Dioptric strength: a scalar representation of dioptric power.
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Lenses↗
Biomedical subjects
W F Harris
Publications and source records attributed to W F Harris.
Astigmatic optical systems with separated and prismatic or noncoaxial elements: system matrices and system vectors.
A general system of noncoaxial separated astigmatic optical elements is examined. Allowance is made for prismatic elements including decentered lenses and refracting surfaces, prisms, and plane interfaces. The directions and positions of a ray entering and emerging from the system are related by a system matrix and a system vector. Given an incident ray one can readily obtain the emergent ray. The analysis is of fundamental importance for visual optics. Two numerical examples are presented. One is a model eye with astigmatic and obliquely crossing astigmatic and decentered surfaces.
Astigmatism↗
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.
Humans↗
Ocular rotation factor and orientational demand with astigmatic lenses.
Ocular movements required in most visual functions are affected by lenses in front of the eye. Fundamental to an understanding of these movements are concepts such as the ocular rotation factor and the ocular orientational demand. In this paper the ocular rotation factor for thin spherical lenses is generalized for astigmatic lenses. A compact matrix equation is obtained for the orientational demand in the case of such lenses. The equations are satisfactory in ordinary situations but they break down under certain circumstances. Equations are derived for these special circumstances in particular and also for all circumstances in general.
Eye Movements↗
Relation between meridional power and the dioptric power matrix.
The components of meridional power in any meridian can be expressed elegantly in terms of the dioptric power matrix. The form of the expressions is that which is well known in the statistical literature as a quadratic form; there are important statistical implications for dioptric power and its measurement. The relation forms the basis of the calculation and least-squares estimation of dioptric power and surface curvature from meridional measurements.
Humans↗
Estimation of dioptric power from measurements of meridional power and curvature, sagitta, lens thickness, and prismatic effect.
A method published elsewhere for estimating dioptric power from meridional measurements is generalized here to allow for measurements of sagitta, lens thickness, and prismatic effect as well. Measurements may be of only one type or of combinations of types. Any number of measurements may be used. Within the usual limitations of first-order optics the method always works and gives all solutions when solutions exist. When no solutions exist it gives approximate solutions called least-squares estimates. At least one least-squares estimate always exists. A wide class of ostensibly distinct problems reduces to a single standard routine that is easy to execute with matrix-handling software. If there is reason to believe that the dioptric power is of some particular form then constrained estimation may be applied to find that form.
Eyeglasses↗
Joint distribution of sphere, cylinder, and axis.
A mathematical expression for the joint probability density function of sphere, cylinder, and axis is presented for the first time. It holds under certain circumstances. A relation is derived which may be useful in obtaining the joint probability function when that expression does not hold. The joint probability density function is of fundamental importance in all quantitative studies of dioptric power in the form sphere, cylinder, and axis. Some of its properties are described. It represents a hypersurface in a four-dimensional space with a maximum, a saddle point, and variety of ridges.
Adult↗
Testing hypotheses on dioptric power.
Transformation of dioptric power from the conventional clinical representation as sphere, cylinder, and axis to the vectorial representation known as the vector h makes it possible to apply formal multivariate statistical methods to dioptric power. Methods are described for testing hypotheses on mean dioptric power and on variance-covariance of dioptric power for one and more than one population. A number of numerical examples are presented. Attention is drawn to the underlying assumption of multivariate normality, to other limitations of the methods, and to pitfalls in the use of the methods. Three-dimensional scatter plots, shown by means of stereo-pair drawings, are useful for detecting departures from normality.
Humans↗
Estimation of dioptric power from measurements that may include measurements of dioptric power.
A general routine exists for estimating dioptric power from any number of measurements of sagitta, lens thickness, meridional curvature or power, and prismatic effect. This paper shows how the routine is modified when among the measurements there are measurements of dioptric power itself.
Humans↗
Trajectories of changing refractive status.
Refractive status can be represented as a point plotted on a three-dimensional graph. As the refractive status changes a curve is traced out on the graph. Trajectories of changing refractive status are illustrated in a number of representative ideal cases. The three-dimensional graphs are shown by means of stereo-pairs. Trajectories are constructed from clinical data and interpreted with reference to the ideal examples. They show changing refractive status with age, after cataract extraction, and after radial keratotomy. The trajectories may prove to be a useful tool for both clinician and researcher. They show overall trends and allow prediction of future changes. Certain types of trajectories could prove to be characteristic of certain conditions and thus be useful for diagnosis. Trajectories allow monitoring of change after intervention.
Adolescent↗
Meridional profiles of variance-covariance of dioptric power. Part 1. The basic theory.
It has recently become possible to calculate and represent variation or spread of dioptric power in a meaningful way. This is important for the proper analysis and interpretation of data on dioptric power in a number of areas of the vision sciences. The representation takes the form of a symmetric matrix of six (usually) variances and covariances. Although the matrix is satisfactory for several formal statistical purposes, such as the testing of hypotheses, it does not give an intuitively satisfactory picture of the extent and nature of the variation, nor is it easy to interpret in a way that could be useful to the researcher or clinician. A useful graphical representation of variation can be constructed from the variance-covariance matrix. It consists of curves that show the meridional dependence of the variation. These meridional profiles of variation give a complete and intuitively satisfactory picture of the nature and extent of the variation of power and are potentially of general use to researcher and clinician. A complete theoretical basis is provided for the construction of meridional profiles of variation of dioptric power. An accompanying paper employs the theory to construct profiles for a number of representative samples of dioptric power.
Analysis of Variance↗
Meridional profiles of variance-covariance of dioptric power. Part 2. Profiles representing variation in one or more of sphere, cylinder and axis.
Meridional profiles of variation of dioptric power are constructed. Using the basic theory developed in an accompanying paper, samples are selected in a systematic way to illustrate variation only in sphere, only in cylinder and only in axis, and in all possible combinations of sphere, cylinder and axis. For each of the seven samples, scatter plots are constructed together with ellipsoids that represent the estimated distribution of powers in the population from which the sample was taken. The surfaces of the ellipsoids are surfaces of constant probability density within which 95% of the population is calculated to lie. The scatter plots and distribution ellipsoids are plotted in a three-dimensional space called h-space. Meridional profiles of variation are constructed for each of the samples. Properties of the profiles are discussed. Meridional profiles are also presented for eyes before and after radial keratotomy. Among other things, the profiles show meridians of greatest and least variation and are intuitively satisfying. They are potentially useful for the researcher and clinician including the surgeon. They may help to improve surgical and therapeutic techniques. Certain patterns may prove to be characteristic of physiological or pathological conditions, in which case meridional profiles may have use as a diagnostic tool.
Analysis of Variance↗
Analysis of refractive status of Hong Kong Chinese infants.
Recently developed methods of quantifying changing dioptric power are applied to the refractive status of Hong Kong Chinese infants from about 10 to about 40 weeks of age. The analysis confirms that hyperopia decreases during the period and that with-the-rule astigmatism predominates. Change, however, is not constant over the period. The rate of decrease of hyperopia slows down and the refractive status becomes more spherical. The mean refractive status is calculated at 10, 20, 30 and 40 weeks. The spread of refractive status is represented by variance-covariance matrices calculated for each group and 95% confidence ellipsoids on the mean are constructed. The variance-covariance exhibits little change over the period. The right and left eyes show similar behaviour.
Aging↗
Keratoreformation by contact lenses after radial keratotomy: a re-analysis.
A recent paper analysed change in refraction and corneal curvature associated with contact lens wear in patients who had had radial keratotomy at least a year beforehand. Conventional methods of analysis were used. This paper applies methods that have only recently become available. The results are clearer and less ambiguous. Formally, the analysis shows that there are significant mean changes in refraction (estimated to be +1.69/-0.50 x 105) and corneal power (-0.66/-0.25 x 155). In spite of the difference between these two mean changes, the analysis shows that the change in refraction may be associated with change in the corneal power alone. There is no reason to believe that any other change is occurring in the eye. Care needs to be executed in assigning a causative role to contact lenses in changing refraction and corneal curvature. Almost certainly the lenses do have such a role but the analysis does not formally allow an unequivocal conclusion concerning that role.
Anthropometry↗
Calculation and least-squares estimation of surface curvature and dioptric power from meridional measurements.
The calculation or estimation of dioptric power or surface curvature from meridional measurements is an important problem with applications in a number of areas. By phrasing the problem in terms of matrices and applying new results in the mathematics of matrices one is able to reformulate the problem into a standard form that is well known in mathematics and statistics. As a result the solution can be written down directly. The method copes with any number of measurements along any number of meridians, including repeated measurements along the same meridian, and is more general than methods previously proposed. It gives least-squares and best estimates of the true surface curvature or dioptric power. Being phrased in standard statistical terms the method lends itself readily to extension to related types of problems such as least-squares estimation under certain types of constraints. Matrix methods employed in this paper are likely to find wider application in optometry and the vision sciences.
Mathematics↗
Constrained least-squares estimation of surface curvature and dioptric power from meridional measurements.
A dioptric power or surface curvature may be presumed or known to be of some particular type, for example spherical, or astigmatic and with a particular nearest equivalent sphere. Such knowledge or presumption represents a constraint on the power or curvature. This paper shows how meridional measurements can be used to obtain least-squares estimates of surface curvature or dioptric power under the constraint.
Mathematics↗
Summary of the workshop on Drug Development, Biological Diversity, and Economic Growth.
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Developing Countries↗