Wavefront aberrations: analytical method to convert Zernike coefficients from a pupil to a scaled arbitrarily decentered one (original) (raw)
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Journal of The Optical Society of America A-optics Image Science and Vision, 2006
In eye aberrometry it is often necessary to transform the aberration coefficients in order to express them in a scaled, rotated, and/or displaced pupil. This is usually done by applying to the original coefficients vector a set of matrices accounting for each elementary transformation. We describe an equivalent algebraic approach that allows us to perform this conversion in a single step and in a straightforward way. This approach can be applied to any particular definition, normalization, and ordering of the Zernike polynomials, and can handle a wide range of pupil transformations, including, but not restricted to, anisotropic scalings. It may also be used to transform the aberration coefficients between different polynomial basis sets.
Journal of Optics A: Pure and Applied Optics, 2009
Aberrations of the eye and other image-forming systems are often analyzed by expanding the wavefront aberration function for a given pupil in Zernike polynomials. In previous articles explicit analytical formulae to transform Zernike coefficients of up to seventh order corresponding to wavefront aberrations for an original pupil into those related to a contracted transversally displaced new pupil are obtained. In the present paper, selection rules for the direct and inverse coefficients' transformation are given and missing modes associated with certain displacement directions are analyzed. Taking these rules into account, a graphical method to qualitatively identify which are the elements of the transformation matrix and their characteristic dependence on pupil parameters is presented. This method is applied to fictitious systems having only one non-zero original coefficient and, for completeness, the new coefficient values are also analytically evaluated.
Eye aberration analysis with Zernike polynomials
Proceedings of SPIE, 1998
New horizons for accurate photorefractive sight correction, afforded by novel flying spot technologies, require adequate measurements of photorefractive properties of an eye. Proposed techniques of eye refraction mapping present results of measurements for finite number of points of eye aperture, requiring to approximate these data by 3D surface. A technique of wave front approximation with Zernike polynomials is described, using optimization of the number of polynomial coefficients. Criterion of optimization is the nearest proximity of the resulted continuous surface to the values calculated for given discrete points. Methodology includes statistical evaluation of minimal root mean square deviation (RMSD) of transverse aberrations, in particular, varying consecutively the values of maximal coefficient indices of Zernike polynomials, recalculating the coefficients, and computing the value of RMSD. Optimization is finished at minimal value of RMSD. Formulas are given for computing ametropia, size of the spot of light on retina, caused by spherical aberratio n, coma, and astigmatism. Results are illustrated by experimental data, that could be of interest for other applications, where detailed evaluation of eye parameters is needed.
Eye aberration analysis with Zernike polynomials
Ophthalmic Technologies VIII, 1998
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Journal of Optics A: Pure and Applied Optics, 2007
An analytical method to convert the set of Zernike coefficients that fits the wavefront aberration for a pupil into another corresponding to a contracted and horizontally translated pupil is proposed. The underlying selection rules are provided and the resulting conversion formulae for a seventh-order expansion are given. These formulae are applied to calculate corneal aberrations referred to a given pupil centre in terms of those referred to the keratometric vertex supplied by the SN CT1000 topographer. Four typical cases are considered: a sphere and three eyes-normal, keratoconic and post-LASIK. When the pupil centre is fixed and the pupil diameter decreases from 6 mm to the photopic natural one, leaving aside piston, tilt and defocus, the difference between the root mean square wavefront error computed with the formulae and the topographer is less than 0.04 μm. When the pupil diameter is kept equal to the natural one and the pupil centre is displaced, coefficients vary according to the eye. For a 0.3 mm pupil shift, the variation of coma is at most 0.35 μm and that of spherical aberration 0.01 μm.
Representation of wavefronts in free-form transmission pupils with Complex Zernike Polynomials
Journal of Optometry, 2011
To propose and evaluat e Complex Zernike polynomials (CZPs) t o represent general wavefront s wit h non uniform int ensit y (amplit ude) in free-from t ransmission pupils. Met hods: They consist of t hree st ages: (1) t heoret ical formulat ion; (2) numerical implement at ion; and (3) t wo st udies of t he fi delit y of t he reconst ruct ion obt ained as a funct ion of t he number of Zernike modes used (36 or 91). In t he fi rst st udy, we generat ed complex wavefront s merging wave aberrat ion dat a from a group of 11 eyes, wit h a generic Gaussian model of t he St iles-Crawford effect ive pupil t ransmission. In t he second st udy we simulat ed t he wavefront passing t hrough different pupil st op shapes (annular, semicircular, ellipt ical and t riangular). Result s: The reconst ruct ions of t he wave aberrat ion (phase of t he generalized pupil funct ion) were always good, t he reconst ruct ion RMS error was of t he order of 10-4 wave lengt hs, no mat t er t he number of modes used. However, t he reconst ruct ion of t he amplit ude (effect ive t ransmission) was highly dependent of t he number of modes used. In part icular, a high number of modes is necessary t o reconst ruct sharp edges, due t o t heir high frequency cont ent. Conclusions: CZPs provide a complet e ort hogonal basis able t o represent generalized pupil funct ions (or complex wavefront s). This provides a unifi ed general framework in cont rast t o t he previous variet y of ad oc solut ions. Our result s suggest t hat complex wavefront s require a higher number of CZP , but t hey seem especially well-suit ed for inhomogeneous beams, pupil apodizat ion, et c. Resumen Obj et ivo: Proponer y evaluar los polinomios de Zernike complej os (CZP) para represent ar frent es de onda de int ensidad (amplit ud) no uniforme a t ravés de pupilas con cualquier t ipo de t ransmisión. Mét odos: Consist en en t res et apas: a) formulación t eórica; b) implement ación numérica, y c) realización de dos est udios evaluando la fi delidad de las reconst rucciones obt enidas en función del número de modos de Zernike usados (36 o 91). En el primer est udio generamos frent es de onda complej os usando aberraciones de onda reales de un grupo de 11 oj os, e incorporando (en t odos los casos) un modelo genérico gaussiano de la t ransmisión efect iva a t ravés de la pupila debida al efect o St iles-Crawford. El segundo est udio consist ió en simular el frent e de onda a t ravés de apert uras de diferent es formas (anular, semicircular, elípt ica y t riangular). Result ados: La reconst rucción de la aberración de onda (fase de la función pupila generalizada) fue sat isfact oria en t odos los casos; el error RMS fue siempre del orden de 10-4 longit udes de onda, independient ement e del número de modos usados. La reconst rucción de la amplit ud (t ransmisión), sin embargo, es muy dependient e de la complej idad del frent e de ondas y del número de modos usados. En part icular, se necesit an muchos modos de Zernike para reconst ruir los bordes abrupt os de las apert uras, debido a su elevado cont enido en alt as frecuencias espaciales. Conclusiones: Los CZP const it uyen una base complet a ort ogonal capaz de represent ar funciones pupila generalizadas (o frent es de onda complej os). Est o proporciona un marco general, en cont ra de la variedad de soluciones ad oc propuest as previament e. Los result ados muest ran que si aument a la complej idad del frent e de onda es t ambién necesario increment ar el número de modos. En est e sent ido, los CZP parecen especialment e int eresant es para frent es de onda inhomogéneos, pupilas apodizadas, et c.
Validating Theoretical Pupil Size Scaling Formula For The Estimation Of Ocular Wavefront Aberrations
Purpose: To validate the mathematical pupil size scaling formula by comparing the estimates of the Zernike coefficients with corresponding clinical measurements obtained at different pupil sizes. Methods: The iProfiler aberometer (Carl Zeiss, Germany) was used to measure the wavefront aberrations and it provides Zernike coefficients for two pupil sizes (3mm and the maximum natural pupil size). 81 eyes (40 OD, 41 OS) of 49 visually normal subjects (mean age 57±7 yrs) whose maximum pupil size was ≥4.8mm were enrolled. For those subjects with pupil size >4.8mm, Zernike coefficients were recalculated from the measured data for a pupil size of 4.8mm.1 To validate a scaling procedure, Zernike coefficients were estimated for a 4.8mm pupil size using the measured data for the 3mm pupil size. The conversion matrix [C] derived by Lundstrom and Unsbo2 was used to generate the estimated Zernike coefficients. MATLAB software version (R2010b) was used to code the procedure. The estimated coeff...
Zernike phase spatial filter for measuring the aberrations of the optical structures of the eye
Journal of Biomedical Photonics & Engineering, 2015
To measure directly the wavefront aberration coefficients, we propose to use the multi-order diffractive element fitted with the set of Zernike polynomials. Polynomials of lowest degree describe defocusing (ametropy) and astigmatism. Coefficients of highest degree correspond to the spherical aberration of oblique rays that occurs as a consequence of misalignment of the crystalline lens and foveola, as well as deflection at the periphery of the crystalline lens. Multi-order elements allow several tens of expansion coefficients to be measured simultaneously, which will enable to investigate insufficiently known high-order aberrations for the differentiated diagnostics of eye diseases.