Practical three-dimensional computer vision techniques for full-field surface measurement Y. Y. Hung L. Lin H. M. Shang B. G. Park Oakland University Mechanical Engineering Department Rochester Hills, Michigan 48309

Abstract. Two practical computer-aided optical techniques for full-field surface shape measurement are presented, one for diffuse surfaces and the other for specularly reflective surfaces. The former technique is based on projecting a computer-generated fringe pattern onto a diffuse surface, and the latter is based on reflecting the fringe pattern from a specularly reflective surface. The fringe pattern is perturbed in accordance with the object surface with the fringe phase bearing information on the depth/slope of the object surface. The computer-generated fringe pattern conveniently enables the fringe phase to be manipulated, and hence the determination of the phase distribution using a phase extraction algorithm. Instead of deriving the mathematical relationship between the fringe phase distribution and the surface depth/slope, this relationship is obtained by calibration. The techniques described can be easily implemented for rapid measurement of 3-D surface shapes in an industrial setting. © 2000 Society of Photo-Optical Instrumentation Engineers. [S0091-3286(00)01601-9]

Subject terms: three-dimensional shape measurement; machine vision; profilometry; surface quality inspection; nondestructive testing. Paper SM-16 received June 20, 1999; revised manuscript received Aug. 11, 1999; accepted for publication Aug. 11, 1999.

1

Introduction

The quest for improved product quality and process efficiency is ongoing. Associated with this is the need for new and improved methods of measuring 3-D surface geometry. Presently, 3-D surface coordinates are often measured using a coordinate measurement machine 共CMM兲. The use of a CMM requires point-by-point scanning of the object surface, which is slow and generally cannot meet the speed requirement in the production environment. Laser scanning techniques are also available for noncontact measurement. However, they also require scanning on a point-by-point or line-by-line basis. Recently interest has been growing in exploring full-field optical methods because of their highspeed capability. Full-field optical methods for 3-D shape measurement can be divided into two categories: coherent light methods and incoherent light methods. The coherent light methods are based on interferometric principles. A review of holography and shearography for this application is given in Ref. 1. The coherent light methods are generally not practical for industrial applications as they generally require a delicate optical setup and stringent environmental stability. The incoherent light methods can be further classified into moire´ methods and projected/reflected grating techniques. Moire´ is referred to as the ‘‘beat phenomenon’’ that produces a wavy pattern resulting from the optical interference of two slightly mismatched 2-D spatial signals 共such as gratings兲. The wavy pattern is known as a moire´ fringe pattern. Shadow moire´2–5 is the oldest technique utilizing the moire´ phenomenon for measuring 3-D shapes. A grating is placed in front of a 3-D object. Upon illuminating the grating with a strong light source, the shadow of the grating Opt. Eng. 39(1) 143–149 (January 2000)

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is cast on the object surface. The moire´ fringe pattern formed by the grating and its shadow depicts the surface contours. A drawback of the shadow moire´ technique is that it is not sensitive enough for practical applications and it requires a grating screen larger than the size of the object. In the projection grating moire´ technique,6–9 where the use of a large grating is not required, a grating is projected onto the object using a projector or a two-point source interferometer. The images of the grating projected on two different objects are recorded separately, and the moire´ fringe pattern subsequently formed by the two grating images represents the contour difference of the two objects. If one of the objects is planar, the moire´ fringe pattern is a contour map of the 3-D object. An alternative approach to moire´ contouring is the projected/reflected grating techniques,10–12 which does not make use of the moire´ phenomenon. Instead, the 3-D surface information is directly extracted from the projected/ reflected grating. For objects with diffuse surface, a grating is projected onto the surface 共by a projector or a two-point source interferometer.兲 The grating image is digitized and the phase distribution in the grating is obtained using a phase extraction technique 共such as digital Fourier transformation, phase stepping technique, etc.兲 and the depth information is determined. The reflected grating technique is used for objects with specularly reflective surfaces. In this case, a grating is reflected by the object surface, which acts like a mirror, and the phase distribution of the reflected grating is related to the surface slope. This paper presents a practical approach referred to as the 3-D computer vision technique, based on computercontrolled grating projection/reflection techniques.13–20 The © 2000 Society of Photo-Optical Instrumentation Engineers

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Fig. 2 Projected fringe pattern of a 3-D car model.

When the fringe pattern expressed by Eq. 共1兲 is projected on an object surface, the distorted fringe pattern that is received by a CCD camera can be expressed as Fig. 1 Schematic diagram of computer-controlled fringe projection technique.

computer-controlled grating offers the following advantages: 共1兲 the grating pitch can be altered at will; 共2兲 the phase in a grating can be precisely manipulated; 共3兲 a grating intensity variation of any form can be created; and 共4兲 a grating of any shape can be generated to match the object surface, thus optimizing the dynamic range of measurement. Furthermore, a simple calibration technique is introduced to relate the grating phase and the surface depth/ slope, thus eliminating the use of complicated mathematical relationships. The measurement process is fully automated, and the technique has the great potential of being developed into a practical tool for measurement in a manufacturing environment. 2

Digitally Controlled Fringe Projection Technique This technique is for measuring objects with diffuse surfaces. The optical arrangement of the technique is shown in Fig. 1. A computer generates a fringe pattern consisting of a set of sinusoidal linear and parallel fringes, and this fringe pattern is projected onto the surface of the object with a video projector. The fringe pattern received by a video camera is digitized into the computer via a frame grabber. The phase distribution of the fringe pattern is deduced, and the 3-D shape is determined. The intensity distribution of a computer-generated fringe pattern can be expressed as I g ⫽1⫹cos 关共 2 ␲ / P 兲 x 兴 ,

共1兲

where I g is the normalized intensity and (2 ␲ / P)x is the fringe phase. Equation 共1兲 represents a set of linear and parallel fringes with fringe spacing P 共pitch兲 and fringe lines perpendicular to the x axis. Here the intensity of the fringe pattern varies sinusoidally. 144

I⫽a 共 x,y 兲 ⫹b 共 x,y 兲 cos ⌬,

共2兲

where a(x,y) is the dc and b(x,y) is the amplitude of modulation. Both a(x,y) and b(x,y) represent intensity distortions that depend on the variations arising from nonuniform illumination and object surface reflectivity, as well as the nonuniform response of the camera sensor. The fringe phase is distorted by the surface topography, and the distorted fringe phase ⌬ is given by ⌬⫽ 共 2 ␲ /m P 兲共 x⫺kZ 兲 ,

共3兲

where m is the pitch distortion factor arising from noncollimated illumination and the distortion in the optical system; k is a sensitivity factor related to the angle between the projection direction and the camera direction; and Z(x,y) is the surface depth, which defines its shape. Equation 共3兲 infers that ⌬, the fringe phase, carries information on Z(x,y). An example of such a fringe pattern observed on a 3-D car model is shown in Fig. 2. Note that m and k in Eq. 共3兲 depend on the optical setup and are a function of the position of surface points. Derivation of equations for m⫽m(x,y) and k⫽k(x,y,Z) is very difficult; in particular, k is a function of the surface coordinates in which Z is unknown. Furthermore, the equations are difficult to be implemented as they require knowledge of the precise positions of the camera and the projector relative to the object. A calibration process is presented as follows, which enables the measurement be performed without the knowledge of m⫽m(x,y) and k(x,y,Z). If the object surface is perfectly flat 关i.e., Z(x,y)⫽0], the phase distribution in Eq. 共3兲 is reduced to ⌬ R ⫽ 共 2 ␲ /m P 兲 x.

共4兲

Figure 3 shows a typical example of a fringe pattern that is observed on a planar surface. We can see that the fringe spacing is not uniform and the fringe lines are not perfectly straight. The distortion is attributed to m⫽m(x,y) even if Z

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Hung et al.: Practical three-dimensional computer vision techniques . . .

Fig. 3 Projected fringe pattern on a flat surface.

Fig. 4 Four fringe segments with phase shifts of 0, 90, 180, and 270 deg as observed on a planar object.

is equal to zero. The fringe phase ⌬ R of the planar object will serve as the reference phase for shape measurement of 3-D objects. Comparing Fig. 2 with Fig. 3, we see that the fringe lines on a 3-D surface are further distorted. A comparison of Eqs. 共3兲 and 共4兲 shows that this additional distortion is due to the phase angle, which is the kZ(x,y) term. Therefore, the problem of measuring Z(x,y) is reduced to the determination of this phase angle of distortion. The phase extraction algorithm described in the following can be used to determine the fringe phase with precision. 3 Phase Extraction Algorithm Here we describe a phase extraction algorithm to determine the fringe phase, which is the argument of the cosine function in Eq. 共2兲. Three unknowns appear in the equation, namely, a(x,y), b(x,y), and ⌬. At least three equations are required to separate these unknowns from fringe patterns observed on the object. With computer-generated fringes, a known phase could easily be superimposed onto a fringe pattern to generate additional equations. This process is known as phase shifting. Several phase determination algorithms21 are available for determining the phase distribution ⌬. Here a four-frame phase determination algorithm is used. This requires projecting four fringe patterns with phase shifts of 0, 90, 180, and 270 deg sequentially onto the object surface and their respective images are digitized via a video camera and a frame grabber. Figure 4 illustrates the phase shift phenomenon. The fringe pattern projected on a flat surface is divided into four sections and their positions on the object surface are displaced by an amount proportional to the degree of phase shift. The phase-shifted fringe patterns illustrated are to be read in a top-down manner from 0 to 270 deg. The four digitized fringe patterns can therefore be represented by the following equations: I 1 ⫽a⫹b cos ⌬, I 2 ⫽a⫹b cos 共 ⌬⫹90 deg兲 , I 3 ⫽a⫹b cos 共 ⌬⫹180 deg兲 , I 4 ⫽a⫹b cos 共 ⌬⫹270 deg兲 .

共5兲

By solving these equations, the phase distribution ⌬ can be computed using the following equation: ⌬⫽tan⫺1

冉 冊

I 4 ⫺I 2 . I 1 ⫺I 3

共6兲

This procedure is also applicable to any other object shapes. Note that the phase is precisely determined at each pixel point of the digitized image. Since the phase is wrapped between 0 to 2␲, it is necessary to unwrap it. In general, a phase unwrapping technique22 is required to identify phase reversal in the fringe pattern. With the use of computer-generated fringe patterns, however, the phase in the projected fringe pattern can be easily programmed to be monotonically increasing 共or decreasing兲 from the left-hand side of the object to its right-hand side to avoid phase reversal. This has greatly simplified the phase unwrapping process. During measurement, the phase shift technique is first used on a planar object to determine the reference fringephase distribution ⌬ R , which is then stored in a computer for subsequent measurement of any 3-D objects. Subsequent use of Eqs. 共3兲 and 共4兲 enables the object shape Z(x,y) to be determined from ⌬ R and ⌬ using the following equation: Z 共 x,y 兲 ⫹ 共 ⌬ R ⫺⌬ 兲 /K,

共7兲

where K⫽(2 ␲ k/m P) is a constant of the optical setup but is a function of the position on the object. Here, m and k are combined into one unknown function K, which can be predetermined by the following calibration procedure. 4 Calibration Procedure To determine K, the 3-D object is given a rigid-body translation through a known distance ␦ Z parallel with the z axis. Alternatively, the same result is achieved by translating the projector through a distance ␦ Z. Hence, Eq. 共3兲 becomes ⌬ c ⫽ 共 2 ␲ /m P 兲关 x⫺k 共 Z⫹ ␦ Z 兲兴 ,

共8兲

where ⌬ c is the calibrated phase due to ␦ Z, which is also determined by the phase shift technique. By combining Optical Engineering, Vol. 39 No. 1, January 2000 Downloaded from SPIE Digital Library on 30 Jun 2011 to 130.215.125.7. Terms of Use: http://spiedl.org/terms

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Fig. 5 Three-dimensional shape of the car model of Fig. 2.

Eqs. 共3兲 and 共8兲, we obtain the following equation: K⫽ 共 ⌬⫺⌬ c 兲 / ␦ Z.

共9兲

5 Experimental Demonstration To determine the distorted phase in the fringe pattern for the car model shown in Fig. 2, fringe patterns with phase shifts of 0, 90, 180, and 270 deg are projected and imaged. The fringe-phase distribution ⌬ of the car model is thus determine using Eq. 共6兲. The surface coordinates of the car model Z(x,y), determined using Eq. 共7兲, is plotted in Fig. 5. Figure 6 shows another example in which the surface has holes. In this case, the boundaries of the holes were first determined by an edge detection technique,23 and a computer program was written to unwrap the phase around the holes. That this technique can be applied to surface quality inspection is obvious. Normally, a defective surface, such as one with a dent on it, would cause a localized anomaly in the fringe pattern that is readily visible. As an example, the localized fringe anomaly due to three surface dents is shown in Fig. 7共a兲, and a 3-D plot of the dented surface is shown in Fig. 7共b兲. The measurement speed of this machine vision technique is very high. In this investigation, each measurement 共including image acquisition and phase determination兲 takes only less than a minute with the use of a Pentium 133-MHz microcomputer. The use of a computer with faster processing capability would considerably increase the measurement speed. 6

Digitally Controlled Reflected-Fringe Technique The reflected fringe technique is used to gauge objects with specularly reflective surfaces. The surface of a painted automotive body is one such surface. Surfaces not fully specularly reflective can become so by the application of a coating of oil. Figure 8 shows a schematic diagram of the technique. The specularly reflective object surface acts as a mirror. A computer-generated fringe pattern displayed on a TV monitor is reflected off the object surface, and a virtual 146

Fig. 6 (a) Wrapped phase distribution on an object with holes, (b) unwrapped phase distribution, and (c) 3-D plot of the object shape.

image of the fringe pattern is received by the video camera and digitized into a computer. The fringe pattern is distorted in accordance with the slope of the object surface. In this case, the fringe phase ⌬ is related to the surface derivative ⳵ Z/ ⳵ x, and Eq. 共3兲 is modified to become ⌬⫽ 共 2 ␲ /m P 兲关 x⫺k 共 ⳵ Z/ ⳵ x 兲兴 .

共10兲

Should the computer-generated fringe lines be made perpendicular to the y axiz, ⳵ Z/ ⳵ x in Eq. 共10兲 is replaced by ⳵ Z/ ⳵ y, the surface slope in the y direction. Like the projected fringe technique, the fringe phase is determined by the phase determination algorithm using four fringe patterns with phase shift of 0, 90, 180, and 270

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Hung et al.: Practical three-dimensional computer vision techniques . . .

Fig. 7 (a) Fringe pattern on a surface with three dents and (b) 3-D plot of the dents.

deg. The calibration procedure, however, is slightly different from that of the projection technique. Instead of translating the object, the object is slightly tilted through a known calibration angle ␪ c . Tilting the object introduces a uniform slope change in the object and Eq. 共11兲 becomes ⌬ c ⫽ 共 2 ␲ /m P 兲关 x⫺k 共 ⳵ Z/ ⳵ x⫹ ␪ c 兲兴 .

共11兲

Combining Eqs. 共10兲 and 共11兲 yields K⫽(2 ␲ k/m P), the constant of the optical setup. Thus, K⫽ 共 ⌬⫺⌬ c 兲 / ␪ c .

共12兲

A flat mirror can be used to obtain the reference phase ⌬ R as follows: ⌬ R ⫽ 共 2 ␲ /m P 兲 x.

共13兲

Hence from Eqs. 共10兲 and 共13兲, the surface slope is calculated using the following equation:

⳵ Z/ ⳵ x⫽ 共 ⌬ R ⫺⌬ 兲 /K.

Fig. 8 Schematic diagram of computer-controlled reflected-fringe technique.

共14兲

If the surface shape Z(x,y) is desired, it can be obtained by integrating the surface slope. The surface slope can also be differentiated for the determination of surface curvatures1 that are related to the second derivative of Z(x,y). Note that for highly curved surfaces, this technique yields only semiquantitative results, as the preceding derivations are valid only for relatively planar objects. The reflected fringe technique is particularly suited for surface quality inspection, particularly for painted automotive bodies. In this application, generally a semiquantitative result is sufficient. Surface imperfection is normally characterized by an abrupt change in the surface slope. Figure 9 shows surface imperfection including a local surface waviness and two tiny dents: Fig. 9共a兲 is the fringe pattern, Fig. 9共b兲 is the unwrapped phase distribution depicting the surface slope, and Fig. 9共c兲 is a 3-D plot of the surface slope.

7

Other Applications

7.1 Deformation Measurement The projected/reflected fringe techniques can be applied to the measurement of surface deformation. It simply involves in the measurement of the surface before and after deformation, and their difference depicts the surface deformation. 7.2 Nondestructive Testing The projected/reflected fringe techniques may also be developed for nondestructive inspection of subsurface defects. In this application, the test object is deformed and the deformation is measured. The presence of subsurface deformation will exhibit abnormal deformation in the area of the surface above the defect. As an example, Fig. 10 shows two interfacial debonds in an adhesive joint when vacuum stressed. The surfaces of the joint are specularly reflective. The fringe pattern shown in Fig. 10 is reconstructed from the phase distribution depicting surface deformation. The reflected fringe technique is used in nondestructive testing 共NDT兲 as it is more sensitive than the projected fringe technique. 8

Resolution of Technique

Two separate issues are involved in the resolution of this technique: the 共x,y兲 spatial resolution and Z or ⳵ Z/ ⳵ x resolution. The 共x,y兲 resolution depends on the spatial resolution of the image digitization hardware 共combination of camera, Optical Engineering, Vol. 39 No. 1, January 2000

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Fig. 10 (a) Reconstructed fringe pattern revealing two interfacial debonds in an adhesive joint and (b) 3-D plot of the surface deformation.

higher resolution digitization hardware could lead to higher electronic noise, and, consequently, would set a limit on the resolution. 9

Fig. 9 (a) Fringe pattern reflected from a part of a painted car panel, (b) unwrapped phase distribution of the fringe phase, and (c) 3-D plot depicting the surface slope.

frame grabber, and projector兲. In this investigation, the camera and the frame grabber has each a spatial resolution of 512 ⫻ 512 pixels, which means that a total of 250,000 surface points can be measured. Should hardware with higher spatial resolution be used, more points could be measured simultaneously. For example, a digitization resolution of 2000 ⫻ 2000 pixels, which is commercially available, would facilitate measurement of 4 million points. The Z and ⳵ Z/ ⳵ x resolution, however, depends on the gray-level resolution of the digitization hardware. The hardware used in this investigation has a resolution of 256 gray levels, which yields a depth resolution of around 0.5 mm for the projected fringe technique and a slope resolution of 0.001 for the reflected fringe technique. The resolution would be proportionately increased with the use of a higher gray-level image digitizer. However, the use of 148

Conclusion

Two practical 3-D machine vision techniques were presented: one for diffuse surfaces and the other for specularly reflective surfaces. Both techniques are full-field, noncontact, simple, and rapid and can be automated. These techniques have great potential for development into a practical tool for mass measurement and inspection of components in an industrial setting.

Acknowledgments This investigation is supported by the National Science Foundation under Grant No. CMS9601778. The encouragement of Dr. Ken Chong is greatly appreciated.

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Biographies of the authors not available.

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