## 2 Developed 3-D Profilometry Using Dual Sensing
### Measuring Principle  
#### 2.2.1 Fundamentals in PSP  
Using PSP technique for 3-D shape measurement involves several procedures, including projecting sinusoidal fringe patterns onto the object surface, taking the images of the reflected distorted pattern on the object surface, and then calculating the phase map from these captured images. Height information is extracted from the phase map by phase unwrapping and phase-to-height transformation. The calibration coefficients are then employed to transform the 3-D coordinates in image coordinates into world coordinates \((x,y,z)\) of each measured point. The phase \(\phi(x,y)\) can be computed using light intensities of captured images set by phase-shifted fringe projection. The five-step phase-shifting technique is commonly used because of its accuracy and efficiency. The intensities of the five channels in each pattern can be modeled as follows:  
\[I_{i}(x,y)=I_{b}(x,y)+I_{m}(x,y)\cos[\phi(x,y)+\delta_{i}]_{(i=1-5)} \tag{1}\]  
where \(\phi(x,y)\) is the phase value used by the phase-shifting algorithm in \((x,y)\) location, \(\delta_{i}\) is the phase-shift, \(I_{i}(x,y)\) denotes the intensities of the five captured images for each pixel position, \(I_{m}(x,y)\) and \(I_{b}(x,y)\) are the intensity modulation and the average intensity for each pixel, respectively.  
The phase value of each pixel \(\phi(x,y)\) can be resolved using Equation (1) as:  
\[\phi(x,y)=-\tan^{-1}\left[\begin{matrix}\sum\limits_{i=1}^{N}I_{i}(x,y)\sin( \delta_{i})\\ \sum\limits_{i=1}^{N}I_{i}(x,y)\cos(\delta_{i})\end{matrix}\right] \tag{2}\]  
where \(\delta_{i},i=0,\;1,\;2,\;\dots\;N\), are the phase-shift values of each channel.  
For the five-step phase-shifting technique (\(N=5\)) and \(\delta_{1}=0^{\circ},\delta_{2}=90^{\circ},\delta_{3}=180^{\circ},\delta_{4}=270 ^{\circ},\delta_{5}=360^{\circ}\), the phase value \(\phi(x,y)\) can be computed using the intensity of each channel and the corresponding phase-shift \(\delta_{i}\) from the five-step phase-shifting technique shown in Equation (2).  
With the phase value \(\phi(x,y)\) being a function of arctangent, the phase value can be obtained in a range from \(-\pi\) to \(\pi\) for phase wrapping. Consequently, if the pixel-based phase difference is larger than \(2\pi\), the wrapping phase becomes discontinuous and ambiguous. To obtain a continuous phase map, a phase unwrapping process for subtracting or adding the multiple values of \(2\pi\) is needed to reconstruct the non-ambiguous phase map. The Gold-Stein phase unwrapping algorithm was employed to extract the unwrapped phase map [3]. Then, the unwrapped phase difference can be transferred to a height map using the following phase to height transformation.  
The phase-to-height transformation is performed using the triangulation technique. The height information of object surface \(h(x,y)\) of each pixel is carried by its phase difference value \(\Delta\phi(x,y)\), which is measured by the phase difference between the measured phase on the object surface and the reference plane. Since the measured height is generally much smaller than the projection height of the optical system and the projecting objective is telecentric, the object height has a linear relation with the unwrapped phase difference as shown in Equation (3). Thus, the phase value can be transformed by a linear transformation coefficient \(K\), and the height information of the object is then given as:\[h(x,y)=K\Delta\phi(x,y) \tag{3}\]