# Special techniques dealing with a large number of subapertures

For large and steep convex surfaces, a large number of subapertures are required to cover the full aperture with high lateral resolution. Consequently, a large number of measurement points are to be treated simultaneously in the stitching algorithm. Special techniques are utilized to solve the possible problems of computational efficiency and insufficient memory.

## Sparse technique

In the simultaneous stitching model described either in Eq. (30) or in Eq. (37), the matrix A is typically a sparse matrix. It has *No* rows and L columns, where *L = Is. *Each block corresponding to one subaperture has l columns, and l depends on the compensators used for stitching optimization. For example, Eq. (30) defines compensators including piston, tip-tilt, defocus, lateral shift, clocking, and Zernike polynomial coefficients. Therefore, matrix A has a lot of zeros, and only those elements corresponding to the overlapping block of two neighboring subapertures are nonzeros. The number of nonzero elements is * ^{lk}N_{o}* x 2l for each block.

The insufficient-memory problem probably arises in the storage and treatment of matrix A. A sparse technique is suggested to save space and improve the computational efficiency. The sparse storage basically records the indices of nonzero elements. Most of the arithmetic operations involved, such as SVD and QR decomposition, can be applied to sparse matrices with well-developed algorithms and functions available in commercial software, e.g., MATLAB™.