Orthogonal basis for the optical transfer function

Abstract

We propose systems of orthogonal functions q_n to represent optical transfer functions (OTF) characterized by including the diffraction-limited OTF as the first basis function q₀=OTF_perfect. To this end, we apply a powerful and rigorous theoretical framework based on applying the appropriate change of variables to well-known orthogonal systems. Here we depart from Legendre polynomials for the particular case of rotationally symmetric OTF and from spherical harmonics for the general case. Numerical experiments with different examples show that the number of terms necessary to obtain an accurate linear expansion of the OTF mainly depends on the image quality. In the rotationally symmetric case we obtained a reasonable accuracy with approximately 10 basis functions, but in general, for cases of poor image quality, the number of basis functions may increase and hence affect the efficiency of the method. Other potential applications, such as new image quality metrics are also discussed.

Interbasis expansions in the Zernike system

“…Finally, we underline the fact that the original Zernike polynomials have a great practical importance in phase-contrast microscopy and in the correction of wavefronts in circular pupils. Recent work [4,5,6,15] has extended this technique to pupils of essentially arbitrary shape through diffeomorphisms that conserve their basic properties. This has been applied to describe wavefronts in sectorial, annular and polygonal-shaped pupils, the latter specifically tailored to the hexagonal components of large astronomial mirrors.…”

Image quality eigenfunctions for the human eye

“…2, omitting some (6, 9, 10, 14, 15, 19 and 20) that were either duplicates or rotated versions of those already depicted. It is seen that the first basis function (#1) corresponds to the OTF of a perfect diffraction-limited system [19] (i.e. the autocorrelation of a circle), while the other functions show an ever-increasing degree of complexity.…”

“…It is worth remarking that the resulting orthogonal basis functions are general, meaning they can be used for modelling the OTF of any optical system with a circular pupil, such as human eyes. This basis consists of numerical samples, rather than analytical functions reported before [19], but both approaches are equivalent. The coefficients can be computed through standard methods, by either determining the projections on each basis function (inner product), or through a least-squares fit.…”

“…Figure 8 compares the original versus reconstructed OTF modulus (MTF) for two subjects (upper and lower panels respectively). As the number of modes necessary to approximate OTFs increases, the OTF departs from the diffraction limit [19]. The wavier the OTF, the harder the reconstruction.…”

“…Different authors proposed sets of analytical functions for OTF expansions before [16][17][18], but each suffered from important drawbacks, such as the lack of orthogonality, etc. Another work proposed an orthogonal expansion in which the first function was the OTF of the perfect system [19], causing OTFs close to the diffraction limit to be highly compressed, whereas OTFs of low-quality optical systems would require a large number of modes (basis functions).…”