# Reconstruction of Isolated Moving Objects by Motion-Induced Phase Shift Based on PSP
## 3 The Proposed Method  
### Phase Retrieval Based on Motion-Induced Phase Shift  
This paper proposes to project two fringe patterns based on the dual-frequency phase unwrapping method to reconstruct the isolated object. The fringe pattern with high frequency is projected first, and a set of fringe pattern images is captured; then, the fringe pattern with low frequency is projected and another set of fringe pattern images is captured. When the object with motion is reconstructed, multiple fringe patterns are captured for each frequency, and the captured object fringe patterns are:  
\[I_{n}^{h}(x,y)=a+b\cos(\phi^{h}(x,y)+\phi^{h}(x,y)) \tag{5}\]  
\[I_{n}^{l}(x,y)=a+b\cos(\phi^{l}(x,y)+\phi^{l}(x,y)) \tag{6}\]  
where \(I_{n}^{h}(x,y)\) and \(I_{n}^{l}(x,y)\) are the captured fringe patterns of high frequency and low frequency, respectively; \(\phi^{h}(x,y)\) and \(\phi^{l}(x,y)\) are the phase map on the reference plane of the two frequencies; \(\phi^{h}(x,y)\) and \(\phi^{l}(x,y)\) are the phase variation caused by the height of the object in different frequencies. Different from the traditional PSP in Equations (1) and (2), there is no phase shift among the projected fringe patterns. The same fringe patterns will be captured. However, for the object with motion, the fringe patterns can be described as [17]:  
\[\tilde{I}_{n}^{h}(f_{n}^{h}(x,y),g_{n}^{h}(x,y))=a+b\cos(\phi^{h}(f_{n}^{h}(x, y),g_{n}^{h}(x,y))+\phi^{h}(x,y)) \tag{7}\]  
\[\tilde{I}_{n}^{l}(f_{n}^{l}(x,y),g_{n}^{l}(x,y))=a+b\cos(\phi^{l}(f_{n}^{l}(x, y),g_{n}^{l}(x,y))+\phi^{l}(x,y)) \tag{8}\]  
where \(\tilde{I}_{n}^{h}\) and \(\tilde{I}_{n}^{l}\) are the fringe patterns with movement of the high and low frequency, respectively. \((f_{n}^{h}(x,y),g_{n}^{h}(x,y))\) and \((f_{n}^{l}(x,y),g_{n}^{l}(x,y))\) are the coordinates after movement, which can be calculated with the rotation matrix and translation vector by movement tracking [18]. In Equations (7) and (8), \(\tilde{I}_{n}^{h}\), \(\tilde{I}_{n}^{l}\), \(\tilde{\phi}^{h}(x,y)\) and \(\phi^{l}(x,y)\) can be obtained in advance, and \(a\), \(b\), \(\phi^{h}(x,y)\) and \(\phi^{l}(x,y)\) are the unknown parameters. The phase variation of high frequency \(\phi^{h}(x,y)\) and low frequency \(\phi^{l}(x,y)\) can be calculated by the least-squaremethod. Take the high frequency fringe patterns as the example to illustrate the derivation of the phase information. Rewrite Equation (7) as:  
\[\tilde{I}_{n}^{h}(f_{n}^{h}(x,y),g_{n}^{h}(x,y))=a+B(x,y)\cos\delta+C(x,y)\sin\delta \tag{9}\]  
where \(B(x,y)=b\cos\phi^{h}(x,y)\), \(C(x,y)=-b\sin\phi^{h}(x,y)\) and \(\delta=\phi^{h}(f_{n}^{h}(x,y),g_{n}^{h}(x,y))\). Define \(\tilde{I}_{n}^{hT}(f_{n}^{h}(x,y),g_{n}^{h}(x,y))\) as the real captured fringe pattern. the sum of the squared error for each pixel between the fringe pattern distribution in theory \(\tilde{I}_{n}^{h}\) and the real captured fringe pattern \(\tilde{I}_{n}^{hT}\) can be described as:  
\[S(x,y)=\sum\limits_{n=1}^{N}(\tilde{I}_{n}^{h}(f_{n}^{h}(x,y),g_{n}^{h}(x,y))- \tilde{I}_{n}^{hT}(f_{n}^{h}(x,y),g_{n}^{h}(x,y)))^{2} \tag{10}\]  
Employing the least-square criteria to minimizing Equation (10), the following relationship can be obtained:  
\[\textbf{X}(x,y)=\textbf{A}^{-1}(x,y)\textbf{B}(x,y) \tag{11}\]  
where  
\[\textbf{A}(x,y)=\left[ \begin{array}{ccc}N&\sum\limits_{n=1}^{N}\cos\delta& \sum\limits_{n=1}^{N}\sin\delta\\ \sum\limits_{n=1}^{N}\cos\delta&\sum\limits_{n=1}^{N}\cos^{2}\delta&\frac{1}{ 2}\sum\limits_{n=1}^{N}\sin 2\delta\\ \sum\limits_{n=1}^{N}\sin\delta&\frac{1}{2}\sum\limits_{n=1}^{N}\sin 2\delta& \sum\limits_{n=1}^{N}\sin^{2}\delta\\ \end{array} \right] \tag{12}\]  
\[\textbf{X}(x,y)=[ \begin{array}{ccc}a&B(x,y)&C(x,y)\\ \end{array} ]^{T} \tag{13}\]  
\[\textbf{B}(x,y)=[ \begin{array}{ccc}\sum\limits_{n=1}^{N}\tilde{I}_{n}^{h}(f _{n}^{h}(x,y),g_{n}^{h}(x,y))&\sum\limits_{n=1}^{N}\cos\delta\times\tilde{I}_ {n}^{h}(f_{n}^{h}(x,y),g_{n}^{h}(x,y))&\sum\limits_{n=1}^{N}\sin\delta\times \tilde{I}_{n}^{h}(f_{n}^{h}(x,y),g_{n}^{h}(x,y))\\ \end{array}