splitters is 0.5.
FIGS. 3a through d illustrate intensity inversion achieved by effectively
removing transparency T.sub.2 in FIG. 1 by setting .vertline.T.sub.2
.vertline..sup.2 =1. FIGS. 3a and 3b are the phase conjugate images of
uniform illumination and a resolution chart, respectively. FIG. 3c is the
intensity inverted image detected by detector D.sub.1, and FIG. 3d is the
image addition observed by detector D.sub.2, which is proportional to
[3+.vertline.T.sub.1 .vertline..sup.2 ].sup.2.
FIG. 4 illustrates an arrangement for first order image differentiation
utilizing a PPCM.
FIG. 5a illustrates the image of a rectangle after phase conjugation, and
FIG. 5b illustrates the image of FIG. 5a after first order differentiation
at the recombining beam splitter.
FIG. 6a illustrates the image of a square after phase conjugation, and FIG.
6b illustrates the image of FIG. 6a after summing first order
differentiation with respect to the x and the y axis obtained by shifting
the images by .DELTA.x and .DELTA.y. FIGS. 6c and 6d illustrate the way
the images are shifted by .DELTA.x and .DELTA.y.
FIG. 7 illustrates an arrangement for second order image differentiation
utilizing two beam splitters to form three beams which cross at a point
and displacing the image transparency from the crossing point so that
beams on each side of a central beam will read T(x+.DELTA.x,y) and
T(x-.DELTA.x,y) while the center beam reads T(x,y).
FIG. 8 illustrates the image of a rectangle as shown in FIG. 5a after
second order differentiation.
The novel features that are considered characteristic of this invention are
set forth with particularity in the appended claims. The invention will
best be understood from the following description when read in connection
with the accompanying drawings.
DESCRIPTION OF PREFERRED EMBODIMENTS
Image Subtraction
Referring to FIG. 1, a real time image subtraction, hereinafter referred to
as "exclusive or" operation, is obtained with an interferometer
arrangement using a passive phase conjugate mirror (PPCM) 10. A wave with
amplitude E.sub.in is divided by beam splitter BS.sub.1 whose reflection
and transmission coefficients are equal to r and t, respectively, but are
applied to the split beams transmitted to and retroreflected from the PPCM
in reverse order, namely, first t and then r for beam 1, and first r and
then t for beam 2. The two beams are passed through respective
transparencies 11 and 12 with amplitude transmittance T.sub.1 for beam 1
and T.sub.2 for beam 2. Both beams 1 and 2 are retroreflected by the PPCM
with phase conjugate reflectivities R.sub.1 and R.sub.2, respectively.
(R.sub.1 and R.sub.2 are in general not the same.)
The reflected beams recombine interferometrically at beam splitter BS.sub.1
to form an output field intensity I.sub.out at a detector D.sub.1 given by
I.sub.out =[r't*R.sub.1 .vertline.T.sub.1 .vertline..sup.2 +tr*R.sub.2
.vertline.T.sub.2 .vertline..sup.2 ].sup.2 I.sub.in (1)
where I.sub.in =.vertline.E.sub.in .vertline..sup.2.
From Stokes' principle of the time reversibility of light
r't*+r*t=0 (2)
so that
I.sub.out =[R.sub.1 .vertline.T.sub.1 .vertline..sup.2 -R.sub.2
.vertline.T.sub.2 .vertline..sup.2 ].sup.2 .vertline.r*t.vertline..sup.2
I.sub.in (3)
If the two phase conjugate mirrors are identical, i.e., R.sub.1 =R.sub.2
=R, then in practice only one crystal is required to serve the function of
two phase conjugate mirrors
I.sub.out =[.vertline.T.sub.1.vertline..sup.2 -.vertline.T.sub.2
.vertline..sup.2 ].sup.2 .vertline.rt*R.vertline..sup.2 I.sub.in (4)
I.sub.out .varies..vertline.T.vertline..sup.2
.sym..vertline.T.vertline..sup.2 (5)
where .sym. represents the Boolean "exclusive or" operation. Similarly, the
field intensity I' measured by detector D.sub.2 is
I'.varies.[.vertline.T.sub.1 .vertline..sup.2 +.vertline.r.vertline..sup.2
(.vertline.T.sub.2 .vertline..sup.2 -.vertline.T.sub.1
.vertline..sup.2)].sup.2 .vertline.R.vertline..sup.2 I.sub.in (6)
Note that the .pi. phase shift between the complex fields of the two is
introduced naturally by the principle of time reversibility of light,
which for this invention provides that if the sequence of reflection (R)
and transmission (T) for the two beams is reversed, the phase of one beam
will be inverted relative to the other. This is an essential difference
between this method and other methods by Ja (Opt. Comm. 42, 377, 1982) and
others in which the .pi. phase shift is artificially provided by a
piezoelectric mirror or an electro-optical modulator. The arrangement of
the present invention is thus only sensitive to intensity differences of
the two transparencies and is independent of the phase information of the
transparencies or the optical path lengths of the two arms. This is
because each beam is retroreflected without any phase shift between the
incident beam and the reflected beam. Consequently, at the beam splitter
the beams are in phase regardless of any difference in path lengths for
the two beams. This makes the system accurate and simple because there are
no critical adjustments to be made as to the positions of the components.
In the experimental arrangement shown in FIG. 1, a single TEM.sub.00 mode
Argon laser 12 produced a beam (5145 A, 50 mW) that was expanded by a lens
L.sub.1 and aperture 14, and split into two beams (1 and 2) by 50% beam
splitter BS.sub.1. Each beam was then reflected by mirrors M.sub.1 and
M.sub.2 to pass through separte transparencies T.sub.1 and T.sub.2 to the
PPCM comprised of a poled BaTiO.sub.3 crystal. A lens L.sub.2 (f=30 cm)
was used to focus the two expanded beams which were adjusted to overlap
completely inside thew poled BaTiO.sub.3 crystal. The crystal was aligned
to form the passive (self-pumped) phase conjugate mirror 10 by setting the
angles between the beams and the crystal C-axis to .theta..sub.1
=50.degree. and .theta..sub.2 =40.degree..
The two image bearing beams were phase conjugated simultaneously with no
cross talk. The magnitude of the phase conjugate reflectivities of beam 1
and beam 2 were approximately the same and equal to 25%. The phases of the
complex phase conjugate reflections coefficients of the two beams are also
the same. Since the PPCM regards the combination of the two input beams as
a single complex input wave, and due to the beams' overlap in the crystal
they are both reflected from the same set of gratings. (Another method for
obtaining phase locking between the two phase conjugate beams is described
by M. D. Eubank, et al., Opt. Lett. 10, 282 (1985), in which a
self-induced oscillation locks the relative phase between the two phase
conjugate beams.) The phase conjugate reflected image bearing beams were
then combined interferometrically at beam splitter BS.sub.1. The two
transparencies and the detectors were placed close to the beam splitters
to reduce diffraction aberration.
The transparencies T.sub.1 and T.sub.2 used in the experiment were pictures
of a colon and a semicolon respectively. The phase conjugate images of
these two transparencies are shown in FIGS. 2a and 2b, respectively. FIG.
2c is the image detected by detector D.sub.1, which represents the
"exclusive or" operation between the two images, .vertline.T.sub.1
.vertline..sup.2 .sym..vertline.T.sub.2 .vertline..sup.2. FIG. 2d is the
image recorded by detector D.sub.2, which represents [.vertline.T.sub.1
.vertline..sup.2 +.vertline.r.vertline..sup.2 (.vertline.T.sub.2
.vertline..sup.2 -.vertline.T.sub.1 .vertline..sup.2)].sup.2 and it is
proportional to the sum of intensities, [.vertline.T.sub.1
.vertline..sup.2 +.vertline.T.sub.2 .vertline..sup.2 ].sup.2, when
.vertline.r.vertline..sup.2 =0.5. Slight edge enhancement effects were
also observed in these figures which are probably due to large object beam
intensities as compared to the weaker pump beam intensities. These results
are independent of the optical path lengths of either beam between the
beam splitter BS.sub.1 and the PPCM crystal.
The response time of the PPCM obeyed approximately the relation
.tau..about.10/I where I is the total intensity of the interaction beams
in mW/mm.sup.2.
Intensity Inversion
Optical intensity inversion was also observed by simply removing
transparency T.sub.2, i.e., making T.sub.2 =1, so that the intensity
detected by detector D.sub.1 is proportional to [1-.vertline.T.sub.1
.vertline..sup.2 which result follows from Equation (3) when
.vertline.T.sub.2 .vertline..sup.2 =1. FIGS. 3a and 3b are the phase
conjugate images of a uniform illumination and a resolution chart,
respectively. FIG. 3c is the intensity inverted image detected by detector
D.sub.1. FIG. 3d is the image addition observed by detector D.sub.2, which
is proportional to [1+.vertline.T.sub.1 .vertline..sup.2 ].sup.2.
Intensity inversion by a different method which uses four wave mixing was
reported by Ochoa, et al., Appl. Opt.,supra. In their method, the object
beam intensity is required to be much larger than that of the reference
beam in order to ensure that the diffraction efficiency of the index
grating is inversely proportional to the object beam intensity.
Image Differentiation
Differentiation of a function I(x,y) can be approximated to any degree of
accuracy by using finite differences. Using such a method, the first and
second-order differentials are given by
##EQU1##
Therefore, the differential of any order can be obtained by adding and
subtracting various shifted images of a pattern function.
The experimental arrangement used to perform first-order differentiation,
which is very similar to the experimental arrangement for FIG. 1, but with
two beams 1 and 2 crossing, is shown in FIG. 4. In order to provide
continuity, the same reference numerals are employed for elements common
to the arrangement of FIG. 1. An image transparency T is placed not at the
intersection of the two beams 1 and 2, but at some small distance from the
intersection such that each beam reads a slightly shifted image of the
other. The two images are then focused down into the BaTiO.sub.3 crystal
which, through total internal reflection, forms a self-pumped phase
conjugate mirror 10. Note that while two mirrors, M.sub.1 and M.sub.2, are
still employed, they are now deployed to cause both beams to cross and
pass through the same transparency, but offset by .DELTA.x as noted above,
and the lens L.sub.2 is again used to focus the two beams to overlap
completely inside the poled BaTiO.sub.3 crystal aligned to form the PPCM.
The phase conjugate images are thus phase locked and can recombine at the
beam splitter BS.sub.1 , giving an output amplitude which is proportional
to the first order differential. Results are given in FIGS. 5a and 5b.
FIG. 5a shows a rectangle image from the transparency T.sub.1 after phase
conjugation i.e., without the beam splitter BS.sub.1, to detect only the
beam 2, and FIG. 5b shows the image of FIG. 5a after first order
differentiation .differential./.differential.x by this technique.
FIG. 6a shows a square image from the transparency T.sub.1 after phase
conjugation and FIG. 6b shows the differential
(.differential./.differential.x+.differential./.differential.y) obtained
by shifting the beams through the image transparencies by
.DELTA.x+.DELTA.y.
Shifting the image for beams 1 and 2 by .DELTA.x and .DELTA.y is achieved
in the following way. First position the transparency T.sub.1 at a
position displayed from the crossing point of the two beams to provide a
displacement placement .DELTA.x between them on the transparency, as shown
in FIG. 6c, where the axis of the beam 1 is represented by a dot in the
center of that beam, and the displacement is indicated by the distance
between a dot in the center of beam 2. Then to displace the two beams at
the transparency by a distance .DELTA.y in the plane of the transparency,
rotate the beam splitter BS.sub.1 and the position of the mirror M.sub.2
about the axis of the beam 1 as shown in FIG. 6d, and adjust the position
of the lens L.sub.12.
Second-order image differentiation is also possible. FIG. 7 illustrates an
arrangement for second-order image differentiation. It is the same as for
the arrangement of FIG. 6 for a first-order image differentiation, and so
the same reference numerals are used for the elements common to both in
the processing of beams 1 and 2. It is the processing of a beam 3 on the
outside opposite beam 1 that is added. The two outside beams 1 and 3 read
T(x+.DELTA.x,y) and T(x-.DELTA.x,y), while the center beam 2 reads T(x,y).
Reading T(x-.DELTA.x,y) with beam 3 is achieved by positioning the mirror
M.sub.3 on the opposite side of the axis of beam 2 from beam 1 by the same
angle. This yields a .DELTA.x of a sign (-) opposite the .DELTA.x from
beam 1 with respect to beam 2. The reflectivity of beam splitter BS.sub.2
is 50%, giving an output amplitude at BS.sub.1 proportional to the
second-order differential.
I.sub.out .varies.[.vertline.T(x+.DELTA.x,y).vertline..sup.2
+.vertline.T(x-.DELTA.x,y).vertline..sup.2
-2.vertline.T(x,y).vertline..sup.2 ].sup.2 (9)
Results are shown in FIG. 8 of a rectangle.
Although particular embodiments of the invention have been described, it is
evident that the underlying techniques for computing and processing images
may be used as subsystems of a larger system embodying combinations of the
various underlying techniques. For example, extensions to higher order
image differentiation can be obtained by adding up appropriately shifted
images in groups, the sums of which are then subtracted at the final beam
splitter. While the final beam splitter reflectivity can be arbitrary, the
remaining ones must have their reflectivities chosen to perform the image
additions in the correct proportions. The many shifted images required for
higher order derivatives may cause difficulties in focusing all the images
into a single crystal. In that case, two or more crystals phase locked
through spontaneous oscillations could be used to handle the many images.
Another possibility is to couple the multiple images into a single PPCM
crystal using a single optical fiber by focusing all beams onto the input
end of the optical fiber.
As another example, using methods similar to those shown for image
differentiation, a device can be constructed whose output amplitude is
proportional to the Laplacian function .gradient..sup.2 I(x,y), where
I=.vertline.T(x,y).vertline..sup.2. Once a beam is expanded and split into
two beams, one of the beams can be used to read T(x,y) while the other
beam is split into four beams of equal intensity using 50% beam splitters.
These four beams can then be used to read images of T(x,y) shifted by
.+-..DELTA.x and .+-..DELTA.y. After retroreflection by PPCM, these beams
will then recombine at the first beam splitter giving an output
proportional to .gradient..sup.2 I(x,y), where .gradient..sup.2 is a
Laplacian operator. Consequently, it is intended that the claims be so
interpreted as to include application of the techniques described with
reference to particular embodiments to such other more complex systems.
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