WO2001007864A2

WO2001007864A2 - Method and apparatus for performing calibration-based absolute shape measurement using phase-shifting projected fringe profilometry

Info

Publication number: WO2001007864A2
Application number: PCT/US2000/019834
Authority: WO
Inventors: Hongyu Liu; Benjamin A. Bard
Original assignee: Hongyu Liu; Bard Benjamin A
Priority date: 1999-07-21
Filing date: 2000-07-21
Publication date: 2001-02-01
Also published as: WO2001007864A3; AU6115700A

Abstract

A method and apparatus for performing absolute shape measurement using phase-shifting projected fringe profilometry. Techniques are disclosed to permit calibration of the optical system's distortion characteristics in both phase and lateral directions. Such a calibration utilizes specialized examination surfaces, methods to examine those surfaces and algorithms to process the acquired data, all of which are provided by the present invention. A novel calibration tool incorporating improved examination surfaces is also disclosed.

Description

METHOD AND APPARATUS FOR PERFORMING CALIBRATION- BASED ABSOLUTE SHAPE MEASUREMENT USING PHASE- SHIFTING PROJECTED FRINGE PROFILOMETRY

Background of the Invention The present invention is related to a non- contact method of profiling a surface. More particularly, the invention relates to absolute shape measurement by phase-shifting projected fringe profilometry . Development, manufacturing, and repair can all require high precision profile measurements. Some of the most demanding examples include precision gear manufacturing and refurbishment, turbine blade inspection, and optical disk deformation measurement. Large surfaces, such as airplane wings, ship propellers, rotors, and stators, sometimes require such measurements as well. Profile-based applications, such as reverse engineering, rapid prototyping, and 3-D animation, also drive the demand for this process. The most well-known measurement method is the use of a coordinate measurement machine (CM ) which uses a contact probe to physically trace out a surface profile. While accurate, these devices require physical contact with the surface being measured. Optical profilometry methods have been considered as an alternative to the use of contact probes. In such techniques, a one-dimensional periodic pattern or grating image is projected onto a surface under test . The image of the pattern is observed from another direction as that image is deformed by the surface. The deformed periodic pattern or grating image is captured and analyzed to determine the surface profile. As described in U.S. Patent No. 4,641,972 to Halioua and U.S. Patent No. 6,040,910 to Wu et al , both incorporated herein by reference, one well- known optical profilometry method is referred to as phase-shifting projected fringe profilometry (PSPFP) . In PSPFP, a sinusoidal fringe pattern projected on the object under test is phase-shifted between successive images. The plurality of images thus obtained can be used to precisely calculate the phase at each point on the interference pattern. The calculated phase then serves as the basis for obtaining a depth measurement at that point. PSPFP is commonly used in either a comparative mode, in which a surface is compared to another surface and the difference is measured, or an absolute mode, in which the depth of a single surface is desired.

The sinusoidal grating projected onto the object under test is typically produced using one of two techniques. The first approach is to use a laser to generate interference fringes, and the second is to project an image of a sinusoidal grating. Both of these techniques have given rise to certain problems in their application. For example, it has been very difficult to produce a suitable sinusoidal grating for use when the second approach is employed. Prior art gratings used for this purpose have generally had poor contrast, and the waveform produced by the grating has tended to be inaccurate . The art has also recognized the need to provide accurate calibration data, particularly when PSPFP is performed in the absolute mode. For example, when divergent illumination is used in fringe projection, the conversion coefficient between measured phase and depth is field and depth dependent and is difficult to characterize. A global conversion factor frequently used in some previous works is inadequate to cope with such a spatially varying nature of the transformation from phase to depth. Even with a phase-mapping approach as proposed by Srinivasan et al . , only surface depth can be accurately measured, but not the lateral geometry of the object. Because of magnification variations and distortions in the imaging system, the lateral geometry of the object can be considerably deformed. Another drawback of the conventional PSPFP technique is also evident in many applications, such as reverse engineering and high- precision gear gauging. In these applications, the shape of objects is referred to relative to a predefined object-oriented reference system, which is usually called world coordinate system in photogrammetry terminology. Nevertheless, conventional PSPFP techniques have expressed measured profiles in terms of detection-plane coordinates. Applying conventional PSPFP techniques to those problems suffers from lack of common language. Because of these deficiencies, the applications of conventional PSPFP techniques have been limited to simple pass-fail type of depth inspections .

Summary of the Invention The present invention recognizes and addresses the foregoing disadvantages, and others, of prior art constructions and methods. Accordingly, it is an object of the present invention to provide an improved method of performing non-contact measurements in the absolute mode.

It is a more particular object of the present invention to provide an improved three-dimensional profile measurement method which is capable of providing an object-oriented description of the measured profile.

It is a further object of the present invention to provide an improved method of calibrating an optical system for achieving high- accuracy PSPFP profile measurements.

It is a more particular object of the present invention to provide a method of calibrating such an optical system which provides both lateral distortion calibration and phase calibration.

It is a still further object of the present invention to provide an improved calibration tool for use in calibrating an optical system for measuring a profile of an unknown object. It is also an object of the present invention to provide a grating for use in an optical system for measuring a profile of an unknown object.

Some of these objects are achieved by a method of calibrating an optical system for measuring a profile of an unknown object. One step of the method involves providing a first known grating and a second known grating in which first grating lines of the first known grating are orthogonal to second grating lines of the second known grating. According to another step of the method, the first known grating is situated in a predetermined starting location relative to the optical system. Next, the optical system is used to detect at least one image of the first known grating at each of a plurality of selected depth positions to produce first imaged data.

Another step of the method involves situating the second known grating in a predetermined starting location relative to the optical system, which is preferably the same starting location in which the first known grating is started. Next, the optical system is used to detect at least one image of the second known grating at each of a plurality of selected depth positions to produce second imaged data. The first imaged data and the second imaged data are processed to produce calibration data for the optical system. By repeating the above procedure over several depth positions, a full characterization of lateral distortions is obtained over a small volume that would be occupied by measured objects.

According to exemplary methodology, a plurality of images of the first known grating and the second known grating may be detected at each of the selected depth positions. Preferably, the first known grating and the second known grating may each be laterally moved by a predetermined phase shift between detecting of respective images at each selected depth position. For example, the predetermined phase shift may be a phase shift of π/2 and a total of four phase-shifted images may be detected at each of the selected depth positions. According to another aspect of the present invention, the first imaged grating data and the second imaged grating data may be processed according to a Fourier transform method to produce the calibration data.

Preferably, the first known grating and the second known grating may be fixed to a common calibration tool. For example, the first known grating and the second known grating may be fixed to the calibration tool so as to be coplanar. Alternatively, the calibration tool may have a rotary axis about which the first known grating and the second known grating are angularly separated.

Further calibration data, characterizing phase to depth relation, may be obtained by providing a flat surface having light-diffusing characteristics. The flat surface is situated in a predetermined starting location relative to the optical system. Next, a grating pattern is projected onto the flat surface. The optical system is used to detect at least one image of the grating pattern as reflected from the flat surface at each of a plurality of selected depth positions to produce third imaged data. The third imaged data is processed to produce the further calibration data, as desired. Preferably, the first known grating, the second known grating and the flat surface may be fixed to a common calibration tool.

Other objects of the present invention are achieved by a method of calibrating an optical system for measuring a profile of an unknown object. First, a flat surface having light - diffusing characteristics is provided. The flat surface is then situated in a predetermined starting location relative to the optical system. At each of a plurality of depth positions, a predetermined number of successively phase- shif ed grating patterns are projected onto the flat surface. In some cases, it will be desirable to sequentially select the depth positions according to a bisection technique. Using the optical system, respective images of the grating pattern as reflected from the flat surface are detected to produce imaged data. The imaged data is processed to produce calibration data for the optical system. In accordance with exemplary methodology, the flat surface may be constructed as a multilayer composite structure. For example, the multilayer composite structure may comprise a substrate layer underlying a reflective layer. The reflective layer will preferably define a predetermined pattern of pits so as to provide the light - diffusing characteristics. In some cases, the reflective layer will be formed of metal, with the predetermined pattern of pits being produced in the reflective layer by surface etching. Often, the composite structure may further comprise a protective layer overlying the reflective layer. Still further objects of the present invention are achieved by a calibration tool for use in calibrating an optical system for measuring a profile of an unknown object. The calibration tool comprises a first known grating having first grating lines. A second known grating having second grating lines is also provided. The first grating lines of the first known grating are orthogonal to the second grating lines of the second known grating. The calibration tool further comprises a flat surface having light-diffusing characteristics .

In some exemplary embodiments, the first known grating, the second known grating and the flat surface will be fixed to a common mounting structure so as to be coplanar. In such cases, the calibration tool will preferably include at least one translation stage such that the mounting structure will be movable in at least a depth dimension. For example, the calibration tool may include a mounting structure having three translation stages to be movable in horizontal and vertical dimensions as well as the depth dimension. Alternatively, the calibration tool may have a rotary axis such that the first known grating, the second known grating and the flat surface are angularly separated about the rotary axis .

Still further objects of the present invention are achieved by a grating for use in an optical system for measuring a profile of an unknown object. The grating comprises a plurality of parallel grating lines having sinusoidally varying transmittance characteristics. The sinusoidally varying transmittance characteristics are produced by multiple intervals in a direction normal to the grating lines each having a predetermined level of transmittance. The predetermined level of transmittance is related to a respective number of transparent cells in the particular interval. Preferably, each of the intervals will have an equal width. Moreover, the number of transparent cells in each interval may be produced by a laser lithography process.

Additional objects of the present invention are achieved by an apparatus for profiling a surface of an object. The apparatus comprises illumination means for directing an incident beam of light having a varying intensity pattern onto the surface. Means are provided for spatially phase shifting the varying intensity pattern of the incident beam of light . Detector means or receiving deformed grating images of the surface are also provided. Processing means are operatively coupled to the detector means for determining a profile of the surface based on stored calibration data and a number of deformed grating images at different spatial phases. The stored calibration data includes both lateral distortion and phase calibration data.

Other objects, features and aspects of the present invention are provided by various combinations and subcombinations of the disclosed elements, as well as methods of utilizing same, which are discussed in greater detail below. Brief Description of the Drawings A full and enabling disclosure of the present invention, including the best mode thereof, to one of ordinary skill in the art, is set forth more particularly in the remainder of the specification, including reference to the accompanying drawings, in which:

Figure 1 is a perspective view of a gear measuring apparatus incorporating an optical system of the present invention;

Figure 2 is a diagrammatic representation of the optical system employed in the gear measuring apparatus of Figure 1;

Figure 3 is a schematic diagram illustrating the principles of deformed grating profilometry;

Figure 4 is a diagrammatic representation of a first embodiment of a calibration tool constructed in accordance with the present invention; Figure 5 is a diagrammatic representation of a second embodiment of a calibration tool constructed in accordance with the present invention;

Figure 6 is a graphical representation of a quantization scheme that can be used in the production of a stepped sinusoidal grating (SSG) in accordance with the present invention;

Figure 7 is a diagrammatic representation of a technique with may be used to produce simulated grayscale in a SSG constructed in accordance with the present invention;

Figure 8 illustrates the grating lines of a SSG constructed in accordance with the present invention as viewed from three different resolutions of observation;

Figure 9 is a diagrammatic elevation of an exemplary diffusive flat constructed in accordance with the present invention as viewed from the side; Figure 10 is an enlarged plan view of a portion of a diffusive flat constructed in accordance with the present invention, with a subportion thereof being further enlarged; and

Figure 11 is a diagrammatic representation of the manner in which a point on a SSG used for calibration is translated to a corresponding point on a detector array.

Repeat use of reference characters in the present specification and drawings is intended to represent same or analogous features or elements of the invention.

Detailed Description of Preferred Embodiments It is to be understood by one of ordinary skill in the art that the present discussion is a description of exemplary embodiments only, and is not intended as limiting the broader aspects of the present invention, which broader aspects are embodied in the exemplary constructions.

A device utilizing the teachings of the present invention is illustrated in Figure 1. As shown, a gear measuring apparatus 10 includes a mandrel 12 for supporting a gear 14 to be evaluated. Mandrel 12 is maintained between a top "center" 16 and a bottom "center" 18, with bottom center 18 being located on a rotating table 20. Table 20 is, in turn, supported on a base 22. Measuring system 10 includes an optical projection arm 24 and an optical imaging arm 26.

Figure 2 is a diagrammatic representation of the optical system utilized in measuring system 10. As shown, projection arm 24 includes a light source 28 having an associated optical fiber 30. Light source 28 projects a beam of light, via fiber 30, towards a condenser lens 32 which condenses the light beam. The light is them directed to a grating 34 to create a light beam having varying intensity patterns, i.e., a grating image. In this case, projection optics 36 then direct the image to a mirror 38 which is employed to direct the grating image onto a flank surface 40 of gear 14 at a predetermined angle. A detector array, in this case a CCD camera 42, receives the deformed grating image reflected from surface 40 of gear 12 through its lens 44. Data associated with the deformed grating image is then received by a computer 46 or other suitable processing means. Computer 46 processes the data to produce a profile of surface 40. Some general principles of the deformed image profilometry technique that can be used to determine surface profile can be most easily explained with reference to Figure 3. As can be seen, fringes projected onto the inspected object from one direction and viewed from another direction are deformed due to the non-planar relief of the inspected surface. The departure of the fringes from straight, equally spaced lines reflects the departure of the inspected surface from a plane reference surface in this comparative mode example .

Consider the projected fringe Fi which intersects the reference surface at point P but the inspected surface at point P' . Surface depth h(x,y) at point P' can be found from the lateral shift d(x,y) of the fringe by

h(x,y) =d(x,y)tanθ. (1)

Accuracy is enhanced when a phase-shifting technique is applied to the conventional projected fringe method. In this technique, a sinusoidal grating is used for producing the fringes and the intensity distribution on the object surface is

l{x, y) = (2)

where p is the nominal period of the fringes, and a(x,y) and b(x,y) are the bias and the modulation of fringe intensities caused by surface reflectivity variations, background illumination variations, the offset of electronic circuits, and the local variations of the grating contrast ratio. These two terms represent the additive noise and the multiplicative noise in the image that are responsible for the degradation in measurement accuracy. The phase term φ(x,y) represents the local shift of fringes resulted from surface relief variations. From this phase shift, lateral shift d(x,y) can be found through the phase calibration. For the accurate determination of phase shift φ(x,y), several images are captured with a known phase step introduced between consecutive images. In the four-step algorithm with a phase-step of π/2, four images captured can be represented by

/, {x, y)

where i = 1,2,3,4. The phase of term of Equation (2) can be calculated as

After eliminating the linear term 2π/p, phase shift φ(x,y) can be obtained at high accuracy. Since a(x,y) and b(x,y) are cancelled out by the subtraction and division in this calculation, their effects are minimized. This technique is therefore insensitive to both additive and multiplicative noise. In addition, because of the independent determination of the phase shift at each detection cell, this technique enables pure pixel-to-pixel high-accuracy measurements.

While PSPFP alone can provide relatively accurate depth measurements, a careful calibration of the optical system's distortion characteristics must be performed when accurate shape m the two lateral (non depth) dimensions is also required. This calibration requires specialized surfaces to examine, methods with which to examine them, and algorithms with which to process the acquired data. Once the calibration is performed, accurate absolute shape measurements using PSPFP can be realized.

As will now be explained, the present invention provides improved calibration tools for absolute profile measurements. Preferably, these tools utilize three calibration surfaces: a diffusive flat surface, a horizontal grating and a vertical grating. The flat surface is used for phase calibration, whereas the horizontal and vertical gratings are used for lateral distortion calibrations .

Figure 4 illustrates one embodiment of a calibration tool constructed in accordance with the present invention which is particularly useful where the object to be measured is of arbitrary size and shape. As can be seen, calibration tool 50 has a first surface 52 constructed as a stepped sinusoidal grating (SSG) with horizontally oriented fringes. A second surface 54 of tool 50 is constructed as a SSG having vertically oriented fringes. The third surface 56 is a diffusive flat for phase calibrations. In this embodiments, the three surfaces are coplanar, preferably to a hign degree of accuracy.

For reasons that will become apparent below, calibration tool 50 is configured so that the coplanar calibration surfaces can be moved in three dimensions, i.e. along axes x, y and z. Toward this end, calibration tool 50 is equipped with a horizontal translation stage 58, a vertical translation stage 60 and a depth translation stage 62. As will be explained more fully below, lateral calibrations are performed using a phase-shifting approach. In the calibrations, the two SSGs with mutually orthogonal grating lines are directly imaged by the imaging system. Several images of these two gratings are acquired at several phase shifts for phase calculations. The calculated phase maps, which are deformed solely due to the distortions of the imaging system, are then used to characterize the lateral distortions of the imaging system.

Figure 5 illustrates a calibration tool 64 that may be particularly useful where the surface to be measured is one flank (face) of a given gear tooth. In contrast with the calibration tool described in U.S. Patent 6,040,910, calibration tool 64 utilizes two sinusoidal gratings for lateral calibration. Specifically, calibration tool 64 has a first face 66 having a vertical grating and a second face 68 having a horizontal grating. A third face 70 includes a diffusive flat surface. It can be seen that the three faces are angularly separated about a rotary axis A.

Like those of calibration tool 50, the two gratings of calibration tool 64 are used together with a phase-shifting measurement technique to greatly improve coordinate measurement accuracy. While the design of calibration tool 64 is very suitable for spur gear measurement, its usefulness is not necessarily limited to gear inspection. For example, it can be applied to the absolute measurements of a large variety of objects with rotary geometry.

When the calibration tools are used to calibrate lateral distortions, it will be appreciated that the vertical and horizontal gratings serve as metric standards. The geometric and photometric accuracy of the gratings is thus crucial to accurate lateral calibrations. Preferably, the geometric accuracy of the gratings will be about 1 μm to get a lateral calibration accuracy of several microns. In addition, non- sinusoidal distortions present in the grating transmittance functions need to be sufficiently small to minimize phase errors. In addition, the contrast ratio of the fringes is preferably as close to one as possible to make a full use of a CCD's effective dynamic range. All of these requirements present great challenges to sinusoidal gratings made with conventional techniques.

In accordance with the present invention, a design and fabrication process are provided for creating stepped sinusoidal gratings (SSGs) of proper spatial periodicity, high contrast ratio, and a sufficient number of grayscales.

Specifically, it has been found that a quantatized version of an ideal sinusoidal grating may be fabricated using the lithography technique.

A quantization scheme that may be used for generating such a grating is illustrated in Figure 6. As shown, one period of an ideal sinusoid grating is evenly divided into N intervals along the normal to the grating lines. The width of each interval is chosen to be equal to the width of CCD pixels when the imaging system operates at a designated magnification factor. The averaged transmittance of the nth interval is calculated according to the following formula,

where d represents the grating period, w=d/N is the width of the intervals, and sine (x)= (sin x)/x. Digitizing grating transmittance ranging from zero to full transmission into M levels, the quantatized transmittance level L_n for the nth interval can be found as

L_n = [Mϊ(n)+ 0.5], (6)

where [ ] denotes rounding to the nearest integer. Since lithography can only deal with binary values, an area-encoding scheme simulates grayscales to represent different quantatized transmittance levels. In this method, M cells are used as a group to represent a single transmittance level . The total number of transparent cells within a cell group is set equal to the transmittance level to be represented. Figure 7 shows some patterns resulting from such an encoding method for several transmittance levels.

A dithering optimization process is preferably used to optimize the number and spatial layout of transparent cells within each cell group under the designated magnification factor. In such a case, if we require that the digitized grating represents correct averaged transmittance levels when CCD pixels are aligned with the boundaries of the cells, the total number of transparent cells within a column of a cell group satisfies the following condition.

where Li represents the desired transmittance level when a CCD pixel is aligned with the 1th column of a cell group, and cj represents the total number of transparent cells in the 1th column of the cell group. Shifting the detection position i M times to align a CCD pixel with all columns of a cell group, AΛ/M equations similar to Equation (7) can be obtained. The total numbers of transparent cells in ivM columns can then be determined from these i\Λ/M equations by a matrix inversion method. The elementary pattern of one grating period can be duplicate along the row and the column directions to span the entire grating pattern. Figure 8 shows an example of a grating resulting from the above design procedure as viewed from three different resolutions of observation. As desired, the spectra of its image will contain no noticeable high-order harmonics.

To produce the SSG of Figure 8, a pattern design was created and fractured (converted to e- beam language) with CATS (Transcription Enterprises Ltd.) using an IBM RS 6000. The pattern was written onto a blank maskplate using EBR9 resist (Hoya, Inc.) using a Leica EBPG5 electron beam lithography system running under a DEC VAX computer. This system is capable of writing to less than 20nm resolution. Masks used for subsequent photolithography applications are generally limited to 1 um resolution due to the wavelength of light (~4l0nm) . The exposed mask plate was developed in MIBK:IPA (methyl isobutyl ketone: isopropyl alcohol), rinsed in IPA, and chrome etched using wet chemistry EBR9. This is just one specific example of the lithography process that may be used for fabricating a simulated-grayscale SSG of the present invention. The grating of Figure 8 is advantageous over existing sinusoidal gratings in its high geometrical accuracy (better than 1 μm) , high contrast ratio (near unity) , and very low harmonic distortion (third-order harmonic is less than 2% after low-pass filtering by the imaging system) . Because of the periodicity of the elementary pattern along the row direction, the output of a CCD pixel is shift-invariant along the row direction when the designated magnification factor is used. The arrangement of the transparent cells within a cell group allows correct CCD output when the grating image is aligned with the boundaries of the columns in a cell group. As a result, excellent performance is achieved even when the grating is used under a magnification factor that is slightly different from that which is designated, or a certain degree of misalignment exists between a CCD pixel and the image of the grating.

A preferred manner of constructing the surface used as the "diffusive flat" will be described. To minimize phase errors introduced by the surface itself this surface must be very flat . On the other hand, this surface also needs to have sufficient diffusivity to provide adequate and uniform diffusive reflection. This imposes stringent constraints on the surface microstructure .

First, the standard deviation of the surface roughness should be larger than the wavelength of the light source. Second, the lateral correlation length of surface depth variations should be small compared with the resolution of the imaging optics. There have been many techniques available for making a flat surface within a rather tight tolerance. Desired surface roughness can also be achieved by suitable machining methods. However, accurate control over the lateral correlation length is a strikingly difficult task for most machining technologies, especially mechanical machining technologies .

According to the present invention, a flat surface fabricated with VLSI -based technologies has been found to overcome the difficulties of the prior art. Referring now to Figure 9, a "diffusive flat" 74 of the present invention preferably comprises three distinct layers 76, 78 and 80. Bottom layer 76 is a suitable substrate with a tight flatness tolerance, such as a suitable silicon (or glass) substrate. Layer 78 may be a thin layer of aluminum, such as a layer having a thickness of 4000A, which is deposited on the substrate by means of physical vapor deposition (PVD) . As one skilled in the art will appreciate, layer 78 is mainly responsible for the surface reflection. Layer 80 is preferably a protective coating formed as a thin oxidation layer on the top of the aluminum layer.

After deposition, layer 78 is preferably processed through surface etching to make the surface microscopically rough, thereby yielding pits 82. First, a thin layer of photoresist is evenly applied to the aluminum. A photomask with random black-white patterns is then replicated on the photoresist-covered substrate via contact optical lithography. During the replication, photoresist beneath the black patterns remains unexposed. Following exposure, the unexposed photoresist is then removed in the developer solution (negative resist) .

The remaining resist serves as a protective mask during the subsequent etching step. Surface etching is achieved through chemical reaction. The chemistry makes the etch attack some materials much more vigorously than others. The resulting etch is primarily in the vertical direction with little lateral etching. The depth of pits 82 can be controlled by choosing suitable operation parameters. For white light illumination with an average wavelength of 550 nm, the depth of the pits should be larger that 2750A. Upon the completion of the etching, a large number of pits, whose distribution closely follows the random pattern on the photomask, are then formed on the aluminum surface. The averaged spacing between adjacent pits can be as small as 2μm with a state-of-the-art lithography system. Figure 10 shows a random pattern produced within a 1mm by 1mm area on the flat surface .

Now that the construction of the calibration surfaces has been described, the manner in which those surfaces may be used to calibrate an optical measurement system will be described. As noted above, a phase-shifting numerical lateral calibration may be utilized for calibration of lateral distortion. At least two separate techniques may be utilized in such a calibration: one is the phase-shifting coordinate measurement technique and the other is the numerical calibration algorithm used for data processing. In the phase-shifting coordinate measurement technique, two mutually orthogonal sinusoidal gratings are used as known objects. These two gratings are sequentially placed at the plane z=z₁ in front of the imaging system to form a virtual grid pattern. The normal directions of the gratings' fringes are chosen as the horizontal and the vertical directions of the world coordinate system. The position of an arbitrary point P in the plane z=z_x is characterized by the phase values obtained from the gratings at P.

where ψv (u , v) and φ/. ( u, v) represent the unwrapped phases measured at pixel location (u,v) when the gratings with vertical and horizontal fringes are placed in front of the imaging system. Coordinate { u, v) is the detection plane coordinate of the center of the pixel of interest. After the measurement, a point in the object space (x,y,z) is then related to its corresponding image point (u,v) in the detection plane. This is diagrammatically illustrated in Figure 11.

A phase-shifting detection scheme can be used for measuring the phases in Equation (8) to enhance measurement accuracy. Four images are captured with a phase shift of π/2 introduced between two successive ones. Phases can be calculated from Equation (4) as a function of the measured intensities .

Several features of this method are worth noting. First, a sinusoidal pattern is highly localized in the spatial frequency domain while it may extend infinitely in the spatial domain. The Nyquist criterion can be easily met without sacrificing the available bandwidth of the imaging system. Second, this method is insensitive to slowly varying additive and multiplicative noise. Third, phase calculations are performed independently at each pixel in this method. As a result, a pixel-by-pixel accurate measurement can be realized with extremely high sampling density.

For example, with a 1024x1024 CCD camera, more than 10^s data points can be obtained. The resultant data redundancy can be used to suppress random measurement errors by a factor of ten with suitable post-processing algorithms.

An attractive alternative to the coordinate measurement technique described above is a Fourier transform method. This technique eliminates the needs for shifting the gratings transversally, allowing world coordinates to be obtained from a single snapshot of the grating. In this method, the deformed grating images are first Fourier transformed and then bandpass filtered in the spectral domain to pick up the first order harmonic of the grating images on the positive frequency axis. The filtered images are then inverse transformed back to the space domain to form a complex-valued image s_h(u,v) and s_v(u,v) . Phases φ_h(u,v) and φ_v(u,v) can then be calculated from the following equations.

and

where Im[] and Re [] represent extracting the imaginary and the real parts of a complex value. Similar immunity to additive and multiplicative noise is achieved in this method through the spectral domain filtering process and the division in Equations (9) and (10) .

One skilled in the art will appreciate that the spectra of the modulated signal and the DC term must be suitably separated in the frequency domain. In addition, the period of the gratings needs to be properly chosen to achieve suitable separation between the spectra. The largest measurement error of this method arises from energy leakage. Such a non-shifting detection technique is advantageous when accurate phase shifts are difficult to realize or a large number of phase- shifting measurements are needed. In the latter case, measurement accuracy of the phase-shifting approach becomes problematic when positioning errors of moving parts accumulate quickly. The task of lateral calibration is to establish a nonlinear mapping from the detection plane coordinate (u,v) to its corresponding world coordinate (x,y) at depth z. There have existed some lateral calibration methods in the photogrammetry and 3-D machine vision techniques. These methods, however, have required a physical model of the imaging system. For some PSPFP applications, such a model could be very difficult to obtain. Benefiting from the spatially rich, high precision data set offered by the proposed coordinate measurement technique, a numerical lateral calibration scheme can be adopted to form a numerical representation of the nonlinear coordinate mapping. Sufficient sampling density along the depth direction can be achieved by scanning the entire depth range at a suitable stepsize.

Specifically, assume the depth range occupied by the measured object is from zero to z_raax. The phase-shifting coordinate measurement is performed at several evenly spaced planes z=z₃ (j = 0,1, •• • N- 1) . Calibrations follow the same procedure as the bisection method used in the root-finding of nonlinear functions. First, two planes at z=0 and z=Zmax are calibrated. Then the plane at z=(l/2)z_max is calibrated. At each pixel location (u,v), three measured surface points, which are imaged to the pixel at (u,v) , are used to fit piecewise linear functions through linear interpolation,

and

where x(u,v) and y(u,v) represent the horizontal and vertical world coordinates corresponding to pixel location (u,v) . For the bisected regions [0, (1/2) z_max] and [ (1/2 ) z_max, z_max] , coordinate measurements are performed in the same manner. After the second level of bisections, five planes are calibrated. The newly obtained points are then used to examine the validity of the fitted piecewise linear functions. The overall measure of validity is defined as

where the summations are performed over all of the pixels and the depth positions. If the calculated error is less than the required tolerance, the calibration stops. Otherwise, all of the measured data are used for fitting new piecewise linear functions, which have similar forms as the functions shown in Equations (11) and (12) . Such a cycle of bisection, examination, and fitting repeats until the allowed maximal number of bisections is reached or the required calibration accuracy is achieved.

Such a calibration technique eliminates the necessities of a physical model of the imaging system, which makes it very suitable for the PSPFP system calibration. Furthermore, due to the high sampling density of the phase-shifting coordinate measurements, no interpolation is required in the lateral dimensions. Thus, additional errors caused by the interpolation process are avoided. This calibration technique can also adaptively find the required sampling density along the depth direction for predetermined calibration accuracy.

As noted above, phase calibration is performed using the "diffusive flat" of the calibration tool. The phase calibration can be formulated as follows: at a fixed location (u,v), given N phase values 0_j (u, v) measured at different depth positions z=z-, (j = 0,1,2, •••N-l), find a transformation that maps an arbitrary 0(u,v) to the desired depth z at required accuracy. With a numerical phase calibration approach, the nonlinear mappings from 0(u,v) to z can be found independently for each pixel .

Upon the completion of the phase calibration, a pixel will "see" a trace of phase values varying with depth positions. When the measurement sensitivity of a PSPFP system is sufficiently high, phase values measured at a pixel location varies monotonically with depth positions. Because of this monotonicity, the depth calibration can be performed independently of the lateral calibration at a pixel location.

Assume the depth range occupied by the measured object is from zero to z_max. The phase calibration is performed at several evenly spaced planes z=z-, (j = 0,1, •••N-l). In the phase calibration, the grating measured is projected onto the flat surface with the projection optics. Calibrations follow the same bisection procedure as that used in the lateral calibration.

First, two planes at the z=0 and z=z_max are calibrated. Then the plane at z=(l/2)z_max is calibrated. At each pixel location (u,v), three measured phases are used to fit a piecewise linear function through linear interpolation,

*(«.v)= Λ„(z). (14)

where 0(u,v) represents the phase measured at pixel location (u,v) . For the bisected regions

[0, (1/2) z_max] and [ (1/2 ) z_max, z_max] , phase calibrations are performed m the same manner. After the second level of bisections, five planes are calibrated.

The newly obtained phases are then used to examine the validity of the fitted piecewise linear function. The overall measure of validity is defined as

where the summations are performed over all of the pixels and the depth positions. If the calculated error is within the required tolerance, the calibration stops. Otherwise, all of the measured data are used for fitting new piecewise linear functions, which have the similar forms as the functions shown m Equation (14) . Such a cycle of bisection, examination, and fitting repeats until the allowed maximal number of bisections is reached or the required calibration accuracy is achieved. An absolute PSPFP measurement yields a phase- map ø(u,v) . Based on the calibration data and the measured phase map, the object shape can be retrieved. The monotonicity of phase changes, as already noted above, essentially decouples the depth measurement from the lateral position measurement. With this simplification, depth position z can be found from the measured phase value independently of x and y. Once the depth position z is known, the lateral positions x and y can be found from lateral calibration data.

When the small volume Ω occupied by the measured object is calibrated at sufficiently fine intervals along the depth direction, the process of finding depth position z can be simplified to the following two steps. First, for a given measured phase φ_m at a pixel location, find the two adjacent phase values φ_x and φ₂ measured at the same pixel location which satisfy the following condition:

Depth position z_m can then be found from linear interpolation .

2-. = Z. φ₂ -φ (17)

Assume the lateral coordinates associated with φi and φ₂ are x_i# x₂,

, and y₂. The lateral coordinates x_ra and y_m can be calculated as

In operation, the functionality of the measurement system can be divided into the calibration phase and the measurement phase. Typically, calibrations will be required when the system is first installed or reconfigured. The calibrated system is then ready for measurements. Due to various external and internal changes, the system may need to be recalibrated after a certain number of measuring cycles to maintain high measurement accuracy. As stated previously, system calibrations are performed sequentially at several depth positions (or several angular positions if a rotary calibration tool such as tool 64 is used) . At each depth position, the horizontal calibration is first performed, then the vertical calibration, and finally the phase calibration. During the horizontal calibration, one image is first captured with the grating at its initial position. The calibration tool is then shifted by translation stage 58 by one fifth of a grating period, and the second image is captured. After four phase steps, five images are captured. The captured image data is sent to the computer for phase calculations. The vertical calibration and phase calibration follow the same data acquisition and phase calculation sequence.

The gratings used to perform the lateral calibrations are illuminated from behind without using the projection optics of the measurement system. In the phase calibration, however, grating lines from the projection arm's grating are projected on the flat surface with projection optics. Phase shifts are generated by translating the projection grating using the translation stage of the projection arm. Since different calibration processes use different parts of the calibration tool, the calibration tool needs to be translated horizontally during the calibration process.

After all of the calibrations are finished at one depth position, the calibration tool is then shifted to the next depth position to perform the same set of calibrations. This process repeats until the entire range of depths is completely swept. The calibration data is first unwrapped and filtered by the processing algorithms, and then passed to the calibration algorithms for further processing. The processed calibration data is stored on the computer's disk for future use.

It should be distinctly understood that the order of performing the horizontal, vertical and flat surface calibrations is a matter of convenience for the user. One can choose to complete them one by one at one depth position and then move to the next depth position. Alternatively, one type of calibration may be completed at all depth positions before the next calibration is performed.

In the measurement phase, the calibration tool is removed from the mounting fixtures. The object to be measured is then placed within the calibrated volume. The grating in the projection arm is projected on the object to perform phase measurements. The measured phases are first unwrapped and filtered and then used for shape retrieval. The retrieved object profile data is finally displayed or evaluated as desired.

While preferred embodiments of the invention and preferred methods of practicing same have been shown and described, modifications and variations may be made by those of ordinary skill in the art without departing from the spirit and scope of the present invention, which is more particularly set forth in the appended claims. In addition, it should be understood that aspects of the various embodiments may be interchanged both in whole or in part. Furthermore, those of ordinary skill in the art will appreciate that the foregoing description is by way of example only, and is not intended to be limitative of the invention so further described in such appended claims.

Claims

WHAT IS CLAIMED IS:

1. A method of calibrating an optical system for measuring a profile of an unknown object, said method comprising steps of: providing a first known grating and a second known grating, first grating lines of said first known grating being orthogonal to second grating lines of said second known grating; situating said first known grating in a predetermined starting location relative to said optical system; using said optical system, detecting at least one image of said first known grating at each of a plurality of selected depth positions to produce first imaged data; situating said second known grating in a predetermined starting location relative to said optical system; using said optical system, detecting at least one image of said second known grating at each of a plurality of selected depth positions to produce second imaged data; and processing said first imaged data and said second imaged data to produce calibration data for said optical system.

2. A method as set forth in claim 1, wherein a plurality of images of said first known grating and said second known grating are detected at each of said selected depth positions.

3. A method as set forth in claim 2, wherein said first known grating and said second known grating are each laterally moved by a predetermined phase shift between detecting of said plurality of images at each of said selected depth positions.

4. A method as set forth in claim 3, wherein said predetermined phase shift is a phase shift of π/2.

5. A method as set forth in claim 4, wherein a total of four phase-shifted images are detected at each of said selected depth positions.

6. A method as set forth in claim 1, wherein said first imaged grating data and said second imaged grating data are processed according to a Fourier transform method to produce said calibration data.

7. A method as set forth in claim 1, wherein said first known grating and said second known grating are fixed to a common calibration tool.

8. A method as set forth in claim 7, wherein said first known grating and said second known grating are fixed to said calibration tool so as to be coplanar.

9. A method as set forth in claim 8, wherein said calibration tool has a rotary axis, said first known grating and said second known grating being angularly separated about said rotary axis.

10. A method as set forth in claim 1, wherein each of said first known grating and said second known grating is a stepped sinusoidal grating (SSG) .

11. A method as set forth in claim 1, wherein said depth positions are sequentially selected according to a bisection technique.

12. A method as set forth in claim 1, further comprising steps of : providing a flat surface having light- diffusing characteristics; situating said flat surface in a predetermined starting location relative to said optical system; projecting a grating pattern onto said flat surface; using said optical system, detecting at least one image of said grating pattern as reflected from said flat surface at each of a plurality of selected depth positions to produce third imaged data; and processing said third imaged data to produce further calibration data for said optical system.

13. A method as set forth in claim 12, wherein said first known grating, said second known grating and said flat surface are fixed to a common calibration tool.

14. A method as set forth in claim 13, wherein said first known grating, said second known grating and said flat surface are fixed to said calibration tool so as to be coplanar.

15. A method as set forth in claim 13 , wherein said calibration tool has a rotary axis, said first known grating, said second known grating and said flat surface are angularly separated about said rotary axis.

16. A method as set forth in claim 1, further comprising the step of using said calibration data in determining profile of an unknown object.

17. A method of calibrating an optical system for measuring a profile of an unknown object, said method comprising steps of: providing a flat surface having light- diffusing characteristics; situating said flat surface in a predetermined starting location relative to said optical system; at each of a plurality of depth positions, projecting a predetermined number of successively phase-shifted grating patterns onto said flat surface; using said optical system, detecting respective images of said grating pattern as reflected from said flat surface to produce imaged data ; and processing said imaged data to produce calibration data for said optical system.

18. A method as set forth in claim 17, wherein said flat surface is constructed as a multilayer composite structure.

19. A method as set forth in claim 18, wherein said multilayer composite structure comprises a substrate layer underlying a reflective layer.

20. A method as set forth in claim 19, wherein said reflective layer defines a predetermined pattern of pits so as to provide said light-diffusing characteristics.

21. A method as set forth in claim 20, wherein said reflective layer is formed of metal.

22. A method as set forth in claim 21, wherein said predetermined pattern of pits is produced in said reflective layer by surface etching.

23. A method as set forth in claim 19, wherein said composite structure further comprises a protective layer overlying said reflective layer.

24. A method as set forth in claim 17, wherein said depth positions are sequentially selected according to a bisection technique.

25. A method as set forth in claim 17, further comprising the step of using said calibration data in determining profile of an unknown object.

26. A calibration tool for use in calibrating an optical system for measuring a profile of an unknown object, said calibration tool comprising: a first known grating having first grating lines ; a second known grating having second known grating lines, said first grating lines of said first known grating being orthogonal to second grating lines of said second known grating; and a flat surface having light-diffusing characteristics .

27. A calibration tool as set forth in claim 26, wherein said first known grating, said second known grating and said flat surface are fixed to a common mounting structure so as to be coplanar.

28. A calibration tool as set forth in claim

27, wherein said calibration tool includes at least one translation stage such that said mounting structure is movable in at least a depth dimension.

29. A calibration tool as set forth in claim

28, wherein said mounting structure includes three translation stages such that said mounting structure is movable in horizontal and vertical dimensions as well as said depth dimension.

30. A calibration tool as set forth in claim 27, wherein said calibration tool is movable in at least a depth dimension so that said first known grating, said second known grating and said flat surface can be situated in different depth planes for performing a calibration.

31. A calibration tool as set forth in claim 26, wherein said calibration tool has a rotary axis, said first known grating, said second known grating and said flat surface being angularly separated about said rotary axis.

32. A calibration tool as set forth in claim 26, wherein each of said first known grating and said second known grating is a stepped sinusoidal grating (SSG) .

33. A calibration tool as set forth in claim

32, wherein said stepped sinusoidal grating is produced by multiple intervals in a direction normal direction to said grating lines thereof, each of said intervals having a predetermined level of transmittance related to a respective number of transparent cells therein.

34. A calibration tool as set forth in claim

33, wherein said intervals have an equal width.

35. A calibration tool as set forth in claim 33, wherein said number of transparent cells in each of said intervals is produced by a laser lithography process .

36. A calibration tool as set forth in claim 26, wherein said multilayer composite structure comprises a substrate layer underlying a reflective layer.

37. A calibration tool as set forth in claim 36, wherein said reflective layer defines a predetermined pattern of pits so as to provide said light-diffusing characteristics.

38. A method as set forth in claim 37, wherein said reflective layer is formed of metal.

39. A calibration tool as set forth in claim 38, wherein said predetermined pattern of pits is produced in said reflective layer by surface etching.

40. A calibration tool as set forth in claim 36, wherein said composite structure further comprises a protective layer overlying said reflective layer.

41. A grating for use in an optical system for measuring a profile of an unknown object, said grating comprising: a plurality of parallel grating lines having sinusoidally varying transmittance characteristics; said sinusoidally varying transmittance characteristics being produced by multiple intervals in a direction normal to said grating lines each having a predetermined level of transmittance; and said predetermined level of transmittance being related to a respective number of transparent cells in said interval.

42. A grating as set forth in claim 41, wherein said intervals have an equal width.

43. A grating as set forth in claim 41, wherein said number of transparent cells in each of said intervals is produced by a laser lithography process .

44. An apparatus for profiling a surface of an object comprising: illumination means for directing an incident beam of light having a varying intensity pattern onto said surface; means for spatially phase shifting said varying intensity pattern of said incident beam of light; detector means for receiving deformed grating images of said surface; and processing means, operatively coupled to said detector means, for determining a profile of said surface based on a number of said deformed grating images at different spatial phases and stored calibration data, said stored calibration data including both lateral distortion and phase calibration data.