## 3 Procedures of InSAR Image Co-registration
### Selection of Transformation Models  
Another issue for image co-registration are the mathematical models used for transformations. The transformation models can also affect the reliability of co-registration. In InSAR, two complex images are used, which cover the same scene but are taken from slightly different positions. Geometrically, they are in different coordinate systems. Their orientations could also be quite different. Therefore, an operation is required to bring them down to an identical coordinate system so as to have an identical orientation. There are two possible solutions: either to bring both down to an absolute ground coordinate or to fit one image into the coordinate system of the other. Normally, co-registration refers to the latter. In order to bring one image to fit into the coordinate system of the other, the relationship between these two images needs to be established. In InSAR practice, some sorts of polynomials are used as approximate models for the transformation between them. The models include geometric transformation models, and radiometric transformation models for re-sampling. Figure 11: Components of the auto-correlation of the Tai Lam image decomposed by wave db1. \(d_{1}\sim d_{4}\) are the high-frequency components and \(a_{4}\) is the lowest-frequency component. In order to enhance the accuracy of the results, different transformation models could be adopted in different situations. In the case of a photograph, one frame of an image is taken at one time. For each frame, there are six orientation elements (three rotations and three translations). The elements are the same for all image pixels on the same frame. If the image coordinate system is that the origin being the center of the image and the x-axis being along the flight direction, an arbitrary Cartesian coordinate system could possibly be used in the image plane. It is as follows [75]:  
\[x=\frac{A_{1}X+A_{2}Y+A_{3}Z+A_{4}}{A_{9}X+A_{10}Y+A_{11}Z+1} \tag{14a}\] \[y=\frac{A_{5}X+A_{6}Y+A_{7}Z+A_{8}}{A_{9}X+A_{10}Y+A_{11}Z+1} \tag{14b}\]  
where, \(A_{1}\)\(\sim\)\(A_{11}\) are the coefficients; \(x\) and \(y\) are the coordinates of an image point; \(X\), \(Y\) and \(Z\) are the coordinates of its position on the ground. In the case of scanner images, image pixels are acquired scan by scan. A frame of an image may consist of thousands of scan lines. In this case, there are six orientation elements and there is a need of at least three control points for the computation of coefficients for each scan line. This is not feasible. The problem could be simplified if the relationship among the scan lines is known. However, this is normally not the situation. In this case, a single polynomial could be used to approximate the geometric model transformation for all scan lines [75]:  
\[x=a_{0}+a_{1}X+a_{2}Y+a_{3}X^{2}+a_{4}Y^{2}+a_{5}XY+\cdots \tag{15a}\] \[y=b_{0}+b_{1}X+b_{2}Y+b_{3}X^{2}+b_{4}Y^{2}+b_{5}XY+\cdots \tag{15b}\]  
where, \(a_{0}\)\(\sim\)\(a_{5}\) and \(b_{0}\)\(\sim\)\(b_{5}\) are the coefficients; \(x\) and \(y\) are the coordinates of an image point; \(X\) and \(Y\) are the coordinates of its position on the ground. Indeed, in this case, the errors caused by distortions are re-distributed rather than removed by the polynomial function. The latter case is suitable for InSAR practice. There are several functions used as the polynomial. They are known as nearest-neighbor interpolation, bilinear interpolation, and bicubic interpolation [76]. The bicubic function is adopted as the polynomial for the transformation between the master and slave images [44; 45; 68]. In InSAR image co-registration, \((x,y)\) is the pixel's coordinate in the slave image while \((X,Y)\) is the pixel's coordinate in the master image. [PERSON] _et al_. presented a method for an accurate co-registration of spacebore repeat-pass InSAR image based on Matrix Transformation, which considers the offsets errors generated by sensor trajectory excursion in repeat-pass style [77].