Improved geometrical model of fringe projection profilometry

Zhengrong Huang; Jiangtao Xi; Yanguang Yu; Qinghua Guo; Limei Song

doi:10.1364/OE.22.032220

1. Introduction

Fringe Projection Profilometry (FPP) has been considered as an enabling technology for non-contact 3-D shape measurement due to such advantages as simple structure and fast measurement [1–4]. A typical FPP system consists of a digital projector, a camera, a computer and a reference plane. The digital projector generates a group of image patterns of periodic fringes, which are projected respectively onto the reference plane and the object surface to be measured. The camera captures the image patterns reflected from the reference plane and the object surface, the latter of which are deformed version of former. The 3-D information of the object surface shape can be extracted by analyzing the phases of the projected patterns acquired by means of a phase-to-height (PTH) mapping relationship. The PTH mapping is based on the triangulation relationship among the projector, the camera, and the corresponding point on the patterns acquired from the reference plane and the object surface. The effectiveness of the PTH mapping depends on if it matches the structure of FPP system, that is, the positions of the camera, the projector and the reference plane. In early years, the PTH mapping proposed by Takeda, et al. [1] based on a simple model, where the FPP system is assumed to have an ideal structure meeting three conditions, including (1) the optical centers of camera and projector are located at the same distance from the reference plane; (2) the optical axes of the camera and projector are coplanar and the plane is perpendicular to the reference plane; and (3) the optical axis of camera is vertical to reference plane. However, these conditions are not always met in practice, and measurement error will occur if the PTH [1] is employed to recover the 3-D shape. In order to remedy this problem, an improved PTH mapping relationship was proposed by Mao, et al. [2], where the first condition is removed in that camera and projector can be positioned with different distances from the reference plane. However, the second and third conditions are still required. Recently in 2012, further effort for solving the problem was reported in [3], where a PTH mapping was proposed to remove the first and third condition in that camera and projector are permitted to locate at different distance from the reference plane, and the optical axis of camera is not required to be vertical to the reference plane. However, the second condition remains. To the best of our knowledge, there is not a PTH mapping reported in literature where all these three conditions can be lifted, allowing a flexible positioning of camera and projector.

As the PTH mapping relationship is determined by the system structure, accurate evaluation of the parameters associated with the structure plays an important role, which is carried out by means of system calibration. These parameters include the intrinsic parameters of the camera and the projector, as well as extrinsic parameters associated with the geometrical structure of the FPP. Over the past decades, a number of approaches for system calibration are proposed, e.g [5–11]. Recently in 2013, a method to calibrate five essential parameters associated with the camera and the projector is presented by Song, et al. [12]. However, all the existing work reported in literature was based on the FPP structure meeting the three conditions. To the best of our knowledge, there is not work reported for system calibration with a flexible positioning of the camera and projector, that is, without the restrictions of the three conditions.

In this paper, we firstly propose a model to describe a general structure of FPP, where the three conditions are not required. Such a model involves seven parameters related to the system structure which are all required to be obtained. In order to achieve this, we propose a system calibration method. Before the seven parameters are calibrated, the camera and projector have been calibrated simultaneously. Camera calibration has been widely used for 3-D measuring system. A highly accurate and robust camera calibration method was proposed by Zhang [13],. Hence this paper adopts Zhang’s method [13]. Since the projector cannot take the picture as camera, the camera is used to help it to capture image. The phase-shifting method proposed by Zhang, et al. [14] in 2008 is adopted to transform the points in camera image into projector image. Then Zhang’s method [13] is used to calibrate projector by considering it as an inverse camera, since camera and projector share the same optical principle. The innovations of this paper are improving the existing PTH mapping, and presenting the calibration method for our improved PTH mapping. The experiments demonstrate the accuracy and flexibility of 3-D measuring system have been improved based on our method. Such an improved algorithm will be highly suitable to the practical application of 3-D measuring system.

2. Existing work and problem statement

In this section, we will introduce in the order of the PTH mapping relationship proposed by Takeda, et al. [1], an improved PTH mapping by Mao, et al. [2], the work by Xiao, et al. [3], and then problem statement.

2.1 Geometric model proposed in [1]

Figure 1 shows an ideal geometric model proposed in [1]. The fringe patterns are projected onto the reference plane by the projector, and then the fringe patterns reflected are captured by the camera. Without loss of generality, we consider a beam of light $p_{1} (u_{p}, v_{p})$ projected by the projector onto the reference plane on the point $p (x, y, z)$ , which is reflected via $p_{2} (u_{c}, v_{c})$ and captured by the camera. When the object is placed over the reference plane, the same light beam is projected onto the surface of object on point $p_{1} (u_{p}, v_{p})$ , and reflection is taken by the camera via $p (x, y, z)$ . The PTH mapping relationship is following [1]:

h (x, y) = \frac{- L_{C} Δ φ_{D C} (x, y)}{2 π f_{0} d - Δ φ_{D C} (x, y)}

where

f_{0}

is the frequency of fringe pattern on reference plane.

L_{C}

is the distance between point

O

and optical center of camera ;

d

is the distance between optical centers of camera and projector;

Δ φ_{D C} (x, y)

is the difference between the phase of point

D

and that of point

C

; and

h (x, y)

is the height of object.

Fig. 1 The ideal geometric modal.

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The Eq. (1) is derived based on three conditions, including (1) the line $O_{P} O_{C}$ is parallel to reference plane; (2) the line $O_{p} O$ and the line $O O_{C}$ are coplanar, and the plane is vertical to reference plane; and (3) the line $O_{C} O$ is vertical to reference plane. If the three conditions are not satisfied at the same time, the PTH mapping will lead to measuring error.

2.2 Geometric model proposed in [2]

Figure 2 shows an improved geometric model proposed in [2], which allows the camera to move along z-axis direction, and hence is more flexible than the ideal one in [1]. The PTH mapping in [2] is as follows:

h (x, y) = - \frac{Δ φ_{D C} (x, y) L_{C} (L_{C} + S_{1} \sin α_{1})}{2 π f_{0} L_{C} r - L_{C} Δ φ_{D C} (x, y) - φ_{D} (x, y) S_{1} \sin α_{1}}

where, in addition to the parameters associated with the model in Eq. (1), another three parameters were introduced, including

S_{1}

the distance between the optical centers of camera and projector,

α_{1}

the angle between the linear

O_{p} F

and

O_{P} O_{C}

, and

r

the distance between point

O

and

K

.

Fig. 2 (a). An improved geometric model (b). Simplified geometric model.

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When $α_{1} = 0$ and $r = d$ , Eq. (2) becomes the same as Eq. (1), and hence the PTH mapping in [1] can be considered as a special case of the PTH mapping proposed in [2]. In fact, since $S_{1} \sin α_{1}$ only represents the difference between the heights of camera and projector relative to reference plane, Fig. 2(a) can be simplified as Fig. 2(b). The geometric model proposed in [2] only removes the first condition that projector and camera must be located at the same height relative to the reference plane. Because the line $O_{c} O$ is vertical to reference plane, the plane including the lines $O_{c} O$ and $O_{p} O$ is vertical to reference plane. Although the geometric model proposed in [2] improves the ideal geometric model, it is still limited by the second and third condition. If these two conditions are not satisfied, the PTH mapping proposed in [2] will still lead to error for measuring object.

2.3 Geometric model proposed in [3]

Another improved geometric model in Fig. 3 is proposed in [3], where camera and projector are allowed to be different height relative to reference plane, and optical axis of camera is permitted not to be vertical to reference plane. Hence, compared to the simplified geometric model as Fig. 2 (b), this improved geometric model is more flexible. The PTH mapping in [3] can be given as:

h (x, y) = \frac{- Δ ϕ_{D C} (x, y) L_{c} \cos θ_{2} (L_{c} \cos θ_{2} + S_{1} \sin α_{1})}{2 π f_{0} L_{c} (r \cos θ_{2} + S_{1} \sin α_{1} \sin θ_{2}) - ϕ_{D} (x, y) S_{1} \sin α_{1} - ϕ_{D C} (x, y) L_{c} \cos θ_{2}}

where, in addition to the parameters associated to the model in Eq. (2), another one parameter was introduced, that is,

θ_{2}

the angle between the linear

O_{C} O

and

O_{C} O_{1}

.

Fig. 3 Another improved geometric model in [3].

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when $θ_{2} = 0$ , Eq. (3) can be the same as Eq. (2) indicating that the PTH mapping proposed in [2] is a specific case of the PTH mapping proposed in [3]. Although geometric model proposed in [3] improves the geometric model proposed in [2], it requires the lines $O_{C} O$ and $O_{p} O$ are coplanar, and the plane is perpendicular to reference plane.

2.4 Problem statement

When the plane formed by the axes of camera and projector is perpendicular to reference plane, the fringe patterns captured by camera are shown as Fig. 4(a). However, it is difficult to satisfy this condition in practice. When this condition is not met, the fringe patterns taken by the camera can be the form in Fig. 4(b), where the fringe patterns are not orthogonal to $u_{c}$ -axis or parallel to $v_{c}$ -axis. If the PTH mapping in [3] is used for such fringe patterns, error will occur for 3-D shape measurement.

Fig. 4 (a). Captured ideal fringes (b). Captured actual fringes.

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3. Improved phase-to-height (PTH) mapping

Now let us introduce a new PTH mapping which does not require any limitation of these three conditions listed in the section of introduction. As shown in Fig. 5, plane $O_{P} O_{C} O$ is not perpendicular to the reference plane, and hence $θ_{0}$ is not $0$ .

Fig. 5 The proposed geometric model for FPP.

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Considering triangles $O O_{3} O_{C}$ , $O_{2} O_{3} O_{C}$ and $O_{p} F O_{C}$ , we have $\bar{O_{3} O_{C}} = L_{C} \cos θ_{0}$ , $\bar{O_{2} O_{C}} = L_{C} \cos θ_{0} \cos θ_{2}$ , $\bar{O_{2} O_{3}} = L_{C} \cos θ_{0} \sin θ_{2}$ and $\bar{F O_{C}} = S_{1} \sin α_{1}$ .

Hence, $\bar{K O_{p}}$ can be obtained as

\bar{K O_{p}} = \bar{O_{1} O_{C T}} + \bar{E O_{C T}} = L_{C} \cos θ_{0} \cos θ_{2} + S_{1} \sin a_{1}

From triangles

A O_{C T} O_{1}

and

K O O_{p}

,

\tan δ

and

\tan η

can be respectively expressed as

\tan δ = \frac{\bar{O_{1} O_{C T}}}{\bar{A O} + \bar{O O_{1}}} and \tan η = \frac{\bar{O_{P} K}}{\bar{K O} - \bar{C O}}

From triangles

A B D

and

B C D

,

\bar{A B}

and

\bar{B C}

can be respectively described as

\bar{A B} = \frac{\bar{B D}}{\tan δ} and \bar{B C} = \frac{\bar{B D}}{\tan η}

Hence,

\bar{C A} = \bar{A B} + \bar{B C}

can be given as

\bar{C A} = \bar{B D} (\frac{1}{\tan δ} + \frac{1}{\tan η})

Then,

h (x, y) = \bar{B D}

can be presented as

h (x, y) = \frac{\bar{C A} \tan δ \tan η}{\tan δ + \tan η}

Substituting Eq. (5) into Eq. (8) yields the following:

h (x, y) = \frac{\bar{C A} L_{c} \cos θ_{0} \cos θ_{2} (L_{c} \cos θ_{0} \cos θ_{2} + S_{1} \sin α_{1})}{L_{c} \cos θ_{0} (r \cos θ_{2} + S_{1} \sin α_{1} \sin θ_{2}) + \bar{A O} S_{1} \sin α_{1} + \bar{C A} L_{c} \cos θ_{0} \cos θ_{2}}

According to [2],

\bar{C A}

and

\bar{A O}

can be presented as

\bar{C A} = \frac{- Δ ϕ_{D C} (x, y)}{2 π f_{0}} and \bar{A O} = \frac{- ϕ_{D} (x, y)}{2 π f_{0}}

Substituting Eq. (10) into Eq. (9), we have

h (x, y) = \frac{- Δ ϕ_{D C} (x, y) L_{c} \cos θ_{0} \cos θ_{2} (L_{c} \cos θ_{0} \cos θ_{2} + S_{1} \sin α_{1})}{2 π f_{0} L_{c} \cos θ_{0} (r \cos θ_{2} + S_{1} \sin α_{1} \sin θ_{2}) - ϕ_{D} (x, y) S_{1} \sin α_{1} - Δ ϕ_{D C} (x, y) L_{c} \cos θ_{0} \cos θ_{2}}

where, in addition to parameters associated to the model in [3], a new parameter

θ_{0}

is introduced, which is the angle between the lines

O O_{C}

and

O_{3} O_{C}

. When

θ_{0} = 0

, Eq. (11) is the same as Eq. (3). Hence, the geometric model proposed in [3] is a specific case of our geometric model.

4. System calibration

With the introduction of $θ_{0}$ in the model proposed in Fig. 5, the system must be calibrated in order to determine all the seven parameters associated, including $L_{C}$ , $r$ , $f_{0}$ , $θ_{0}$ , $S_{1}$ , $α_{1}$ and $θ_{2}$ . Before the seven parameters are estimated, camera and projector should be calibrated. Hence, this section introduces camera calibration, projector calibration and calculation of systematic parameters.

4.1 Calibration of camera and projector

The camera and projector can be calibrated using the system in Fig. 6 with the aid of calibration plane. In order to describe the mapping relationship among 2-D points on the DMD of projector, 3-D points on calibration board, and 2-D points on the CCD of camera, a number of coordinate system are required, including world coordinate system (WCS), camera coordinate system (CCS), camera image coordinate system (CICS), projector coordinate system (PCS) and projector image coordinate system (PICS). Let $P_{1} (u_{p}, v_{p})$ denote the point on the projector DMD, $P (x, y, z)$ the corresponding point on the calibration board, and $P_{2} (u_{c}, v_{c})$ the corresponding point on camera CCD. The relationship between $P_{2} (u_{c}, v_{c})$ and $P (x, y, z)$ can be described as:

s_{c} \underset{{\tilde{m}}_{c}}{\underset{︸}{[\begin{matrix} u_{c} \\ v_{c} \\ 1 \end{matrix}]}} = \underset{A_{c}}{\underset{︸}{[\begin{matrix} α_{c} & γ_{c} & u_{c 0} \\ 0 & β_{c} & v_{c 0} \\ 0 & 0 & 1 \end{matrix}]}} \underset{[\begin{matrix} R_{c} & T_{c} \end{matrix}]}{\underset{︸}{[\begin{matrix} r_{11} & r_{12} & \begin{matrix} r_{13} & t_{1} \end{matrix} \\ r_{21} & r_{22} & \begin{matrix} r_{23} & t_{2} \end{matrix} \\ r_{31} & r_{32} & \begin{matrix} r_{33} & t_{3} \end{matrix} \end{matrix}]}} \underset{\tilde{M}}{\underset{︸}{[\begin{matrix} x \\ y \\ z \\ 1 \end{matrix}]}}

where

A_{c}

is the matrix containing all the intrinsic parameters of camera, in which

(u_{c 0}, v_{c 0})

is the coordinates of principle point of the camera;

α_{c}

and

β_{c}

are the focal length along

u_{c}

-axis and

v_{c}

-axis of the CCD;

γ_{c}

is the skewness of

u_{c}

-axis and

v_{c}

-axis;

R_{c}

and

T_{c}

are the rotation matrix and translation vector containing extrinsic;

s_{c} (μ m / p i x e l)

is the scale factor which denotes the ratio of the physical dimension of an object (in microns) to its size (in pixel). Since camera calibration has been studied extensively with many effective methods developed. In this paper we employ the technique proposed in [13] to calibrate the camera.

Fig. 6 Schematic illustration of systematic calibration.

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Since the projector can be considered as an inverse camera, the pinhole model can be used to describe the projector. As shown in Fig. 6, the relationship between $P_{1} (u_{p}, v_{p})$ and $P (x, y, z)$ can be described as

s_{p} \underset{{\tilde{m}}_{p}}{\underset{︸}{[\begin{matrix} u_{p} \\ v_{p} \\ 1 \end{matrix}]}} = \underset{A_{p}}{\underset{︸}{[\begin{matrix} α_{p} & γ_{p} & u_{p 0} \\ 0 & β_{p} & v_{p 0} \\ 0 & 0 & 1 \end{matrix}]}} \underset{[\begin{matrix} R_{p} & T_{p} \end{matrix}]}{\underset{︸}{[\begin{matrix} g_{11} & g_{12} & \begin{matrix} g_{13} & e_{1} \end{matrix} \\ g_{21} & g_{22} & \begin{matrix} g_{23} & e_{2} \end{matrix} \\ g_{31} & g_{32} & \begin{matrix} g_{33} & e_{3} \end{matrix} \end{matrix}]}} \underset{\tilde{M}}{\underset{︸}{[\begin{matrix} x \\ y \\ z \\ 1 \end{matrix}]}}

where

A_{p}

is the matrix containing all the intrinsic parameters of projector, in which

(u_{p 0}, v_{p 0})

is the coordinates of principle point of the projector;

α_{p}

and

β_{p}

are the focal length along

u_{p}

-axis and

v_{p}

-axis of the DMD;

γ_{p}

is the skewness of

u_{p}

-axis and

v_{p}

-axis;

R_{p}

and

T_{p}

are the rotation matrix and translation vector containing extrinsic;

s_{p (μ m / p i x e l)}

is the scale factor which denotes the ratio of the physical dimension of an object (in microns) to its size (in pixel).

As the projector cannot take a picture by itself, its calibration must be with the help of camera, and the mapping relationship between the pixels on the DMD and CCD should be determined. In order to establish the mapping between the pixels on the DMD and CCD, two sets of phase shifted sinusoidal fringe patterns are generated by the projector, one set with vertical patterns and the other with horizontal patterns. Assuming that the resolution of the DMD is $N_{p} \times M_{p}$ pixels, and each individual fringe spans $T_{V}$ and $T_{H}$ pixels on the vertical patterns and horizontal patterns, respectively, these patterns can be expressed by the following:

I_{V_{l}} (u_{p}, v_{p}) = I_{1} + I_{2} \cos (2 π \frac{u_{p}}{T_{V}} + \frac{l - 4}{3} π), l = 1, 2, ..., 6 and u_{p} = 0, 1, 2, ..., M_{p}

and

I_{H_{l}} (u_{p}, v_{p}) = I_{1} + I_{2} \cos (2 π \frac{v_{p}}{T_{H}} + \frac{l - 4}{3} π), l = 1, 2, ..., 6 and v_{p} = 0, 1, 2, ..., N_{p}

where

I_{1}

is average intensity,

I_{2}

is the intensity modulation. These patterns are projected onto the calibration board, and are captured by a camera. Assuming that the resolution of the camera CCD is

N_{c} \times M_{c}

pixels, the captured vertical fringe patterns can be expressed as follows:

I_{V_{n}} (u_{c}, v_{c}) = I_{1} + I_{2} \cos (ϕ_{V} (u_{c}, v_{c}) + \frac{n - 4}{3} π), n = 1, 2, ..., 6 and u_{c} = 0, 1, 2, ..., M_{c}

Similarly, the captured horizontal fringe patterns are:

I_{H_{n}} (u_{c}, v_{c}) = I_{1} + I_{2} \cos (ϕ_{H} (u_{c}, v_{c}) + \frac{n - 4}{3} π), n = 1, 2, ..., 6 and v_{c} = 0, 1, 2, ..., N_{c}

where

ϕ_{V} (u_{c}, v_{c})

and

ϕ_{H} (u_{c}, v_{c})

are the phases, which can be retrieved by the following:

ϕ_{V} (u_{c}, v_{c}) = arc \tan [\frac{\sum_{n = 1}^{6} I_{V_{n}} (u_{c}, v_{c}) \sin (2 π n / 6)}{\sum_{n = 1}^{6} I_{V_{n}} (u_{c}, v_{c}) \cos (2 π n / 6)}] and ϕ_{H} (u_{c}, v_{c}) = arc \tan [\frac{\sum_{n = 1}^{6} I_{H_{n}} (u_{c}, v_{c}) \sin (2 π n / 6)}{\sum_{n = 1}^{6} I_{H_{n}} (u_{c}, v_{c}) \cos (2 π n / 6)}]

Note that, due to the arctangent operation in Eq. (18), the values of

ϕ_{V} (u_{c}, v_{c})

and

ϕ_{H} (u_{c}, v_{c})

are wrapped into the range between

0

and

2 π

. In order to have a unique mapping between the DMD and the CCD pixels, both of them are required to be unwrapped. To this end two sets of the gray code fringe images are projected, one vertical and the other horizontal respectively. These Gray code patterns have the same period as the sinusoidal fringe pattern, but the number of patterns is determined by

n = \log (2^{M})

, where

M

is the number of fringes in the corresponding sinusoidal pattern. The sinusoidal patterns have 64 fringes, and hence 6 gray code patterns should be used. For a point

(u_{c}, v_{c})

on

ϕ_{V} (u_{c}, v_{c})

the six vertical gray code patterns provide a code by means of the light intensity on the corresponding point, which allows unique determination of the

m_{V} (u_{c}, v_{c})

, which can be used to unwrap

ϕ_{V} (u_{c}, v_{c})

as follows:

Φ_{V} (u_{c}, v_{c}) = ϕ_{V} (u_{c}, v_{c}) + 2 π m_{V} (u_{c}, v_{c})

In the same way, the six horizontal gray code patterns can be used to unwrap

ϕ_{H} (u_{c}, v_{c})

, yielding the following:

Φ_{H} (u_{c}, v_{c}) = ϕ_{H} (u_{c}, v_{c}) + 2 π m_{H} (u_{c}, v_{c})

where

Φ_{V} (u_{c}, v_{c})

and

Φ_{H} (u_{c}, v_{c})

are monotonic with respect to

v_{c}

and

u_{c}

, respectively. Compared them with the phases in Eq. (14) and Eq. (15), a unique point-to-point mapping between the CCD pixels and the DMD pixels can be determined as follows:

u_{p} = \frac{Φ_{V} (u_{c}, v_{c})}{2 π} T_{V} and v_{p} = \frac{Φ_{H} (u_{c}, v_{c})}{2 π} T_{H}

Hence the mapping relationship between the CCD and DMD can be established. The same procedure will be applied to all the circle of the DMD image, resulting in a set of corresponding points on the board and the DMD, which can be used to calibrate the projector based on method in [13]. Note that in order to have enough number of independent equations the calibration board needs to be rotated twice. Although the camera and projector have been calibrated simultaneously, the relationship between systematic parameters and the extrinsic and intrinsic parameters of both camera and projector is not clear. Hence, this relationship must be proposed. The whole derived procedure of this relationship is given as the section of calculation of systematic parameters.

4.2 Calculation of systematic parameters

With the camera and projector calibrated, we are now able to determine the seven parameters associated model in Fig. 5, including $L_{c}$ , $S_{1}$ , $f_{0}$ , $θ_{0}$ , $α_{1}$ , $r$ and $θ_{2}$ . To achieve this, we set the last position of calibration plane as the position of reference plane. The relationship between coordinates $θ_{0}$ of one point in the CCS and the coordinates $(x_{w c}, y_{w c}, z_{w c})$ of the same point in the WCS can be described as

[\begin{matrix} x_{c} \\ y_{c} \\ z_{c} \end{matrix}] = R_{c} [\begin{matrix} x_{w c} \\ y_{w c} \\ z_{w c} \end{matrix}] + T_{c}

Similarly, the relationship between coordinates

(x_{p}, y_{p}, z_{p})

of one point in the PCS and the coordinates

(x_{w p}, y_{w p}, z_{w p})

of the same point in WCS can be described as

[\begin{matrix} x_{p} \\ y_{p} \\ z_{p} \end{matrix}] = R_{p} [\begin{matrix} x_{w p} \\ y_{w p} \\ z_{w p} \end{matrix}] + T_{p}

where

R_{c}

is

3 \times 3

rotation matrix of camera, and

T_{c}

is

3 \times 1

the translation vector of camera.

R_{p}

is

3 \times 3

rotation matrix of projector, and

T_{p}

is

3 \times 1

translation vector of projector. When

{[\begin{matrix} x_{c} & y_{c} & z_{c} \end{matrix}]}^{T} = {[\begin{matrix} x_{p} & y_{p} & z_{p} \end{matrix}]}^{T} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T}

, the optical centers of the camera and the projector in world coordinate system can be calculated as:

[\begin{matrix} x_{w c} \\ y_{w c} \\ z_{w c} \end{matrix}] = - R_{c}^{- 1} T_{c} and [\begin{matrix} x_{w p} \\ y_{w p} \\ z_{w p} \end{matrix}] = - R_{p}^{- 1} T_{p}

Where

R_{c}^{- 1}

and

R_{p}^{- 1}

are the inverse of

R_{c}

and

R_{p}

, respectively. From Eq. (24), parameters

S_{1}

,

a_{1}

and

r

can be expressed as

S_{1} = \sqrt{{(x_{w p} - x_{w c})}^{2} + {(y_{w p} - y_{w c})}^{2} + {(z_{w p} - z_{w c})}^{2}}

and

α_{1} = arc \sin (\frac{| z_{w p} - z_{w c} |}{S_{1}})

and

r = | x_{w p} - x_{w c} |

In order to determine

θ_{0}

,

L_{c}

and

θ_{2}

, the coordinates

(x_{O}, y_{O}, z_{O})

of point

O

in WCS should be determined. Since point

O

corresponds to the principle point

(u_{c 0}, v_{c 0})

in the camera image plane,

x_{O}

and

y_{O}

can be expressed by

u_{c 0}

and

v_{c 0}

by Eq. (12), which ignores the lens distortion of camera. Hence, by assuming

z = 0

, we have following:

s_{c} {\tilde{m}}_{c} = H \tilde{M}

where

{\tilde{m}}_{c} = {[\begin{matrix} u_{c 0} & v_{c 0} & 1 \end{matrix}]}^{T}

,

\tilde{M} = {[\begin{matrix} x_{o} & y_{o} & 1 \end{matrix}]}^{T}

and

H = [\begin{matrix} α_{c} & r_{c} & u_{c 0} \\ 0 & β_{c} & v_{c 0} \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} r_{11} & r_{12} & t_{1} \\ r_{21} & r_{22} & t_{2} \\ r_{31} & r_{32} & t_{3} \end{matrix}]

.

Therefore, we have

\tilde{M} = s_{c} H^{- 1} {\tilde{m}}_{c}

where

H^{- 1}

is the inverse of

H

.

When the coordinates of point $O$ are obtained, parameters $L_{c}$ , $θ_{2}$ and $θ_{0}$ can be expressed as:

L_{c} = \sqrt{{(x_{O} - x_{w c})}^{2} + {(y_{O} - y_{w c})}^{2} + z_{w c}^{2}}

and

θ_{2} = arc \tan (\frac{| x_{O} - x_{w c} |}{| z_{w c} |})

and

θ_{0} = arc \tan (\frac{| y_{O} - y_{w c} |}{\sqrt{{(x_{O} - x_{w c})}^{2} + z_{w c}^{2}}})

Finally, the frequency

f_{0}

of projected fringe patterns on reference plane can be calculated using the method in [12].

5. Experiments

The experiments are conducted to verify the performance of proposed geometrical model and the calibration approach presented in Section 4. The experimental setup in our lab is shown in Fig. 7, consisting of a computer, a camera, a projector, and calibration board. The resolution of projector is 768 pixels by 1024 pixels, and that of camera is 1024 pixels by 1280 pixels. The calibration board is a black metal plane with 99 engraved circles as shown in Fig. 7.

Fig. 7 System calibration equipment in our lab.

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The procedure of calibration experiment is as follows. Firstly, a white paper is stuck on the surface of calibration board, and then six vertical patterns and six horizontal patterns are projected onto the covered calibration board. These projected patterns are captured by a camera. Then six vertical gray code patterns and six horizontal gray code patterns are projected onto the covered calibration board, which are then taken by the camera. After these 24 images are projected, we removed the white paper and capture an image of calibration board. Since camera calibration in [13] needs at least three different views of calibration board, the calibration board has been viewed from three different positions, and the gray code phase shifting is used to every view of calibration board. The captured CCD images of calibration board are shown in Fig. 8.

Fig. 8 (a). image of calibration board on first position (b). that on second position (c). that on the third position.

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When both camera and projector are calibrated, the last position of calibration board is chosen as the reference plane. Then their corresponding extrinsic parameters are used to estimate the seven parameters using the method described in Section 4.2. All the obtained parameters on the new PTH mapping are shown in Table 1.

Table 1. Parameters of our Proposed PTH Mapping

View Table

Let us now look at the accuracy of the parameters obtained in Table 1. Because their true values are unknown, we employ an indirect method. We measure a cuboid with a flat top surface with its known a priori (i.e., 14.23mm) and hence we can compare the measurement result against the true value. Figure 9 shows the reconstruction results using the ideal geometrical model in [1], the model proposed in [3] and the model in Fig. 5 incorporating the parameter values in Table 1. It is seen that the reconstructed results using the proposed method are much smoother than the models in [1] and [3], and hence the proposed method is the most accurate. Also, the standard deviation of the measurement associated with the proposed method is 0.1238mm (or 0.87%), implying that the parameters obtained in Table 1. is also very accurate. In contrast, standard deviation of results in Fig. 9(a) and Fig. 9(b) are 0.49 mm, 0.29 mm respectively, which are much higher. Therefore, we can say that a noticeable improvement in the measurement accuracy can be achieved by the proposed model.

Fig. 9 (a) Reconstruction based on ideal geometric model (b) Reconstruction based on model in [3] (c) Reconstruction based on proposed model.

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6. Conclusion

In this paper, we proposed a new geometric model for FPP where the plane formed by the axes of camera and projector is not necessarily perpendicular to the reference plane, thus making it much easier for implementing a FPP. Based on the new model, we presented a new PTH mapping relationship in order to improve the measurement accuracy. The new model involves seven parameters, for which we also proposed a new system calibration method to determine the values. Experiments are conducted to verify the performance of the proposed technique, showing a noticeable improvement in the accuracy of 3D shape measurement.

References and links

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4. B. Li, N. Karpinsky, and S. Zhang, “Novel calibration method for structured-light system with an out-of-focus projector,” Appl. Opt. 53(16), 3415–3426 (2014). [CrossRef] [PubMed]

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10. E. Zappa and G. Busca, “Fourier-transform profilometry calibration based on an exhaustive geometric model of the system,” Opt. Lasers 47(7-8), 754–767 (2009). [CrossRef]

11. E. Zappa, G. Busca, and P. Sala, “Innovative calibration technique for fringe projection based 3D scanner,” Opt. Lasers 49(3), 331–340 (2011). [CrossRef]

12. L. M. Song, C. M. Chen, Z. Chen, J. T. Xi, and Y. G. Yu, “Essential parameter calibration for the 3D scanner with only single camera and projector,” Optoelectron. Lett. 9(2), 143–147 (2013). [CrossRef]

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14. S. Zhang, “Novel method for structured light system calibration,” Opt. Eng. 45(8), 083601 (2006). [CrossRef]

Parameter	Value	Parameter	Value
$S_{1}$ $(m m)$	165.3877	$θ_{2}$ $(. \deg)$	2.28
$L_{c}$ $(m m)$	1166.7035	$f_{0}$ $(1 / m m)$	0.0506
$α_{1}$ $(. \deg)$	22.02	$r$ $(m m)$	153.3260
$θ_{0}$ $(. \deg)$	2.62

Improved geometrical model of fringe projection profilometry

Abstract

1. Introduction

2. Existing work and problem statement

2.1 Geometric model proposed in [1]

2.2 Geometric model proposed in [2]

2.3 Geometric model proposed in [3]

2.4 Problem statement

3. Improved phase-to-height (PTH) mapping

4. System calibration

4.1 Calibration of camera and projector

4.2 Calculation of systematic parameters

5. Experiments

6. Conclusion

References and links

Cited By

Figures (9)

Tables (1)

Equations (32)

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