# Active Optics and Aberration Correction Technology for Sparse Aperture Segmented Mirrors
## 2 Sparse Aperture Segmented Mirror Aberration Evaluation and Active Optical Control
### Segmented Mirror Aberration Mode  
A Zernike polynomial defined over a single aperture is a set of continuous complete orthogonal bases, denoted as \(Z_{i}(r,\theta)\). Suppose there exists a linear irrelevant vector group \(B\) in a Euclidean space \(V\) such that for all other vectors in the space, it can be represented linearly by \(B\). So, the \(B\) is the basis of \(V\). \(Gram-Schimdt\) orthogonalization is the method of transforming a set of irrelevant vector groups in Euclidean space into an orthogonal vector group. The orthogonality of Zernike polynomials in the unit circle region of the sub-aperture is analyzed, and orthogonal normalization is performed to obtain a new set of standard orthogonal polynomials given as follows [15].  
\[U_{k}=\left\{\begin{array}{c}Z_{1},k=1\\ Z_{k}-\sum\limits_{i=1}^{k-1}\frac{(Z_{k}U_{i-1})}{(U_{i-1},U_{i-1})}U_{i}\ (i=2,3, \cdots,k)\end{array}\right. \tag{1}\]Clearly, \(U_{k}\) can be calculated by iterations. However, this iterative process is computationally difficult and time-consuming. Observing Equation (1), we can see that \(U_{k}\) can be linearly combined with \(Z_{k}\) to form the following relation.  
\[U_{k}=\sum_{i}^{k-1}M_{ki}Z_{i} \tag{2}\]  
where \(M_{ki}\) is the transformation matrix, and \(k\) is the number of polynomials determined for a particular aberration in representing the aberration function. If the set \(Z_{i}\) is known, it is possible to determine the transformation matrix \(M_{ki}\), which will make it possible to acquire the orthogonal polynomial \(U_{k}\). The inner product and orthogonality of the polynomials \(U_{i}\) and \(U_{j}\) is defined as follows.  
\[\left\langle\left.U_{k}\right|\left.U_{j}\right\rangle\right.=\frac{1}{A} \int_{\Sigma}U_{k}U_{j}dS=\delta_{kj} \tag{3}\]  
where \(A\) is the area of the integral domain, \(dS\) is the integral element on the domain, and \(\delta_{kj}\) is the Kronecker delta function. Equation (4) is the result of multiplying both sides of Equation (2) with \(Z_{j}\), and Equation (5) is the transpose of Equation (4).  
\[\left\langle\left.U_{k}\right|\left.Z_{j}\right\rangle\right.=\sum_{i}^{k-1}M_ {ki}\langle\left.Z_{i}\right|\left.Z_{j}\right\rangle \tag{4}\]  
\[\left\langle\left.Z_{j}\right|\left.U_{k}\right\rangle\right.=\sum_{i}^{k-1} \left[M_{ki}\langle\left.Z_{i}\right|\left.Z_{j}\right\rangle\right]^{T}= \sum_{i}^{k-1}\left\langle\left.Z_{j}\right|\left.Z_{i}\right]\left[M_{ki} \right]^{T} \tag{5}\]  
Equation (3) can be written as follows.  
\[\left\langle\left.U_{k}\right|\left.U_{j}\right\rangle\right.=\sum_{i}^{k-1}M_ {ki}\langle\left.Z_{i}\right|\left.U_{j}\right\rangle\right.=\delta_{ij} \tag{6}\]  
Equations (4)-(6) can be written in the following matrix form.  
\[T_{ZU}=T_{ZZ}M^{T} \tag{7}\]  
Solving Equation (7) and substituting \(Q=\left(M^{T}\right)^{-1}\) yields the following relation.  
\[T_{ZZ}=Q^{T}Q \tag{8}\]  
From the Zernike polynomial property of the unit circle, it is known that \(T_{ZZ}\) is a symmetric positive definite. Hence, the Cholesky decomposition can be solved to obtain the unique lower triangular matrix \(Q\) such that Equation (8) holds, which means \(U_{k}\) can be obtained.