## 3 Procedures of InSAR Image Co-registration
### Determination of Window Size for Tie Point Matching  
The window size is another important factor to be considered for tie point matching, which can greatly affect the result of tie point matching. Too large a window size would result in not only a dramatic calculation but also the appearance of pseudo-tie point. However, too small a window size would make the co-registration unreliable. A multi-window sized 21\(\times\)21 pixels based on maximum cross-correlation is developed by [PERSON] and [PERSON] [74]. The window size, 33\(\times\)33 pixels, 9\(\times\)9 pixels and 43\(\times\)43 pixels have been adopted by [PERSON] _et al_. [55], [PERSON] [53] and [PERSON] [63]. However, no theory has been developed to guide the selection of window size.  
Recently a method for automated determination of an optimum window size for tie point matching was presented [45], which aims to find a mathematical method to determinate an optimum window size. The method is based on the decomposition of auto-correlation of the SAR image with wavelet transform to select an optimum window size for tie point matching. In this method, the auto-correlation of the SAR images, which reflects the similarity between image pixels, is used as a basis for analysis. To measure such a similarity, auto-correlation coefficient is the most widely used measure. It is defined as:  
\[R\left(d\right)=\frac{Cov\ \left(d\right)}{V} \tag{10}\]  
where \(R(d)\) is the auto-correlation coefficient of all pixels with a distance of \(d\) part; \(Cov(d)\) is the covariance of all pixels with a distance of \(d\) part, and \(V\) is the variance of all pixels. Respectively, they are as follows:  
\[V=\frac{\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}\left(\left|Z_{i,j}\right|-K \right)^{2}}{m\ n-1} \tag{11}\]  
\[Cov\left(d\right)=\frac{\sum\limits_{i=1}^{m}\ \ \sum\limits_{j=1}^{n}\left[\left|Z_{i,j} \right|-K\right)\ \left(\left|Q_{i,j+d}\right|-K\right)+\sum\limits_{i=1}^{n}\ \ \sum\limits_{j=1}^{n}\left[\left|Z_{i,j}\right|-K\right)\ \left(\left|Q_{i+d,j}\right|-K\right)\right]}{2\ m\ n-1} \tag{12}\]  
and  
\[K=\frac{\sum\limits_{i=1}^{M}\ \ \sum\limits_{j=1}^{N}\left|Z_{i,j}\right|}{M\ N} \tag{13}\]  
where \(m=0\), 1, 2, \(\Lambda\), \(M\)-1, \(n=1\), 2, \(\Lambda\), \(N\)-1, and the size of the image is \(M\times N\); \(Z_{i,j}\) is one pixel with a coordinate \((i,j)\) (expressed in pixel) in the image, its grey value is \(\left|Z_{i,j}\right|\); \(Q_{i+d}\) is another pixel with a distance \(d\) apart from the pixel \(Z_{i,j}\) in column direction, \(\left|Q_{i,j+d}\right|\) is its gray value; \(Q_{i+dj}\) is another pixel with a distance \(d\) apart from the pixel \(Z_{i,j}\) in row direction, \(\left|Q_{i+dj}\right|\) is its gray value; \(K\) is the average grey value of all pixels. A pair of SAR images and its curve of auto-correlation coefficient are shown in Figures 9 and 10, respectively.  
Figure 10: Auto-correlation coefficient of the Tai Lam image.  
Figure 9: A pair of SAR images (Tai Lam in Hong Kong). (a) the master image; (b) the slave image.  
Then, the auto-correlation function is decomposed into waves of various frequencies by wavelet transform as shown in Figure 11. By a combined analysis of the variations of wave amplitudes with frequency, the optimum window size for tie point matching can be determined. When wavelet db1 is used to decompose the auto-correlation coefficient, the jumping point (on the lowest-frequency component), at which the percentage amplitude is smaller than 15% of the amplitude at \(d=0\) and the difference of amplitude in percentage is smaller than 10%, can be selected as the optimum window size. In this way, an optimum window size could be determined for any pair of SAR images.