Popperian falsification of methods of assessing astigmatism.
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Astigmatism↗
Biomedical subjects
W F Harris
Publications and source records attributed to W F Harris.
Meaningful representation of power.
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Cornea↗
Interconverting the matrix and principal meridional representations of dioptric power in general including powers with nonorthogonal and complex principal meridians.
The principal meridians of the powers of thick astigmatic systems, like the eye, are not necessarily at right angles. The consequence is a class of phenomena included in the category commonly described as irregular astigmatism. The conventional principal meridional representation of power, however, is unsuited to quantitative analysis. This paper presents equations for converting from the principal meridional form of power to a representation, the dioptric power matrix, which is amenable to quantitative analysis. It generalizes an earlier paper which treated only powers of a conventional form in which the principal meridians are always at right angles. It copes in particular with what are known as asymmetric powers. A routine is also presented for converting in the reverse direction, from the power matrix to the principal meridional form of power. The principal meridional form of power turns out not always to be unique, there being distinct powers (they are asymmetric) with the same principal powers and meridians. Thus, in general, the dioptric power matrix is a satisfactory representation of power while the principal meridional representation is not.
Data Interpretation, Statistical↗
Interpretating nonorthogonal keratometric measurements.
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Cornea↗
Closed surfaces of constant visual acuity in symmetric dioptric power space.
This paper demonstrates a multivariate approach to understanding the complicated relations of visual acuity to refractive state or ametropia. Other approaches, as previously used, included graphical representations of lines or profiles of iso-oxyopia (Peters, 1961). But one limitation of Peters' method is that cylinder axis was ignored. However, here the relationship between visual acuity and refractive power will be represented by estimated closed surfaces of constant visual acuity in symmetric dioptric power space. At or near the common center (of several closed surfaces, for example) is the refractive compensation. Coming outwards from such a center, the visual acuity drops in all directions in the space. The primary purpose of this paper was to present estimated closed surfaces of constant visual acuity for several eyes. Various procedures were performed on several subjects including measurement of iris aperture diameter, subjective refraction, and autorefraction. Thereafter, an automated phoropter and either Jackson cross-cylinders or spheres were used to influence dioptric blur or defocus in the subjects. The visual stimulus was a computer-generated nondirectional or meridionally independent letter O. Ovoidal surfaces fit the measurements obtained (with Jackson cross-cylinders and spheres) better than ellipsoidal surfaces. The cross-section, in symmetric dioptric power space, at powers with the same nearest equivalent sphere as the refractive compensation is elliptical in many cases and reflects a dependence of visual acuity on cylinder axis. The surfaces differ when powers are changed so that one is moving away from (decompensation surfaces) or toward (accompensation surfaces) the refractive compensation. The multivariate and graphical methods used in this paper probably have implications for the direction of future research in a number of areas involving measures of vision function such as autorefraction, retinoscopy, subjective refraction, and visual acuity.
Accommodation, Ocular↗
Comparing optical systems, and the concept of the converter system.
The purpose of this article was to present a complete and general method for comparing the first-order optical character of optical systems. The method provides a common basis for quantifying the difference between systems of all kinds including thin lenses, ophthalmic prisms, eyes before and after accommodation, eyes before and after refractive surgery, etc. Systems may be astigmatic or stigmatic, coaxial or noncoaxial. In special cases, the method reduces to being equivalent in essence to ostensibly incommensurate comparisons implicit or explicit in current optometric and ophthalmological usage (difference in power for refractions, corneas, and thin lenses, difference in prismatic power for prisms, ratio of magnifications for afocal telescopes, etc.). The method uses the concept of a converter system that when placed in front of or behind one system, converts its first-order optical character to the equivalent of a second system. Equations are presented for the ray transferences of the anterior and posterior converter systems for pair-wise comparisons in general. For any two systems, the transferences of the converter systems always exist and are unique. Numerical examples are presented; they illustrate converter systems that may be thin in special cases but thick otherwise. The transference of a converter system embodies and quantifies the optical difference between systems or characterizes the change from one state of a system (presurgical or preaccommodative, for example) to another (postsurgical or postaccommodative). The method provides a rational and uniform methodology for research and clinical applications in many areas of optometry and ophthalmology.
Accommodation, Ocular↗
Clinical measurement, artifact, and data analysis in dioptric power space.
It appears now to be recognized that traditional clinical representations of astigmatic power, including sphere, cylinder, and axis, in particular, do not lend themselves directly to satisfactory quantitative analysis. For purposes of analysis, the clinical representations need first to be transformed into representations in dioptric power space. It turns out, however, that characteristics of the clinical measurements carried over into dioptric power space can be a source of spurious conclusions reached in such studies. The source of the problem lies in the nature of sphere, cylinder, and axis and in the discreteness (multiples of 0.25 D, usually, in sphere and cylinder and 1 or 5 degrees in axis) of the clinical measurements. As a consequence, scatter plots of clinical measurements in dioptric power space may show patterns and structures, including clusters, arcs, and moiré effects. All the structures are artifacts of the discreteness in sphere, cylinder, and axis; they have no other physical basis. Furthermore, the clinical measurements can show departures from normality that are also purely artifact. By learning to recognize artifact in the scatter plots, the researcher can overcome some of the problems. But because of the assumption of normality underlying many statistical procedures, the problem of distortions in the distributions remains. The distortions may weaken the confidence with which inferences can be made or even lead to erroneous conclusions. The purpose of this paper is to draw attention to the problems of artifact in analyses of clinical data and to suggest ways of overcoming it or avoiding it at least in part.
Artifacts↗
Magnification, blur, and ray state at the retina for the general eye with and without a general optical instrument in front of it: 1. Distant objects.
PURPOSE: To derive general equations that characterize rays, magnification, and blur at the retina in the case of distant object points for a naked eye and for an eye looking through an arbitrary optical instrument. The eye and optical instrument may be astigmatic and noncoaxial. METHOD: The derivation is based on linear optics and makes use of the concept of the augmented ray transference of an optical system. Because the transference completely characterizes the linear optics, the analysis can claim completeness. RESULTS: Equations are presented for position and direction of rays at the retina from distant object points. They lead naturally to the definition of six properties that characterize blur, shape, size, orientation, and position of images of distant objects viewed by the naked eye and by the eye looking through an instrument. By way of example, the general equations are applied to the simple examples of a thin contact lens and a thin spectacle lens in particular. CONCLUSION: The analysis provides a framework, complete as far as linear optics is concerned, for the analysis of light arriving at the retina through any instrument from a distant point. In so doing, it unifies and generalizes concepts like blur and spectacle magnification, which, in the past, have been treated separately.
Contact Lenses↗
Magnification, blur, and ray state at the retina for the general eye with and without a general optical instrument in front of it: 2. Near objects.
PURPOSE: To derive general equations that characterize rays, magnification, and blur at the retina in the case of near object points for the naked eye and for the eye in combination with a general optical instrument. METHOD: The paper draws on results obtained via linear optics in an accompanying paper. RESULTS: Equations are presented that completely characterize the state of rays at the retina from objects at any distance. They allow quantification of blur, size, shape, orientation, and magnification at the retina. CONCLUSION: The analysis provides a framework, complete in linear optics, for the analysis of light arriving at the retina of a general eye through any instrument from any object point.
Contact Lenses↗
Criteria for assessing autorefractor accuracy and accommodation.
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Accommodation, Ocular↗
Prentice's equation and generalizations.
The purpose of this note is to clarify confusion over the range of applicability of Prentice's equation for the calculation of prismatic effects in lenses. Making use of the concept of the ray transference in Gaussian and linear optics, the paper obtains generalized forms of Prentice's equation that apply in various situations. Redefinition of prismatic deviation as change in reduced direction of a ray makes Prentice's equation valid for thin systems in media of any index of refraction. In particular, the equation then holds for a single refracting surface. It does not hold in general for thick systems, although there are some special cases in which it does apply. The appropriate equation is presented for thick systems in general. The correct form of power to use in the equations is equivalent power and not back-vertex power. A generalized form of Prentice's equation applies to astigmatic systems, including single refracting surfaces and thin lenses. An analog of Prentice's equation gives the transverse displacement of a ray across a system as a function of the incident-reduced direction of the ray.
Eyeglasses↗
Astigmatism.
The quantitative representation and analysis of astigmatism present difficulties for the researcher. The case is made that the difficulties arise because of the way astigmatism is conceived and defined. In most cases astigmatism is regarded as cylinder. Cylinder, however, is not invariant under spherocylindrical transposition and, hence, cannot strictly be regarded as meaningful. The purpose of the paper is to find a rational, context-free, invariant and universally-applicable definition of astigmatism for all quantitative analyses. One is led to definitions of astigmatism and its components some of which have already appeared in the literature but which are not in use in analyses of astigmatism. Astigmatism is defined with respect to pure sphere. In the case of thin systems (including keratometric measurements and refraction) astigmatism turns out to be Jacksonian power, that is, the power of a Jackson crossed cylinder. The power of every thin system can be regarded as consisting of two orthogonal components, sphere and astigmatism. The astigmatism of thin systems itself further decomposes naturally into two orthogonal components called ortho- and oblique astigmatism. In the case of thick systems, like the eye itself, astigmatism decomposes naturally into three orthogonal components, ortho-, oblique and antisymmetric astigmatism. The approach is based on the general definition of power in paraxial optics, the dioptric power matrix, and leads to useful graphical representations. Because of its mathematical foundation the analysis can claim completeness and contextual independence. Furthermore it is also directly applicable to the four fundamental paraxial properties of optical systems.
Astigmatism↗
Step-along vergence procedures in stigmatic and astigmatic systems.
The traditional step-along vergence procedure applies to stigmatic systems, that is, systems that are not astigmatic. Computation is disrupted when a focus coincides with a thin lens or refracting surface. A small change to the procedure results in a modified procedure which overcomes the computational problems. The modified procedure is easier to execute than the traditional procedure and allows one to write down useful equations directly. Among the formulae are those for back-vertex power. A step-along vergence procedure also exists for astigmatic systems. It makes use of the dioptric power matrix and the reduced vergence matrix. Computational problems arise when a point or line focus coincides with a thin lens or refracting surface; however they are not overcome by an analogous modification to the procedure. Nevertheless the modified procedure has some advantages including the fact that, as for stigmatic systems, it allows one to write down useful formulae directly. Stepwise calculations of vergence are sometimes performed backward through a system; the advantages and disadvantages described for step-along procedures holds for such step-back procedures as well.
Astigmatism↗
Interconverting the matrix and principal-meridional representations of dioptric power and reduced vergence.
Converting the traditional representation of power as sphere, cylinder and axis to the dioptric power matrix F is usually performed by means of Long's equations and the reverse process by means of Keating's equations. It is sometimes useful to be able to convert directly between the matrix and power expressed in terms of principal powers F1 and F2 along corresponding principal meridians at angles a1 and a2. The equations for interconverting F and the principal-meridional representation expressed as F1(a1)F2 are presented here. Equivalent equations allow direct interconversion of the reduced vergence matrix L and the principal-meridional representation of vergence L1(a1)L2. Vergence becomes infinite at line and point focuses. Similarly effective power and back- and front-vertex power are infinite for some systems. Nevertheless it is possible unambiguously to represent infinite vergence and vertex power in principal-meridional form. However, information is usually lost in these infinite cases when the principal-meridional representation is converted to the matrix representation, and the former is not recoverable from the latter. As a consequence the matrix representation is usually unsatisfactory for vergences and vertex powers that are infinite. On the other hand, the principal-meridional representation of vergence and power is always satisfactory. If one adopts the position that effective powers and vertex powers are really vergences rather than powers then one concludes that the matrix provides a satisfactory representation for powers of thin systems in general but not for vergences. Implied by a vergence at a point is an interval of Sturm. The equations for characterizing the interval from the reduced vergence are presented.
Finite Element Analysis↗
A unified paraxial approach to astigmatic optics.
In Gaussian optics properties such as dioptric power, lateral and angular magnification and thickness are simple scalar concepts. In linear optics, the optics of thick astigmatic systems, however, these concepts generalize to three-dimensional concepts in some cases (the dioptric power of thin systems, for example) and to four-dimensional concepts in general. As a result, the quantitative treatment of these properties in astigmatic systems presents challenges to the researcher in optometry, ophthalmology, and vision science. Considerable progress has been made only in the case of dioptric power. This paper presents a generalized approach to astigmatic optics which allows different physical properties to be treated in the same way: the theory is unified and, in a sense, complete. Mathematical and statistical methods developed for treating one concept become directly applicable to others. The paraxial optical properties of any optical system are completely defined by the 4 x 4 ray transfer matrix, called here the (ray) transference. The transference defines four fundamental properties of an optical system, tentatively called here positional magnification, optical thickness, divergence, and directional magnification. They are the four 2 x 2 submatrices A, B, C, and D of the transference. Each fundamental property is a modification of a familiar concept. Divergence is the negative of dioptric power expressed as the dioptric power matrix F. The four fundamental optical properties A, B, C, and D, and the derived property F, despite being different physically, all have the same underlying mathematical structure. This fact is exploited in developing a unified theory. The theory is complete in the sense that the fundamental properties fully characterize the paraxial optics of any system. The paper presents a general treatment that applies to any of the five properties. The implications are far reaching and extend beyond what can be described in the paper. Dioptric power of thin systems is treated as a particular application of the general theory. The result is the resolution of a number of issues of current interest to the researcher. It is shown, for example, that root-mean-squared (curvital) power, root-mean-squared torsional power, and length of the power vector (or dioptric strength) have a Pythagorean relationship, the power vector being the hypotenuse. Mean-squared curvital and torsional powers are in effect the area enclosed by polar profiles of curvital and torsional power, respectively. The full character of dioptric power cannot be represented by a single vector in the usual sense of the term. Two vectors are required: they are the meridional (vector) power and the orthogonal (vector) power, both of which are associated with the reference meridian. The power along a meridian (often thought of as a scalar or as two scalars) is a vector, the meridional power. This meridional power has components along (the meridional component of the meridional power) and perpendicular to (the orthogonal component of the meridional power) the meridian. In the literature, these components are the curvital power and the negative of the torsional power, respectively. The paper also examines the generalization of these results to the dioptric power of thick systems. Dioptric power is not a fundamental optical property but a derived property. Divergence, the negative of dioptric power, is the corresponding fundamental property. The theory described here is ray-based. The concept of the wavefront is unnecessary. The many formulas and concepts that apply in the context of dioptric power apply directly to the fundamental properties as well. The theory has the potential to provide a complete framework for future studies of astigmatic systems and could systematize the approach to and enhance the knowledge of astigmatism.
Astigmatism↗
Signs of surface torsion and torsional dioptric power.
Although there is agreement in the literature over the magnitude of torsion and torsional dioptric power, there is ambiguity over the signs of those quantities. The purpose of this paper is to define terms in such a way that the ambiguity is removed. Explicit equations are presented for torsion and torsional power along a meridian of a surface. In keeping with common practice in other disciplines, right-handed torsion is chosen to be positive. The components of the dioptric power matrix of thin systems and of the reduced vergence matrix are reinterpreted in terms of curvital and torsional power. In this reinterpretation the off-diagonal components of the matrices remain the torsional power and the reduced torsion along the meridian orthogonal to the reference meridian. However, they become the negatives of those quantities along the reference meridian. In particular, the top-right component can be interpreted as the reduced torsion or the torsional power along the meridian orthogonal to the reference meridian and the bottom-left as the negative of those quantities along the reference meridian. Torsion and torsional power along a meridian, as well as curvature and curvital power, are invariant under change of reference meridian and under spherocylindrical transposition.
Eyeglasses↗
Ray vector fields, prentice's equation, and spherocylindrical lenses.
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Astigmatism↗