# Improved Least Squares Phase Unwrapping Method Based on Chebyshev Filter
## 2 Methods
### The Improved Least Squares Phase Unwrapping Method  
Assuming there are no phase discontinuities in the wrapped phase, the unwrapped phase is accurate. However, when noise and steep phase gradient are present, there will be phase discontinuities in the wrapped phase [27]. Therefore, the phase gradient exists:  
\[\Delta\phi_{i,j}\
eq\Delta\varphi_{i,j}, \tag{6}\]  
where \(\Delta\phi_{i,j}\) represents the gradient of the unwrapped phase, and \(\Delta\varphi_{i,j}\) represents the gradient of the wrapped phase; thus, there must inevitably exist  
\[\delta_{i,j}=W\big{(}\phi_{i,j}-\varphi_{i,j}\big{)}, \tag{7}\]  
where \(\delta_{i,j}\) represents the errors of the unwrapped phase. Using the LS phase unwrapping method in this case does not generate good results. In other words, when there are multiple phase discontinuities, the LS phase unwrapping method will produce larger errors [26; 27]. Considering that the LS phase unwrapping method has a fast and continuous phase unwrapping effect under low noise and flat phase and that it is equivalent to multiple low flat phase maps using the LS phase unwrapping method for phase unwrapping under high noise and steep phase gradient, it can be expressed that  
\[\varphi_{i,j}=W\big{(}\phi_{i,j}\big{)}=W\Bigg{(}\sum_{k=1}^{N}\phi_{i,j}^{k} \Bigg{)} \tag{8}\]  
However, how to equalize the low phase gradient and how to determine the number of low phase gradients is a difficult problem. To solve this problem, we think of the phase gradient as a form of a discrete signal, employing a Chebyshev low-pass filter to filter the phase gradient, and use an iterative method to determine the number of low phase gradients. The Chebyshev low-pass filter has a steep transition band, which can effectively change the steep phase gradient into a flat phase gradient for the stopband phase gradient. In addition, to preserve the phase characteristics, the passband phase gradient is not filtered. When the type I second-order Chebyshev filter is improved and applied to phase gradient filtering, the transfer function can be represented as follows [28]:  
\[H\big{(}\Delta_{i,j}\big{)}=\Bigg{\{}\sqrt{\frac{1}{1+2^{c}T_{2}^{2}\Big{(} \frac{\Delta_{ij}}{\Delta_{c}}\Big{)}}}\quad\big{(}\big{|}\Delta_{i,j}\big{|} >\Delta_{c}\big{)} \tag{9}\]  
where \(\Delta_{i,j}\) represents the unwrapped phase gradient of \(x\) or \(y\) direction, which is \(\Delta_{i,j}^{x}\) or \(\Delta_{i,j}^{y}\) and \(\varepsilon\) represents the fluctuation coefficient of the ripple magnitude within the passband. We set the maximum allowable attenuation in the passband to be 1 dB; it is about 0.707. \(\Delta_{c}\) represents the cut-off phase of the passband, which is set to the standard deviation of the phase gradient, and \(T_{2}(\ldots)\) represents the second-order Chebyshev polynomial. These formulas can be expressed as follows:  
\[ \begin{array}{c}\frac{1}{1+\varepsilon^{2}}\approx 0.707\\ \Delta_{c}=\sqrt{\frac{\sum_{i=1}^{M}\left(\Delta_{ij}-\mu\right)^{2}}{M}}\\ \mu=\frac{\sum_{i=1}^{M}\Delta_{ij}}{M}\\ T_{2}(\ldots)=\cosh(2 arccosh(\ldots)),\end{array}  \tag{10}\]  
where \(1/\left(1+\varepsilon^{2}\right)\) represents the maximum allowable attenuation in the passband, _cosh_ and _arccosh_ represent hyperbolic cosine and inverse hyperbolic cosine functions, and in the direction of \(j\) is the same equation.