Wave-front interpretation with Zernike polynomials
TL;DR: 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.
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 determine their coefficients is also discussed.
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TL;DR: An approach to adaptive optics in microscopy wherein the rear pupil of an objective lens is segmented into subregions, and light is directed individually to each subregion to measure, by image shift, the deflection faced by each group of rays as they emerge from the objective and travel through the specimen toward the focus.
Abstract: Biological specimens are rife with optical inhomogeneities that seriously degrade imaging performance under all but the most ideal conditions. Measuring and then correcting for these inhomogeneities is the province of adaptive optics. Here we introduce an approach to adaptive optics in microscopy wherein the rear pupil of an objective lens is segmented into subregions, and light is directed individually to each subregion to measure, by image shift, the deflection faced by each group of rays as they emerge from the objective and travel through the specimen toward the focus. Applying our method to two-photon microscopy, we could recover near-diffraction-limited performance from a variety of biological and nonbiological samples exhibiting aberrations large or small and smoothly varying or abruptly changing. In particular, results from fixed mouse cortical slices illustrate our ability to improve signal and resolution to depths of 400 microm.
559 citations
TL;DR: The authors show that the new method of deriving Zernike moment invariants along with the new normalization scheme yield the best overall performance even when the data are degraded by additive noise.
Abstract: Moment invariants have been proposed as pattern sensitive features in classification and recognition applications. In this paper, the authors present a comprehensive study of the effectiveness of different moment invariants in pattern recognition applications by considering two sets of data: handwritten numerals and aircrafts. The authors also present a detailed study of Zernike and pseudo Zernike moment invariants including a new procedure for deriving the moment invariants. In addition, the authors introduce a new normalization scheme that reduces the large dynamic range of these invariants as well as implicit redundancies in these invariants. Based on a comprehensive study with both handwritten numerals and aircraft data, the authors show that the new method of deriving Zernike moment invariants along with the new normalization scheme yield the best overall performance even when the data are degraded by additive noise.
532 citations
TL;DR: A new IOL with a prolate anterior surface, designed to partially compensate for the average spherical aberration of the cornea, is intended to improve the ocular optical quality of pseudophakic patients.
Abstract: PURPOSE: The aim of this study was to design and evaluate in the laboratory a new intraocular lens (IOL) intended to provide superior ocular optical quality by reducing spherical aberration. METHODS: Corneal topography measurements were performed on 71 cataract patients using an Orbscan I. The measured corneal surface shapes were used to determine the wavefront aberration of each cornea. A model cornea was then designed to reproduce the measured average spherical aberration. This model cornea was used to design IOLs having a fixed amount of negative spherical aberration that partially compensates for the average positive spherical aberration of the cornea. Theoretical and physical eye models were used to assess the expected improvement in optical quality of an eye implanted with this lens. RESULTS: Measurements of optical quality provided evidence that if this modified prolate IOL was centered within 0.4 mm and tilted less than 7 degrees, it would exceed the optical performance of a conventional spherical IOL. This improvement occurred without an apparent loss in depth of focus. CONCLUSION: A new IOL with a prolate anterior surface, designed to partially compensate for the average spherical aberration of the cornea, is intended to improve the ocular optical quality of pseudophakic patients. [J Refract Surg 2002;18: 683-691] T
498 citations
TL;DR: The results in a keratoconic subject showed that important benefits are obtained despite decentrations in highly aberrated eyes, and provided practical rules to implement a selective correction depending on the amount of decentration.
Abstract: An ideal correcting method, such as a customized contact lens, laser refractive surgery, or adaptive optics, that corrects higher-order aberrations as well as defocus and astigmatism could improve vision The benefit achieved with this ideal method will be limited by decentration To estimate the significance of this potential limitation we studied the effect on image quality expected when an ideal correcting method translates or rotates with respect to the eye’s pupil Actual wave aberrations were obtained from ten human eyes for a 73-mm pupil with a Shack–Hartmann sensor We computed the residual aberrations that appear as a result of translation or rotation of an otherwise ideal correction The model is valid for adaptive optics, contact lenses, and phase plates, but it constitutes only a first approximation to the laser refractive surgery case where tissue removal occurs Calculations suggest that the typical decentrations will reduce only slightly the optical benefits expected from an ideal correcting method For typical decentrations the ideal correcting method offers a benefit in modulation 2–4 times higher (15–2 times in white light) than with a standard correction of defocus and astigmatism We obtained analytical expressions that show the impact of translation and rotation on individual Zernike terms These calculations also reveal which aberrations are most beneficial to correct We provided practical rules to implement a selective correction depending on the amount of decentration An experimental study was performed with an aberrated artificial eye corrected with an adaptive optics system, validating the theoretical predictions The results in a keratoconic subject, also corrected with adaptive optics, showed that important benefits are obtained despite decentrations in highly aberrated eyes
291 citations
Patent•
23 May 2001
TL;DR: In this article, the authors presented methods of obtaining an ophthalmic lens capable of reducing the aberrations of the eye comprising the steps of characterizing at least one corneal surface as a mathematical model, calculating the resulting aberration of said corNEal surface(s) by employing said mathematical model and selecting the optical power of the intraocular lens.
Abstract: The present invention discloses methods of obtaining ophthalmic lens capable of reducing the aberrations of the eye comprising the steps of characterizing at least one corneal surface as a mathematical model, calculating the resulting aberrations of said corneal surface(s) by employing said mathematical model, selecting the optical power of the intraocular lens. From this information, an ophthalmic lens is modeled so a wavefront arriving from an optical system comprising said lens and corneal model obtains reduced aberrations in the eye. Also disclosed are ophthalmic lenses as obtained by the methods which are capable reducing aberrations of the eye.
249 citations
References
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TL;DR: In this paper, a Zernike representation of the Kolmogoroff spectrum of turbulence is given that provides a complete analytical description of the number of independent corrections required in a wave-front compensation system.
Abstract: This paper discusses some general properties of Zernike polynomials, such as their Fourier transforms, integral representations, and derivatives. A Zernike representation of the Kolmogoroff spectrum of turbulence is given that provides a complete analytical description of the number of independent corrections required in a wave-front compensation system.
2,741 citations
TL;DR: In this paper, a statistical definition for the geometric shape of a randomly distorted wavefront was established and relationships between the phase-structure function and the statistics governing the shape were derived.
Abstract: A precise statistical definition is established for the geometric “shape” of a randomly distorted wavefront. Relationships between the phase-structure function and the statistics governing the shape are derived. The most significant portion of wavefront distortion caused by atmospheric turbulence is a random tilting of the plane-wave front. A procedure is outlined for calculating the influence of wavefront distortion on optical systems. Estimates are formed of the effect of wave-front distortion on photographic resolution and optical heterodyne efficiency.
694 citations
Journal Article•
627 citations
TL;DR: In this article, the minimax norm is used to deal with the competing requirements of data-fitting problems, and the two norms most frequently considered are the minimization norm and the minimisation norm 11 e 11 for the vector.
Abstract: the yk(X,) f,. cannot ordinarily all be made 0 simultaneously. When they cannot, the m conditions (2) compete with one another, and the numerical analyst must somehow take account of this in order to formulate a problem of data-fitting. The m numbers e, = Yk(X,) are the m components of an error vector e. Since the x,, and f,, are regarded as fixed, the vector e depends only on the parameters t(7k), , tk. Each common method for dealing with the competing requirements (2) corresponds to the selection of a norm 11 e 11 for the vector e. The two norms most frequently considered are the minimax norm
426 citations
TL;DR: In this paper, it is shown that it is desirable to minimize the number of modes estimated and the amount of measurements used to maintain the quality of the estimates of low-order modes, and that cross coupling occurs when one tries to estimate modes higher than astigmatism.
Abstract: Modal estimation of wave-front phase from phase derivatives is discussed. It is shown that it is desirable to minimize the number of modes estimated and the number of measurements used to maintain the quality of the estimates of low-order modes. It is also shown that mode cross coupling occurs when one tries to estimate modes higher than astigmatism.
212 citations