Dioptric Power Matrices Research Articles

A review of multivariate methods of analysing refractive data with dioptric power matrices

Background: There are three components to refractive state, namely sphere, cylinder and axis. Similarly, central corneal power is also composed of three components, namely the power along the flat meridian, the power along the steep meridian and the axis of the flat meridian. Most studies that investigate refractive data and corneal power analyse each of the three components individually rather than as a trivariate entity. In doing so, pertinent information may inadvertently be omitted. Aim: The purpose of this review is to provide a brief overview of the multivariate statistics that are available to analyse multivariate data such as dioptric power. This will enable readers to better understand research that is analysed using these methods. Method: An extensive review of databases such as Google Scholar, Science Direct and ResearchGate was done to gather publications on the topic of multivariate statistical analysis. Keywords such as multivariate statistical analysis, dioptric power, stereo-pairs, polar profiles and hypothesis testing were used to conduct the search. Results: The debate for the need to analyse dioptric power using multivariate statistical methods has been a long-standing one. For this review, more than 40 publications were analysed to provide a simplified overview of the multivariate statistical methods that can be used to analyse dioptric power. Conclusion: The use of multivariate statistical methods is a valuable tool in analysing and understanding dioptric power holistically and may provide more insights for research involving refractive error and corneal power.

Quantitative analysis of eyes and other optical systems in linear optics.

To show that 14-dimensional spaces of augmented point P and angle Q characteristics, matrices obtained from the ray transference, are suitable for quantitative analysis although only the latter define an inner-product space and only on it can one define distances and angles. The paper examines the nature of the spaces and their relationships to other spaces including symmetric dioptric power space. The paper makes use of linear optics, a three-dimensional generalization of Gaussian optics. Symmetric 2 × 2 dioptric power matrices F define a three-dimensional inner-product space which provides a sound basis for quantitative analysis (calculation of changes, arithmetic means, etc.) of refractive errors and thin systems. For general systems the optical character is defined by the dimensionally-heterogeneous 4 × 4 symplectic matrix S, the transference, or if explicit allowance is made for heterocentricity, the 5 × 5 augmented symplectic matrix T. Ordinary quantitative analysis cannot be performed on them because matrices of neither of these types constitute vector spaces. Suitable transformations have been proposed but because the transforms are dimensionally heterogeneous the spaces are not naturally inner-product spaces. The paper obtains 14-dimensional spaces of augmented point P and angle Q characteristics. The 14-dimensional space defined by the augmented angle characteristics Q is dimensionally homogenous and an inner-product space. A 10-dimensional subspace of the space of augmented point characteristics P is also an inner-product space. The spaces are suitable for quantitative analysis of the optical character of eyes and many other systems. Distances and angles can be defined in the inner-product spaces. The optical systems may have multiple separated astigmatic and decentred refracting elements.

Measurements and an adjoining corneal zone: Effects on the power matrix of a regular astigmatic cornea

A circular zone around the corneal cap is magnified in order to observe the regularity on this anterior surface element of an eye and measure its principal refractive powers. Experiment confirms that powers on smaller zones on the same cornea are augmented for slightly larger zones. This article aims to perturb powers from the smaller zone to associate its exact and paraxial measurements with a central neighbouring zone small enough to reflect paraxial rays. Readings estimated for this slightly larger zone are propagated to their own 2×2 matrix which can be expressed as components that systematically supplement a matrix from the smaller zone. Orders of perturbation parameters distinguish a matrix from the smaller central zone from the other power components and their natures. Matrix components with interesting structures represent the sensitivity of the power matrix to changes in principal powers only or to principal meridians only or to both. An analytic method for bounds estimated around a host of clinical standard measurements to be propagated to their dioptric power matrices is established to also facilitate calculations for quality in clinical measurements. A numerical example supports the techniques discussed.

Inner-product spaces for quantitative analysis of eyes and other optical systems

Because dioptric power matrices of thin systems constitute a (three-dimensional) inner-product space, it is possible to define distances and angles in the space and so do quantitative analyses on dioptric power for thin systems. That includes astigmatic corneal powers and refractive errors. The purpose of this study is to generalise to thick systems. The paper begins with the ray transference of a system. Two 10-dimensional inner-product spaces are devised for the holistic quantitative analysis of the linear optical character of optical systems. One is based on the point characteristic and the other on the angle characteristic; the first has distances with the physical dimension L−1 and the second has the physical dimension L. A numerical example calculates the locations, distances from the origin and angles subtended at the origin in the 10-dimensional space for two arbitrary astigmatic eyes.

The short-term effects of PMMA contact lens wear on keratometric behaviour: results for a single keratoconic cornea*

Keratometric behaviour is a multifaceted issue that many researchers have investigated for years. Many internal and external influences can have an effect on the cornea’s keratometric behaviour. This investigation forms a small part of a larger study that aims at determining the effects that rigid contact lenses might have on keratometric behaviour. This pilot study examined the keratometric behaviour of a single, mildly keratoconic cornea that was fitted with a polymethyl methacrylate (PMMA) contact lens. Sixty successive auto-keratometric measurements were taken immediately before and immediately after three hours of contact lens wear. The data obtained was transformed to dioptric power matrices and were analysed using multivariate statistical methods. This study showed that, at least in one keratoconic cornea, there appeared to be a statistically significant change in corneal curvature under the influence of a PMMA contact lens. The contact lens had also appeared to decrease variation in corneal curvature. There was no control study done on this eye therefore the effects of diurnal variation, if any, could not be established. (S Afr Optom 2010 69(2) 69-76)

Thin lenses of asymmetric power

It is generally supposed that thin systems, including refracting surfaces and thin lenses, have powers that are necessarily symmetric. In other words they have powers which can be represented assymmetric dioptric power matrices and in the familar spherocylindrical form used in optometry and ophthalmology. This paper shows that this is not correct and that it is indeed possible for a thin system to have a power that is not symmetric and which cannot be expressed in spherocylindrical form. Thin systems of asymmetric power are illustratedby means of a thin lens that is modelled with small prisms and is chosen to have a dioptric power ma-trix that is antisymmetric. Similar models can be devised for a thin system whose dioptric power matrix is any 2 2 × matrix. Thus any power, symmetric, asymmetric or antisymmetric, is possible for a thin system. In this sense our understanding of the power of thin systems is now complete.

Optical Magnification Matrix: Near Objects in the Paraxial Case

For appropriate patient management understanding, the problems due to the change of image size in the near magnification matrix of spectacles are important. The purpose here is to derive an equation for the paraxial 2 x 2 magnification matrix of spherocylindrical two-surface optical instruments (spectacles) in front of an eye at a finite object distance. A 2 x 2 stepalong method for the contribution factors to the magnification matrix is developed, starting from a definition within a ray-tracing picture before translating it into quadratic-order wave-tracing concepts. Extensive use is made of the wave-tracing description of propagating wavefronts by 2 x 2 vergence matrices for deriving an expression for the magnification matrix in ophthalmic terms. A 2 x 2 magnification matrix as a function of the dioptric power matrices of the interfaces as well as the object distance is given explicitly. The expression is an intuitive product representation possessing the familiar appearance of a shape factor times a power factor. Additionally, however, the result contains now a scaling prefactor depending on the finite object distance, which would not be present for objects at infinity. The result is a generalization of the formerly known result for the paraxial magnification matrix for infinite object distance. The effect of the finite object distance correction is small but not negligible. Moreover, the methods for deriving this result serve as a starting point for an even more general treatment of oblique incidence in a future work.

Optical Magnification Matrix: Near Objects and Strongly Oblique Incidence

In every day spectacle use relevant parts of a spectacle lens involve large gaze angles and lie outside the paraxial region. The purpose here is to derive a general equation for the local 2 x 2 magnification matrix of an arbitrary K-surface optical instrument in front of an eye for any given gaze direction. The general case comprises a finite object distance as well as astigmatic surfaces and strongly oblique incidence angles of the involved rays. A 2 x 2 stepalong method for the contribution factors to the magnification matrix is developed that allows to treat oblique rays, which propagate through a system possessing tilted refracting interfaces. Extensive use is made of the wave-tracing method to describe propagating wavefronts by 2 x 2 vergence matrices for the corresponding general oblique case. An analytical 2 x 2 matrix as a function of the ray geometry and the dioptric power matrices of the interfaces in the intersection points is given explicitly. The expression is an intuitive product representation in which all the usual factors occur, which also in the paraxial case would correspond to the gaps between two subsequent surfaces. Additionally, however, the result contains now exact matrix-valued correction factors, which take the oblique incidences at each interface into account, and which would not be present in the paraxial case. The result is a generalization of any of the formerly known results for the magnification matrix in the paraxial case. The treatment is complete in the sense that it covers any situation that can be encountered when light propagates through an optical system with a finite number of refracting surfaces to an eye. In particular, the oblique case is treated exactly even if the chief ray lies beyond the domain in which the transference theory of linear optics could be applied.

Dihedral representations and statistical geometric optics II: elementary optical instruments

The interpretation of the Stokes parameters as radiation field data indexed by the algebra associated with the Pauli matrices is extended to two-dimensional representations of the dihedral group algebras with the purpose of utilizing the underlying vector space and the tools of dihedral Fourier analysis for statistical and analytical interpretations of the data indexed by the elements (planar rotations and reversals) of the group algebra. The one-to-one correspondence between a coherence matrix and its Stokes parameters determined by the Pauli algebra is carried out to a one-to-one correspondence between any 2×2 linear operator and the coefficients of rotations and reversals via dihedral Fourier calculus. Examples of applying the group algebra formulation to coherence and dioptric power matrices and other linear optics instruments are discussed. A graphical method for displaying and interpreting data indexed by rotations and reversals is introduced and applied to dioptric power matrices.

The concept of the average eye

For most quantitative studies one needs to calculate an average. In the case of refraction an average is readily computed as the arithmetic average of dioptric power matrices. Refraction, however, is only one aspect of the first-order optical character of an eye. The question is: How does one determine an average that rep-resents the average optical character of a set of eyes completely to first order? The exponen-tial-mean-log transference has been proposed recently but it is not without its difficulties. There are four matrices, naturally related to the transference and called the characteristics or characteristic matrices, whose mathematical features suggest that they may provide alterna-tive solutions to the problem of the average eye. Accordingly the purpose of this paper is to propose averages based on these characteristics, to examine their nature and to calculate and compare them in the case of a particular sample of 30 eyes. The eyes may be stigmatic or astig-matic and component elements may be centred or decentred. None turns out to be a perfect average. One of the four averages (that based on one of the two mixed characteristics) is proba-bly of little or no use in the context of eyes. The other three, particularly the point-characteristic average, seem to be potentially useful.

The average eye

For statistical and other purposes one needs to be able to determine an average eye. An average of refractive errors is readily calculated as an average of dioptric power matrices. A refractive error, however, is not so much a property of the eye as a property of the compensating lens in front of the eye. As such, it ignores other aspects of the optical character of the eye. This paper discusses the difficulties of finding a suitable average that fully accounts for the first-order optics of a set of optical systems. It proposes an average based on ray transferences and logarithms and exponentials of matrices. Application to eyes in particular is discussed.

Asymmetric dioptric power matrices and corresponding thick lenses.

When thickness or separation is taken into account, the dioptric power of systems with obliquely crossed toric surfaces (OCTS), including some corrected and uncorrected human eyes, can be represented by a four-dimensional vector space, provided that an arbitrary 2 x 2 asymmetric matrix is the equivalent power matrix of some optical system. The provision is crucial for the statistical validity of dioptric power matrix methods that take account of thickness. Previous efforts to verify the provision have not been satisfactory. To verify the provision. Prove that an arbitrary 2 x 2 asymmetric matrix is the equivalent power for a thick bitoric lens with obliquely crossed surface. The procedure and new equations needed to find the lens. The provision holds, which is an important foundation block for the validity of the statistical matrix methods for thick astigmatic systems. The resulting equations also have optical uses other than statistics.

Keating??s Asymmetric Dioptric Power Matrices Expressed in Terms of Sphere, Cylinder, Axis, and Asymmetry

Twelve years ago Keating pointed out that dioptric powers existed which could not be represented by the familiar three parameters sphere, cylinder, and axis. They are the equivalent powers of optical systems (including many eyes) with separated obliquely crossing astigmatic elements. Four parameters are required to represent such powers, and all four are unfamiliar to most clinicians and researchers. This note shows that it is, in fact, possible to transform the four parameters so that the three familiar parameters are retained and only one (called asymmetry) remains unfamiliar. The consequence is that it is always possible to represent a power by means of sphere, cylinder, axis, and asymmetry. Powers commonly used in practice all have asymmetry equal to zero which is why only the first three are usually necessary. Powers, however, do exist, and are of potential interest in optometry, for which asymmetry is not zero and cannot be omitted from the representation. Two numerical examples are given, including Keating's model eye.

Statistical inference on mean dioptric power: asymmetric powers and singular covariance

Methods have been developed recently for testing hypotheses on mean dioptric power and for constructing confidence regions in situations that are most likely to be encountered. In this paper the methods are extended to make the analysis complete. A new situation covered specifically is that of dioptric power not of the form sphere/cylinder x axis. Such powers, termed asymmetric powers because the dioptric power matrices are asymmetric, include the equivalent power of a thick obliquely crossed bitoric lens. A second situation is that in which the covariance matrix of the sample of powers is singular. Symmetric dioptric power (the more familiar form of power) can be represented by a point in three-dimensional space. In general, however, dioptric power is four dimensional in character. Singularity of covariance arises when variation in the sample is limited to a subspace of dimension less than the full three or four. The space spanned by the sample is called the range space of the sample. The dimension of the range space may be four, three, two, one or zero. Each case is considered in turn. Numerical examples of hypothesis testing are presented in range spaces of dimension four to one. The test statistic devised for each case also gives the equation of the confidence region about the mean of a sample of dioptric powers. Singularity can sometimes be avoided merely by taking larger samples and by taking more accurate readings. The problem of near singularity is briefly discussed. The paper allows basic hypothesis testing on mean dioptric power and the construction of confidence regions in all possible circumstances.

Lens effectivity in terms of dioptric power matrices.

An effective dioptric power matrix is defined. The effective dioptric power matrix can be calculated directly from the lens dioptric power matrix, its trace, and its determinant, without using the sphere, cylinder, and axis parameters of the lens. This greatly simplifies any calculations in which dioptric power matrices are obtained in intermediate steps, including separated obliquely crossed spherocylindrical lens calculations. The generalization of the effective dioptric power matrix to a reduced vergence matrix is also discussed.