哪位好人帮小弟翻译下
Fourier transform profilometry for the automatic
measurement of 3-D object shapes
Mitsuo Takeda and Kazuhiro Mutoh (Applied Optics, Vol.22, No.24,(1983))
A new computer-based technique for automatic 3-D shape measurement is proposed and verified by experiments. In contrast to the moire contouring technique, a grating pattern projected onto the object surface is Fourier-transformed and processed in its spatial frequency domain as well as in its space-signal domain.This technique has a much higher sensitivity than the conventional moire technique and is capable of fully automatic distinction between a depression and an elevation on the object surface. There is no requirement for assigning fringe orders and interpolating data in the regions between contour fringes. The technique is free from errors caused by spurious moire fringes generated by the higher harmonic components of the grating pattern.
I. Introduction
The method of moire contouring is a well-known technique for 3-D shape measurement.1 Recent interest in the technique has been an automatic measurement based on data processing by computer.2-4
For this purpose it is essential to have means to (1) make automatic distinction between a depression and an elevation from a contour map of the object, (2) assign fringe orders automatically including those separated by discontinuities, (3) locate the center lines of broad fringes by correcting unwanted irradiance variations caused by nonuniform light reflection on the object surface, and (4) interpolate the regions lying between the contour lines. To satisfy these requirements, Idesawa et al.2 proposed the scanning moire method, and Moore and Truax4 proposed the phase-locked moire method. Further, we proposed another method called Fourier-transform profilometry (FTP) which is better for automatic measurement by computer processing.5
The idea of the FTP stemmed from the observation that all these cumbersome requirements mentioned above arise merely from attempting computer-based automatic measurement by means of a moire contouring technique that was originally developed for fringe analysis by human observation rather than by computer processing. Since FTP does not use moire fringes, it is free from all the difficulties associated with the moire contouring technique. Another great advantage of FTP is that it has a much higher sensitivity than the conventional moire technique. It can detect a shape variation much less than one contour fringe in moire topography. Our previous paper5 described a general principle applicable to both profilometry and interferometry, but experiments were given only for interferometry. The purpose of this paper is to give a more specific description of the principle of FTP and to present experimental results of 3-D shape measurement by FTP.
II. Optical Geometry
Optical geometry is similar to that of projection moire topography, 6,7 but in FTP the grating image projected on an object surface is put directly into the computer and processed without using the second grating to generate moire fringes. Two different optical geometries have been proposed and used in moire topography; one has some merit over the other as well as some demerit. In crossed-optical-axes geometry, the optical axes of a projector and a camera lie in the same plane and intersect a point near the center of the object. This geometry is easy to construct because both a grating and an image sensor can be placed on the optical axes of theprojector and the camera, respectively, but it gives planar contours only when the optics are telecentric. 8,9
In parallel-optical-axes geometry, the optical axes of a projector and a camera lie in the same plane and are parallel. This geometry gives planar contours but is somewhat awkward because the grating must be placed far off the optical axis of the projector to ensure that the grating image is formed within the field of view of the observation camera.2,6,7 These two options of optical geometry are also available in FTP, and in addition FTP can solve the problem of nonplanar contours in crossed - optical-axes geometry.
Fig. 1. Crossed-optical-axes geometry
A. Crossed-Optical-Axes Geometry
Figure 1 shows a geometry in which the optical axes Ep' Ep of a projector lens crosses the other optical axis Ec' Ec of a camera lens at point O on a reference plane R, which is a fictitious plane normal to Ec' Ec and serves as a reference from which object height h(x,y) is measured. Grating G has its lines normal to the plane of the figure, and its conjugate image (with period p) is formed by the projector lens on planeI through point O, Ep' and Ep denote, respectively, the centers of the entrance and the exit pupils of the projector lens. The camera lens, with the centers of the entrance and the exit pupils at Ec and Ec', images reference plane R onto the imagesensor plane S. Ep and Ec are located at the same distance lo from plane R. It should be noted that Ep and Ec are the centers of the pupils,10 not the nodal points of the lenses as is so often confused in the literature.2,6,8 When the object is a flat and uniform plane on R, i.e., h(x,y) = 0, and if Ep is at infinity (as denoted by E for a telecentric projector), the grating image projected on the object surface and observed through point E is a regular grating pattern which can be expressed by a Fourier series expansion:
where
is the fundamental frequency of the observed grating image. The x axis is chosen as in the figure and the y axis is normal to the plane of the figure. If Ep is at finite distance, we observe on the image sensor plane a deformed grating image with a pitch increasing with x, even for h(x,y) = 0. Noting that a principal ray through a conjugate image point A strikes reference plane R at point B in the telecentric case and at point C in the nontelecentric case, we write the deformed grating image for h(x,y) = 0 as
where so(x) = is a function of x and has a positive sign when C is to the right of B as in the figure. For the convenience of later discussion, we express Eq. (3) as a spatially phase-modulated signal
where
Since the grating image is deformed and phase-modulated even for h(x,y) = 0, the crossed-optical-axes geometry, when used in moire topography, gives nonplanar contours unless the pupils are at infinity, i.e., the case of telecentric optics.9 This has imposed a great restriction on the application of the nontelecentric crossed-optical-axes geometry to moire topography, in spite of its easy-to-construct merit. In FTP, this initial phase modulation is automatically corrected as will be shown in the next section.
For a general object with varying h(x,y), the principal ray EpA strikes the object surface at point H, and point H will be seen to be a point D on plane R when observed through Ec. Hence,the deformed grating image for a general object is given by
or
where
and r(x,y) is a nonuniform distribution of reflectivity on the object surface.
B. Parallel-Optical-Axes Geometry
Figure 2 shows a geometry in which the optical axis Ep'Ep of a projector lens and that of a camera lens Ec'Ec are parallel and are normal to reference plane R. The conjugate image of grating G is formed on plane R, and the three points A, B, and C in Fig.1 degenerate into point C in Fig.2, so that Eqs. (5) and (8) become
Fig. 2. Parallel-optical-axes geometry.
Hence, the grating image projected on the plane h(x,y) = 0 remains as a regular grating pattern regardless of the position of the pupil of the projector. This brings a merit of generating planar contours in moire topography, but the optical setup becomes more or less awkward because the grating must be translated far off the optical axis of the projector lens in order to form its image within the field of view of the camera lens.
III. Fourier Transform Method
The deformed grating image given by Eq. (7) can be interpreted as multiple signals with spatial carrier frequencies nfo modulated both in phase Ø(x,y) and amplitude r(x,y). Since the phase carries information about the 3-D shape to be measured, the problem is how to obtain Ø(x,y) separately from the unwanted amplitude variation r(x,y) caused by nonuniform reflectivity on the object surface. We rewrite Eq. (7) as
Where
By using a FFT algorithm, we compute the 1-D Fourier transform of Eq. (12) for the variable x only, with y being fixed:
where G(f,y) and Qn(f,y) are the l-D Fourier spectra of g(x,y) and qn (x,y), respectively, computed with respect only to the variable x, and the other variable y being treated as a fixed parameter. Since in most cases r(x,y) and $(x,y) vary very slowly compared to the frequency fo of the grating pattern, all the spectra Qn (f- nfo,y) are separated from each other by the carrier frequency fo, as shown in Fig.3. We select only one spectrum Q1(f-fo,y) dotted in the figure and compute its inverse Fourier transform to obtain a complex signal
Fig. 3. Spatial frequency spectra of deformed grating image
for a fixed value. Only a spectrum Q1(dotted) is selected
by the filtering operation.
In the crossed-optical-axes case, we do the same filtering operation for Eq. (4) to obtain
and generate from Eqs. (14) and (15) a new signal
where
Since the initial phase modulation Ø0(x) for h(x,y) = 0 is now subtracted, Ø(x,y) in Eqs. (16) and (17) givesthe phase modulation due to the object-height distribution. In principle, this operation is not necessary in parallel-optical-axes geometry, but we apply it also to parallel-optical-axes geometry because by phase subtraction we can cancel errors caused by misalignments and/or distortion of the lenses. For example, a misalignment which rotates the grating by an angle 3a around the optical axis gives a nonzero initial phase distribution
which gives rise to a mismatching error in moire topography. We therefore use the formulas of Eqs. (16) and (17) in both cases.
Now our task is to obtain the phase distribution Ø(x,y) in Eq. (16), separating it from the unwanted amplitude variation r(x,y). Noting that both |A1|2·r(x,y) and Ø(x,y) in Eq. (16) are real functions, we compute a complex logarithm of Eq. (16):
We obtain the phase distribution Ø(x,y) in the imaginary part, completely separated from the unwanted variation of reflectivity r(x,y) in the real part. Since the phase calculation by computer gives principal values ranging from - to , the phase distribution is wrapped into this range and consequently has discontinuities with 2 -phase jumps for variations more than 2 .
These discontinuities can be corrected easily by adding or subtracting 2 according to the phase jump ranging from to - or vice versa. A 2-D phase distribution can be obtained simply by repeating the same procedure for the y sections. Details of the phase-unwrapping algorithm are described in Ref.5. Since the loci of the discontinuities with 2 -phase jumps correspond to contour fringes in moire topography, the phase-unwrapping process plays the role of the fringe order assignment algorithm of conventional moire topography.
IV. Phase-to-Height Conversion
In this section we derive a formula for converting the measured phase distribution into the physical height distribution. Noting that EpHEc CHD in both Figs. 1 and 2, we can write
where the object height h(x,y) is defined positive when measured upward from reference plane R. Substituting Eq. (20) into Eq. (17) and solving it for h(x,y), we obtain the conversion formula
We can express this formula in another form which can be directly compared to the well-known formula of moire topography. Substituting Eq. (2) into Eq. (21), we have
When we note that Ø(x,y) / 2 gives the number N of the fringe order in moire topography, we see that Eq.(22) is exactly the same as the formula of moire topography.1 However, the difference should be noted that, whereas in the moire technique the height distribution information is given only along a discrete set of contour lines, our technique, FTP, gives the height information at all picture elements regardless of whether
Ø(x,y) / 2 is an integer or not. This is the reason that FTP does not need fringe interpolation as is necessary in moire topography.
V. Maximum Range of Measurement
Since FTP is based on filtering for selecting only a single spectrum of the fundamental frequency component, the carrier frequency fo must separate this spectrum from all other spectra. This condition limits the maximum range measurable by FTP. Noting that r(x,y) varies much slower than fo in Eq. (7), we define for the nth spectrum component a local spatial frequency11 fn analogous to an instantaneous frequency of FM signal12:
For the fundamental spectrum to be separated from all other spectra, it is necessary that
and that
Fig. 4. Condition for separating fundamental frequency
spectrum Q1(dotted)from other spectra.
Where fb,( fn)max and ( fn )min are show in Fig.4 for n = 1,2,and3. Substituting Eq. (23) into Eqs. (24) and (25) we have
A safer and more practical condition can be set by
where |( Ø) / ( x )|max denotes the maximum absolute value which is a larger value of |[( Ø) / x )]max| and
|[( Ø ) / ( x )]min|. From Eqs. (28) and (29) we have
Since in most cases fb is much smaller than fo / 2 and ( n -1 ) / ( n + 1) increases monotonically with n, the limit is set by Eq . (30) for n = 2:
When discussing the maximum range of measurement, we can assume that Ø(x,y) is much larger than Ø0(x), and from Eq. (21) we can write
where we have also assumed that lo >> h (x,y). Substituting Eq. (33) into Eq. (32), we finally obtain
This condition states that the maximum range of measurement is not limited by the height distribution h(x,y) itself but by its derivative in the direction normal to the line of the grating. The maximum range of measurement can be extended by employing a geometry in which lo / d is large to prevent the phase from being overmodulated. This corresponds to reducing the fringe sensitivity in the case of moire topography, but FTP retains a sensitivity sufficient for most applications since it can detect a phase distribution much less than 2 . This will we demonstrated by experiments in Sec.VI.
VI. Experiments
Figure 5 shows a schematic diagram of the experimental setup. A crossed-optical-axes geometry was employed because of its easy-to-construct merit. A300-W slide projector with an 85-mm focal length projecting lens was used to project a Ronchi grating of 150 lines/in. onto an object surface. The object is a whiskey bottle embedded in a uniform plane plate which serves as a reference plane in the background.
Fig. 5. Schematic diagram of experimental setup.
The deformed grating pattern was observed by a low distortion TV camera(Hamamatsu C-1000) with a 55-mm focal length Micro-Nikkor lens. An analog video output signal is converted into an 8-bit digital signal and stored in a frame memory in the form of a picture with 512 X 512 pixels which can be monitored through a TV monitor.
The picture in the frame memory is DMA transferred to the memory of a Digital LSI-11/23 microcomputer to make a temporal file on a disk which is then transferred through a communication lineto a faster Digital PDP-11/44 minicomputer and processed. The final result is sent back and displayed on an X-Y plotter.
Figure 6 shows a picture of a deformed or phase-modulated grating pattern, where the straight grating lines in the background serve as reference signals for determining the absolute phase values to be converted into a height distribution. Figure 7 shows an example of the irradiance profile along a horizontal line in the direction of the x axis. Note that the reflectivity r(x,y) is strongly nonuniform over the object surface. Figure 8 shows Fourier spectra of Eq. (13) computed by using a FFT algorithm. In the figure, the large spectrum (n = 0) is clipped to get an enhanced view of other spectra. Note that the spectrum(n = 1) is completely separated from other spectra satisfying the condition of Eq. (34).
Figure 9 shows a wrapped phase distribution Ø(x,y) computed from the imaginary part of Eq. (19). Noting that from Eq. (22) the line along the discontinuities with 2 -phase jumps corresponds to one contour fringe in moire topography, we can see that the height variation less than the amount of one fringe is clearly detected in the figure. Figure 10 shows the rewrapped phase distribution which has the form of a whiskey bottle. This phase distribution is converted into the height distribution using the formula of Eq. (21) and compared with the result of direct measurement by the contact method. Figure 11 shows three examples of the object profiles, where the lines and circles represent the results obtained by FTP and the contact method, respectively.
Fig. 6. Deformed grating pattern with straight lines
in the background for reference signals.
Fig. 7. Irradiance profile of deformed grating pattern
for a fixed y value
Fig. 8. Spatial frequency spectra of deformed grating pattern
Fig. 9. Wrapped phase distribution. The line along discontinuities
with 2 -phase jumps corresponds to a fringe contour in moire topography.
Fig. 10. Unwrapped phase distribution.
Fig. 11. Comparison with contact measurement. Solid lines denote
measurements by FTP, and circles denote measurements by the contact method.
