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Some practical considerations in fringe projection profilometry
ARTICLE IN PRESS Optics and Lasers in Engineering 48 (2010) 218–225
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Some practical considerations in fringe projection profilometry Zhaoyang Wang , Dung A. Nguyen, John C. Barnes Department of Mechanical Engineering, The Catholic University of America, Washington, DC 20064, USA
a r t i c l e in f o
a b s t r a c t
Available online 4 July 2009
As technologies evolve, there have been high demands for the three-dimensional (3D) shape measurement techniques to posses the following combined technical features: high accuracy, fast speed, easy implementation, capability of measuring multiple objects as well as measuring complex shapes. Generally, the existing techniques can satisfy some of the requirements, but not all of them. This paper presents four practical considerations in fringe projection profilometry (FPP) based 3D shape measurements, along with simple but robust solutions, including gamma correction of digital projection, arbitrary setup of system components, phase unwrapping with multi-frequency fringes, and system calibration with a least-squares inverse approach. The validity and practicability of the FPPbased 3D shape measurement technique using the four corresponding technical approaches have been verified by experiments. The presented technique is capable of satisfying the various critical demands in enormous scientific and engineering applications. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Fringe projection 3D shape measurement 3D imaging High accuracy Fast speed
1. Introduction
2. Practical considerations in FPP
Fringe projection profilometry (FPP), also called fringe projection technique, is one of the most widely used techniques in practical applications of three-dimensional (3D) shape measurements, including object detection, digital model generation, object replication, reverse engineering, rapid prototyping, product inspection, quality control, etc. Despite that numerous FPP-based 3D shape measurement approaches have been developed in the past two decades [1–11], the literature survey conducted by the authors reveals that some important issues have not been well addressed in the practice of FPP-based 3D shape measurements. Accordingly, this paper aims to point out a few important considerations in FPP applications, along with simple yet practical solutions that give great promise of satisfying the various high demands of enormous applications in various fields. With the proposed considerations and schemes, the FPP measurement will be capable of detecting 3D shape of objects with the following combined technical features: high accuracy, fast speed, easy implementation, capability of measuring multiple objects and measuring complex shapes.
2.1. Gamma correction of digital projection
Corresponding author. Tel.: +1 202 319 6703; fax: +1 202 319 5173.
E-mail address: [email protected] (Z. Wang). 0143-8166/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2009.06.005
A basic FPP-based 3D shape measurement system usually contains a digital projector, a digital camera, and a computer (Fig. 1). During the 3D shape measurement, a set of fringe patterns are projected onto the surfaces of the objects of interest. The surface height/depth information is naturally encoded into the distorted fringe patterns, which will be captured by the camera for further processing. Among the various fringe analysis techniques, the one employing sinusoidal fringes and phase shifting approach is one of the most widely used due to its automatic, full-field, and fast processing. To extract the 3D surface profiles easily, in most cases, the initial fringe patterns being projected are straight, vertically (or horizontally) oriented, and equally spaced. Fig. 2 shows a few representative fringe patterns with different numbers of fringes in the images, and the fringe patterns (for vertical fringes) are normally generated in a numerical way with the following sinusoidal function: h i x I ¼ I0 1 þ cos 2pk þ d , w
(1)
where I is the pattern intensity at the point whose horizontal pixel coordinate is x, w is the width of the pattern image, k is the number of fringes in the image, d is the phase shifting amount, and I0 is normally set to 127.5 to obtain a desired intensity in the range of 0–255 for gray scale images. In reality, a digital projector often applies gamma decoding correction to the images and videos to enhance the visual effect.
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The gamma correction is defined as g
Iout ¼ Iin ,
(2)
where the input intensity Iin and output intensity Iout are nonnegative values. In gamma correction process, the gamma encoding uses a go1, whereas the decoding uses a g41. Since the images and videos stored in the computers are usually gamma
Fig. 1. Schematic setup of a FPP system.
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encoded, the modern projectors usually apply a nonlinear gamma decoding processing automatically with g ¼ 2.2–2.6. Nevertheless, for the FPP-based 3D shape measurement, the gamma decoding will generate unwanted intensity changes to the fringe patterns and make the projector fail to produce the ideal sinusoidal intensity distributions. Fig. 3(a) shows an example of captured fringe image where the initial fringe pattern is generated based on Eq. (1), and the intensity distributions along two lines AA0 and BB0 marked in figure are plotted in Fig. 3(b). It can be seen from the figure that the intensity distortion due to the undesired gamma correction is evident. In spite of the various ways of solving the undesired gamma correction problem [11–14], a simple yet robust scheme is to encode the initial fringe patterns before projection, similar to the case of most images and videos. By combining Eqs. (1) and (2), the theoretical fringe patterns to be numerically generated can be expressed as 1=g 1 1 x . (3) I ¼ 2I0 þ cos 2pk þ d 2 2 w As a comparison with Fig. (3), Fig. 4(a) shows a captured fringe image where the initial fringe pattern is generated according to Eq. (3) with g ¼ 2.2, and Fig. 4(b) plots the intensity distributions along lines AA0 and BB0 marked in Fig. 4(a). It is obvious that the gamma correction based on Eq. (3) is very effective. The existing gamma correction approaches [11–14] usually use statistical analysis to cope with the gamma issue of digital projection. A main disadvantage of these approaches is that such statistical analysis must be carried out for each measurement, and additional mathematical calculations are involved in each determination of phase distributions. Our new approach deals with the gamma problem at the source, i.e., the digital projection part; therefore, it provides a direct, simple yet effective solution, and does not introduce any additional calculation in the 3D shape determination part. It is noted that the actual gamma correction functions used in digital projectors are more complicated than the one shown by Eq. (2), and the gamma value is usually not a fixed constant.
Fig. 2. Projection fringe patterns with (a) 1, (b) 3, (c) 9, and (d) 36 fringes.
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Fig. 3. Fringe patterns projected without applying gamma correction encoding. (a) Fringe patterns captured by camera and (b) normalized intensity distributions along AA0 and BB0 .
Fig. 4. Fringe patterns projected with applying gamma correction encoding. (a) Fringe patterns captured by camera and (b) normalized intensity distributions along AA0 and BB0 .
Despite that, since the phase shifting algorithm to be employed in further image analysis is quite insensitive to small distortions of sinusoidal distributions, using dynamic gamma values for different intensity levels or choosing different fixed gamma values from 2.2 to 2.6 yields negligible difference to the final results of 3D shape determinations. In practice, the errors due to other minor effects such as noise and lens distortion [15,16] may be even larger.
ing algorithm will be presented in the paragraph below), the geometrical parameters of the system setup can be arbitrarily adjusted without affecting the 3D shape measurement results as long as they are within the illumination and capture capacities of the system components. In other words, the technique does not require physically adjusting or measuring the geometrical parameters, and the locations and orientations of the system components (projector and camera units) can be arbitrarily set. Accordingly, the measurement range can be substantially broadened and the targeted objects can vary largely in size and shape. As a consequence, the generalized setup approach has become of great interest to the practical application of FPP [17–22].
2.2. Arbitrary setup of system components The existing FPP-based 3D shape measurement system generally relies on a certain setup where the system components must be specifically located and oriented; moreover, the geometrical and other parameters must be precisely determined in advance. In practice, however, many of those parameters, such as the projection and capture orientations (pitch, yaw, and roll), the focal point location of a lens, and the distance between two components, are subject to excessive uncertainties in physical adjustments and measurements. Consequently, the corresponding shape measurement accuracies are inevitably limited. To cope with the geometrical uncertainties of system setup of components, the technique based on generalized setups can be adopted (e.g., Fig. 1). With the generalized setup (the correspond-
2.2.1. 3D shape determination algorithm To obtain the 3D shapes of an object or object system, both the in-plane and out-of plane dimensions must be determined. Due to the fact the 2D in-plane dimension can be directly calculated from the corresponding digital image through a simple transformation with acknowledgement of the camera–object distance, the primary task of the 3D shape determination algorithm is actually to rigorously determine the out-of-plane height and depth information. The mathematical derivation of the governing equation for the 3D shape determination based on an arbitrary or generalized
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setup involves a very tedious procedure. The derivation has been elaborated in Ref. [22], and this paper puts emphasis on the practical implementation of the technique. Basically, the governing equation of the out-of-plane shape determination can be simplified as z¼
1 þ c1 f þ ðc2 þ c3 fÞx þ ðc4 þ c5 fÞy , d0 þ d1 f þ ðd2 þ d3 fÞx þ ðd4 þ d5 fÞy
(4)
where z is the out-of-reference-plane height or depth at point (x,y), and f is the projection fringe phase at the same point. In the equation, the coefficients C1–C5 and D0–D5 are constants determined by the geometrical and other system parameters. Since the out-of-plane dimension determination is based on digital image, it is necessary to express z as a function of pixel coordinates instead of physical coordinates for calculation purpose. Considering that the relationship between the in-plane physical coordinates (x, y) and the corresponding pixel coordinates (i, j) in the captured digital image can be express as x ¼ ðk1 zk2 Þi;
y ¼ ðk1 zk2 Þj,
(5)
where k1 and k2 are the system-related coefficients, Eq. (4) can be rewritten as a function in terms of pixel coordinates. After neglecting the high order small terms, the out-of-plane height or depth with respect to the reference plane can be further simplified as
z¼
manner. At present, a few similar multi-frequency fringe schemes have been developed, including the so-called temporal unwrapping and hierarchical unwrapping [23–30]. These schemes typically use a series of fringes with specific pre-defined frequencies to achieve full-field phase unwrapping. The novel algorithm described in the next paragraph is a substantial extension and enhancement of the original algorithms. Compared with the existing algorithms, the new algorithm always uses a fringe pattern with one and only one fringe in the image as the lowest-frequency fringe pattern, and all other frequencies can be arbitrary as long as the ratio of fringe frequencies between two adjacent patterns are not large (e.g., no larger than 5). This simple extension makes the algorithm more flexible and capable of correctly detecting the full-field unwrapped phase distributions in any typical measurements of multiple separate objects with complex shapes; moreover, it is very easy to use. The basic principle of the algorithm is as follows. The lowestfrequency fringe pattern with one single-fringe in the entire field can yield full-field phase distributions without a phase-unwrapping process. The accuracies of such single-fringe-determined phase distributions are insufficient for being directly used in accurate 3D shape constructions; however, they can provide the required integer fringe order offsets for the fringes of higher frequency. Consequently, the unwrapped phase distributions of
1 þ C 1 f þ ðC 2 þ C 3 fÞi þ ðC 4 þ C 5 fÞj þ ðC 6 þ C 7 fÞi2 þ ðC 8 þ C 9 fÞj2 þ ðC 10 þ C 11 fÞij D0 þ D1 f þ ðD2 þ D3 fÞi þ ðD4 þ D5 fÞj þ ðD6 þ D7 fÞi2 þ ðD8 þ D9 fÞj2 þ ðD10 þ D11 fÞij
To calculate the out-of-reference-plane height z with Eq. (6), the new coefficients C1–C11 and D0–D11, which are associated with the geometrical and other relevant system parameters, must be determined first. The relevant details will be presented in a later section. Compared with the existing FPP techniques which use specific setups, the above approach based on arbitrary and generalized setup of system components is much easier to implement; more importantly, it can cope with the numerous uncertainties in practice of FPP. In addition, the new approach involves a single, simple and rigorous governing equation for the 3D shape determination; therefore, it does not require a subtraction of the reference phases, as other existing techniques do.
2.3. Phase unwrapping with multi-frequency fringes When the object or object system of interest involves complex shapes and/or multiple separated objects, the phases of the projection fringes on each object and among different objects are often discontinuous. For a correct phase determination, such discontinuities of fringe phases must be detected correctly. FPP-based techniques generally employ phase shifting scheme to obtain the full-field wrapped phase distributions of the projection fringes. The wrapped phase must then be unwrapped to obtain the real phase distributions, which are essential for the 3D shape determination. In actual applications, a notable challenge is how to correctly and quickly perform the phase unwrapping when fringe discontinuities are present. This issue can be well addressed by the following approach based on multifrequency fringe projection. The multi-frequency fringe projection approach uses fringes of various frequencies (e.g., from one to tens of fringes in the field), as shown in Fig. 2, to automatically determine the unwrapped full-field phase distributions from the wrapped ones in a fast
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.
(6)
the higher frequency fringes can be readily calculated without a general phase unwrapping, no matter how complex the fringe patterns are. The algorithm can be expressed as uw fi1 ðf i =f i1 Þ fw w i 2p ði ¼ 2; 3; . . . ; nÞ, (7) fuw ¼ f þ INT i i 2p where the subscript i indicates the ith projection fringe pattern, and the superscripts uw and w denote unwrapped phase and wrapped phase, respectively. In the equation, n is the number of fringe frequencies and nZ2; f is the relative fringe frequency or the number of fringes in the projection pattern, and fn4fn14 ? 4f1 ¼ 1; INT represents an operator to take the rounding integer of a decimal number; the wrapped phase fw is obtained from the w traditional or advanced phase shifting algorithm, and fuw 1 ¼ f1 for the lowest-frequency fringe pattern with one fringe or less in the entire field. Since the algorithm involves only one single governing equation, i.e., Eq. (7), and it can handle arbitrary fringe frequencies, the algorithm is very simple and easy to use. In practice, due to the noise effect, the ratio of the two adjacent fringe frequencies should not be too large; for instance, a ratio less than 5 is desired to ensure high reliability of measurements. The direct phase-unwrapping approach based on multifrequency fringe projection can obtain the full-field unwrapped phase distributions in an ultrafast manner. Furthermore, the approach is suitable for measuring multiple objects with complex shapes without any additional processing. 2.4. System calibration with least-squares inverse approach To calculate the absolute out-of-reference-plane height or depth z using Eq. (6), the coefficients C1–C11 and D0–D11, which are determined by geometrical and other relevant parameters of the measurement system, must be determined in advance. Since
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physically measuring those parameters should be avoided to ensure high measurement accuracy and practicability, the coefficients will be determined through using a least-squares inverse approach based on the reference plane and two or more gage blocks whose heights are precisely known. The reference plane and gage objects are used for the system calibration (i.e., determination of the coefficients) only; they are not required for actual 3D shape measurements. The calibration approach involves minimizing the linear least-squares error defined as S¼
m X ðF C F D zgk Þ2 ,
(8)
k¼1
where F C ¼ 1 þ C 1 fk þ ðC 2 þ C 3 fk Þik þ ðC 4 þ C 5 fk Þjk
The least-squares criterion [31,32] requires 8 @S ¼ 0; p ¼ 1; 2; . . . 11 < @C p @S : @D ¼ 0; q
q ¼ 0; 1; . . . 11:
(10)
Eq. (10) yields a group of linear equations, which will be employed to solve for the coefficients C1–C11 and D0–D11. It is noted that although the above FPP-based 3D shape measurement technique provides measurements of the out-ofplane dimension relative to a reference plane, the reference plane does not have to physically exist in actual applications except for the system calibration purpose; additionally, the reference plane is not necessarily the background plane nor necessarily located behind the objects of interest. The above feature indicates that a rigid-body translation and rotation of the entire camera–projector system will not affect the governing parameters and the virtual reference plane.
þ ðC 6 þ C 7 fk Þi2k þ ðC 8 þ C 9 fk Þj2k þ ðC 10 þ C 11 fk Þik jk F C ¼ D0 þ D1 fk þ ðD2 þ D3 fk Þik þ ðD4 þ D5 fk Þjk þ ðD6 þ D7 fk Þi2k þ ðD8 þ D9 fk Þj2k þ ðD10 þ D11 fk Þik jk
3. Experiment (9)
and zkg denotes the absolute out-of-reference-plane heights of the reference plane (height is zero) and gage blocks, k is the ordinal number of each valid point, m is the total number of datum points on the reference plane and gage blocks used in the calculation, and a larger m generally yields more reliable results.
To demonstrate the validity and applicability of the described four practical considerations in FPP implementation, a few experiments have been carried out and two are presented here. In the experiments, an FPP system with arbitrarily arranged components was used; the distance between the camera and the reference plane is about 1.5 m, and the valid field of view has a
Fig. 5. System calibration. (a) Calibration plate, (b) a representative fringe pattern, (c) calibration regions, (d) 2D shape map and (e) 3D shape map.
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width of around 300 mm. If necessary, the field of view can be easily and directly extended by decreasing the imaging magnification and increasing the distance between the camera and the reference plane. In the two particular experiments, the FPP measurement system was calibrated with a calibration plate shown in Fig. 5(a). The flat plate has seven calibration gage blocks and one testing gage block attached on. The in-plane dimensions of the calibration gage blocks are 50.80 by 50.80 mm, and the outof-plane heights vary from 6.35 to 50.80 mm. As mentioned previously, in addition to the reference plane, the minimum number of gage blocks is two. It has been noticed that more blocks do not always lead to better results; however, a detailed theoretical investigation on this matter is not easy, and thus has not been conducted yet. Five different fringe frequencies, with 1, 3, 9, 36, and 72 fringes respectively included in the projection images, were utilized in the experiments. This gives n ¼ 5, f1 ¼ 1, f2 ¼ 3, f3 ¼ 9, f4 ¼ 36, and f5 ¼ 72 to Eq. (7). Fig. 5(b) shows a representative experimental image with f5 ¼ 72. The datum points used for the calibration are illustrated as white regions in Fig. 5(c). It should be pointed out that the small gage block of nominal height 25.40 mm was not adopted as a calibration gage; instead, it was employed to check the accuracy of the calibration. The 3D shape measurement results based on the calibration indicate that the height of the small block is 25.4270.18 mm with a standard deviation of 0.07 mm over the block surface. This small error helps to verify the validity of the calibration. The final detected 2D and 3D shape maps are shown in Fig. 5(d) and (e), respectively. To check the effect of gamma correction, a comparison has been performed. When the gamma correction is not applied to the fringe projection, the height of the small block is detected to be 25.4270.26 mm with a standard deviation of 0.08 mm. The small result difference, i.e., 70.18 mm vs. 70.26 mm, is due to the relatively large number of fringes (72 fringes) used in the measurements. A further comparison shows that when the highest-frequency pattern has 36 fringes in the image, the results obtained from measurements with and without gamma corrections are 24.9870.70 mm with a standard deviation of 0.27 mm and 23.6671.95 mm with a standard deviation of 0.75 mm, respectively. The above comparisons reveal that the gamma correction is necessary when the number of fringes utilized in FPP is relatively small. This is in accordance with the work presented in literature [11–14] where a small number of fringes had to be employed to ensure a correct phase unwrapping. With the phase-unwrapping scheme described in this paper, the gamma issues can be alleviated since a large number of fringes are typically used; however, it is still desired to use gamma correction at the projection end. The first experiment was carried out to simultaneously measure the 3D shapes of multiple objects which are shown in Fig. 6(a). The experimentally determined 2D and 3D shapes of objects are shown in Fig. 6(b) and (c), respectively. In this experiment, the measurement accuracy was confirmed by the height differences of the two square gage blocks which were put against the back plane and located near the center of the view. The actual and experimentally measured average height differences are 17.78 mm and 17.8670.20 mm, respectively; again, this denotes a relatively high accuracy. This experiment also demonstrates the effectiveness of the direct phase unwrapping with multi-frequency fringes. When using a conventional phaseunwrapping scheme instead of the proposed one, it takes a long time to obtain a correct full-field unwrapped phase map for the fringe image shown in Fig. 6(a) because identifying the fringe order offsets in each separate regions relies on an time-consuming and ambiguous manual analysis. The 3D shape measurement technique has also been employed to construct a complete 3601 3D image of a rabbit model which
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Fig. 6. 3D shape measurements of multiple objects: (a) objects with a typical fringe pattern, (b) 2D shape map, (c) 3D shape map.
has a complex, very rough and textured surface in dark color. In the experiment, the 3D shapes of the rabbit model at five different views are determined first; they are then connected together to form a complete 3601 3D image. Fig. 7(a) shows a typical image captured from a typical view, and Fig. 7(b) illustrates the constructed complete 3601 3D image. A visual inspection shows that the 3D image has a very good match with the actual object. This experiment demonstrates the practicability of the presented FPP scheme for 3D imaging of object(s) with complex shapes. Finally, it is noteworthy that for each of the experiments performed, it took only around two seconds with a regular personal computer to analyze 20 images (four phase-shifted
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The validity and practicability of the FPP-based 3D shape measurement technique using the four corresponding technical approaches have been verified by experiments. As technologies evolve, there have been high demands for the 3D shape measurement techniques to posses a number of advanced technical features, and the technique presented in this paper is capable of satisfying the various critical demands in enormous scientific and engineering applications.
Acknowledgments This work was supported by Burns Fellowship at the Catholic University of America, National Collegiate Inventors and Innovators Alliance (under Grant 5205-07), and National Science Foundation (under Grant 0825806). References
Fig. 7. 3D shape measurement of a rabbit model. (a) A representative fringe image pattern from a typical view and (b) illustration of the complete 3601 3D image.
images for each of the five frequencies) to obtain the 3D shape information even though the images are relatively large (2048 pixels by 1536 pixels). As described previously, the fast processing speed mainly originates from the presented ultrafast phase unwrapping.
4. Conclusion This paper presents four practical considerations in FPP-based 3D shape measurements, along with simple but robust solutions that give great promise of providing measurements with high accuracy, fast speed, easy implementation, capability of measuring multiple objects, and capacity of measuring complex shapes.
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Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Some practical considerations in fringe projection profilometry Zhaoyang Wang , Dung A. Nguyen, John C. Barnes Department of Mechanical Engineering, The Catholic University of America, Washington, DC 20064, USA
a r t i c l e in f o
a b s t r a c t
Available online 4 July 2009
As technologies evolve, there have been high demands for the three-dimensional (3D) shape measurement techniques to posses the following combined technical features: high accuracy, fast speed, easy implementation, capability of measuring multiple objects as well as measuring complex shapes. Generally, the existing techniques can satisfy some of the requirements, but not all of them. This paper presents four practical considerations in fringe projection profilometry (FPP) based 3D shape measurements, along with simple but robust solutions, including gamma correction of digital projection, arbitrary setup of system components, phase unwrapping with multi-frequency fringes, and system calibration with a least-squares inverse approach. The validity and practicability of the FPPbased 3D shape measurement technique using the four corresponding technical approaches have been verified by experiments. The presented technique is capable of satisfying the various critical demands in enormous scientific and engineering applications. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Fringe projection 3D shape measurement 3D imaging High accuracy Fast speed
1. Introduction
2. Practical considerations in FPP
Fringe projection profilometry (FPP), also called fringe projection technique, is one of the most widely used techniques in practical applications of three-dimensional (3D) shape measurements, including object detection, digital model generation, object replication, reverse engineering, rapid prototyping, product inspection, quality control, etc. Despite that numerous FPP-based 3D shape measurement approaches have been developed in the past two decades [1–11], the literature survey conducted by the authors reveals that some important issues have not been well addressed in the practice of FPP-based 3D shape measurements. Accordingly, this paper aims to point out a few important considerations in FPP applications, along with simple yet practical solutions that give great promise of satisfying the various high demands of enormous applications in various fields. With the proposed considerations and schemes, the FPP measurement will be capable of detecting 3D shape of objects with the following combined technical features: high accuracy, fast speed, easy implementation, capability of measuring multiple objects and measuring complex shapes.
2.1. Gamma correction of digital projection
Corresponding author. Tel.: +1 202 319 6703; fax: +1 202 319 5173.
E-mail address: [email protected] (Z. Wang). 0143-8166/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2009.06.005
A basic FPP-based 3D shape measurement system usually contains a digital projector, a digital camera, and a computer (Fig. 1). During the 3D shape measurement, a set of fringe patterns are projected onto the surfaces of the objects of interest. The surface height/depth information is naturally encoded into the distorted fringe patterns, which will be captured by the camera for further processing. Among the various fringe analysis techniques, the one employing sinusoidal fringes and phase shifting approach is one of the most widely used due to its automatic, full-field, and fast processing. To extract the 3D surface profiles easily, in most cases, the initial fringe patterns being projected are straight, vertically (or horizontally) oriented, and equally spaced. Fig. 2 shows a few representative fringe patterns with different numbers of fringes in the images, and the fringe patterns (for vertical fringes) are normally generated in a numerical way with the following sinusoidal function: h i x I ¼ I0 1 þ cos 2pk þ d , w
(1)
where I is the pattern intensity at the point whose horizontal pixel coordinate is x, w is the width of the pattern image, k is the number of fringes in the image, d is the phase shifting amount, and I0 is normally set to 127.5 to obtain a desired intensity in the range of 0–255 for gray scale images. In reality, a digital projector often applies gamma decoding correction to the images and videos to enhance the visual effect.
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The gamma correction is defined as g
Iout ¼ Iin ,
(2)
where the input intensity Iin and output intensity Iout are nonnegative values. In gamma correction process, the gamma encoding uses a go1, whereas the decoding uses a g41. Since the images and videos stored in the computers are usually gamma
Fig. 1. Schematic setup of a FPP system.
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encoded, the modern projectors usually apply a nonlinear gamma decoding processing automatically with g ¼ 2.2–2.6. Nevertheless, for the FPP-based 3D shape measurement, the gamma decoding will generate unwanted intensity changes to the fringe patterns and make the projector fail to produce the ideal sinusoidal intensity distributions. Fig. 3(a) shows an example of captured fringe image where the initial fringe pattern is generated based on Eq. (1), and the intensity distributions along two lines AA0 and BB0 marked in figure are plotted in Fig. 3(b). It can be seen from the figure that the intensity distortion due to the undesired gamma correction is evident. In spite of the various ways of solving the undesired gamma correction problem [11–14], a simple yet robust scheme is to encode the initial fringe patterns before projection, similar to the case of most images and videos. By combining Eqs. (1) and (2), the theoretical fringe patterns to be numerically generated can be expressed as 1=g 1 1 x . (3) I ¼ 2I0 þ cos 2pk þ d 2 2 w As a comparison with Fig. (3), Fig. 4(a) shows a captured fringe image where the initial fringe pattern is generated according to Eq. (3) with g ¼ 2.2, and Fig. 4(b) plots the intensity distributions along lines AA0 and BB0 marked in Fig. 4(a). It is obvious that the gamma correction based on Eq. (3) is very effective. The existing gamma correction approaches [11–14] usually use statistical analysis to cope with the gamma issue of digital projection. A main disadvantage of these approaches is that such statistical analysis must be carried out for each measurement, and additional mathematical calculations are involved in each determination of phase distributions. Our new approach deals with the gamma problem at the source, i.e., the digital projection part; therefore, it provides a direct, simple yet effective solution, and does not introduce any additional calculation in the 3D shape determination part. It is noted that the actual gamma correction functions used in digital projectors are more complicated than the one shown by Eq. (2), and the gamma value is usually not a fixed constant.
Fig. 2. Projection fringe patterns with (a) 1, (b) 3, (c) 9, and (d) 36 fringes.
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Fig. 3. Fringe patterns projected without applying gamma correction encoding. (a) Fringe patterns captured by camera and (b) normalized intensity distributions along AA0 and BB0 .
Fig. 4. Fringe patterns projected with applying gamma correction encoding. (a) Fringe patterns captured by camera and (b) normalized intensity distributions along AA0 and BB0 .
Despite that, since the phase shifting algorithm to be employed in further image analysis is quite insensitive to small distortions of sinusoidal distributions, using dynamic gamma values for different intensity levels or choosing different fixed gamma values from 2.2 to 2.6 yields negligible difference to the final results of 3D shape determinations. In practice, the errors due to other minor effects such as noise and lens distortion [15,16] may be even larger.
ing algorithm will be presented in the paragraph below), the geometrical parameters of the system setup can be arbitrarily adjusted without affecting the 3D shape measurement results as long as they are within the illumination and capture capacities of the system components. In other words, the technique does not require physically adjusting or measuring the geometrical parameters, and the locations and orientations of the system components (projector and camera units) can be arbitrarily set. Accordingly, the measurement range can be substantially broadened and the targeted objects can vary largely in size and shape. As a consequence, the generalized setup approach has become of great interest to the practical application of FPP [17–22].
2.2. Arbitrary setup of system components The existing FPP-based 3D shape measurement system generally relies on a certain setup where the system components must be specifically located and oriented; moreover, the geometrical and other parameters must be precisely determined in advance. In practice, however, many of those parameters, such as the projection and capture orientations (pitch, yaw, and roll), the focal point location of a lens, and the distance between two components, are subject to excessive uncertainties in physical adjustments and measurements. Consequently, the corresponding shape measurement accuracies are inevitably limited. To cope with the geometrical uncertainties of system setup of components, the technique based on generalized setups can be adopted (e.g., Fig. 1). With the generalized setup (the correspond-
2.2.1. 3D shape determination algorithm To obtain the 3D shapes of an object or object system, both the in-plane and out-of plane dimensions must be determined. Due to the fact the 2D in-plane dimension can be directly calculated from the corresponding digital image through a simple transformation with acknowledgement of the camera–object distance, the primary task of the 3D shape determination algorithm is actually to rigorously determine the out-of-plane height and depth information. The mathematical derivation of the governing equation for the 3D shape determination based on an arbitrary or generalized
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setup involves a very tedious procedure. The derivation has been elaborated in Ref. [22], and this paper puts emphasis on the practical implementation of the technique. Basically, the governing equation of the out-of-plane shape determination can be simplified as z¼
1 þ c1 f þ ðc2 þ c3 fÞx þ ðc4 þ c5 fÞy , d0 þ d1 f þ ðd2 þ d3 fÞx þ ðd4 þ d5 fÞy
(4)
where z is the out-of-reference-plane height or depth at point (x,y), and f is the projection fringe phase at the same point. In the equation, the coefficients C1–C5 and D0–D5 are constants determined by the geometrical and other system parameters. Since the out-of-plane dimension determination is based on digital image, it is necessary to express z as a function of pixel coordinates instead of physical coordinates for calculation purpose. Considering that the relationship between the in-plane physical coordinates (x, y) and the corresponding pixel coordinates (i, j) in the captured digital image can be express as x ¼ ðk1 zk2 Þi;
y ¼ ðk1 zk2 Þj,
(5)
where k1 and k2 are the system-related coefficients, Eq. (4) can be rewritten as a function in terms of pixel coordinates. After neglecting the high order small terms, the out-of-plane height or depth with respect to the reference plane can be further simplified as
z¼
manner. At present, a few similar multi-frequency fringe schemes have been developed, including the so-called temporal unwrapping and hierarchical unwrapping [23–30]. These schemes typically use a series of fringes with specific pre-defined frequencies to achieve full-field phase unwrapping. The novel algorithm described in the next paragraph is a substantial extension and enhancement of the original algorithms. Compared with the existing algorithms, the new algorithm always uses a fringe pattern with one and only one fringe in the image as the lowest-frequency fringe pattern, and all other frequencies can be arbitrary as long as the ratio of fringe frequencies between two adjacent patterns are not large (e.g., no larger than 5). This simple extension makes the algorithm more flexible and capable of correctly detecting the full-field unwrapped phase distributions in any typical measurements of multiple separate objects with complex shapes; moreover, it is very easy to use. The basic principle of the algorithm is as follows. The lowestfrequency fringe pattern with one single-fringe in the entire field can yield full-field phase distributions without a phase-unwrapping process. The accuracies of such single-fringe-determined phase distributions are insufficient for being directly used in accurate 3D shape constructions; however, they can provide the required integer fringe order offsets for the fringes of higher frequency. Consequently, the unwrapped phase distributions of
1 þ C 1 f þ ðC 2 þ C 3 fÞi þ ðC 4 þ C 5 fÞj þ ðC 6 þ C 7 fÞi2 þ ðC 8 þ C 9 fÞj2 þ ðC 10 þ C 11 fÞij D0 þ D1 f þ ðD2 þ D3 fÞi þ ðD4 þ D5 fÞj þ ðD6 þ D7 fÞi2 þ ðD8 þ D9 fÞj2 þ ðD10 þ D11 fÞij
To calculate the out-of-reference-plane height z with Eq. (6), the new coefficients C1–C11 and D0–D11, which are associated with the geometrical and other relevant system parameters, must be determined first. The relevant details will be presented in a later section. Compared with the existing FPP techniques which use specific setups, the above approach based on arbitrary and generalized setup of system components is much easier to implement; more importantly, it can cope with the numerous uncertainties in practice of FPP. In addition, the new approach involves a single, simple and rigorous governing equation for the 3D shape determination; therefore, it does not require a subtraction of the reference phases, as other existing techniques do.
2.3. Phase unwrapping with multi-frequency fringes When the object or object system of interest involves complex shapes and/or multiple separated objects, the phases of the projection fringes on each object and among different objects are often discontinuous. For a correct phase determination, such discontinuities of fringe phases must be detected correctly. FPP-based techniques generally employ phase shifting scheme to obtain the full-field wrapped phase distributions of the projection fringes. The wrapped phase must then be unwrapped to obtain the real phase distributions, which are essential for the 3D shape determination. In actual applications, a notable challenge is how to correctly and quickly perform the phase unwrapping when fringe discontinuities are present. This issue can be well addressed by the following approach based on multifrequency fringe projection. The multi-frequency fringe projection approach uses fringes of various frequencies (e.g., from one to tens of fringes in the field), as shown in Fig. 2, to automatically determine the unwrapped full-field phase distributions from the wrapped ones in a fast
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.
(6)
the higher frequency fringes can be readily calculated without a general phase unwrapping, no matter how complex the fringe patterns are. The algorithm can be expressed as uw fi1 ðf i =f i1 Þ fw w i 2p ði ¼ 2; 3; . . . ; nÞ, (7) fuw ¼ f þ INT i i 2p where the subscript i indicates the ith projection fringe pattern, and the superscripts uw and w denote unwrapped phase and wrapped phase, respectively. In the equation, n is the number of fringe frequencies and nZ2; f is the relative fringe frequency or the number of fringes in the projection pattern, and fn4fn14 ? 4f1 ¼ 1; INT represents an operator to take the rounding integer of a decimal number; the wrapped phase fw is obtained from the w traditional or advanced phase shifting algorithm, and fuw 1 ¼ f1 for the lowest-frequency fringe pattern with one fringe or less in the entire field. Since the algorithm involves only one single governing equation, i.e., Eq. (7), and it can handle arbitrary fringe frequencies, the algorithm is very simple and easy to use. In practice, due to the noise effect, the ratio of the two adjacent fringe frequencies should not be too large; for instance, a ratio less than 5 is desired to ensure high reliability of measurements. The direct phase-unwrapping approach based on multifrequency fringe projection can obtain the full-field unwrapped phase distributions in an ultrafast manner. Furthermore, the approach is suitable for measuring multiple objects with complex shapes without any additional processing. 2.4. System calibration with least-squares inverse approach To calculate the absolute out-of-reference-plane height or depth z using Eq. (6), the coefficients C1–C11 and D0–D11, which are determined by geometrical and other relevant parameters of the measurement system, must be determined in advance. Since
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physically measuring those parameters should be avoided to ensure high measurement accuracy and practicability, the coefficients will be determined through using a least-squares inverse approach based on the reference plane and two or more gage blocks whose heights are precisely known. The reference plane and gage objects are used for the system calibration (i.e., determination of the coefficients) only; they are not required for actual 3D shape measurements. The calibration approach involves minimizing the linear least-squares error defined as S¼
m X ðF C F D zgk Þ2 ,
(8)
k¼1
where F C ¼ 1 þ C 1 fk þ ðC 2 þ C 3 fk Þik þ ðC 4 þ C 5 fk Þjk
The least-squares criterion [31,32] requires 8 @S ¼ 0; p ¼ 1; 2; . . . 11 < @C p @S : @D ¼ 0; q
q ¼ 0; 1; . . . 11:
(10)
Eq. (10) yields a group of linear equations, which will be employed to solve for the coefficients C1–C11 and D0–D11. It is noted that although the above FPP-based 3D shape measurement technique provides measurements of the out-ofplane dimension relative to a reference plane, the reference plane does not have to physically exist in actual applications except for the system calibration purpose; additionally, the reference plane is not necessarily the background plane nor necessarily located behind the objects of interest. The above feature indicates that a rigid-body translation and rotation of the entire camera–projector system will not affect the governing parameters and the virtual reference plane.
þ ðC 6 þ C 7 fk Þi2k þ ðC 8 þ C 9 fk Þj2k þ ðC 10 þ C 11 fk Þik jk F C ¼ D0 þ D1 fk þ ðD2 þ D3 fk Þik þ ðD4 þ D5 fk Þjk þ ðD6 þ D7 fk Þi2k þ ðD8 þ D9 fk Þj2k þ ðD10 þ D11 fk Þik jk
3. Experiment (9)
and zkg denotes the absolute out-of-reference-plane heights of the reference plane (height is zero) and gage blocks, k is the ordinal number of each valid point, m is the total number of datum points on the reference plane and gage blocks used in the calculation, and a larger m generally yields more reliable results.
To demonstrate the validity and applicability of the described four practical considerations in FPP implementation, a few experiments have been carried out and two are presented here. In the experiments, an FPP system with arbitrarily arranged components was used; the distance between the camera and the reference plane is about 1.5 m, and the valid field of view has a
Fig. 5. System calibration. (a) Calibration plate, (b) a representative fringe pattern, (c) calibration regions, (d) 2D shape map and (e) 3D shape map.
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width of around 300 mm. If necessary, the field of view can be easily and directly extended by decreasing the imaging magnification and increasing the distance between the camera and the reference plane. In the two particular experiments, the FPP measurement system was calibrated with a calibration plate shown in Fig. 5(a). The flat plate has seven calibration gage blocks and one testing gage block attached on. The in-plane dimensions of the calibration gage blocks are 50.80 by 50.80 mm, and the outof-plane heights vary from 6.35 to 50.80 mm. As mentioned previously, in addition to the reference plane, the minimum number of gage blocks is two. It has been noticed that more blocks do not always lead to better results; however, a detailed theoretical investigation on this matter is not easy, and thus has not been conducted yet. Five different fringe frequencies, with 1, 3, 9, 36, and 72 fringes respectively included in the projection images, were utilized in the experiments. This gives n ¼ 5, f1 ¼ 1, f2 ¼ 3, f3 ¼ 9, f4 ¼ 36, and f5 ¼ 72 to Eq. (7). Fig. 5(b) shows a representative experimental image with f5 ¼ 72. The datum points used for the calibration are illustrated as white regions in Fig. 5(c). It should be pointed out that the small gage block of nominal height 25.40 mm was not adopted as a calibration gage; instead, it was employed to check the accuracy of the calibration. The 3D shape measurement results based on the calibration indicate that the height of the small block is 25.4270.18 mm with a standard deviation of 0.07 mm over the block surface. This small error helps to verify the validity of the calibration. The final detected 2D and 3D shape maps are shown in Fig. 5(d) and (e), respectively. To check the effect of gamma correction, a comparison has been performed. When the gamma correction is not applied to the fringe projection, the height of the small block is detected to be 25.4270.26 mm with a standard deviation of 0.08 mm. The small result difference, i.e., 70.18 mm vs. 70.26 mm, is due to the relatively large number of fringes (72 fringes) used in the measurements. A further comparison shows that when the highest-frequency pattern has 36 fringes in the image, the results obtained from measurements with and without gamma corrections are 24.9870.70 mm with a standard deviation of 0.27 mm and 23.6671.95 mm with a standard deviation of 0.75 mm, respectively. The above comparisons reveal that the gamma correction is necessary when the number of fringes utilized in FPP is relatively small. This is in accordance with the work presented in literature [11–14] where a small number of fringes had to be employed to ensure a correct phase unwrapping. With the phase-unwrapping scheme described in this paper, the gamma issues can be alleviated since a large number of fringes are typically used; however, it is still desired to use gamma correction at the projection end. The first experiment was carried out to simultaneously measure the 3D shapes of multiple objects which are shown in Fig. 6(a). The experimentally determined 2D and 3D shapes of objects are shown in Fig. 6(b) and (c), respectively. In this experiment, the measurement accuracy was confirmed by the height differences of the two square gage blocks which were put against the back plane and located near the center of the view. The actual and experimentally measured average height differences are 17.78 mm and 17.8670.20 mm, respectively; again, this denotes a relatively high accuracy. This experiment also demonstrates the effectiveness of the direct phase unwrapping with multi-frequency fringes. When using a conventional phaseunwrapping scheme instead of the proposed one, it takes a long time to obtain a correct full-field unwrapped phase map for the fringe image shown in Fig. 6(a) because identifying the fringe order offsets in each separate regions relies on an time-consuming and ambiguous manual analysis. The 3D shape measurement technique has also been employed to construct a complete 3601 3D image of a rabbit model which
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Fig. 6. 3D shape measurements of multiple objects: (a) objects with a typical fringe pattern, (b) 2D shape map, (c) 3D shape map.
has a complex, very rough and textured surface in dark color. In the experiment, the 3D shapes of the rabbit model at five different views are determined first; they are then connected together to form a complete 3601 3D image. Fig. 7(a) shows a typical image captured from a typical view, and Fig. 7(b) illustrates the constructed complete 3601 3D image. A visual inspection shows that the 3D image has a very good match with the actual object. This experiment demonstrates the practicability of the presented FPP scheme for 3D imaging of object(s) with complex shapes. Finally, it is noteworthy that for each of the experiments performed, it took only around two seconds with a regular personal computer to analyze 20 images (four phase-shifted
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The validity and practicability of the FPP-based 3D shape measurement technique using the four corresponding technical approaches have been verified by experiments. As technologies evolve, there have been high demands for the 3D shape measurement techniques to posses a number of advanced technical features, and the technique presented in this paper is capable of satisfying the various critical demands in enormous scientific and engineering applications.
Acknowledgments This work was supported by Burns Fellowship at the Catholic University of America, National Collegiate Inventors and Innovators Alliance (under Grant 5205-07), and National Science Foundation (under Grant 0825806). References
Fig. 7. 3D shape measurement of a rabbit model. (a) A representative fringe image pattern from a typical view and (b) illustration of the complete 3601 3D image.
images for each of the five frequencies) to obtain the 3D shape information even though the images are relatively large (2048 pixels by 1536 pixels). As described previously, the fast processing speed mainly originates from the presented ultrafast phase unwrapping.
4. Conclusion This paper presents four practical considerations in FPP-based 3D shape measurements, along with simple but robust solutions that give great promise of providing measurements with high accuracy, fast speed, easy implementation, capability of measuring multiple objects, and capacity of measuring complex shapes.
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