title: Wave-front interpretation with Zernike polynomials.
abstract: Several low-order Zernike modes are photographed for visualization. These polynomials are extended to include both circular and annular pupils through a Gram-Schmidt orthogonalization procedure. Contrary to the traditional understanding, the classical least-squares method of determining the Zernike coefficients from a sampled wave front with measurement noise has been found numerically stable. Furthermore, numerical analysis indicates that the so-called Gram-Schmidt method and the least-squares method give practically identical results. An alternate method using the orthogonal property of the polynomials to determinem their coefficients is also discussed.
authors.forename: J Y
authors.initials: JY
authors.lastname: Wang
authors.forename: D E
authors.initials: DE
authors.lastname: Silva