A perspective from Mopane.
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Congresses as Topic↗
Biomedical subjects
W F Harris
Publications and source records attributed to W F Harris.
Dioptric power: its nature and its representation in three- and four-dimensional space.
Dioptric power expressed in the familiar three-component form of sphere, cylinder, and axis is unsuited to mathematical and statistical treatments; there is a particular class of power that cannot be represented in the familiar form; and it is possible that sphere, cylinder, and axis will prove inadequate in future clinical and research applications in optometry and ophthalmology. Dioptric power expressed as the four-component dioptric power matrix, however, overcomes these shortcomings. The intention in this paper is to provide a definitive statement on the nature, function, and mathematical representation of dioptric power in terms of the matrix and within the limitations of paraxial or linear optics. The approach is universal in the sense that its point of departure is not power of the familiar form (that is, of thin systems) but of systems in general (thick or thin). Familiar types of power are then seen within the context of power in general. Dioptric power is defined, for systems that may be thick and astigmatic, in terms of the ray transfer matrix. A functional definition is presented for dioptric power and its components: it defines the additive contribution of incident position to emergent direction of a ray passing through the system. For systems that are thin (or thin-equivalent) it becomes possible to describe an alternative and more familiar function; for such systems dioptric power can be regarded as the increase in reduced surface curvature of a wavefront brought about by the system as the wavefront passes through it. The curvital and torsional components of the power are explored in some detail. Dioptric power, at its most general, defines a four-dimensional inner product space called dioptric power space. The familiar types of power define a three-dimensional subspace called symmetric dioptric power space. For completeness a one-dimensional antisymmetric power space is also defined: it is orthogonal in four dimensions to symmetric dioptric power space. Various bases are defined for the spaces as are coordinate vectors with respect to them. Vectorial representations of power in the literature apply only to thin systems and are not obviously generalizable to systems in general. They are shown to be merely different coordinate representations of the same subspace, the space of symmetric powers. Some of the uses and disadvantages of the different representations are described. None of the coordinate vectors fully represent, by themselves, the essential character of dioptric power. Their use is limited to applications, such as finding a mean, where addition and scalar multiplication are involved. The full character of power is represented by the dioptric power matrix; it is in this form that power is appropriate for all mathematical relationships.
Eyeglasses↗
Meridional profiles of variance-covariance of symmetric dioptric power: classes of variation that are uniform across the meridians of the eye.
Conventional dioptric power, including refractive status and keratometric measurements, can vary in a large variety of ways. The three-dimensional character of this type of power implies that six numbers are required for the complete representation of its variance: three variances and three covariances. Curves (called profiles) that show how these numbers compare in different meridians (or transverse directions) of the eye provide a useful graphical picture of the variation and its nature. This paper explores an important subclass of types of variation in which the variation is the same for all meridians. In this subclass the variation of power is said to be uniform across all meridians of the eye. It turns out that the uniformity may be either complete or partial. In the former case all aspects of the variation are the same. In the latter case certain aspects of the variation are the same across all meridians whereas others are not; in particular the variance of the torsional component of power is the same for every meridian whereas the variance of the curvital component changes from meridian to meridian. There is a range of types of completely uniform variation from spherical variation at one extreme to Jacksonian variation at the other. All types of uniform variation (partial or complete) are characterized by two indices called the jacksonian index and the completeness index. The types can be represented geometrically as points on the triangle of uniformity. Samples of measurements of dioptric power are selected to illustrate various types of uniform variation. Methods are presented for detecting and classifying uniform variation from profiles of variation and also from the variance-covariance matrix. Examples are given of uniform variation of refractive status and of keratometry of an eye. The variation is analyzed and classified. The concepts and methods are proving to be of fundamental importance in the study of the nature and underlying causes of fluctuations of refractive status and keratometric measurements.
Analysis of Variance↗
Variability of the refractive state: meridional profiles and uniform variation.
Variation of the refractive state as measured with autorefraction is the result of many factors which need to be considered if optometry is to develop a more complete understanding of the behavior of the visual process. Various methods can be applied to develop such an understanding including scatter plots, meridional profiles of variance-covariance, and graphs of uniform variation. These methods are used, in this paper, to investigate some results for autorefraction from a sample of 106 university students studying optometry. Some of the eyes in the sample display variation, some or all of whose characteristics are the same in all meridians of the eye. Such eyes are said to exhibit refractive variation that is partially or completely uniform across the meridians of the eye. Most eyes, however, show variation which appears to depart from uniformity. The typical eye in the sample appears to exhibit variation that is mainly spherical in character and of a small magnitude in keeping with the view that the accommodative system mostly is responsible for this variation. Nevertheless, there is an astigmatic component to the variation. The mean variance-covariance matrices for the right and left eyes are presented.
Accommodation, Ocular↗
Short-term keratometric variation in the human eye.
Previous studies of corneal and keratometric variation used statistical methods that were not entirely satisfactory. For the first time, proper multivariate statistical methods are applied to evaluate short-term keratometric variation in human eyes. Keratometric variation is represented graphically by means of stereo-pair scatter plots, ellipsoidal confidence regions for mean dioptric power, and meridional profiles of variation. There is great variability in the keratometric variation displayed by different subjects, although most subjects exhibit greatest variation in the vertical meridian of the eye on most of the measuring occasions. Variance-covariance matrices based on vector h are given. In some cases keratometric variation approaches neutral uniform variation. In many of the subjects, mean keratometric measurements change from morning to afternoon, usually showing an increase in curvature later in the day. Physical activity may increase keratometric variation and mean curvature.
Adult↗
Excess of autorefraction over subjective refraction: dependence on age.
The purpose of this study was to examine the difference between subjective refraction and autorefraction for different age groups. We call the difference (autorefraction minus subjective refraction) the excess of autorefraction over subjective refraction or the autorefractive excess. Five age groups of 50 subjects each were used. Subjects in group 1 were aged between 1 and 10 years, group 2 between 11 and 20 years, group 3 between 21 and 30 years, group 4 between 31 and 40 years, and group 5 consisted of subjects over 41 years of age. Automatic refraction was performed with an Allergan Humphrey model 580 autorefractor. The data were analyzed using recently developed statistical methods for the analyzing of dioptric power. These methods include the use of the coordinate vector h as a representation of dioptric power. The results indicate that there is a statistically significant mean autorefractive excess and that the mean is different for different age groups. The behavior of the left and right eyes appears to be essentially the same. In terms of vector h the mean autorefractive excess for both the left and the right eyes of group 1 (1 to 10 years of age) is approximately (-0.25 0.00-0.43)'. It increases by roughly delta h = (0.10 0.00 0.10)' per decade. In more conventional terms the nearest equivalent sphere of the mean excess for group 1 is approximately -0.34 D for the right and for the left eyes. The mean autorefractive excess for group 1 is approximately -0.25 -0.18 x 180. The astigmatic component appears to be the same for all age groups, whereas the spherical component increases by approximately 0.1 D per decade. The standard deviations of the autorefractive excesses are relatively large for components h1, and h3 of h: they were between approximately 0.4 and 0.7 D, possibly decreasing slightly with age. The standard deviation of h2 remains at 0.2 D or less for all age groups. The greatest variation of autorefractive excess appears to be approximately in the spherical direction in symmetric dioptric power space and appears to be less for the older age groups than the younger age groups.
Adolescent↗
Refractive variation under accommodative demand: curvital and scaled torsional variances and covariance across the meridians of the eye.
Autorefractor measurements were taken on the right eye of 10 students with an external target at vergences -1.00 and -3.00 D. The refractive errors in the form of sphere, cylinder, and axis were converted to vectors h and variance-covariance matrices calculated for different reference meridians. Scatter plots are drawn in symmetric dioptric power space. The profiles of curvital and scaled torsional variances, the scaled torsional fraction, and the scaled torsional-curvital correlation are shown using a polar representation. This form of representation provides a meridional pattern of variation under accommodative demand. The profile for scaled torsional variance is characteristically in the form of a pair of rabbit ears. At both target vergences curvital variance is larger than scaled torsional variance in all the meridians of the eye: the relative magnitudes are quantified by the scaled torsional fraction. An increase in accommodative demand generally results in an increase in variance. The rabbit ears usually become larger but less well divided. The correlation between curvital and torsional powers is usually positive in the first quadrant and negative in the second quadrant. Typical, atypical, and mean typical responses are discussed.
Accommodation, Ocular↗
Refraction and keratometry: departures from and transformations toward multivariate normality.
The multivariate distributional properties of refraction and keratometric data were investigated across eyes with power represented in the coordinate system introduced by Deal and Toop. Normality and departure from normality were assessed with the aid of chi 2 and normal probability plots and by the comparison of multivariate sample skewness and kurtosis with critical values. Two of the three data sets show significant departure from normality in each of the marginal distributions and, therefore, the joint distribution too. The keratometric data were normally distributed along the line of spherical powers but departed from normality in the astigmatic plane. Marginal transformations are used to reduce the departure from normality where necessary. The transformation that was found to be successful is essentially an example of a Box/Cox transformation involving a shift, ci, and an exponent, gamma i, where i = 1,2,3. For two of the data sets, the values of the exponent, gamma i, result in a transformation that is similar to a modified square root transformation.
Adolescent↗
Invariance of ophthalmic properties under spherocylindrical transposition.
Ophthalmic properties expressed as functions of dioptric power cannot depend on the particular spherocylindrical form (positive or negative cylinder) chosen to represent the power: they are necessarily invariant under spherocylindrical transposition. This condition of invariance places restrictions on the mathematical form that valid ophthalmic functions can assume. Tests are presented for checking the validity of proposed ophthalmic functions and properties. Examples from the literature are examined, including Keating's concept of torsional power and Peters' graphs of expected unaided visual acuity vs. ametropia. The former satisfies the condition of invariance but the latter are shown to violate the condition for ametropias which are close to spherical. The analysis shows partly how the graphs need to be refined. Invariance under spherocylindrical transposition can assist the researcher in developing new concepts and relationships that depend on power expressed in terms of sphere, cylinder, and axis.
Analysis of Variance↗
Keratometric variation: the influence of a fluid layer.
Variation of keratometric measurements on an approximately spherical surface on a polymethyl methacrylate (PMMA) button is compared to keratometric variation on the PMMA button after the application of an artificial solution. The variation is shown by means of scatter plots in a three-dimensional space, as well as quantitatively in the form of variance-covariance matrices. The wet button showed variances a factor of more than 1000 times that of the dry button. This suggests that the precorneal tear film could be an important, often-overlooked contributing factor to keratometric variation in the case of the eye.
Humans↗
Ray vector fields, prismatic effect, and thick astigmatic optical systems.
The application of the concept of ray vector fields to optical systems is reexamined. Paraxial or linear optics defines a four-dimensional ray vector field for any optical system: the vector field maps the incident ray vector into the emergent ray vector. In the case of thin systems, including thin astigmatic lenses, one can define a vector field of reduced dimensionality: the vector field is two-dimensional and maps the ray's incident position into the change in reduced direction. When the index of refraction is the same before and after a thin system, the change in reduced direction is the reduced deflection through the system or the reduced prismatic effect. Contrary to what has recently been claimed, this type of two-dimensional vector field does not apply in general to thick systems. However, a number of different types of two-dimensional vector fields can be defined for various particular classes of optical systems. Thick systems differ qualitatively from thin systems. They do not have equivalent thin lenses and cannot generally be replaced by thin lenses. Equations are derived for the change in reduced direction and deflection for a ray through optical systems in general and through separated two- and three-lens systems in particular.
Astigmatism↗
Thickness extrema at the edge of astigmatic lenses including those with straight, circular and elliptical edges.
The problem of locating the points of maximum and minimum thickness at the edge of a lens, and of calculating the thickness at those points, is examined for lens powers and for edge shapes in general. The edge extremal problem, as the problem is called, is solved explicitly for general powers along straight cut edges. The extremal problem is also analysed for lenses with circular and elliptical edges but explicit solutions are obtained only for centrally cut lenses, that is, lenses with coincident optical and geometrical centres. For edges that are neither straight nor centrally cut ellipses (including circles), it appears that explicit solutions cannot be obtained for the edge extrema: one has to resort to numerical solution of implicit equations or to the calculation of thickness at sufficiently many points around the edge. For the centrally cut ellipse the edge extremal problem turns out to be the eigenvalue problem of linear algebra. In general the thickness extrema at the edge do not lie on the principal meridians of the lens, nor do they lie on meridians that are mutually perpendicular. With minor modification the results apply equally well to the edge extrema for sagitta of a surface.
Astigmatism↗
Wavefronts and their propagation in astigmatic optical systems.
The geometry of the astigmatic wavefront is derived from the symplectic nature of linear optics. It is shown to be paraboloidal. Equations are derived that govern the propagation of such wavefronts through astigmatic systems in general and through thin lenses and across refracting interfaces and homogeneous gaps in particular. The equations allow the generalization of the concept of wavefront curvature or vergence to astigmatic systems. In particular they show how the step-along method of calculating wavefront curvature is generalized. Not only are Keating's earlier conclusions on this topic confirmed but also they are shown to hold under more general circumstances. They hold even when the system contains gradient-index elements such as the natural lens of the eye. Some of the premises used in the earlier study are shown not to be necessary: they are a consequence of symplecticity. The analysis also provides a step-along procedure for calculating wave-front direction. A numerical example in the Appendix shows the application of the step-along method to a particular separated astigmatic system: the back-vertex power of the system is determined as is the equation of the emergent wavefront for a distant object point.
Astigmatism↗
Adding obliquely crossed spherocylindrical powers: a nonproblem.
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Mathematics↗
Refractive variation during autorefraction: multivariate distribution of refractive status.
The refractive variation of a sample of 106 university students (63 females and 43 males) studying optometry was examined by means of autorefraction. Stereo-pair scatter plots in Euclidean three-dimensional h-space are used to illustrate the nature of the spread or distribution of data measurements found in particular subjects. A wide variety of different distributions were observed ranging from tightly to loosely clustered arrangements of measurements. Some aspects of departure from multivariate normality, including outliers (atypical measurements in a sample) and polymodal or multimodal distributions, are demonstrated. Outliers appear to be possible anywhere in the space of the scatter plots, although outliers may be more common in the region of h-space corresponding to transitory increases in accommodation. Multimodal distributions may be indicative of changes in ocular fixation during autorefraction or may reflect accommodative or other anomalies. Other departures from multivariate normality such as kurtosis and skewness are also of importance when attempting to form an understanding of variation of refractive state. Measurements made on an artificial or test eye showed very tight clusters in h-space. This suggests that the autorefractor itself contributes little to the variation observed during autorefraction of an eye.
Accommodation, Ocular↗
Coordinate systems and vector spaces for lenses.
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Eyeglasses↗
Paraxial ray tracing through noncoaxial astigmatic optical systems, and a 5 x 5 augmented system matrix.
The matrix method of tracing a paraxial ray through a coaxial optical system that contains spherical refracting elements requires a 2 x 2 system matrix. This paper shows that a suitable 5 x 5 system matrix enables one to apply the same method in the general case of systems that may be noncoaxial and that may contain astigmatic elements. The systems may contain prisms and decentered astigmatic lenses, for example.
Astigmatism↗
Prismatic effect in bifocal lenses and the location of the near optical centre.
A matrix expression is derived for the prismatic effect in the reading portion of a bifocal lens. Using this expression the problem of finding the location of the optical centre of the reading portion is examined. Matrix methods lead to a complete solution that holds in all circumstances within the usual limitations of first-order optics and thin lenses. Both the main and segment lenses may be astigmatic. In the usual case there is a unique near optical centre but situations can occur in which there is no near optical centre or an infinite number of them. These situations are examined systematically. Numerical examples illustrate the method.
Eyeglasses↗