### Our proposed projecting and imaging model

The geometric optical geometry of our setup is shown in Figure 4. *I*
_{p} and *I*
_{c} are the exit pupil and entrance pupil of the projector and camera, respectively. The optical axes *I*
_{p}
*O* and *I*
_{c}
*O* cross the reference plane at point *O* and make angles *θ*
_{1} and *θ*
_{2} with *Z*
_{
W
} axis (i.e., the normal direction of the reference plane), respectively. The baseline between these two optical centers is *I*
_{p}
*I*
_{c} = *b*, which is not parallel to the reference plane. *M* is the perpendicular projection point of *I*
_{p} on the reference plane, and the distance between them is *I*
_{p}
*M* = *L*
_{p}. *N* is the perpendicular projection point of *I*
_{c} on the reference plane, and the distance between them is *I*
_{c}
*N* = *L*
_{c}. Point *A* on the reference plane and point *P* on the object surface correspond to the same image pixel location on the CCD plane. Point *C* on the reference plane and point *P* on the object surface are on the same pixel ray projecting from the projector. We add several dashed lines in the figure as guidelines for analysis. The dashed line *I*
_{p}
*F* is parallel with the reference plane and crosses the extension line of *BP* (*BP* = *h*) at point *D*, which intersects with lines *I*
_{c}
*P* and *I*
_{c}
*N* at points *E* and *F* respectively. In this work, we mainly focus on the measurement performance with respect to the influence of one parameter, which is the baseline *b*. Hence, similarly to the work in [10], we can assume that the fringe patterns formed by the projector are parallel to *Y*
_{
W
}. From the geometry setup in Figure 4, we can get that the triangle *APB* is similar with the triangle *EI*
_{c}
*F*,

\mathit{AB}/\mathit{EF}=\mathit{PB}/{\mathit{I}}_{\mathrm{c}}\mathit{F}=\mathit{PB}/\mathit{b}sin\mathit{\alpha}.

(12)

Similarly, from the fact that triangle *APB* is similar with triangle *ANI*
_{c} and triangle *ACP* is similar with triangle *I*
_{p}
*PE*, we can get

\mathit{AB}/\mathit{AN}=\mathit{PB}/{\mathit{I}}_{\mathrm{c}}\mathit{N}=\mathit{PB}/{\mathit{L}}_{\mathrm{c}},

(13)

\mathit{AC}/\mathit{PB}={\mathit{I}}_{\mathrm{p}}\mathit{E}/\mathit{PD}=\left({\mathit{I}}_{\mathrm{p}}\mathit{F}-\mathit{EF}\right)/\mathit{PD}=\left({\mathit{L}}_{\mathrm{p}}tan{\mathit{\theta}}_{1}+{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}-\mathit{EF}\right)/\left({\mathit{L}}_{\mathrm{p}}-\mathit{PB}\right).

(14)

Submitting Equation 12 into Equation 14, we can get

\mathit{AC}/\mathit{PB}=\left[{\mathit{L}}_{\mathrm{p}}tan{\mathit{\theta}}_{1}+{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}-\left(\mathit{AB}\cdot \mathit{b}sin\mathit{\alpha}/\mathit{PB}\right)\right]/\left({\mathit{L}}_{\mathrm{p}}-\mathit{PB}\right).

(15)

Note that *AN* = *AC* + *OC* + *ON* = *AC* + *x* + *L*
_{c} tan *θ*
_{2}, submitting this relationship and Equation 16 into 18, we obtain

\mathit{PB}={\mathit{L}}_{\mathit{p}}{\mathit{L}}_{\mathit{c}}\mathit{AC}/\left[\mathit{AC}{\mathit{L}}_{\mathit{c}}+{\mathit{L}}_{\mathit{p}}{\mathit{L}}_{\mathit{c}}tan{\mathit{\theta}}_{1}+{\mathit{L}}_{\mathit{c}}^{2}tan{\mathit{\theta}}_{2}-\left(\mathit{AC}+\mathit{x}+{\mathit{L}}_{\mathit{c}}tan{\mathit{\theta}}_{2}\right)\mathit{b}sin\mathit{\alpha}\right]

(16)

where *p* denotes the periodicity of the fringe patterns on the reference plane under divergent illumination. According to the work in [3], we can get

\mathit{AC}=\mathit{p}\cdot \left|{\mathit{\phi}}_{\mathit{C}}-{\mathit{\phi}}_{\mathit{A}}\right|/2\mathit{\pi}=\mathit{p}\cdot \left|\mathrm{\Delta}{\mathit{\phi}}_{\mathit{PA}}\right|/2\mathit{\pi}

(17)

Submitting Equation 17 into Equation 16, we can get the final relationship between the phase distribution *φ*(*x*, *y*) and the height information *h*(*x*, *y*), which is expressed as

\begin{array}{l}\mathit{h}\left(\mathrm{x},\mathrm{y}\right)=\left({\mathit{L}}_{\mathrm{p}}{\mathit{L}}_{\mathrm{c}}\mathit{p}\left|\mathrm{\Delta}{\mathit{\phi}}_{\mathit{PA}}\left(\mathit{x},\mathit{y}\right)\right|\right)/[(\mathit{p}-\mathit{pb}sin\mathit{\alpha})\left|\mathrm{\Delta}{\mathit{\phi}}_{\mathit{PA}}\left(\mathit{x},\mathit{y}\right)\right|-2\mathit{\pi bx}sin\mathit{\alpha}\\ \phantom{\rule{7.5em}{0ex}}+2\mathit{\pi}{\mathit{L}}_{\mathrm{p}}{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{1}+2\mathit{\pi}{\mathit{L}}_{\mathrm{c}}^{2}tan{\mathit{\theta}}_{2}-2\mathit{\pi}{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}\mathit{b}sin\mathit{\alpha}].\end{array}

(18)

It can also be written in a concise form as

\mathit{h}\left(\mathit{x},\mathit{y}\right)={\mathit{c}}_{1}\left|\mathrm{\Delta}{\mathit{\phi}}_{\mathit{PA}}\left(\mathit{x},\mathit{y}\right)\right|/\left[{\mathit{c}}_{2}\left|\mathrm{\Delta}{\mathit{\phi}}_{\mathit{PA}}\left(\mathit{x},\mathit{y}\right)\right|+{\mathit{c}}_{3}\mathit{x}+{\mathit{c}}_{4}\right],

(19)

where parameters *c*
_{1}, *c*
_{2}, *c*
_{3}, *c*
_{4} are related with geometric parameters *L*
_{p}, *L*
_{c}, *p*, *b*, *α*, *θ*
_{1}, *θ*
_{2} and can be denoted as

\left\{\begin{array}{c}\hfill {\mathit{c}}_{1}={\mathit{L}}_{\mathrm{p}}{\mathit{L}}_{\mathrm{c}}\mathit{p},{\mathit{c}}_{2}=\mathit{p}-\mathit{pb}sin\mathit{\alpha}\hfill \\ \hfill {\mathit{c}}_{3}=-2\mathit{\pi b}sin\mathit{\alpha},{\mathit{c}}_{4}=2\mathit{\pi}{\mathit{L}}_{\mathrm{p}}{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{1}+2\mathit{\pi}{\mathit{L}}_{\mathrm{c}}^{2}tan{\mathit{\theta}}_{2}-2\mathit{\pi}{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}\mathit{b}sin\mathit{\alpha}\hfill \end{array}\right..

(20)

### Performance analysis

#### Influence of the length of baseline *b*

The geometric parameters of the system setup include the angle between the optical axis of the projector and the camera, the distance between optical center of camera system and reference plane, the focal length of camera system and the periodicity of projected fringe patterns, etc. In this paper, we regard the baseline’s length as the priority factor and focus on the length of baseline’s influence on the final measurement result. From Equation 19, we can transform it into another form, which takes the baseline *b* as an input variable and is expressed as follows:

\mathit{h}\left(\mathit{x},\mathit{y}\right)={\mathit{K}}_{1}\left(\mathit{x},\mathit{y}\right)/\left({\mathit{K}}_{2}\left(\mathit{x},\mathit{y}\right)-{\mathit{K}}_{3}\left(\mathit{x},\mathit{y}\right)\cdot \mathit{b}\right)

(21)

where \left\{\begin{array}{c}\hfill {\mathit{K}}_{1}\left(\mathit{x},\mathit{y}\right)={\mathit{L}}_{\mathrm{p}}{\mathit{L}}_{\mathrm{c}}\mathit{p}\left|\mathrm{\Delta}{\mathit{\phi}}_{\mathit{PA}}\left(\mathit{x},\mathit{y}\right)\right|\hfill \\ \hfill {\mathit{K}}_{2}\left(\mathit{x},\mathit{y}\right)=\mathit{p}\left|\mathrm{\Delta}{\mathit{\phi}}_{\mathit{PA}}\left(\mathit{x},\mathit{y}\right)\right|+2\mathit{\pi}{\mathit{L}}_{\mathrm{p}}{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{1}+2\mathit{\pi}{\mathit{L}}_{\mathrm{c}}^{2}tan{\mathit{\theta}}_{2}\hfill \\ \hfill {\mathit{K}}_{3}\left(\mathit{x},\mathit{y}\right)=\mathit{p}sin\mathit{\alpha}\left|\mathrm{\Delta}{\mathit{\phi}}_{\mathit{PA}}\left(\mathit{x},\mathit{y}\right)\right|+2\mathit{\pi x}sin\mathit{\alpha}+2\mathit{\pi}{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}sin\mathit{\alpha}.\hfill \end{array}\right.

From Equation 19, we get the relationship between the phase difference and the height information:

\left|\mathrm{\Delta}{\mathit{\phi}}_{\mathit{PA}}\left(\mathit{x},\mathit{y}\right)\right|=\frac{\mathit{h}\left(\mathrm{x},\mathrm{y}\right)\left({\mathit{c}}_{3}\mathit{x}+{\mathit{c}}_{4}\right)}{\mathit{h}{\mathit{c}}_{2}-{\mathit{c}}_{1}}.

(22)

Similar to the derivative method in the work in [10], the partial derivative of Equation 23 with respect to the baseline *b* is calculated, and Equations 18 and 19 are submitted into the result. We get

\partial \mathit{h}\left(\mathrm{x},\mathrm{y}\right)/\partial \mathit{b}=\left({\mathit{Q}}_{1}{\mathit{h}}^{2}\left(\mathit{x},\mathit{y}\right)-{\mathit{Q}}_{2}{\mathit{h}}^{2}\left(\mathit{x},\mathit{y}\right)\mathit{b}-{\mathit{Q}}_{3}\mathit{h}\left(\mathit{x},\mathit{y}\right)\right)/\left({\mathit{Q}}_{4}-{\mathit{Q}}_{5}\mathit{b}\right)

(23)

where \left\{\begin{array}{l}{\mathit{Q}}_{1}=2\mathit{\pi p}sin\mathit{\alpha}\left({\mathit{L}}_{\mathrm{p}}{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{1}+{\mathit{L}}_{\mathrm{c}}^{2}tan{\mathit{\theta}}_{2}+\mathit{x}-\mathit{xb}sin\mathit{\alpha}+{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}-{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}\mathit{b}sin\mathit{\alpha}\right)\\ {\mathit{Q}}_{2}=2\mathit{\pi p}{sin}^{2}\mathit{\alpha}\left({\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}+\mathit{xb}\right)\\ {\mathit{Q}}_{3}=2\mathit{\pi p}{\mathit{L}}_{\mathrm{p}}{\mathit{L}}_{\mathrm{c}}\left(\mathit{x}sin\mathit{\alpha}+{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}sin\mathit{\alpha}\right)\\ {\mathit{Q}}_{4}=2\mathit{\pi p}{\mathit{L}}_{\mathrm{p}}{{\mathit{L}}_{\mathrm{c}}}^{2}\left({\mathit{L}}_{\mathrm{p}}tan{\mathit{\theta}}_{1}+{\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}\right)\\ {\mathit{Q}}_{5}=2\mathit{\pi p}{\mathit{L}}_{\mathrm{p}}{\mathit{L}}_{\mathrm{c}}\left({\mathit{L}}_{\mathrm{c}}tan{\mathit{\theta}}_{2}sin\mathit{\alpha}+\mathit{x}sin\mathit{\alpha}\right).\end{array}\right.

Equation 23 shows that the height error ∂*h*(*x*, *y*)/∂*b* is a function of the parameters *Q*
_{1}, *Q*
_{2}, *Q*
_{3}, *Q*
_{4}, *Q*
_{5}, *b*, *h*. The dependence of ∂*h*(*x*, *y*)/∂*b* on *h* is shown in Figure 5 with respect to the variation of other parameters *Q*
_{1}, *Q*
_{2}, *Q*
_{3}, *Q*
_{4}, *Q*
_{5}. The red curve and blue curve in Figure 5 indicate the lengths of baseline *b* = 30 mm and *b* = 120 mm, respectively. A real experimental system setup consists of a pico-projector (Optoma PK301; Optoma USA, Fremont, CA, USA) and a mini-camera (Point Grey FL3-U3-13S2M-CS; Point Grey Research KK, Chiyoda-ku, Tokyo, Japan**)** with a 6-mm focal length. Hence, the parameters variation are in the following ranges: *L*
_{p} from 390 to 420 mm, *L*
_{c} from 400 to 450 mm, *p* from 10 to 20 mm, *θ*
_{1} from 0° to 15°, *θ*
_{2} from 0° to 10°, *α* from 0° to 30°, *x* from −150 to 150 mm.

It is important to note that the parameters in the following analysis are also falling into this range. The results shown in Figure 5 indicate that in the two cases (the baseline *b* = 30 mm and the baseline *b* = 120 mm), the measurement error becomes larger as the height of target object increases. However, in the red curve which indicates the shorter baseline case (*b* = 30 mm), the measurement error is smaller (less than 0.1); meanwhile, the relationship between ∂*h*(*x*, *y*)/∂*b* and *h* is almost linear. On the other hand, when the baseline changes to a larger value (*b* = 120 mm), the relationship between the measurement error ∂*h*(*x*, *y*)/∂*b* and *h* is nonlinear and as the height of the object increases the measurement error increases faster than the shorter baseline (*b* = 30 mm). In particular, when the height of target object equals to 100 mm, the maximum value of the measurement error is larger than 0.5.

### Relative measurement error analysis

A relative measurement error analysis is also conducted with respect to other parameters *K*
_{1}, *K*
_{2}, *K*
_{3} while the baseline is assumed fixed. Suppose there are small errors *δK*
_{1}, *δK*
_{2}, *δK*
_{3} existing in the parameters *K*
_{1}, *K*
_{2}, *K*
_{3}, respectively. The relative measurement error of height *δh*/*h* can be expressed by the following generic approximation:

\mathit{\delta h}/\mathit{h}=\sqrt{{\left(\frac{\partial \mathit{h}}{\partial {\mathit{K}}_{1}}\mathit{\delta}{\mathit{K}}_{1}\right)}^{2}+{\left(\frac{\partial \mathit{h}}{\partial {\mathit{K}}_{2}}\mathit{\delta}{\mathit{K}}_{2}\right)}^{2}+{\left(\frac{\partial \mathit{h}}{\partial {\mathit{K}}_{3}}\mathit{\delta}{\mathit{K}}_{3}\right)}^{2}}/\mathit{h}

(24)

Submitting Equation 21 into Equation 24, we can obtain

\mathit{\delta h}/\mathit{h}=\sqrt{\mathit{\delta}{{\mathit{K}}_{1}}^{2}+{\mathit{h}}^{2}\mathit{\delta}{{\mathit{K}}_{2}}^{2}+{\mathit{b}}^{2}{\mathit{h}}^{2}\mathit{\delta}{{\mathit{K}}_{3}}^{2}}/{\mathit{K}}_{1}.

(25)

Equation 25 indicates that the relative measurement error *δh*/*h* is a function of the length of baseline *b*, the parameters *K*
_{1}, *K*
_{2}, *K*
_{3}, and their variation *δK*
_{1}, *δK*
_{2}, *δK*
_{3}.

The results are shown in Figure 6. Without loss of generality, we set the variation of parameters *δK*
_{1}, *δK*
_{2}, *δK*
_{3} = 0.01. The results tell us that when the height of the target object increases to 100 mm, the yellow curve which represents the largest baseline configuration system (*b* = 120 mm) yields the biggest relative measurement error, which is 25%, 0.0005% for minimizing and maximizing the parameters *K*
_{1}, *K*
_{2}, *K*
_{3} respectively. Meanwhile, the red curve (*b* = 30 mm) presents the smallest relative measurement error, which is 5%, 0.0001% for minimizing and maximizing the parameters *K*
_{1}, *K*
_{2}, *K*
_{3} respectively. Hence, we can get the same conclusion that if the length of the baseline increases, the relative error of the measured height becomes bigger, when the same errors *δK*
_{1}, *δK*
_{2}, *δK*
_{3} are introduced into the system.

### Sensitivity analysis

Furthermore, we take a sensitivity analysis with respect to the baseline *b*. In the following part, *b*
_{e} represents the estimates of *b* and Δ*b*/*b* = (*b*
_{e} − *b*)/*b* indicates the relative discrepancy with respect to the nominal values. The error ∆*h* can be expressed as the difference between the depth values calculated by substituting the two values *b*
_{e} and *b* into Equation 21:

\mathrm{\Delta}\mathit{h}=\frac{{\mathit{K}}_{1}}{{\mathit{K}}_{2}-{\mathit{K}}_{3}{\mathit{b}}_{\mathrm{e}}}-\frac{{\mathit{K}}_{1}}{{\mathit{K}}_{2}-{\mathit{K}}_{3}\mathit{b}}.

(26)

When Equations 22 and 26 are combined, relative error ∆*h*/*h* results:

\mathrm{\Delta}\mathit{h}/\mathit{h}=\frac{{\mathit{K}}_{1}}{{\mathit{K}}_{2}-{\mathit{K}}_{3}{\mathit{b}}_{\mathrm{e}}}/\frac{{\mathit{K}}_{1}}{{\mathit{K}}_{2}-{\mathit{K}}_{3}\mathit{b}}-1=-\frac{\mathit{\Delta b}}{\mathit{b}}\frac{1}{1+\frac{\mathit{\Delta b}}{\mathit{b}}-\frac{{\mathit{K}}_{2}/{\mathit{K}}_{3}}{\mathit{b}}}

(27)

Equation 27 expresses ∆*h*/*h* as a hyperbolic function of ∆*b*/*b*, but for small values of ∆*b*/*b*, the function is almost linear, as shown in Figure 7. The yellow curve which represents the largest baseline configuration system (*b* = 120 mm) yields the smallest relative variation of height with respect to the same relative discrepancy of baseline, while the red curve which represents the shortest baseline arrangement system (*b* = 30 *mm*) presents the biggest relative variation of height. This means that for a system with a shorter baseline, the proposed model is more sensitive to the small variation of other parameters. In other words, we can get the conclusion that the larger the baseline, the less sensitive is the system with respect to the same bias in the calibrated parameters. This conclusion is the same as presented in the discussion part of the work [11].