1. Introduction

High accuracy three-dimensional (3D) optical measurement is increasingly important in industrial applications [1]. Among various optical 3D shape measurement techniques, fringe projection profilometry (FPP) has the merits of high accuracy, high-speed and non-contact measurement, and has found wide applications in the fields of manufacturing [2], biomedicine [3] and forensic science [4].

As shown in Fig. 1, a typical FPP system is composed of a projector, a camera and a computer, where the projector projects the fringes onto an object surface, the camera captures the distorted fringes, and the computer analyzes the data, extracts phase and reconstructs 3D geometry of the object. Currently, there are two major reconstruction methods to obtain 3D geometry from the phase distribution. One is the phase-to-height conversion method [5] which establishes a mapping relationship between the height and phase. The other is the stereo-vision-based model, or simply called Zhang’s method [6], where the camera and the projector (treated as an inverse camera) form a stereo-vision system. By establishing correspondence between a camera pixel $({{u^c},{v^c}} )$ and the projector pixel $({{u^p},{v^p}} )$, a 3D object point $({{x^w},{y^w},{z^w}} )$ can be subsequently obtained. Compared with the phase-to-height method, Zhang’s method is more widely used in FPP [7] and thus is the focus of this paper.

 

Fig. 1. A typical FPP system.

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The typical measurement principle and procedure are as follows [6],

In these four steps, the first (calibration) step has been intensively studied [8]. The second (phase measurement) step has the largest body of literature [9–12], including the current strong interest for fast measurement [13,14]. However, the projected fringe patterns are designed to be horizontal or vertical, until recently, Wang and Zhang [15] insightfully described the optimal fringe angle and proposed to measure it by projecting two sets of fringe patterns, one horizontal and one vertical, onto a step-height object. The optimal angle phenomenon was further analyzed and verified by Zhou et al. [16] and Zhang et al. [17]. However, there is a lack of theoretical analysis of the optimal fringe angle for convenient use.

In the third (correspondence) and fourth (reconstruction) steps, given $({{u^c},{v^c}} )$, $\mathrm{\Phi }({{u^c},{v^c}} )$ is known from the second step. If the vertical fringes are projected, the equal-phase line is vertical, from which, ${u^p}$ alone is immediately determined. Since ${u^c}$, ${v^c}$ and ${u^p}$ provide three equations, they are sufficient to reconstruct $({{x^w},{y^w},{z^w}} )$. We call it the Ver3 method to highlight that a vertical fringe orientation is used for correspondence and three equations are used for reconstruction. This idea can be extended to projecting horizontal fringe patterns, which is called the Hor3 method. One can also project both vertical and horizontal fringe patterns to obtained both ${u^p}$ and ${v^p}$, so that four equations will be used for 3D reconstruction, which is called the HorVer4 method. In practice, especially in existing literature, the Hor3 and Ver3 methods are seen more widely used because they project only one set of fringe patterns (either horizontal or vertical, but not both) and thus achieve a higher speed than the HorVer4 method. Furthermore, it is noticeable that, when projecting fringe patterns with the optimal fringe angle, the equal-phase line is tilted, i.e., it is no longer horizontal or vertical. Thus, neither ${u^p}$ nor ${v^p}$ can be immediately determined. To this end, Li [18] developed a method to project two sets of fringe patterns, one rotates the optimal angle by $\pi /4$ counterclockwise so that the fringe angle is ${\theta _{opt}} + \pi /4$, while the other rotates clockwise and the fringe angle is ${\theta _{opt}} - \pi /4$. Two sets of fringe patterns provided two equal-phase lines to solve $({{u^p},{v^p}} )$ as the intersection point, which is then used for 3D reconstruction. We abbreviate it as an OptR4 method where Opt and R highlight the involvement of the optimal fringe angle and the angle rotation, respectively. Unfortunately, although we have seen the interesting variety of methods for correspondence and reconstruction, there is no theoretical analysis on them.

Knowing the extreme importance of theoretical understanding in precision measurement, this paper interrogates the optimal fringe angle, the correspondence and the reconstruction equations in a theoretical manner. Interesting results will be discovered, based on which, a novel FPP measurement method is proposed. Our contributions are as follows:

We believe that our theoretical results are fundamental for the study of FPP, and our proposed method is effective in practice.

The rest of the paper is arranged as follows: the basic introduction of FPP and the problem statement are given in Sec. 2; the theoretical aspects of general FPP is analyzed in Sec. 3, based on which, our proposed method is described in Sec. 4 and verified in Sec. 5.

2. Basics of an FPP system and the problem statement

For the FPP system shown in Fig. 1, the camera and the projector play an important role. Both of them are usually modeled by a pinhole model and the epipolar line can also be derived, which is described in Sec. 2.1. Theoretical questions to interrogate the phase measurement, correspondence and reconstruction steps will be raised in Sec. 2.2 as our problem statement.

2.1 Pinhole model and the epipolar geometry

The camera essentially realizes the conversion from a world coordinate system to the camera’s physical coordinate system and further to the camera’s pixel coordinate system, which can be described by a pinhole model as follows [19],

For convenience, we select the camera’s physical coordinate system to be the same as the world coordinate system so that ${R^c}$ is an identity matrix and ${t^c}\; $ is a zero vector [20]. By substituting the special ${R^c}$ and ${t^c}$ and Eq. (2) into Eq. (1), we can readily get the following simple relationship between $({{x^w},{y^w},{z^w}} )$ and $({{u^c},{v^c}} )$,

Similarly, the pinhole model can be used to describe the projector, and is written below by merely changing the subscript c in Eq. (1) to a subscript p,

Further canceling ${z^w}$ in Eqs. (7) and (8) leads to the following epipolar line equation,

The epipolar line can also be derived by fundamental matrix [21,22]. This line indicates the constraint of $({{u^p},{v^p}} )$, i.e., given $({{u^c},{v^c}} )$, the corresponding $({{u^p},{v^p}} )$ is on a line as illustrated by Fig. 2. We have listed the complicated Eqs. (7), (8) and (10–12) as a preparation, which will lead to neat and interesting results later.

 

Fig. 2. Epipolar geometry model of the projector and the camera, where ${l_p}$ is the epipolar line.

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2.2 Problem statement

We have mentioned that the FPP system involves four steps: calibration, phase measurement, correspondence and 3D reconstruction. However, there is a lack of interrogation on their theoretical aspects. In detail,

3. Theoretical analysis on the FPP system

To answer the questions in Sec. 2.2, we perform a series of theoretical analysis on the FPP system in this section. The optimal and worst fringe angles are derived and their relationships with the epipolar line are discovered in Sec. 3.1. The best selection of a pair of fringe angles in correspondence step is analyzed in Sec. 3.2. The effect of the number of equations on 3D reconstruction precision is analyzed in Sec.3.3.

3.1 Analysis on phase measurement: the optimal fringe angle and worst fringe angles

We provide a theoretical analysis for the optimal and worst fringe angles, and then examine their relationships with the epipolar line.

To start with, the sinusoidal fringe pattern with an arbitrary fringe angle $\theta $ to be projected are designed as follows,

The optimal fringe angle $\theta $ maximizes the sensitivity of the phase $\mathrm{\Phi }({{u^c},{v^c}} )$ in the captured fringe pattern with respect to the surface profile. Nevertheless, when $({{u^c},{v^c}} )$ and $({{u^p},{v^p}} )$ correspond to the same object point $({{x^w},{y^w},{z^w}} )$, we have

The above sensitivity can be expressed as,

In both cases, we have the following relationship,

In order to calculate the optimal angle in Eq. (18), it is necessary to evaluate the partial derivatives in the right hand of this equation, which can be easily derived based on Eqs. (7) and (8) as,

By substituting the Eqs. (20) and (21) into Eq. (19), the optimal angle can be obtained as follow,

This equation clearly indicates that the optimal angle is not only a function of both the internal and external parameters of the system setup but also the camera pixel coordinates $({{u^c},{v^c}} )$. Strictly speaking, this means that the optimal angle is not a constant, but varying with $({{u^c},{v^c}} )$. In the meantime, for a typical FPP system with a camera and a projector placed on the left and right, ${r_{11}}$, ${r_{22}}\; $ and ${r_{33}}$ are close to 1, and the other entries of the rotation matrix R are much smaller. In addition, the focal lengths of the camera and the projector are larger than the respective detector resolutions, and thus $({{u^c} - u_0^c} )/f_u^c$, $({{v^c} - v_0^c} )/f_v^c$, $({{u^p} - u_0^p} )/f_u^p\; $ and $({{u^p} - u_0^p} )/f_v^p$ are all relatively small values. We will show such system details in our experiments in Sec. 5.1. Another such system with detailed calibration results is given in Ref. [20]. The optimal angle can then be approximated as,

On the other hand, the worst angle, i.e., the angle gives the least sensitivity, can be determined from $H = 0$. By referring to Eq. (16), we obtain

From Eq. (18) and (24), we have $[{ - tan ({{\theta_{opt}}} )} ]\times [{ - tan({{\theta_{worst}}} )} ]={-} 1$, which leads to an interesting and important theoretical result on projecting fringe patterns with an angle:

In additional, from Eqs. (10) and (11), the slope of the epipolar line can be calculated as,

Thus, we also have $[{ - tan ({{\theta_{opt}}} )} ]\times {k_1} ={-} 1$, and thus we have

After we have discovered this interesting property, we find that this result is rather intuitive. Referring to Fig. 2, it is clearly that $({{u^c},{v^c}} )$ correspond to multiple candidates of $({{u^p},{v^p}} )$ which are all at the epipolar line. To differentiate different object points with high sensitivity is the same as to differentiate different $({{u^p},{v^p}} )$ on the epipolar line. Since we rely on phase for correspondence, and we know that there is no phase change along the epipolar line, we can conclude that the epipolar line orientation is the worst fringe orientation; on the contrary, the fringe orientation perpendicular to the epipolar line gives the highest sensitivity and hence becomes the optimal angle. Both our solid theoretical analysis and the intuitive explanation highlight the essential role of the epipolar line.

3.2 Analysis on correspondence: selection of a pair of fringe angles

If the optimal fringe angle is an arbitrary value, i.e. neither horizontal nor vertical, we have to project two sets of fringes with different fringe angles. As an example, in OptR4, ${\theta _{opt}} + \pi /4$ and ${\theta _{opt}} - \pi /4$ are used as a pair, which are perpendicular to each other. We can also use 0 and $\pi /2$ as a pair, as in HorVer4. In general, an arbitrary pair of (${\theta _1}$, ${\theta _2}$) with ${\theta _1} \ne {\theta _2}$ is qualified for finding $({{u^p},{v^p}} )$. We examine the influence of different (${\theta _1}$, ${\theta _2}$).

As we know, the purpose of using two sets of fringe patterns is to determine $({{u^p},{v^p}} )$ when given $({{u^c},{v^c}} )$, and the result of $({{u^p},{v^p}} )$ affects the 3D reconstruction result according to Eqs. (3), (4), (7) and (8). Thus we focus on the accuracy and precision of $({{u^p},{v^p}} )$ which is the intersection point of the following two equal-phase lines according to Eq. (14),

Based on these two equations, if ${\mathrm{\Phi }_1}$ and ${\mathrm{\Phi }_2}$ have zero means [13], ${u^p}$ and ${v^p}$ also have zero means. Furthermore, the variances of ${u^p}$ and ${v^p}$ can be calculated as follow,

This theoretical analysis result gives two following important conclusions:

Finally, for completeness, we consider Hor3, where ${v^p} = {\mathrm{\Phi }_H}T/2\pi $. The variance of ${v^p}$ can be obtained as $\sigma _{{v^p}}^2 = {({\textrm{T}/2\pi } )^2}\sigma _\mathrm{\Phi }^2$ . We do the same for ${u^p}$ in Ver3 and similarly obtain $\sigma _{{u^p}}^2 = {({\textrm{T}/2\pi } )^2}\sigma _\mathrm{\Phi }^2.$ We have used an earlier assumption that both variances of ${\mathrm{\Phi }_H}$ and ${\mathrm{\Phi }_V}$ are $\sigma _\mathrm{\Phi }^2$.

3.3 Analysis on reconstruction: 4 equations vs 3 equations

Now that we have many FPP measurement methods, such as the Hor3, Ver3, HorVer4 and OptR4 methods. According to the number of equations they use for reconstruction, we group them into two categories: 3-equation methods (Hor3 and Ver3) and 4-equaiton methods (HorVer4 and OptR4). A natural question is whether the number of equations makes a difference in 3D reconstruction and whether this difference is significant.

To answer this question and for convenience, we re-write Eqs. (3), (4), (7) and (8), as follows,

The 3D reconstruction task is simply to solve three unknowns of a 3D point $({{x^w},{y^w},{z^w}} )$ from the linear equations (33-36). As mentioned before, the solution of Eqs. (33–35) corresponds to the Ver3 method, the solution of Eqs. (33), (34), (36) corresponds to the Hor3 method and the solution of Eqs. (33–36) corresponds to the HorVer4 and OptR4 methods, depending on the fringe angles used for correspondence.

In the ideal case, all Eqs. (33–36) are satisfied. However, in practice, the left hand of one or more equations of Eqs. (33–36) is nonzero, resulting in a residual. We use the sum of squared residuals (SSR) to indicate the quality of a solution. For example, for the Ver3 method, the three equations (33-35) give a unique solution $({{x^w},{y^w},{z^w}} )$. When we substitute this solution back to these three equations, the residual is zero. However, when we substitute it into Eq. (36), a residual will be resulted as

Obviously, by comparing the SSRs in Eqs. (39–41), we can immediately conclude that the 3D reconstruction by the 4-equation methods is better than that by the 3-equation methods. In order to more quantitatively estimate the magnitude of the SSRs, we consider a typical FPP system with a camera and a projector placed on the left and right, which has been used in getting Eq. (23) in Sec. 3.1. In such case, we recall that ${r_{11}}$, ${r_{22}}\; $ and ${r_{33}}$ are close to 1, and the other entries of the rotation matrix R are much smaller, and $({{u^c} - u_0^c} )/f_u^c$, $({{v^c} - v_0^c} )/f_v^c$, $({{u^p} - u_0^p} )/f_u^p\; $ and $({{u^p} - u_0^p} )/f_v^p$ are all relatively small values. Thus, we also have $c \approx 1$ and $g \approx 1$. In addition, a product between the two small quantities approaches zeros, e.g., $bd \approx 0$, $af \approx 0$, $df \approx 0$, $dh \approx 0$, $ef \approx 0$. With these facts, we can make the following approximations:

Substituting Eq. (42) into Eq. (41), we have the following neat results,

This result clearly indicates that,

4. Novel FPP measurement method with high accuracy and precision, efficiency and convenience

Based on the theoretical analysis in Sec. 3, we propose a novel FPP measurement method in Sec. 4.1. The role of the epipolar line in our proposed method is analyzed in Sec. 4.2. The accuracy and precision of our proposed method are analyzed in Sec 4.3 and 4.4, respectively.

4.1 Novel FPP measurement method with a set of optimal angle fringes

Based on the theoretical analysis in Sec. 3, we propose a novel FPP measurement method based on the following considerations:

We thus call our method OptE3, where Opt, E and 3 indicate the involvement of the optimal angle, the epipolar line, and the three equations for reconstruction, respectively. Our OptE3 method achieves high precision (being essentially a 4-equation method), high efficiency (by projecting only one set of fringe patterns) and high convenience (by obtaining ${\theta _{opt}}$ effortlessly during calibration, without any extra measurements). To ease the implementation of our OptE3 method, its procedure is illustrated in Fig. 3 with detailed explanations below:

 

Fig. 3. The proposed OptE3 measurement procedure.

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Reconstruct 3D profile $({{x^w},{y^w},{z^w}} )$ using Eqs. (33), (34) and (35) or Eqs. (33), (34) and (36).

The detailed measurement procedure enables us to further elaborate its high convenience:

4.2 Theoretical revealing of the epipolar constraint in 3D reconstruction

We now provide more theoretical analysis on 3D reconstruction in OptE3. We first show that the four equations for reconstruction, Eqs. (33–36), are redundant when the epipolar line is used for correspondence. To prove it, we first re-write the linear Eqs. (33–36) as the following augmented matrix form,

Next, by Gaussian elimination, the matrix becomes,

Finally, we check when the following equation holds:

The checking is straightforward as all the parameters appearing in Eq. (47) are given in Eq. (37). By substituting Eq. (37) into Eq. (47), followed by some rearrangement, Eq. (47) turns into the epipolar line as given by Eq. (9). In other words, this means that one equation in the reconstruction is equivalent to an epipolar line. This is rather interesting but in fact not surprising, because the epipolar constraint can indeed be implied in the triangulation principle for reconstruction. Because in our OptE3, this epipolar constraint has been used in the correspondence step to obtain $({{u^p},{v^p}} )$ from $({{u^c},{v^c}} )$ and thus has already been satisfied, this equation is redundant. Thus, we only need to use three equations, Eqs. (33–35) or Eqs. (33), (34), and (36), for reconstruction, which has been used in our proposed method. We list and highlight this theoretical property of 3D reconstruction below:

An advantage of using three equations to solve three unknowns for 3D reconstruction is that, an elegant closed-form solution with simple algebra computation is now available. From the third row in Eq. (46), we obtain ${z^w}$ as

Submitting the parameters given in Eq. (37) into Eq. (48), we have

${C_4} = \left( { - \frac{{{u^c} - u_0^c}}{{f_u^c}}\frac{{{r_{31}}}}{{f_u^p}} - \frac{{{r_{33}}}}{{f_u^p}} - \frac{{{v^c} - v_0^c}}{{f_v^c}}\frac{{{r_{32}}}}{{f_u^p}}} \right)$. This simple solution increases the reconstruction speed, which will be tested in Sec. 5.3. We list this property below:

4.3 Theoretical analysis on OptE3’s reconstruction accuracy

The proposed OptE3 is unique in the sense that it follows the 4-equation methods for high precision but finally, it only uses 3 equations because of the special involvement of the epipolar line for correspondence. It is of importance to find out how this uniqueness affects the final reconstruction accuracy and precision, which are analyzed in this and next subsections, respectively.

To do so, we first use the OptE3 method for correspondence, i.e., find $({{u^p},{v^p}} )$ by solving Eqs. (9) and (14) as follows,

The depth from OptE3 is given in Eq. (49). When the phase noise presents, ${u^p}$ will be deviated by ${\delta _{{u^p}}}$, and the reconstructed depth becomes,

Combining Eqs. (49) and (54), the depth error is

Finally, in practice, as shown in Appendix B, ${C_4}{\delta _{{u^p}}} \ll {C_3} + {C_4}{u^p}$. We can use which Taylor approximation to approximate ${({{C_3} + {C_4}{u^p} + {C_4}{\delta_{{u^p}}}} )^{ - 1}}$ as ${({{C_3} + {C_4}{u^p}} )^{ - 1}} \times \left[ {1 - \frac{{{C_4}}}{{({{C_3} + {C_4}{u^p}} )}}{\delta_{{u^p}}}} \right]$, so that

Since ${\delta _{{u^p}}}$ has a zero mean, ${\delta _z}$ also has a zero mean. We have the following conclusion:

4.4 Theoretical analysis on OptE3’s reconstruction precision

In Sec. 4.3, we have concluded that the OptE3 method is an unbiased depth estimation. In this Section, we will explore whether the reconstruction precision of OptE3 follows 4-equaiton methods or 3-equation methods. Fortunately, the former is the answer, making OptE3 a precise method. To demonstrate it, we will show that the OptE3 method has approximately the same depth variance as the OptR4 method, where the depth variance is defined as

Since ${v^p}$ is not used, the second term on the righthand of Eq. (57) is zero. Thus, with Eqs. (37), (52) and (57), we have

As for the OptR4 method, $({{u^p},{v^p}} )$ has been obtained in Eqs. (28) and (29), and their variance in Eqs. (30) and (31). More specifically, in the OptR4 method, ${\theta _1} = {\theta _{opt}} - \pi /4$, ${\theta _2} = {\theta _{opt}} + \pi /4$, and thus we have

Next, we obtain the depth from Eqs. (33–36) by the least squares method, which has been used in Sec. 3.3 with details given in Appendix A. Finally, substituting Eq. (59) into Eq. (57), we have,

Now that $\sigma _{{z_{OptR4}}}^2$ and $\sigma _{{z_{OptE3}}}^2$ have been obtained in Eqs. (58) and (60), respectively, the remaining task is to examine their ratio,

As all the variables and parameters in Eqs. (58) and (60) are already available in earlier equations, the calculation of the ratio seems to be straightforward, but turns out to be very difficult because of tens of terms will appear in deriving $\partial {z_{OptR4}}/\partial {u^p}$ and $\partial {z_{OptR4}}/\partial {v^p}$ in Eq. (60). As such, we use a numerical verification. First, we obtain $\partial {z_{OptR4}}/\partial {u^p}$ and $\partial {z_{OptR4}}/\partial {v^p}$ from Eq. (A(1)1) using the symbolic operation in MATLAB. Next, For each $({{u^c},{v^c}} )$ under the projection with the optimal fringe angle, we follow the practical depth range given in Eq. (B.3), based on which, we calculate ${\mathrm{\Phi }_{zmin}}({{u^c},{v^c}} )$ and ${\mathrm{\Phi }_{zmax}}({{u^c},{v^c}} )$ corresponding to ${z_{min}}$ and ${z_{max}}$, respectively, and we have ${\mathrm{\Phi }_{zmin}}({{u^c},{v^c}} )\le \mathrm{\Phi }({{u^c},{v^c}} )\le {\mathrm{\Phi }_{zmax}}({{u^c},{v^c}} )$. We then randomly select 1000 phase values within this range and calculate $({{u^p},{v^p}} )$ by Eqs. (9) and (14). Finally, substituting all $({{u^c},{v^c}} )\; $ and $({{u^p},{v^p}} )$ into Eq. (61), we obtain the ratio $0.9999 \le F({{u^c},{v^c},{u^p},{v^p}} )\le 1$, indicating that the OptE3 method and the OptR4 method indeed give an approximately (maybe theoretically exactly) the same reconstruction precision. We have also tested that the same result is applicable to HorVer4. Hence,

5. Experiments

5.1 System setup and calibration

An FPP system has been set up to verify the performance of the proposed method. The system includes a camera (model: DAHENG MER-131-210U3M-L, resolution: 1024 × 1280 pixels) attached with a 25 mm lens (model: Computar-M2514-MP2) and a digital-light-processing projector (model: PROVIS900, resolution: 1080 × 1920 pixels). The optimum three-frequency method is applied for phase extraction and unwrapping [24], where the nine-, five-, and five-step phase-shifting methods are used with fringe periods of ${T_0} = 21\textrm{pixel}$ s, ${T_1} = 700/33\; \textrm{pixels}$, ${T_2} = 70/3\,\textrm{pixels}$, respectively. In Ref. [24], three primary color channels (RGB) are used to project three- frequency fringe patterns, while we project three set of gray-scale fringe patterns with different frequencies sequentially. To conveniently compare the number of patterns required by different 3D reconstruction methods, we define L as the total number of patterns to obtain one unwrapped phase. L can be expressed as $L = \mathop \sum \limits_{i = 1}^F {M_i}$, where F is the number of fringe total frequencies used and ${M_i}$ represents the phase-shifting steps, i.e. in this paper, $i = 1,2,3$,and ${M_1} = 9$, ${M_2} = {M_3} = 5$.

In step 1A, the system is calibrated by Zhang’s [6] method and the intrinsic and extrinsic matrices are obtained as follow,

In step 1B, we substitute the intrinsic and extrinsic parameters given in Eqs. (62–64) into Eq. (22) and obtain the optimal angle as 1.108 rad.

Subsequently, in Step 1C, we substitute the intrinsic and extrinsic parameters given in Eqs. (62–64) to Eqs. (10 12) and obtain the parameters ${l_1}$, ${l_2}$ and ${l_3}$ of the epipolar line, which are dependent on $({{u^c},{v^c}} )$.

5.2 Validation of the optimal fringe angle

In this section, we verify that the epipolar line is indeed perpendicular to the optimal fringe angle because this property is used in our proposed method as a key ingredient. Furthermore, with this property, the optimal fringe angle can be directly obtained from the epipolar line without measurement. For this purpose, we measure the optimal fringe angle by Wang and Zhang’s method [15] which uses a step-height block. Figures 4(a) and 4(b) show the horizontal and vertical fringe images captured by a camera, respectively. Figure 4(c) shows the measured optimal fringe angle distribution at the position of the step-height block, following the steps detailed in [15]. We also calculate the optimal angle using Eq. (22), which is a field varying with $({{u^c},{v^c}} )$, as shown in Fig. 4(d). Note that Wang and Zhang’s method only provides the optimal angle for the pixels in the step-height area, while our method provides the optimal angle for all the camera pixels. Thus, a red box is shown in Fig. 4(d) to indicate the same step-height block area.

 

Fig. 4. Determining the optimal fringe angle. (a) One of the captured horizontal fringe patterns; (b) one of the captured vertical fringe patterns; (c) the optimal fringe angle distribution by Wang ang Zhang [15]; (d) the optimal fringe angle distribution by our method.

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We do the following verifications:

5.3 Quantitative evaluation with a planer object

With the completion of the calibration in Step 1, we proceed to do the phase measurement in Step 2, the correspondence in Step 3 and the 3D reconstruction in Step 4. All the methods listed in Table 1 are compared for quantitative evaluation. Since these methods are tested in the same calibrated FPP system, they share the same intrinsic and extrinsic metrices as given in Sec. 5.1.

Table 1. Comparison of different 3D reconstruction methods

View Table

A ceramic plate with a size of 300 × 300 $\textrm{m}{\textrm{m}^2}$ and flatness of less than 0.005 mm is used as the measured object. After measuring the plate and obtaining its 3D point cloud, a plane is fitted as

Then the closest distance from each measurement object point $({{x_0},{y_0},{z_0}} )$ to the fitted plane is computed by

 

Fig. 5. Comparison of measurement results of the ceramic plate at 15 different positions. (a) The depth data of the ceramic plate at the first location. (b) The STDE distributions of the ceramic plate at 15 different positions.

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It is worth noting that, in OptE3, we can project an additional set of fringe pattern with the worst fringe angle for correspondence, and then use all four equations (33–36) for reconstruction (abbreviated as OptWor4). The OptWor4 and OptE3 methods are expected to give same results, which has indeed been verified in our experiment. Since OptWor4 is slower and more troublesome than OptE3, we do not encourage readers to use it and thus not listed in Table 1.

5.4 Qualitative evaluation of a complicated object

To visually demonstrate the advantage that the proposed method, we test all the five measurement methods on a Briar Bear as shown in Fig. 6(a). Figures 6(b)-(f) are the 3D results, where the hand portion is magnified. It can be seen that the results by HorVer4 (Fig. 6(d)), OptR4 (Fig. 6(e)), and OptE3 methods (Fig. 6(f)) are very similar, and all of them are smoother than those from the Ver3 method (Fig. 6(c)) and the Hor3 method (Fig. 6(b)), which is consistent with our previous analysis and the plate experiment. Again, since the optimal fringe angle is closer to vertical fringes, the Ver3 method performs better than the Hor3 method.

 

Fig. 6. Measurement results of a complex object. (a) The original picture of the object; (b) Hor3; (c) Ver3; (d) HorVer4; (e) OptR4; (f) OptE3.

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6. Conclusions

This paper performs a series of fundamental theoretical analysis on a FPP system, which is critical in precision measurement. Our investigation reveals that, (i) in phase measurement, the optimal and worst fringe angles are perpendicular and parallel to epipolar line, respectively, and can be considered as system parameters and can be directly made available during traditional calibration, highlighting the significance of the epipolar line; (ii) in correspondence, the variance of ${u^p}$ and ${v^p}$ when using two sets of fringes with a pair of (${\theta _1}$, ${\theta _2}$) is minimized where ${\theta _1} \bot {\theta _2}$, but is independence of ${\theta _1}$ itself, clarifying the angle selection problem; (iii) in reconstruction, the sum of squared residual of equations from the 4-eqation methods is at most half of that from 3-equation method, suggesting the preference of the 4-equation methods for higher precision.

Based on these theoretical analysis results, this paper proposes a novel FPP 3D measurement method (OptE3). Compared with the HorVer4 and OptR4 methods, the OptE3 method only needs to project one set of optimal angle fringes, solve three equations by a simple closed form, but achieve the same precision as the 4-equation methods. This unique and graceful property is also theoretically proved, where the epipolar line plays a critical role again. The proposed OptE3 method was demonstrated to have high efficiency, precision and convenience.