# Configuration optimization subproblem

The configuration optimization subproblem is solved to get the optimal estimation of configurations based on the LS overlapping deviations. It is solved by minimizing the objective function defined in Eq. (37) with fixed overlapping correspondence, i.e., {hj,j},{n,_{:}j} and {^},{%} are fixed. It is inherently a nonlinear LS problem. The first-order Taylor expansion is used to linearize the problem. Therefore, in each iteration, configuration optimization is again simplified as a linear LS problem with additional ball constraints:

where “s.t.” stands for “subject to.” The ball-constrained LS problem is typically solved by constraining the variables in a ball of certain radius, i.e., the norm of the vector m is equal to or less than the magnitude of the ball radius. It can still be solved by SVD. Additional computation is involved to solve a nonlinear equa- tion.^{64} For an unconstrained problem, a is infinity. In our stitching experience for spherical or aspheric surfaces, unconstrained stitching sometimes results in obviously incorrect surface error. Therefore, the spherical bound constraint is imposed on the variables to guarantee the validity of linearization.