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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...
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... [4] Other approaches to mathematically scale Zernike coefficients have been proposed, and progressively simplified. [5][6][7][8][9][10][11][12][13] These formulas use differing algebraic, recursive, or matrix methods, but all are equivalent. [14] In theory, scaling Zernike coefficients to a smaller diameter has no error. ...
... This makes the model an ideal test-case for the formula selected; however, pupil scaling for decentered, tilted, and non-circular pupils has also been described. [7,10,11] The aperture dependent nature of Zernike polynomials requires special consideration when measurements from different instruments or participants are compared, for example, in repeatability and agreement studies. The ability to precisely predict the extent of HOA reduction at specific pupil sizes will enable clinicians to assess the likely efficacy of pinhole piggyback lenses to reduce visual symptoms associated with HOA. ...
Purpose:
Zernike polynomials for describing ocular higher order aberrations are affected by pupil aperture. The current study aimed to validate Mahajan's formula for scaling Zernike polynomials by pupil size.
Methods:
Higher order aberrations for 3 intraocular lens models (AcrySof IQ IOL SN60WF, Technis ZA9003, Adapt Advanced Optics) were measured using the Zywave aberrometer and a purpose-built physical model eye. Zernike coefficients were mathematically scaled from a 5 mm to a 3 mm pupil diameter (5:3 mm), from a 5 mm to a 2 mm pupil diameter (5:2 mm), and from a 3 mm to a 2 mm pupil diameter (3:2 mm). Agreement between the scaled coefficients and the measured coefficients at the same pupil aperture was assessed using the Bland-Altman method in R statistical software.
Results:
No statistically significant mean difference (MD) occurred between the scaled and measured Zernike coefficients for 21 of 23 analyses after Holm-Bonferroni correction (P > 0.05). Mean differences between the scaled and measured Zernike coefficients were clinically insignificant for all aberrations up to the fourth order, and within 0.10 μm. Oblique secondary astigmatism (Z-24) was significantly different in the 5:3 mm comparison (MD = -0.04 μm, P < 0.01). Horizontal coma (Z13) was significantly different in the 3:2 mm comparison (MD = -0.07 μm, P = 0.03). There were borderline statistical differences in both vertical (Z-13) and horizontal coma (Z13) in the 5:3 mm comparison (MD = 0.02 μm, -0.09 μm, P = 0.05, 0.05, respectively).
Conclusion:
A formula for the scaling of higher order aberrations by pupil size is validated as accurate. Pupil scaling enables accurate comparison of individual higher order aberrations in clinical research for situations involving different pupil sizes.
... The limit σ = 0 recovers A n,m (q, σ = 0) = A n,m (q) in (10) as a special case. These integrals are decomposed by multipole expansions around the mid-points of the two sub-pupils that define local circular coordinates s and φ as in Figure 1 (Campbell 2003;Comastri et al. 2007;Lundström & Unsbo 2007;Schwiegerling 2002;Shu et al. 2006): Fig. 3. The basis functions f7 to f12 for q = 1/2. ...
Sets of orthogonal basis functions over circular areas -- pupils in optical applications -- are known in the literature for the full circle (Zernike or Jacobi polynomials) and the annulus. Here, an orthogonal set is proposed if the area is two non-overlapping circles of equal size. The geometric master parameter is the ratio of the pupil radii over the distance between both circles. Increasingly higher order aberrations -- as defined for a virtual larger pupil in which both pupils are embedded -- are fed into a Gram-Schmidt orthogonalization to distill one unique set of basis functions. The key effort is to work out the overlap integrals between a full set of primitive basis functions of hyperspherical type centered at the mid-point between both pupils. Constructed from the same primitive basis, the orthogonal Karhunen-Loève modes of spatially filtered Kolmogorov phase screens are computed for this shape of mask. Matrix elements of the covariance matrix -- an established intra-circle and a special inter-circle category -- are worked out in wavenumber space.
... The noise may be minimized if a dynamic aberrometer is used instead of static to measure the aberrations and the coefficients are extracted by averaging multiple frames. Pupil scaling methods that deal with the scaled pupil being translated, 15 rotated, and noncircular 19,20 have been developed, which can also contribute to inaccuracies in the estimation of Zernike coefficients; the methodology of this study assumed that there was no translation of the pupil center and was unlikely to have contributed to the errors in estimation seen. ...
Purpose:
This study aimed to validate the mathematical Zernike pupil size scaling from bigger pupils to smaller pupils, and vice versa, by comparing the estimates of the Zernike coefficients with corresponding clinical measurements obtained at different pupil sizes.
Methods:
The i.Profiler Plus (Carl Zeiss Vision, Inc, USA) was used to obtain measures of wavefront aberrations for two pupil sizes (3 mm and the maximum natural pupil size) from the right eyes of 28 visually normal subjects (mean [±SD] age, 57 [±7] years) whose maximum pupil size was greater than or equal to 5 mm without pharmacological dilation. Zernike coefficients were estimated for a 3-mm pupil size scaling down from the measured data of the maximum natural pupil size and, similarly, for the maximum pupil size scaling up from the measured data of the 3-mm pupil.
Results:
The differences between the estimated and measured values were not significantly different (repeated-measures analysis of variance; p > 0.05) over the range of pupil sizes examined, irrespective of whether the estimates were made by scaling up from a small pupil or scaling down from a large pupil. However, the difference between the measured and estimated coefficients was more variable and less systematic when scaling to a larger pupil size when compared with scaling to a smaller pupil size.
Conclusions:
Estimation of ocular wavefront aberration coefficients either scaling down from large to smaller pupils or scaling up from smaller to large pupils provides estimates that are not significantly different from clinically measured values. However, when scaling up to a larger pupil size, the estimates are more variable. These findings have implications for pupil scaling on an individual basis, such as in cases of refractive surgery or when using pupil scaling to examine a clinical cohort.
... La luminancia de la carta afecta la claridad subjetiva y el tamaño pupilar mientras que el tamaño pupilar afecta la claridad subjetiva. Si la pupila se dilata manteniendo la luminancia de la carta constante entonces la máxima frecuencia espacial que entra al ojo [Comastri et al., 2007] y la claridad aparente aumentan mientras que la profundidad de foco disminuye . ...
... En general las aberraciones dependen [Born y Wolf, 1980;Comastri et al., 2006Comastri et al., , 2007Comastri et al., , 1999 de la apertura (o sea de cuan alejados están los rayos del centro de la pupila de entrada) y del campo (o sea de cuan alejado del eje óptico está el punto objeto). Para tener en cuenta las aberraciones analíticamente, en principio se considera un sistema óptico formador de imágenes como una caja negra con dos terminales, la pupila de entrada y la pupila de salida (figura A.8 (a)). ...
In this Thesis we develop both the hardware and the software of a pupillometer eye
tracker which we denominate Blick. The hardware comprises an illumination system
(consisting in two LEDs that emit infrared radiation) and a detection system (consisting
in a USB camera, an optical system and a filter). At present, Blick is a table top device
(this meaning that the detection system and the subject’s chin rest are both mounted
on a table) and runs in a Linux environment. Blick captures eye images, processes them
by means of real time computations and supplies plots and tables of pupil diameter and
gaze direction. This development is aimed to be used by the Grupo de Óptica y Visión
(UBA) in investigations dealing with the eye and, also, by other research groups, in
ophthalmic and psychological clinics or by any person. Blick could be used in various
applications related not only to psychophysical vision tests but also to the study of
mental resources employed when performing certain tasks. This is, the device could
be used to detect conscious or unconscious processes, to analyze preferences (sexual
or of other type), to study marketing conveniences of certain products, as a human
computer interaction system (HCI) for people with motor disabilities, etc.
Our goals are, first, to develop a low cost pupillometer eye tracker exhibiting an
adequate performance and, second, to attain pupil diameter and gaze directión data
during psychophysical tests and analyze information concerning certain visual and
mental processes. To help us achieve these objectives, we review the state-of-the-art
concerning commercial and academic pupillometers and eye trackers; the capture and
processing of ocular images; behaviours of the visual and/or mental systems associated
to pupil dilation and contraction or changes in gaze direction and, also, many other
applications employing eye trackers.
Afterwards we briefly describe the visual system, including its optical and neurolo-
gical systems, its performance under different conditions and, in particular, its pupil.
Moreover we deal with some concepts related to the traditional visual acuity test, used
for eye control and refractive correction prescription (ophthalmic and contact lenses).
Then we explain how we develop Blick, first its illumination and detection system
and then its software programmed in C++ using OpenCV and cvblob libraries. We outline
each of the image processing steps required to determine pupil diameter and gaze
direction.
Finally, we design and carry out five tests: visual acuity, homographic calibration,
visual preferences, cognitive tasks, and writing via eye movements. Using Blick, we
capture the right eye of six subjects and analyse the corresponding pupil and eye gaze
plots. This enables us both to investigate some topics regarding the visual and mental
systems which are of interest for our Group and to verify the adequate performance of
Blick and acquire insights concerning its potential improvements.
... The noise may be minimized if a dynamic aberrometer is used instead of static to measure the aberrations and the coefficients are extracted by averaging multiple frames. Pupil scaling methods that deal with the scaled pupil being translated, 15 rotated, and noncircular 19,20 have been developed, which can also contribute to inaccuracies in the estimation of Zernike coefficients; the methodology of this study assumed that there was no translation of the pupil center and was unlikely to have contributed to the errors in estimation seen. ...
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 coefficients were then compared with the measured data for the 4.8mm pupil size. The procedure was repeated to estimate Zernike coefficients for a 3mm pupil size from the measured data of the 4.8mm pupil size. Third and fourth orders Zernike coefficients were used for the analysis.
Results: Repeated measures ANOVA (Factor: Method, Coefficients) was used to evaluate differences between estimated and theoretical Zernike coefficients. No significant differences (all p>0.05) were found between the estimated and measured coefficients overall. Paired t-tests between the estimated and measured individual coefficients were not significant (Tukey; all p>0.05). Correlation coefficients were at the lowest of r=0.87, (p<0.001). A Bland-Altman analysis was used to determine any systematic and spread differences between the estimated and measured Zernike coefficients for each pupil. The limits of agreement (LOA: mean±95% CI) ranged between +0.033 and -0.043 for scaling up to 4.8mm, and +0.128 and -0.124 for scaling down to 3.0mm. In all cases, the differences were distributed symmetrically around the mean difference.
Conclusions: The differences between the estimated and measured values were not significantly different over the range of pupils examined, irrespective if the estimates were made when scaling up from a small pupil or the converse. Scaling to a larger pupil size had larger LOAs than scaling to a smaller pupil size.
... This formula is based on a concise expression for scaled radial polynomials in terms of unscaled radial polynomials, see Eq. (8) below, the proof of which is of similar nature as the one given in [5] and heavily depends on the basic NZ-result. Results on pupil scaling find applications in optical lithography, where the NA of the optical system may be decreased deliberately for imaging enhancement of particular structures, and in ophthalmology, where pupil scaling is studied as a natural attribute of the human eye pupil, see [7] and [20]. ...
... Although Eq. (8) is normally used for ε, ρ ∈ [0, 1], it should be emphasized that they are valid for all complex values of ε and ρ by analyticity. The result in Eq. (8) is of interest to both the lithographic community and the ophthalmological community, see [7], [8], [19]- [20]. ...
... Furthermore, it gives the analytic solution of the transformation problem for the aberration coefficients of an eye pupil when the pupil is scaled and displaced. This problem has a long history in the ophthalmological community, see [7] and [20] for recent work and survey material, but no closed-form solution seems to have been found thus far. Moreover, by expressing the scaled-andshifted polynomials as linear combinations of the orthogonal terms Z m n one has a handle, via the transformation matrix elements K mm nn (a, b), to tackle the important problem of assessing the condition of a finite set of circle polynomials when they are restricted to subdisks of the unit disk. ...
Several quantities related to the Zernike circle polynomials admit an
expression, via the basic identity in the diffraction theory of Nijboer
and Zernike, as an infinite integral involving the product of two or
three Bessel functions. In this paper these integrals are identified and
evaluated explicitly for the cases of (a)~the expansion coefficients of
scaled-and-shifted circle polynomials, (b)~the expansion coefficients of
the correlation of two circle polynomials, (c)~the Fourier coefficients
occurring in the cosine representation of the circle polynomials.
... Various authors have addressed this problem proposing either analytical or numerical methods to transform coefficients [11,12], some only considering pupil scaling [13]. We have developed 2 methods to find a new set of coefficients associated to a circular pupil in terms of an original set evaluated for a dilated shifted circular pupil, one analytical (valid for a 7 th order aberration expansion) and the other graphical (valid for any order of the expansion), and we have also derived selection rules [14][15][16][17]. On the other hand, aberrations of up to 2 nd order are routinely compensated with conventional ophthalmic or contact lenses while the customized correction of higher-order aberrations may yield considerable visual benefit [5] in old or abnormal eyes (post-Lasik [6], suffering keratoconus [5,7,10], implanted with intraocular lenses [8], with corneal transplant [10], etc.) where the aberration balance between cornea and crystalline lens attained in young normal subjects is degraded [5,9]. ...
... In the current article we apply our formulas [14][15][16][17] to transform aberration coefficients corresponding to both corneas of 10 subjects suffering mild to severe keratoconus. In Section 2 we briefly describe our methodology. ...
... Following OSA´s ordering and normalization [2] (adopted by ANSI [3]), Zernike modes are represented in a pyramid ( figure 1 (a)) and, to simplify the notation [2], are henceforth indicated with the single index j=[n(n+2)+m]/2. The root mean square wavefront error is RMS=(Σ C j 2 ) 1/2 and higherorder ocular aberrations (j>5) can be regarded as negligible [14] if RMS HO <0.10 µm (for example, the mean spherical aberration coefficient for a normal population and a pupil semi-diameter of 3mm is ...
Ocular aberrations vary among subjects and under different conditions and are commonly analyzed expanding the wavefront aberration function in Zernike polynomials. In previous articles, explicit analytical formulas to transform Zernike coefficients of up to 7th order corresponding to an original pupil into those related to a contracted displaced new pupil are obtained. In the present paper these formulas are applied to 20 keratoconic corneas of varying severity. Employing the SN CT1000 topographer, aberrations of the anterior corneal surface for a pupil of semi-diameter 3 mm centered on the keratometric axis are evaluated, the relation between the higher-order root mean square wavefront error and the index KISA% characterizing keratoconus is studied and the size and centering of the ocular photopic natural pupil are determined. Using these data and the transformation formulas, new coefficients associated to the photopic pupil size are computed and their variation when coordinates origin is shifted from the keratometric axis to the ocular pupil centre is analyzed.
... Furthermore, it gives the analytic solution of the transformation problem for the aberration coefficients of an eye pupil when the pupil is scaled and displaced. This problem has a long history in the ophthalmological community, see [7] and [20] for recent work and survey material, but no closed-form solution seems to have been found thus far. Moreover, by expressing the scaled-and-shifted polynomials as linear combinations of the orthogonal terms Z m n one has a handle, via the transformation matrix elements K mm nn (a, b), to tackle the important problem of assessing the condition of a finite set of circle polynomials when they are restricted to subdisks of the unit disk. ...
Several quantities related to the Zernike circle polynomials admit an expression as an infinite integral involving the product of two or three Bessel functions. In this paper these integrals are identified and evaluated explicitly for the cases of (a) the expansion coefficients of scaled-and-shifted circle polynomials, (b) the expansion coefficients of the correlation of two circle polynomials, (c) the Fourier coefficients occurring in the cosine representation of the circle polynomials, (d) the transient response of a baffled-piston acoustical radiator due to a non-uniform velocity profile on the piston.
... En la Sección 2 presentamos la notación utilizada para expandir la función aberración del frente de ondas en polinomios Zernike. En la Sección 3 sintetizamos nuestro método analítico que brinda expresiones explícitas de los elementos de la matriz que transforma coeficientes de hasta 7 o orden al contraer y descentrar la pupila [25][26][27] y, considerando al igual que otros autores [19,23] la matriz de rotación, hallamos la matriz de transformación completa (rotación, contracción y descentrado). En la Sección 4 teniendo en cuenta el método gráfico, sintetizamos el implementado previamente para contracción y descentrado pupilar [28] y lo generalizamos para el caso de rotación pupilar. ...
... Considerando n max =7, transformamos coeficientes Zernike originales en nuevos cuando la pupila se rota, contrae y descentra empleando 2 matrices de dimensión 36×36, la de rotación [19,23], U, y la de contracción y traslación, T, cuyos elementos obtenemos analíticamente en artículos previos [25,26]. Los modos originales corresponden a una pupila de radio a, los caracterizamos mediante (n,m) o j y los coeficientes asociados, C j , son los elementos de un vector columna conocido, C. Los modos nuevos corresponden a una pupila de radio b descentrada y rotada siendo b<a, los caracterizamos mediante (n,m) o j y los coeficientes, C j , son los elementos del vector columna, C, a evaluar. ...
... En lo que sigue resumimos nuestra metodología [25,26] para hallar analíticamente, primero, los elementos de la matriz correspondiente a escalado y descentrado horizontal de la pupila, SH, y, luego, la matriz de escalado y descentrado arbitrario, T. Aunque esta matriz es matemáticamente válida para contracción o dilatación, como las aberraciones se determinan por trazado de rayos solo en la pupila original, en situaciones de interés físico consideramos contracción (B≤1) y, además, F<1−B. ...
Zernike polynomials are often used to expand the wavefront aberration of manufactured optical systems and of the eye. In previous articles we present an analytical method to find explicit expressions for the elements of a matrix which transforms Zernike coefficients of up to 7th order computed for a circular pupil into those corresponding to another contracted and decentred pupil and we develop a graphical method to identify new coefficients in terms of original ones or vice versa for any order of the expansion. In the present paper we synthesize fundamental concepts of both methods and, taking into account the rotation matrix, we generalize them to transform coefficients when pupil is contracted, decentred and rotated. We illustrate how to employ our methods showing two applications in which we find the new coefficients and we obtain the corresponding spot diagrams at the image plane.
... In previous papers [18][19][20], considering scaling and decentering of circular pupils and aberrations of up to seventh order, we develop an analytical method that yields the formulae for the new coefficients in terms of pupil parameters and original coefficients and facilitates the qualitative analysis of the interrelationships between coefficients. In [18] we consider pupil scaling and horizontal translation; we give the corresponding selection rules and we apply our formulae to four typical cases (normal, keratoconic and post-Lasik corneas and a sphere). ...
... In previous papers [18][19][20], considering scaling and decentering of circular pupils and aberrations of up to seventh order, we develop an analytical method that yields the formulae for the new coefficients in terms of pupil parameters and original coefficients and facilitates the qualitative analysis of the interrelationships between coefficients. In [18] we consider pupil scaling and horizontal translation; we give the corresponding selection rules and we apply our formulae to four typical cases (normal, keratoconic and post-Lasik corneas and a sphere). In [19], considering scaling and arbitrary transversal pupil shift, we define a transformation matrix, T, of dimension 36 × 36 that converts old coefficients to new ones. ...
... Though analytical [18][19][20] or numerical methods [16] to compute new coefficients in terms of the old ones are useful (for example, to evaluate the optical response to myopic Lasik surgery [11], to study the benefits of compensating ocular highorder aberrations with contact lenses [13], etc), in some cases, it can be sufficient to employ a graphical method to quickly identify whether a certain new coefficient appears or not. Graphical, analytical and numerical methods complement each other [13][14][15][16][17][18][19][20], for example, while a system (ophthalmic lens, optical instrument, etc) is designed, a graphical method can be employed to identify the change in the coefficients' pattern if the pupil is shifted along a given direction whereas during the optimization process, an analytical method has to be used. ...
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.
