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Description  |
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BACKGROUND OF THE INVENTION
This invention relates to the determination of the surface height of
three-dimensional objects and, more particularly, to an improved apparatus
and method of surface profilometry.
Surface profile measurement by non-contact optical methods has been
extensively studied because of its importance in fields such as automated
manufacturing, component quality control, medicine, robotics and solid
modeling applications. In most of these methods a known periodic pattern,
such as a grating, is projected on the surface to be measured and the
image of the grating, deformed by the surface, is analyzed to determine
the profile. "Demodulation" of the deformed grating by means of a matched
reference grating results in the well known Moire fringe patterns, which
are easily interpretable as surface contours by a human observer, but, are
somewhat more complicated for computer analysis. (See, for example, D. M.
Meadows, W. O. Johnson and J. B. Allen, Appl. Opt. 9, 942 (1970); H.
Takasaki, Appl. Opt. 9, 1467 (1970); P. Benoit, E. Mathieu, J. Hormiere
and A. Thomas, Nouv. Rev. Opt. 6, 67 (1975); T. Yatagai, M. Idesawa and S.
Saito, Proc. Soc. Photo-Opt. Instrum. Eng. 361, 81 (1982)). Improvements
to the Moire method, aimed at increasing accuracy and at automating the
measurements have been based, for example, on phase modulation. (See G.
Indebetouw, Appl. Opt. 17, 2930 (1978), D. T. Moore and B. E. Truax, Appl.
Opt. 18, 91 (1979).
An alternative approach to Moire is by an analysis of the deformed grating
itself without the use of a physical or virtual reference grating. Direct
methods based on geometrical analysis of the deformed grating requiring
fringe peak determination are computationally complex, slow, and result in
low accuracy. Another direct method, based on the use of a Fast Fourier
Transform analysis of the deformed grating, has been demonstrated to be
more suitable for automated profilometry (see, for example, M. Takeda and
K. Mutoh, Appl. Opt. 22, 3977 (1983)). Limitations on measurement of steep
object slopes and step discontinuities, the need for high resolution
imaging systems and the need for powerful computing capability are some of
the disadvantages of the Fast Fourier Transform method.
It is among the objects of the invention to provide an improved method and
apparatus for surface profilometry.
SUMMARY OF THE INVENTION
The invention employs, inter alia, phase measuring techniques that have
been used in classical interferometry, but are particularly advantageous
for use in deformed grating profilometry. When a sinusoidal intensity
distribution is projected on a three dimensional diffuse surface, the
mathematical representation of the deformed grating image intensity
distribution is similar to that encountered in conventional optical
interferometry. The surface height distribution can be translated to a
phase distribution, and the accuracy which is characteristic of phase
modulation interferometry, can be used to advantage here. [See, for
example, J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld,
A. D. White and D. J. Brangaccio, Appl. Opt. 13, 2693 (1974); J. C. Wyant,
Appl. Opt. 14, 2622 (1975) for background.] Further, by using several
phase modulated frames of deformed grating image data, a high degree of
precision in the phase measurement can be achieved. By analogy with
phase-measuring interferometry, where phase values can be measured with a
resolution of 1/1000 of a fringe period (versus 1/30 for conventional
single frame interferometry), surface profile measurements with less than
10 micron resolution are possible by the use of an optical system with a
projected grating pitch in the millimeter range.
In accordance with an embodiment of the method of the invention, a
technique is set forth for determining the surface profile of a
three-dimensional object. An incident beam of light, having a sinusoidally
varying intensity pattern, is directed at the object. The phase of the
sinusoidal pattern of the incident beam is modulated. A deformed grating
image of the object is received, at a detector array, for a number of
different modulated phases of the input beam. The height of each point
(i.e., elemental region) of the surface of the object is then determined
with respect to a reference plane, each such height determination
including the combining of the image intensity values at a detector
position corresponding to a respective point of the object.
In the disclosed embodiments of the invention, the combined intensity
values, for the object and for the reference plane, are used to obtain an
object phase for each point and a reference phase for each point. In one
embodiment the difference between the object and reference phases for
corresponding points is used in the height determination. In another
embodiment, a phase mapping as between object and reference phases is used
in the height determination.
Among the advantages of the invention are the following: relatively simple
optical hardware; relatively low frequency grating and low density
detector array; full frame data capture and relatively simple processing.
Further features and advantages of the invention will become more readily
apparent from the following detailed description when taken in conjunction
with the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a block diagram, partially in schematic form, of an apparatus in
accordance with the invention, and which can be used to practice the
method of the invention.
FIG. 2 is a schematic diagram of an embodiment of a projection and phase
shifting system as used in the FIG. 1 embodiment.
FIG. 3 illustrates the optical geometry of a form of the FIG. 1 embodiment,
as used for an analysis of operation.
FIG. 4, which includes FIGS. 4A and 4B placed one below another, is a flow
diagram of a routine for programming the processor of the FIG. 1
embodiment in accordance with a form of the invention.
FIG. 5 shows a deformed grating interferogram of a three dimensional object
as seen by the detector array of the FIG. 1 embodiment.
FIG. 6 is a surface profile block of the three dimensional object of FIG.
4, as generated using an embodiment of the invention.
FIG. 7 is a schematic diagram which illustrates a further embodiment of the
invention, and is used in describing the optical geometry thereof.
FIG. 8, which includes FIGS. 8A and 8B placed one below another, is a flow
diagram of a routine for programming the processor of the FIG. 7
embodiment for operation in accordance with a further form of the
invention.
FIG. 9 shows the result of a calibration experiment, using a technique in
accordance with the embodiment of FIG. 7.
FIG. 10 shows a mannequin face with a projected sinusoidal grating of 15 mm
period.
FIG. 11 is a perspective profile plot of the mannequin face with respect to
a reference plane, as obtained using an embodiment of the invention.
FIG. 12 is a flow diagram of a routine that is used in conjunction with the
routines of FIGS. 4 and 8.
DESCRIPTION OF THE PREFERRED EMBODIMENT
Referring to FIG. 1, there is shown a block diagram of an apparatus in
accordance with the invention, and which can be used to practice the
method of the invention. A sinusoidal grating projection system and phase
shifter 110 is provided, and projects an optical beam at a three
dimensional object 150. The reflected beam is received by a detector array
systems 120 which, in the illustrated embodiment detects frames of data
which are stored in buffer 140 and processed by processor 150. The
processor 150 may be, for example, any suitable analog or digital
computer, processor, or microprocessor and conventional associated memory
and peripherals, programmed consistent with the teachings hereof. In the
experiments described hereinbelow, a model LSI 11/23, made by Digital
Equipment Corp., was utilized. The processed information can be displayed
on a display device 160.
A sinusoidal intensity distribution can be projected on the surface to be
profiled, e.g. by generating an interference pattern between two coherent
plane wavefronts or by projecting an image of a grating with sinusoidal
transmission function distribution illuminated by an incoherent light
source. FIG. 2 illustrates an embodiment of the projection system and
phase shifter 110 (of FIG. 1), which comprises a laser illuminated
shearing polarization interferometer. The linearly polarized output beam
from the laser 111 is spatially filtered by filter 112, which includes
lens 113 and pinhole 114, and then sheared by a Wollaston prism W. The
phase modulator includes a combination of a quarter wave plate Q and a
rotatable polarized P. By rotating the polarizer, the sinusoidal intensity
distribution of the interference pattern can be modulated. A 180.degree.
rotation of the polarizer corresponds to a 2.pi. phase modulation and this
permits precise phase shifts. It will be understood that other types of
phase shifters, for example polarization dependent phase shifters such as
electro-optic modulators, may also be used in this system. The fringe
period can also be easily changed by an axial translation of the Wollaston
prism W. A collimating lens L is used to conserve light and simplify the
optical geometry.
Before further describing operation of the apparatus of this embodiment,
consider the diagram of FIG. 3 in which the height h(x,y) of object 150 is
to be measured relative to the indicated reference plane. The projected
sinusoidal interferometric pattern has a period p.sub.o as seen on the
reference plane, and the intensity that it produces at a point, such as C
on the reference plane, is
I=a(x,y)+b(x,y) cos (2.pi.OC/p.sub.o) (1)
where a(x,y) is the background or DC light level, b(x,y) is the fringe
contrast and O, the intersection of the imaging optical axis with the
reference plane is assumed, for convenience, to coincide with an intensity
maxima of the projected pattern. The argument .phi..sub.R =2.pi.OC/p of
the cosine function in Eq. (1) is defined as the "phase" at C and it
effectively measures the geometric distance OC, if O is taken at the
reference point where the phase is zero. A.sub.n is one of the detectors
in the array, located at the image plane and is used to measure the
intensity at C on the reference plane and at D on the object. An imaging
lens 120a, of the detection system 120, is shown in the FIG. 3 diagram.
The intensity observed at D is the same as that which would have been
observed at A on the reference plane, modified by the object reflectivity
r(x,y), that is
I.sub.D =r(x,y)[a(x,y)+b(x,y) cos (2.pi.OA/p.sub.o)] (2)
The difference .DELTA..phi..sub.CD in phase values for the points C and D,
observed by the same detector A.sub.n, can be related to the geometric
distance AC as follows:
AC=(p.sub.o /2.pi.).multidot..DELTA..phi..sub.CD (3)
AC is related to the surface height BD as follows:
BD=AC tan .theta..sub.o /(1+tan .theta..sub.o /tan .theta..sub.n) (4)
where the angles .theta..sub.o and .theta..sub.n as shown in FIG. 3.
Assuming that .theta..sub.n is nearly 90, as is the case for any practical
system with a large demagnification factor, the relationship (4) can be
simplified to:
BD=AC tan .theta..sub.o (5)
From Eqs. (3) and (5), the effective wavelength of the system is defined as
.lambda..sub.e =p.sub.o tan .theta..sub.o.
To measure the phase of the intensity variation represented by either Eq.
(1) of Eq. (2), the projected pattern can be phase modulated by rotating
the polarizer P (FIG. 2). In the case of Eq. (1), let
.phi..sub.R =2.pi.OC/p.sub.o =2.pi.M+.phi..sub.R ' (6)
where .phi..sub.R ' is the phase angle reduced to the range 0-2.pi. and M
is an integer. If .phi..sub.M represents the phase modulation, then from
Eq. (1),
I.sub.c =a(x,y)+b(x,y) cos (.phi..sub.M +.phi..sub.R ') (7)
N measurements I.sub.1, I.sub.2, . . . I.sub.N of I.sub.C are made with a
phase increment of 2.pi./N following each measurement. From these
measurements, one obtains,
##EQU1##
[See J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A.
D. White and D. J. Brangaccio, Appl. Opt. 13, 2693 (1974).] By recording N
frames of intensity data, the phase seen by each detector in the array can
be computed, both for the reference plane and the object surface. Based on
the continuity of the phase function, starting from a reference location
with zero phase, the integer M of Eq. (6) can also be determined by
monitoring the computed phases between two adjacent detectors and
identifying sharp phase discontinuities which result from the 2.pi.
transitions. Eqs. (3) and (5) can then be used to compute the surface
profile, as is further described below.
Referring to FIG. 4, there is shown a flow diagram of a routine for
programming the processor 150 of FIG. 1 in accordance with a form of the
first embodiment of the invention. The block 411 represents the setting of
the initial phase of the incident beam of sinusoidally varying intensity;
e.g., for example, to a reference phase designated as zero degrees. A
frame of data (intensity information) is then collected and stored from
the detector array to get a first set of intensity values for the object
designated as I.sub.o1 (x,y) values. A determination is then made (diamond
413) as to whether or not the last frame of the sequence has been
collected and stored. If not, the block 414 is entered, this block
representing the shifting of the phase of the incident beam, such as by
rotating the polarizer (FIG. 2). In the present embodiment, three frames
are used in the sequence, so there are two phase shifts of 120 degrees
each. It will be understood that the phase shifting can be implemented
automatically under computer control, semi-automatically, or manually, as
desired. Accordingly, at the completion of the loop 415, three frames of
intensity values for the object, I.sub.on (x,y) are obtained, as follows:
I.sub.o1 =a+b cos (.phi..sub.M +.phi..sub.o '(x,y))
I.sub.o2 =a+b cos (.phi..sub.M +.phi..sub.o '(x,y)+2.pi./3)
I.sub.o3 =a+b cos (.phi..sub.M +.phi..sub.o '(x,y)+4.pi./3)
The procedure of loop 415 is then repeated for the reference plane, as
represented by the block 420. This data can be obtained, for example, by
placing a white reflective reference plane adjacent the object, as
illustrated in FIG. 3, and following the procedure just described to
obtain frames of reference intensity values I.sub.rn (x,y) as follows:
I.sub.r1 =a+b cos (.phi..sub.M +.phi..sub.r '(x,y))
I.sub.r2 =a+b cos (.phi..sub.M +.phi..sub.r '(x,y)+2.pi./3)
I.sub.r3 =a+b cos (.phi..sub.M +.phi..sub.r '(x,y)+4.pi./3)
It will be understood that the data could alternatively be taken in any
desired sequence, such as by interposing the reference plane before each
phase shift so as to obtain both the object and reference intensity
information at each incident beam phase, although this is not preferred.
Also, it will be understood that the reference phase information can be
computed without taking measured reference data (from the known
characteristics of the incident beam and the known system geometry), but
it is preferable, when possible, to obtain the reference data from an
actual reference plane so as to minimize the effects of distortions in the
system, etc.
Next, for a point (x,y) the reference plane phase .phi..sub.r '(x,y) at the
point can be computed, using (8) above, from the three reference plane
intensities as:
##EQU2##
as represented by the block 431. The block 432 is then entered, this block
representing the computation, for point (x,y) of the object phase from the
three object intensity measurements as:
##EQU3##
A determination is made as to whether or not the last point has been
processed (diamond 433), and, if not, the processor increments to the next
point (block 434), and the loop 435 continues until the reference and
object phases have been computed for all desired points (x,y).
The block 441 is then entered. This block represents the procedure of
assigning, for each point (x,y) the appropriate integer m (see equation
(6)) by tracing the number of fringes on the reference plane image, where
m is the fringe number, and then the determination of the reference plane
phase for each such point. The block 442 represents the same operation for
each point of the object. FIG. 12 is a flow diagram of a routine as
represented by the blocks 441 and 442 for tracing the deformed grating
fringes in order to assign appropriate m integer values, so that the
object and reference phases .phi..sub.o (x,y) and .phi..sub.r (x,y),
respectively, can be obtained from .phi..sub.o '(x,y) and .phi..sub.r
'(x,y). A y coordinate index is initialized (block 1211), m is initialized
at zero (block 1212), and the x index is also intialized (block 1213).
Processing then proceeds on a line-by-line basis along the x direction.
For a given line, at each point, the previously computed phase value (for
the object phase or the reference plane phase, depending on which one is
being processed), the phase at each point is compared to the phase at the
previous point, as represented by the diamond 1220. The adjacent phase
values are compared by determining when there is a transition over a 2.pi.
value, and the sign of the transition is also noted. The sign will depend
on the slope of the object profile. Blocks 1221 and 1222, as the case may
be, are then utilized to decrement or increment m, depending upon the sign
of the transition, and the block 1225 is then entered. Also, if there was
no transition at the point being examined, the block 1225 is entered
directly. The block 1225 then represents the computation of the reference
plane or object phase value (as the case may) in accordance with
relationship (6). The x index is then tested (diamond 1230) and
incremented (block 1231), and the loop 1232 continues for processing of an
entire line in the x direction on the detector array. When the line is
complete, the y index is tested (diamond 1240) and incremented (block
1241) and the loop 1233 is continued until all lines of points have been
processed.
Referring again to FIG. 4, the block 451 represents the selection of the
first point (x,y) for height computation. For the particular point (x,y),
the phase difference between the object and reference planes is computed
in accordance with:
.DELTA..phi.(x,y)=.phi..sub.o (x,y)-.phi..sub.r (x,y)
as represented by the block 452. The distance AC (FIG. 3) can then be
computed (block 453) from
##EQU4##
Next, the block 454 represents the conversion into height BD, which is
defined as h(x,y) in accordance with equation (5) as
n(x,y)=BD=AC tan .theta..sub.o
It can be noted that suitable calibration factor and geometric correction
can also be applied to h(x,y). h(x,y) can then be stored for display or
other use, as represented by the block 456. Inquiry is then made (diamond
457) as to whether or not the last point has been procesed and, if not,
the point being processed is incremented (block 458) and the loop 459
continues as the height values h(x,y) are obtained and stored for each
point.
Surface profile measurements were made, using the system of FIGS. 1, 2, on
a general test object (a half cylinder with two sections having different
radii), mounted on a reference plane and illuminated with a sinusoidally
varying beam intensity as previously described. In order to generate a
phase variation in both the horizontal as well as vertical directions an
inclined set of fringes were projected on the object. FIG. 5 shows the
deformed grating as seen by the detector array. Three images each were
recorded for the reference plane and the object surface, with a phase
increment of 120.degree. of the projected fringe pattern following each
recording, and processing was performed as previously described. FIG. 6
shows part of the surface profile plot, which was generated by displaying
h(x,y) using a graphics plotter. The two sections of the object with
different radii and the transition region are seen. The values obtained
were very close to those measured using a contact profilimeter.
In the embodiment of FIG. 7, the collimated laser illumination is replaced
by the projected image of a translatable white light illuminated
sinusoidal transmission grating. This system requires generally less
expensive hardware than the previous embodiment, and is more capable of
handling large objects. The analysis of this system, because of the
divergent nature of the illumination and because the optical axes of the
projection and imaging systems are not necessarily parallel, is somewhat
more complicated. In the FIG. 7 arrangement it is assumed that a buffer,
processor, display and projection and imaging lenses are provided, as in
the previous embodiment. The optical geometry of the projection and
recording systems is represented in FIG. 7. P.sub.1 and P.sub.2 are the
centers of the entrance and exit pupils of the projection optics. I.sub.1
and I.sub.2 are the centers of the exit and entrance pupils of the imaging
optics. G is a grating with pitch p and a sinusoidal intensity
transmission. D.sub.c is one element of the image sensing array. The
intensity variation along x on the reference plane can be described by the
equation:
I=a(x,y)+b(x,y) cos .phi.(x) (9)
where a(x,y) is the background or DC light level and b(x,y) is the fringe
contrast. The phase .phi. in this case is a non linear function of x
because of the divergent nature of the image forming rays. With respect to
a reference point such as O, every point on the reference plane is
characterized by a unique phase value. For example, the point C, observed
by the detector D or the array, has
.phi..sub.c =2.pi.m+.phi..sub.c ' (10)
where m is an integer and 0<.phi..sub.c '<2.pi..
The detector array samples the reference plane (as well as the object) and
is again used to measure the phase at the sampling points by a phase
shifting technique. As before, N frames of intensity data, with N>2, are
recorded and after each frame the grating G is translated by a distance
P.sub.o /N. If I.sub.1, I.sub.2, . . . I.sub.N are the intensity
measurements for a point such as C, then, as previously described
##EQU5##
As the phase function is continuous, it is possible, as previously
described, to determine m in equation (10) by detecting sharp phase
changes of nearly 2.pi. which occur between two neighboring sample points,
whenever a complete grating period has been sampled.
A particular detector such as D.sub.c can measure the phase .phi..sub.c at
a point C on the reference plane as well as .phi..sub.D on the point D of
the object. A mapping procedure is then used to determine a point A on the
reference plane such that .phi..sub.A =.phi..sub.D. This enables a
computation of the geometric distance AC. C is a known detector location
and the position of A, which in general would lie between two sampling
points, can be located by linear interpolation using known phase values.
From similar triangles P.sub.2 DI.sub.2 and ADC, the object height is
h(x,y)=(AC/d).multidot.l.sub.o (1+AC/d).sup.-1 (12)
where d and l.sub.o are distances as shown in FIG. 1. As in most practical
situations d>>AC because of the large magnifications involved, equation
(12) can be simplified:
h(x,y)=(AC/d).multidot.l.sub.o (13)
It can be noted that h(x,y) is the object height at the x coordinate
corresponding to B and not C. From known system geometrical parameters,
one can calculate the distance BC and thus determine the x coordinate
corresponding to the point B.
In the routine of FIG. 8, the block 810 represents the collecting and
storing of three frames of reference plane and object data in a manner
similar to that previously described in conjunction with the loop 415 of
FIG. 4. In this case, however, in the context of the system of FIG. 7, the
phase shifting will be implemented by translation of the grating G. In
this case, the three arrays of object intensity values can be represented
as
I.sub.o1 =a'(x,y)+b'(x,y) cos .phi..sub.o (x,y)
I.sub.02 =a'(x,y)+b'(x,y) cos (.phi..sub.o (x,y)+2.pi./3)
I.sub.o3 =a'(x,y)+b'(x,y) cos (.phi..sub.o (x,y)+4.pi./3)
and the three arrays of reference plane intensity values can be represented
as
I.sub.r1 =a'(x,y)+b'(x,y) cos .phi..sub.r (x,y)
I.sub.r2 =a'(x,y)+b'(x,y) cos (.phi..sub.r (x,y)+2.pi./3)
I.sub.r3 =a'(x,y)+b'(x,y) cos (.phi..sub.r (x,y)+4.pi./3)
For a point (x,y), the object phase .phi..sub.o ' is then computed from
##EQU6##
and the reference phase .phi..sub.r ' is computed in accordance with
##EQU7##
These functions are represented by the blocks 831 and 832, and are
consistent with the equation (11) above. The loop 835 then continues, in
similar manner to that previously described in conjunction with FIG. 4, to
obtain the reference and object phases for all desired points (x,y),
diamond 833 and block 834 being used in the loop in the manner previously
set forth.
Next, as represented by the block 841, for each point (x,y), the
appropriate integer m (see equation (10)) is obtained by tracing the
number of fringes on the reference plane image, where m is the fringe
number, and the reference plane phase for each such point is then
determined. The block 842 represents the same operation for each point of
the object. Reference can again be made to FIG. 12, and the accompanying
description above, for the manner in which the resultant phases
.phi..sub.r (x,y) and .phi..sub.o (x,y) are obtained.
FIG. 8B includes the phase mapping portion of the routine. A first point is
selected for phase mapping, as represented by the block 851. For the point
D on the object, the phase .phi..sub.o (x,y) is compared to the phases of
the sampling points on the reference plane, so as to locate point A on the
reference plane which has the same phase. The geometric distance AC,
between the points having the same phase, is then determined in terms of
detector element spacings (block 853). The object height can then be
computed (equation (12)) as:
h(x,y)=(AC/d)l.sub.o (1+AC/d).sup.-1
as represented by the block 854. Alternatively, as described above,
equation (13) could be used in some situations. Again, suitable
calibration factor and geometric correction can then be applied, and the
height h(x,y) is stored, as represented by block 856. Inquiry is then made
(diamond 857) as to whether or not the last point has been processed. If
not, the point to be processed is incremented (block 858), and the loop
859 is continued to completion.
For experimental measurements, sinusoidal gratings were generated by
photographing out of focus a square wave grating pattern. A conventional
slide projector was modified in order to operate with a grating slide,
mounted on a stepper motor driven translation stage. The period of the
projected grating measured on the reference plane, close to the optical
axis of projection, was about 15 mm. Deformed grating images were recorded
on a 128.times.128 photodiode array detector. Phase measurement was by a
three discrete phase shift implementation of equation (11), and processing
was in accordance with the routine previously described.
The results of a calibration experiment, using a cylindrical test object,
which had been measured by a mechanical contact profilometer, are shown in
FIG. 9. The line profile was generated using the experimental optical
system and the `X` marks indicate measurements made by a manual contact
profilometer. An agreement of better than 1% between the measurements was
observed, except in the regions with steep slopes where mechanical contact
methods are not very reliable.
A more general type of diffuse object, a mannequin face, was also measured.
FIG. 10 shows the deformed grating produced by this object. FIG. 11 is a
graphical representation of the results obtained with the optical system
and shows 97 profile lines with 87 points on each line. This object was
approximately 150 mm wide and the depth at the tip of the nose measured
from a reference plane was about 100 mm. In order to obtain a complete
shadow-free illumination of such objects and obtain a large angular field
of view, two or more projected gratings may be used.
The invention has been described with reference to particular preferred
embodiments, but variations within the spirit and scope of the invention
will occur to those skilled in the art. For example, various alternative
types of projection and detector systems can be used, consistent with the
principles of the invention, and the processor steps can be performed in
various suitable orders.
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