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Description  |
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RELATED APPLICATIONS
This application claims the priority of Japanese Patent Application No.
11-363241 filed on Dec. 21, 1999, which is incorporated herein by
reference.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a fringe analysis method using Fourier
transform; and, in particular, to a fringe analysis method which can
effectively use the Fourier transform method even when analyzing image
data having a closed fringe pattern such as interference fringe pattern.
2. Description of the Prior Art
While the light wave interferometry has conventionally been known as an
important means concerning precise measurement of object surfaces, there
have been urgent demands for developing an interferometry technique
(sub-fringe interferometry) which can read out information items smaller
than a single stripe of interference fringe (one fringe) due to the
necessity for measuring a surface or wavefront aberration at an accuracy
of 1/10 wavelength or higher.
For sub-fringe interferometry techniques, attention has been paid to those
using the Fourier transform technique as described in "Basics of
sub-fringe interferometry," Kogaku, Vol. 13, No. 1, pp. 55-65, February,
1984.
However, though excellent in principle, the Fourier transform method leaves
some problems unsolved, and have not always been effectively used in
practice.
One of such problems lies in how to adapt the Fourier transform method to a
closed fringe pattern.
Namely, while interference fringe pattern exhibits a pattern shaped like
closed concentric circles when an object to be observed has a shape
approximating a spherical or parabolic surface, it is difficult for a
lower frequency signal component and a carrier frequency component (which
is superimposed in the Fourier transform method since the surface to be
observed and the reference surface are relatively tilted with each other)
to be securely separated from each other in Fourier spectra plane when the
Fourier transform method is used for such closed interference fringe
pattern. Therefore, the original phase or wavefront can not be recovered
(by the conventional Fourier transform method).
SUMMARY OF THE INVENTION
In view of the circumstances mentioned above, it is an object of the
present invention to provide a fringe analysis method using Fourier
transform which can securely separate the low frequency signal component
and the carrier frequency component from each other in Fourier spectra
plane when a fringe analysis is carried out for closed fringe image data
by use of the Fourier transform method.
The present invention provides a fringe analysis method comprising the
steps of converting fringe image data of a wavefront to be observed which
are expressed by an orthogonal coordinate system into fringe image data
represented by a different coordinate system; by using the conventional
Fourier transform method, wavefront expressed in the converted coordinate
system can be obtained; and then converting the obtained wavefront in the
converted coordinate system back into a wavefront expressed in the
original orthogonal coordinate system.
Preferably, the different coordinate system is a polar coordinate system.
Preferably, the fringe image data expressed by the orthogonal coordinate
system comprise a closed fringe pattern, the origin of the polar
coordinate system is set within the innermost of the closed fringe
pattern.
Preferably, the origin is located near the center of the closed fringe
pattern.
It will particularly be effective if the fringe analysis method using the
Fourier transform in accordance with the present invention is applied to
fringe image data carrying phase information and including carrier fringes
superimposed thereon.
In particular, the present invention is effective in the case where the
fringe image data are interference fringe image data, and carrier fringes
caused by tilting the surface of the wavefront or the reference surface
relatively with each other is superimposed on the interference fringe
image data. Also, the present invention is considerably effective when the
fringe image data are image data of moire fringe pattern or other fringe
pattern.
Preferably, ineffective data areas of the fringe image data represented by
the converted coordinate system are filled by interpolation, and then the
conventional Fourier transform method is applied to.
Preferably, the fringe image data represented by the converted coordinate
system is multiplied by a predetermined window function.
Preferably, the fringe image data expressed by the orthogonal coordinate
system is divided into a plurality of data areas, each of thus divided
areas is converted into fringe image data represented by a converted
coordinate system, conventional Fourier transform method is used so as to
obtain wavefront corresponding to each area of the wavefront to be
observed in the converted coordinate system, and then wavefronts of the
plurality of areas represented by the converted coordinate system are
combined together.
Preferably, the divided areas comprise a first angle area of
0.ltoreq..theta.<.pi. and a second angle area of
.pi..ltoreq..theta.2.pi..
It is also preferred that the divided areas comprise a first angle area of
##EQU1##
a second angle area of
##EQU2##
a third angle area of
##EQU3##
and a fourth angle area of
##EQU4##
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a flowchart for explaining an embodiment of the present
invention;
FIG. 2 is a schematic view showing a wavefront of an object (shaped like a
spherical waveform) to be observed;
FIG. 3 is a view showing closed interference fringe pattern obtained from
the wavefront shown in FIG. 2;
FIG. 4 is a view showing a Fourier spectrum obtained when the image data of
the interference fringe pattern shown in FIG. 3 are Fourier-transformed as
they are;
FIG. 5 is a view showing the recovered wavefront of the object to be
observed determined by the conventional Fourier transform method from the
Fourier spectrum shown in FIG. 4;
FIG. 6 is a chart showing coordinates of image data in an orthogonal
coordinate system;
FIG. 7 is a chart showing coordinates of image data in a polar coordinate
system;
FIG. 8 is a view showing opened interference fringe pattern obtained when
the closed interference fringe pattern shown in FIG. 3 are converted into
a polar coordinate;
FIG. 9 is a view showing interference fringe pattern obtained when the
ineffective data areas in the interference fringe pattern of FIG. 8 are
filled by interpolation;
FIG. 10 is a view showing the recovered wavefront of the object to be
observed that is obtained when the image data of the interference fringe
pattern shown in FIG. 9 are analized by the conventional Fourier-transform
method;
FIG. 11 is a chart showing a curve indicating an original cross section
profile of the original spherical waveform to be analyzed, a curve showing
the cross section of the profile (shaped like a spherical waveform)
obtained by the method in accordance with an embodiment of the present
invention, and the error therebetween; and
FIG. 12 is a schematic view showing an example of system for carrying out
the method of the present invention.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
In the following, the method in accordance with an embodiment of the
present invention will be explained in detail with reference to the
drawings.
In this method, when analyzing the interferogram of an object to be
observed by using the Fourier transform method with respect to closed
interference fringe image data, the original image data expressed by an
orthogonal coordinate system are once converted into fringe image data
represented by a polar coordinate system; thus converted data are analyzed
by the conventional Fourier transform method; and then the obtained
wavefront is converted back into the orthogonal coordinate system by
inverse polar coordinate transformation so as to obtain the original
wavefront of the object to be observed. As a consequence, the method makes
it possible to securely separate the lower frequency signal components
from mixed signals of low frequency signal components and carrier
frequency components in the Fourier spectrum.
Studied in the following is a case where an object to be observed having a
spherical wavefront such as that shown in FIG. 2 is measured with an
interferometer, whereby interference fringe pattern such as those shown in
FIG. 3 are obtained.
As can be seen from FIG. 3, interference fringe pattern is composed by
closed rings in such a spherical object to be observed. The same holds
when the object has a shape approximating a parabolic surface, for
example. When such interference fringe pattern is analyzed by use of
conventional Fourier transform method, a Fourier spectrum shown in FIG. 4
is obtained. As can be seen from FIG. 4, ring-like patterns are spread out
over the Fourier spectrum as well, whereby it is difficult in principle
for a normal Fourier transform analysis technique to securely separate a
low frequency signal component and a carrier frequency component from each
other.
Namely, a wavefront such as that shown in FIG. 5 will be obtained when
analyzed according to the conventional Fourier transform analysis method.
As can be seen from FIG. 5, the obtained results yield a wavefront totally
different from the original one to be observed.
Therefore, in the method of this embodiment, the fact that the most
effective coordinate system representing a concentric circle pattern is a
polar coordinate system is taken into consideration, and the interference
fringe image data represented by a planar orthogonal coordinate system (x,
y) are once converted into a planar polar coordinate system (r, .theta.),
for example (see FIGS. 6 and 7).
As is well-known, the planar orthogonal coordinate system (x, y) and planar
polar coordinate system (r, .theta.) have the following relationship
therebetween:
x=r.multidot.cos .theta., y=r.multidot.sin .theta. (1)
Here, the Fourier transform method on which the method of this embodiment
is based will be explained.
As shown in FIG. 12, when a Michelson type interferometer 1 is used for
measuring the wavefront of a surface 2 of an object to be observed in
sub-fringe order, for example, a planar mirror acting as a reference
surface 3 is tilted by a slight amount, so as to yield a minute angle
.theta. of tilt. Though the object surface 2 can be tilted instead of the
reference surface 3 in this case, it is preferred that the reference
surface 3 be tilted.
In this case, the intensity distribution g(x, y) of interference fringe
pattern concerning the object surface where the interference fringe
pattern is formed become:
g(x,y)=a(x,y)+b(x,y)cos[2.pi.f.sub.o x+.phi.(x,y)], (2)
thus being those spatially phase-modulated by fine vertical fringes
.phi.(x, y) which are carrier signals with a spatial frequency of f.sub.o
=2 tan .theta./.lambda.. Here, a(x, y) is the term composed of the sum of
intensities of two interfering light waves and indicates the intensity
distribution in the background of interference fringe pattern, whereas
b(x, y) is the term composed of the product of two interfering light waves
and indicates the amplitude of changes in brightness of interference
fringe pattern.
The above-mentioned low frequency signals .phi.(x, y), which are phase
components to be determined, and unnecessary signals a(x, y) and b(x, y)
are both spatial signals as with the carrier signals introduced by
tilting.
The above-mentioned expression (2) is converted into the following
expression (3):
g(x,y)=a(x,y)+c(x,y)exp(2.pi.jf.sub.o x)+c*(x,y)exp(-2.pi.jf.sub.o x) (3)
where
c(x,y)=(1/2)c(x,y)exp[j.phi.(x,y)],
and * indicates a complex conjugate.
When only variable x is subjected to one-dimensional Fourier transform with
y in the intensity distribution of expression (3) being fixed, so as to
calculate the spatial frequency spectrum G(f, y) concerning variable x,
the following expression (4) is obtained:
##EQU5##
where capital letters indicate spatial frequency spectra concerning
variable x.
As compared with the speed of change caused by the carrier spatial
frequency f.sub.o, the change in a(x, y) and b(x, y) is quite slow, and
the change in .phi.(x, y) is also slow on the object surface in sub-fringe
order.
Therefore, the three spectra in the above-mentioned expression (4) are
completely separated from each other by the carrier frequency f.sub.o.
Here, the spectrum C(f-f.sub.o, y) of the signal superimposed on the
positive carrier frequency f.sub.o is taken out alone, and is shifted
toward the origin by f.sub.o, so as to yield C(f, y). This spatial
frequency filtering eliminates one of the unnecessary signals, i.e., a(x,
y); whereas the carrier frequency, i.e., tilt, is eliminated by the
shifting of spectrum toward the origin.
When thus obtained C(f, y) is subjected to one-dimensional inverse Fourier
transform as a variable of f, c(x, y) in the above-mentioned expression
(3) is determined.
Thereafter, the following expression (5) represented by complex logarithms
is calculated, so that the imaginary part is completely separated from
unnecessary terms of the real part, whereby the phase .phi.(x, y) can be
determined:
log[c(x,y)]=log[(1/2)b(x,y)]+j.phi.(x,y) (5)
This embodiment will now be explained with reference to the flowchart shown
in FIG. 1.
In the Michelson type interferometer 1 such as the one shown in FIG. 12,
interference fringe pattern (having a closed fringe pattern such as the
one shown in FIG. 3) formed by respective reflected luminous fluxes from
the object surface 2 (shaped like a spherical surface) and reference
surface 3 are captured by a CCD camera 4, and are displayed on a monitor
screen 6 by way of an image input board 5. The interference fringe image
data from the CCD camera 4 are stored into the memory of a computer.
Here, while observing the monitor screen 6, the position of the object to
be observed is adjusted such that the center of interference fringe
pattern is located at the center of screen (S1).
Subsequently, the interference fringe image formed on the CCD camera 4 is
captured upon photoelectric conversion, and thus obtained data are stored
into the above-mentioned memory (S2).
Then, the center of the closed fringe pattern is taken as the origin of the
polar coordinate system in the fringe image data stored in the memory
(S3), and the data are converted into fringe image data in the polar
coordinate system (S4). Thus obtained fringe image data of the polar
coordinate system yield an opened parallel fringe pattern as shown in FIG.
8, which is not a closed fringe pattern. This fact will be easily
understood in concept when imagining a tree cake being opened from an
incision formed at a side part thereof, so as to deform until its inner
rings become substantially parallel lines.
However, this coordinate transformation is not a linear transformation.
Therefore, when square data suitable for Fourier transform in an
orthogonal coordinate system are converted into a polar coordinate system,
areas without effective data (ineffective data areas) such as those shown
in FIG. 8 may occur at an end part thereof. The ineffective data areas may
be neglected since they generate errors in analysis. For effectively
utilizing these areas, however, they are filled with simulated data by
using extrapolation techniques (S5, S6). FIG. 9 shows an interference
fringe image obtained when the ineffective data areas are thus filled with
by interpolation.
When carrying out the above-mentioned transformation by discrete Fourier
transform (DFT), only integers are used. In order to represent
interference fringe pattern expressed in discrete orthogonal coordinates
by discrete polar coordinates, it is necessary that discrete data items be
filled with simulated data by using interpolation techniques.
Subsequently, if necessary, the image data obtained by the processes of the
above-mentioned steps 5, 6 (S5, S6) are multiplied by a predetermined
window function (S7, S8).
If the carrier frequency of interference fringe pattern is not an integer,
then error will be greater in marginal portions of the analysis image of
the object to be observed. Therefore, the error in analysis can be reduced
if the image data are multiplied by a known window function (e.g., a
window function which yields a region excluding marginal portions of the
image data).
When the interference fringe image data of an orthogonal coordinate system
is converted into a polar coordinate system, the .theta. direction varies
from 0 to 2.pi., thereby corresponding to one period of a trigonometric
function. Therefore, the multiplication by a window function is not
necessary in the .theta. direction. Generally, the carrier frequency is
not an integer in the r direction, therefore, it is desirable to be
multiplied by a window function, so as to reduce the error in analysis.
Thereafter, the interference fringe image data converted into the polar
coordinate system are subjected to the well-known Fourier transform
processing (S9). Since the obtained Fourier spectrum results from open
interference fringe pattern, the lower frequency signal components and
carrier frequency components can securely be separated from each other,
whereby a polar coordinate form generated upon an interference fringe
analysis, such as that shown in FIG. 10, is obtained (S10).
Then, the wavefront expressed in the polar coordinate system obtained at
step 10 is converted into the orthogonal coordinate system by inverse
coordinate transformation process (S11). If the data are filled at the
above-mentioned step 6, then the ineffective data areas subjected to the
filling are eliminated (S12), whereby the wavefront of the object to be
observed (shaped like a spherical waveform) is obtained (S13).
FIG. 11 shows two curves (having a given unit) for comparing the original
cross section profile of the wavefront and that of an object to be
observed (shaped like a spherical waveform) determined by the method of
this embodiment with each other, and the amount of error (%) therebetween.
Here, the abscissa indicates the position (the number of CCD pixels).
As can be seen from FIG. 11, the above-mentioned error is about 0.3% or
less than the spherical wave to be measured.
For further reducing such an error, the interference fringe image data are
divided into two areas of .theta.=0 to .pi., and .theta.=.pi. to 2.pi.,
and each area is converted into a polar coordinate system. Then, the
respective interference fringe images of the two areas are combined
together at a position of r=0. Since the respective interference fringe
intensities of the interference fringe image data of both regions equal
each other at the position of r=0 even after being converted into the
polar coordinate system, thus obtained composite interference fringe image
becomes continuous.
Without being restricted to the two areas mentioned above, the interference
fringe data can be divided into any number of small areas. For example,
closed interference fringe image data of an orthogonal coordinate system
may be divided into four areas of .theta.=0 to .pi./2, .theta.=.pi./2 to
.pi., .theta.=.pi. to 3.pi./2, and .theta.=3.pi./2 to 2.pi., so that each
area is converted into a polar coordinate system. In the case where a
plurality of patterns exist on one interference fringe image, as in the
case where a plurality of fringe centers exist, it is preferred that each
divided area include a respective pattern and that the above-mentioned
transformation be carried out for each area.
In this case, the .theta. direction does not become one period of a
trigonometric function, whereby it is necessary that the .theta. direction
be multiplied by a window function as well. Since the error in marginal
portions of the determined form is still large even after being multiplied
by the window function, influences of the error will remain at the time of
combining. Therefore, it is desirable that areas have overlapping portions
when dividing interference fringe pattern to be analyzed, and that the
marginal portions of each area yielding a large error be eliminated at the
time of combining.
Without being restricted to the above-mentioned embodiment, the fringe
analysis method using Fourier transform in accordance with the present
invention can be modified in various ways.
Though the interference fringe image is captured by a Michelson type
interferometer in the above-mentioned embodiment, the present invention is
similarly applicable to interference fringe image data obtained by
interferometers of Fizeau type or other types as a matter of course.
Furthermore, the method of the present invention is applicable not only to
interference fringe pattern, but also to various fringe images of moire
fringe pattern, speckle fringe pattern, and other fringe patterns in a
similar manner.
Also, the method of the present invention is applicable to a partial
analysis of closed fringe pattern.
Though the above-mentioned embodiment relates to a case using
one-dimensional Fourier transform, two-dimensional Fourier transform can
also be used.
Further, though the above-mentioned embodiment relates to a case where the
object to be observed has a form similar to a spherical surface or
parabolic surface, other surfaces which generate a closed fringe pattern
can also be used as an object to be observed.
Also, though a polar coordinate system is used as a coordinate system
different from the orthogonal coordinate system, various linear or
nonlinear coordinate systems such as logarithmic coordinate systems,
exponential coordinate systems, and others can be used.
In the fringe analysis method using Fourier transform in accordance with
the present invention, fringe image data of an object to be observed
expressed by an orthogonal coordinate system are converted into fringe
image data represented by a polar coordinate system, thus converted data
are subjected to Fourier transform, so that the fringe pattern is analyzed
in the polar coordinate system, and the analysis data are finally
converted back into an orthogonal coordinate system, so as to yield the
wavefront of the object.
Namely, even when the fringe image data expressed by the orthogonal
coordinate system form a closed fringe pattern, they can be turned into an
open fringe pattern when converted into a polar coordinate system, whereby
low frequency signal components and carrier frequency components can be
securely separated from each other by the conventional Fourier transform
method thereafter. When the wavefront data obtained after eliminating the
carrier frequency components are returned to the orthogonal coordinate
system, the original wavefront of the object to be observed can be
obtained favorably.
* * * * *
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