Subaperture Layout Preselection Using the Hessian Field for Freeform Surface Metrology

Interferometric measurement of large aperture freeform surfaces often requires sub-aperture stitching to achieve full aperture measurement. We propose layout preselection based on the Hessian response of the design surface to two planar translations and rotation about the optical axis. After removing the best-fit sphere, first-order Hessian responses form a scaled matrix that quantifies the weakest constraint on these variables. The resulting sensitivity field guides overlap placement toward regions with stronger constraints while weakly sensitive regions are covered within individual subapertures. Responses from adjacent overlaps are then assembled to rank feasible layouts by their weakest collective registration constraint. Global orientation selection and subaperture center adjustment refine the retained layouts. The method was evaluated on 180 design surfaces from the Zernike, XY polynomial, and Q-type families. After registration with three degrees of freedom, adjusted layouts yielded median translation and rotation errors of 5.42 μm and 0.228 mrad, respectively, with a median stitching error root mean square (RMS) of 23.95 nm. All three interquartile ranges were narrower than for regular layouts. Using the recovered planar estimates to initialize registration with six degrees of freedom (DoF) reduced median subsequent iteration counts. These results support Hessian-based overlap planning for improved planar registration and initialization of full aperture reconstruction.