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Description  |
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BACKGROUND OF THE INVENTION
This invention relates to the determination of the surface contours of
objects and, more particularly, to an improved apparatus and method for
determining three dimensional surface profiles of objects, including 360
degree renditions thereof.
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.
An approach to obtaining a complete 360 degree view of a general
three-dimensional object, which has been the subject of intense activity,
especially in computer vision research, is the reconstruction of
three-dimensional shapes from several objects views, as seen from
different points of observation. It generally involves the use of
elaborate computer algorithms to `match` these views in order to unify the
images to determine the complete object shape. [See, for example, B.
Bhanu, "Representation and Shape Matching of 3-D Objects", IEEE Trans.
Anal. Machine Intell. PAMI-6, 340 (1984).] Such a digital image processing
approach is computation intensive and generally offers limited accuracy.
Another approach involves the use of the shadow moire method in conjunction
with a periphery camera. It tends to be mechanically cumbersome and to
require tedious processing of fringe patterns recorded on film. [See, for
example, C. G. Saunders, "Replication from 360-degree Moire Sensing", in
`Moire Topography and Spinal Deformity`, M. S. Moreland, et al., EDS.,
Pergamon Press, New York, (1981), p. 76.]
In the parent application hereof, and as described in an article entitled
"Automated Phase-Measuring Profilimetry of 3-D Diffuse Objects" by V.
Srinivasan, H. C. Liu, and M. Halioua, Applied Optics, Vol. 23, No. 18,
Sept. 15, 1984, there is disclosed a technique which employed, inter alia,
phase measuring techniques that were used in classical interferometry, but
were found to be 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, was used to
advantage. [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, 2696 (1974); J. C. Wyant, Appl. Opt. 14, 2622 (1975) for background.]
It was noted that 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 were noted to be
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 in the parent application
hereof and in the referenced Srinivasan, Liu and Halioua article, a
technique was set forth for determining the surface profile of a
three-dimensional object. In the method, 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.
Among the advantages of the technique are the following: relatively simple
optical hardware; relatively low frequency grating and low density
detector array; full frame data capture and relatively simple processing.
It is an object of the present invention to provide further improvement,
and a technique which is applicable to efficient and automatic obtainment
of a three dimensional surface profile of an object, including the
capability of obtaining a complete 360 degree surface profile.
SUMMARY OF THE INVENTION
In accordance with the improvements of the present invention, a complete
360 degree, or portion thereof, surface contour of an object can be
obtained efficiently and with relatively inexpensive equipment, and
without undue processing complexity, undue data acquisition and storage
requirements, etc.
In accordance with an embodiment of the method of the invention, there is
provided a technique for determining a three-dimensional surface profile
of an object. An incident beam of light is directed at the object, the
beam having a sinusoidally varying intensity pattern. The phase of the
sinusoidal intensity pattern is modulated, such as by moving of a grating
used to obtain the sinusoidally varying intensity pattern. A deformed
grating image of a line profile of the object is detected, preferably at a
linear detector; i.e., a detector having individual detector elements
arranged in a line. For points on the line profile of the object, the
distance of each such point from a reference line is then determined, each
such distance determination including the step of combining intensity
values of the received image at the different modulated phases to obtain
an object phase for each point. In this manner, the coordinate locations
of the points on a line profile of the object can be determined. [In the
preferred embodiment, deformed grating images of the reference line are
also obtained and used to obtain a reference phase for each point on the
reference line, the reference phases then being used in the distance
determinations of the object line profile coordinates with respect to the
reference line.] The object is then rotated by a rotational increment, and
the above steps are repeated to obtain the distances from the reference
line of the object points on the next line profile. This procedure can
then be repeated, as the object is rotated by further rotational
increments, to obtain the distance information (and coordinate
information, if desired) for line profiles covering a full 360 degrees of
rotation of the object. As will be described, the data can be obtained in
various procedural orders.
The technique hereof has several advantages compared with other methods of
360-degree shape reconstruction. In addition the the relative simplicity
in setup and computation, the invention does not involve any complex
view-matching operations and is characterized by high measuring accuracy.
The invention also lends itself more readily to complete automation
without the need for major host system support. The main trade-off is a
relatively low data acquisition rate, so the invention is preferably used
for obtaining surface contours of stationary, rigid objects.
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 as
set forth in the above-referenced parent application hereof.
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.
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.
FIG. 7 is a schematic diagram which illustrates a further embodiment as set
forth in the parent application hereof 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.
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.
FIG. 12 is a flow diagram of a routine that is used in conjunction with the
routines of FIGS. 4 and 8.
FIG. 13 illustrates the geometry used for analysis of operation of a form
of the invention, and shows elements of the invention.
FIG. 14 is a block diagram, partially in schematic form, of an apparatus in
accordance with an embodiment of the invention, and which can be used to
practice the method of the invention.
FIG. 15 is a flow diagram of a routine for programming the processor of the
FIG. 14 embodiment to obtain object and reference data.
FIG. 16 is a flow diagram of a routine for programming the processor of the
FIG. 14 embodiment to obtain object and reference phases to be used in
profile distance determination.
FIG. 17 is a block diagram of a routine for programming the processor of
the FIG. 14 embodiment to obtain distances and coordinates on the surface
profile.
FIG. 18 shows line profiles obtained using an experimental form of the
invention.
FIG. 19 shows the information from FIG. 18, presented in a different
direction.
FIGS. 20, 21, 22 and 23 illustrate three dimensional plots as obtained
using an experimental form of the invention.
DESCRIPTION OF THE PREFERRED EMBODIMENT
Referring to FIG. 1, there is shown a block diagram of an apparatus as set
forth in the above-referenced parent application hereof. 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 system 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 filer 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
maximum 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 are as shown in FIG. 3.
Assuming that .theta..sub.n is nearly 90.degree., 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
described first embodiment. 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)ar 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
represented 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
h(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 processed 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.
As described in the parent application hereof, profile measurements were
made, using the system in 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, as set forth in the abovereferenced parent
application hereof, 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 then the previously described 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 sinusodial 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 of the array, has
.phi..sub.c =2.pi.m+100.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)l.sub.o '(1+AC/d).sup.-1 (12)
where d and l.sub.o are distances as shown in FIG. 7. As in most practical
situations d>> AC because of the large magnifications involved, equation
(12) can be simplified:
h(x,y)=(AC/d)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.o2 =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.4 (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 inlcudes 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.
As described in the parent application hereof, 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 | | |
