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W F Harris

Publications and source records attributed to W F Harris.

At least 91 records · Page 5Linked to original sources
PubMed1991

Generalizing Long's inversion of the matrix form of Prentice's equation and the concept of generalized inverse dioptric power.

The concept of generalized inverse dioptric power is introduced. Its use is illustrated in solving Long's matrix generalization of Prentice's equation for the decentration required to produce a specified prismatic effect. The solution holds in all cases for which a solution exists, including for the pure cylinder, and the result gives all solutions when more than one exists.

Eyeglasses↗
PubMed1991

Solving the matrix form of Prentice's equation for dioptric power.

The matrix form of Prentice's equation is solved completely for dioptric power. Solutions in matrix form are presented. For any specified position on a lens and any specified prismatic effect at that position one can calculate a particular power consistent with the specified information. One can also determine the complete set of consistent powers. The purpose of the paper is essentially theoretical: it is to complete the mathematical basis for answering any conceivable question concerning Prentice's equation.

Eyeglasses↗
PubMed1991

The generalized Prentice equation and the matrix equation for lens thickness solved simultaneously for dioptric power.

This paper solves the problem of finding the power of a lens which has given prismatic effect and thickness at a given point. Depending on the conditions there may be no solution or an infinity of solutions. An equation for all the solutions (when they exist) is derived. The paper makes use of matrix results that are new and that are likely to have more general application in ophthalmic optics and vision science. In particular the matrix Sp of Fang and Xu is introduced.

Humans↗
PubMed1991

Ellipsoidal confidence regions for mean refractive status.

Confidence regions for mean dioptric power can be represented by ellipsoids in a three-dimensional space called h-space. A convenient graphical representation is provided by stereo-pair drawings that, at a glance, show the estimated accuracy with which the mean power is known. As an example, a sample of autorefractor measurements made on one eye were used to construct a 95% confidence ellipsoid for the mean objective refractive status of that eye. Key dimensions of the ellipsoid are compared with those of another eye. The analysis is of fundamental importance for scientific studies of variation and change of refractive status. Its potential clinical importance is also discussed.

Adult↗
PubMed1991

Thickness extrema at the edge of shaped lenses: the general problem and the solution for a straight edge obtained by means of direct elimination and of Lagrange multipliers.

In general thickness varies along the cut edge of a lens. There is some interest in being able to predict the locations of the thickest and thinnest points on the edge. One purpose of this paper was to formulate the general problem mathematically of finding the locations of thickness extrema on the edge of an arbitrary lens cut into an arbitrary shape. A second purpose was to illustrate how the problem can be solved. In particular, the problem is solved completely and explicitly for what is probably the simplest case, the straight cut edge. A component-free matrix expression for the position of the extremum is derived by employing Lagrange multipliers and the concept of the generalized inverse of a matrix. The equation applies to spheres, cylinders and sphero-cylinders and allows for the presence of prism as well. Along some edges the thickness is constant. Along other edges the thickness varies linearly; there are no extrema except for the practical extrema at the two ends of the edge: thickest at the one and thinnest at the other. The mathematical conditions for these two cases are presented. Equivalent to the matrix equation for the position vector of the thickness extremum is a pair of scalar equations expressed in terms of components of the matrices. The pair is useful when locating extrema using manual calculations. On the other hand, the component-free matrix equation is useful in other circumstances such as for mathematical manipulations and when using computer software that handles matrices. The former is used in a number of numerical examples; the latter was used to check the answers by computer. The mathematical techniques described here are likely to find application in other areas of ophthalmic and visual optics.

Lenses↗
PubMed1991

Representation of dioptric power in Euclidean 3-space.

Every dioptric power of the usual form sphere/cylinder x axis may be represented by means of a point in three-dimensional space. Graphical representation of data in this manner is important for statistical analysis. In particular, graphical representation may be used to display confidence regions about a mean power, for example. A disadvantage of these representations, however, is that simply changing the reference meridian for cylinder axes changes distances in the space and, therefore, changes the shapes of the confidence regions. Because the shapes define the nature of the variation and give, in particular, the principal components of variation, a researcher who happens to measure the orientation of cylinder axes from the 20 degree meridian, for example, instead of the conventional horizontal meridian, could be led to different statistical conclusions. The implication is that such conclusions are unlikely to have much physical meaning. A representation is described here in which distances and shapes do not depend on the meridian that happens to be chosen as reference. Each dioptric power is represented by a point in Euclidean 3-space. Several examples of graphical representation are given. The spherical powers occupy a particular line in the space, the Jackson crossed cylinders occupy a plane, the cylindrical powers occupy a cone, and so on, for all types of conventional dioptric power. These lines and surfaces are illustrated. The statistical implications are discussed briefly. The representation satisfies the requirements of the statistics and is proposed as the standard one for future use.

Lenses↗
PubMed1991

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.

Multivariate Analysis↗
PubMed1991

The distribution of dioptric power: ellipsoids of constant probability density.

A sample from a population of dioptric powers may be used to estimate the distribution of dioptric powers in the population itself. This paper describes the method and shows further how one can obtain a graphical representation of the distribution. The graphical representation takes the form of ellipsoids of constant probability density. The centroid of each ellipsoid estimates the mean of the population while the size, shape and orientation show the extent and nature of the spread of the population. For illustrative purposes the theory is applied to measurements of refractive status before and after radial keratotomy. The ellipsoids are presented as stereo-pairs. They are useful for comparative and predictive purposes. Thus the ellipsoid that contains 95% of the population after surgery defines the set of refractive errors within which the refractive error of a particular eye can be expected to fall with a probability of 95%.

Humans↗
PubMed1990

Elements of the dioptric power matrix and the concept of torsional power: a reinterpretation.

The elements of the dioptic power matrix have previously been interpreted in terms of power along particular meridians with the off-diagonal elements interpreted as a new form of dioptric power called torsional power. This paper shows that such new concepts are not necessary. Decomposition of the matrix provides an interpretation in terms that are quite familiar. The elements represent three component powers: the diagonal elements are two cylinders and the off-diagonal elements a crossed-cylinder with axes at 45 degrees and 135 degrees. The component crossed-cylinder obviates the need for the concept of torsional power. The concept of meridional power is avoided.

Mathematics↗
PubMed1990

Statistical inference on mean dioptric power: hypothesis testing and confidence regions.

It has not hitherto been possible to apply formal methods of statistical analysis to data on dioptric powers. The solution to the basic statistical problem is now provided in this paper. Recognition of the matric-variate nature of dioptric power allows calculation of sample means and variance-covariances. These in turn can be used to calculate a statistic for testing hypotheses on population means and for obtaining confidence regions for those means. In a graphical representation of dioptric power the confidence region turns out to be an ellipsoid centred on the mean of the sample of dioptric powers. The theory is illustrated by means of numerical examples. Singularity of the variance-covariance matrix may occur especially when the sample is small. When it does occur it is the cause of some difficulty in applying the statistics. Nevertheless singularity is rare in practical situations and can usually be avoided simply by increasing the size of the sample. Singularity, therefore, is not treated fully in this paper. Dioptric power is essentially four-dimensional in character but in practice a three-dimensional subspace is almost always sufficient. To avoid the difficulty of having to represent four-dimensional shapes and to avoid the complication of singularity (which is the rule rather than the exception in practice in four-space) only the common three-dimensional problem is considered in detail.

Analysis of Variance↗
PubMed1990

Comparison of dioptric power.

The important formal process of inferring properties of populations from statistics of samples requires a method by which variables can be compared or ranked. A suitable method of comparing dioptric powers is described. The method recognizes the essentially four-dimensional nature of dioptric power and is sufficiently general to allow for comparison of powers (for example, equivalent powers of general thick bitorics) that cannot be represented in the usual manner, sphere/cylinder x axis.

Eyeglasses↗
PubMed1990

Direct, vec and other squares, and sample variance-covariance of dioptric power.

Matrices can be multiplied in several ways. As a result one can define a number of distinct squares of the dioptric power matrix. Additional squares can be defined for matrices vectorized by means of the vec and vech operators. These various squares can form the basis for the definition of variance and covariance of samples of dioptric powers. The complete form of the variance of dioptric powers is a 4 x 4-matrix (a variance-covariance matrix) with 10 distinct elements; four of them represent the variances of the four elements of the dioptric power matrix and the other six are the covariances between those four elements. For thin systems there are only six distinct elements of the variance-covariance matrix, three of which are variances and three covariances. Examples are included that show the calculation of the different types of squares and the variance-covariance matrix for a sample of equivalent powers of thick bitoric lenses and for a sample of powers of thin systems, including conventional refractive errors. Variances calculated in the past, including those of nearest equivalent spheres and Gartner's and Churm's variances turn out to be components (or combinations of components) of the generalized variance defined here. They are valid as far as they go but they do not completely represent the dispersion of a sample of dioptric powers. The complete variance-covariance matrix does represent the dispersion fully and thus opens the way for the formal statistical analysis of measurements of dioptric power.

Analysis of Variance↗
PubMed1989

The sagitta and lens thickness: the exact solution and a matrix approximation for lenses with toric, spherical, and cylindrical surfaces.

The exact equation for sagitta of spherical surfaces is generalized to toric surfaces which include spherical and cylindrical surfaces as special cases. Lens thickness, therefore, can be calculated accurately anywhere on a lens even in cases of extreme spherical and cylindrical powers and large diameters. The sagittae of tire- and barrel-form toric surfaces differ off the principal meridians, as is shown by a numerical example. The same holds for pulley- and capstan-form toric surfaces. A general expression is given for thickness at an arbitrary point on a toric lens. Approximate expressions are derived and re-expressed in terms of matrices. The matrix provides an elegant means of generalizing equations for spherical surfaces and lenses to toric surfaces and lenses.

Eyeglasses↗