The statistics of refractive error maps : managing wavefront aberration analysis without Zernike polynomials

The refractive error of a human eye varies across the pupil and therefore may be treated as a random variable. The probability distribution of this random variable provides a means for assessing the main refractive properties of the eye without the necessity of traditional functional representation of wavefront aberrations. To demonstrate this approach, the statistical properties of refractive error maps are investigated. Closed-form expressions are derived for the probability density function (PDF) and its statistical moments for the general case of rotationally-symmetric aberrations. A closed-form expression for a PDF for a general non-rotationally symmetric wavefront aberration is difficult to derive. However, for specific cases, such as astigmatism, a closed-form expression of the PDF can be obtained. Further, interpretation of the distribution of the refractive error map as well as its moments is provided for a range of wavefront aberrations measured in real eyes. These are evaluated using a kernel density and sample moments estimators. It is concluded that the refractive error domain allows non-functional analysis of wavefront aberrations based on simple statistics in the form of its sample moments. Clinicians may find this approach to wavefront analysis easier to interpret due to the clinical familiarity and intuitive appeal of refractive error maps.

Zernike radial slope polynomials for wavefront reconstruction and refraction

Ophthalmic wavefront sensors typically measure wavefront slope, from which wavefront phase is reconstructed. We show that ophthalmic prescriptions (in power-vector format) can be obtained directly from slope measurements without wavefront reconstruction. This is achieved by fitting the measurement data with a new set of orthonormal basis functions called Zernike radial slope polynomials. Coefficients of this expansion can be used to specify the ophthalmic power vector using explicit formulas derived by a variety of methods. Zernike coefficients for wavefront error can be recovered from the coefficients of radial slope polynomials, thereby offering an alternative way to perform wavefront reconstruction.

Permutation polynomials of the form (xp −x +δ)s +L(x)

Recently, several classes of permutation polynomials of the form (x2 + x + δ)s + x over F2m have been discovered. They are related to Kloosterman sums. In this paper, the permutation behavior of polynomials of the form (xp − x + δ)s + L(x) over Fpm is investigated, where L(x) is a linearized polynomial with coefficients in Fp. Six classes of permutation polynomials on F2m are derived. Three classes of permutation polynomials over F3m are also presented.

PAC-Bayes-empirical-Bernstein inequality

We present PAC-Bayes-Empirical-Bernstein inequality. The inequality is based on combination of PAC-Bayesian bounding technique with Empirical Bernstein bound. It allows to take advantage of small empirical variance and is especially useful in regression. We show that when the empirical variance is significantly smaller than the empirical loss PAC-Bayes-Empirical-Bernstein inequality is significantly tighter than PAC-Bayes-kl inequality of Seeger (2002) and otherwise it is comparable. PAC-Bayes-Empirical-Bernstein inequality is an interesting example of application of PAC-Bayesian bounding technique to self-bounding functions. We provide empirical comparison of PAC-Bayes-Empirical-Bernstein inequality with PAC-Bayes-kl inequality on a synthetic example and several UCI datasets.

Revisiting Polya's summation techniques using a spreadsheet : from addition tables to Bernoulli polynomials

Recurrence relations in mathematics form a very powerful and compact way of looking at a wide range of relationships. Traditionally, the concept of recurrence has often been a difficult one for the secondary teacher to convey to students. Closely related to the powerful proof technique of mathematical induction, recurrences are able to capture many relationships in formulas much simpler than so-called direct or closed formulas. In computer science, recursive coding often has a similar compactness property, and, perhaps not surprisingly, suffers from similar problems in the classroom as recurrences: the students often find both the basic concepts and practicalities elusive. Using models designed to illuminate the relevant principles for the students, we offer a range of examples which use the modern spreadsheet environment to powerfully illustrate the great expressive and computational power of recurrences.

Review article : ecological approach in studying the relation of perception-action : review of the theoretical and empirical data

In daily activities people are using a number of available means for the achievement of balance, such as the use of hands and the co-ordination of balance. One of the approaches that explains this relationship between perception and action is the ecological theory that is based on the work of a) Bernstein (1967), who imposed the problem of ‘the degrees of freedom’, b) Gibson (1979), who referred to the theory of perception and the way which the information is received from the environment in order for a certain movement to be achieved, c) Newell (1986), who proposed that movement can derive from the interaction of the constraints that imposed from the environment and the organism and d) Kugler, Kelso and Turvey (1982), who showed the way which “the degrees of freedom” are connected and interact. According to the above mentioned theories, the development of movement co-ordination can result from the different constraints that imposed into the organism-environment system. The close relation between the environmental and organismic constraints, as well as their interaction is responsible for the movement system that will be activated. These constraints apart from shaping the co-ordination of specific movements can be a rate limiting factor, to a certain degree, in the acquisition and mastering of a new skill. This frame of work can be an essential tool for the study of catching an object (e.g., a ball). The importance of this study becomes obvious due to the fact that movements that involved in catching an object are representative of every day actions and characteristic of the interaction between perception and action.

Departing from the virtual community : a discussion of the ways and means to uncover localised order in online games

This paper is part one of a three part study into the collective regulation processes of players in massive multiplayer online games (MMOG). Traditionally game playing has not been classed as problematic, however with introduction of new media technologies and new ways to play games, certain contexts have become obscure, namely the localised order of ‘playing online’ or how players manage and maintain order between each other as opposed to ‘following the rules’. Principally this paper will examine concepts of ‘virtual community’. These will be illustrated as particularly unhelpful when considering how people conduct themselves in these spaces. Thus, ‘virtual community’ will be seen as critical in implicating various online behaviours as superior to other online behaviours causing obscurity and blurring actions. This obscurity is grounded by strong associations in the virtual community as logic of practise in and of itself; behaviours that fall outside this category become common sense and as such are made invisible for investigation. This paper will draw upon the theories of Basil Bernstein and Pierre Bourdieu to produce a distinction between online behaviours and ultimately make them visible for further investigation. In doing so this paper seeks to form a basis for future research where interaction in these spaces can be identified as belonging to a certain framework to inform the design of online games and applications more effectively.

Modeling corneal surfaces with rational functions for high-speed videokeratoscopy data compression

High-speed videokeratoscopy is an emerging technique that enables study of the corneal surface and tear-film dynamics. Unlike its static predecessor, this new technique results in a very large amount of digital data for which storage needs become significant. We aimed to design a compression technique that would use mathematical functions to parsimoniously fit corneal surface data with a minimum number of coefficients. Since the Zernike polynomial functions that have been traditionally used for modeling corneal surfaces may not necessarily correctly represent given corneal surface data in terms of its optical performance, we introduced the concept of Zernike polynomial-based rational functions. Modeling optimality criteria were employed in terms of both the rms surface error as well as the point spread function cross-correlation. The parameters of approximations were estimated using a nonlinear least-squares procedure based on the Levenberg-Marquardt algorithm. A large number of retrospective videokeratoscopic measurements were used to evaluate the performance of the proposed rational-function-based modeling approach. The results indicate that the rational functions almost always outperform the traditional Zernike polynomial approximations with the same number of coefficients.