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.

Describing ocular aberrations with wavefront vergence maps

A common optometric problem is to specify the eye’s ocular aberrations in terms of Zernike coefficients and to reduce that specification to a prescription for the optimum sphero-cylindrical correcting lens. The typical approach is first to reconstruct wavefront phase errors from measurements of wavefront slopes obtained by a wavefront aberrometer. This paper applies a new method to this clinical problem that does not require wavefront reconstruction. Instead, we base our analysis of axial wavefront vergence as inferred directly from wavefront slopes. The result is a wavefront vergence map that is similar to the axial power maps in corneal topography and hence has a potential to be favoured by clinicians. We use our new set of orthogonal Zernike slope polynomials to systematically analyse details of the vergence map analogous to Zernike analysis of wavefront maps. The result is a vector of slope coefficients that describe fundamental aberration components. Three different methods for reducing slope coefficients to a spherocylindrical prescription in power vector forms are compared and contrasted. When the original wavefront contains only second order aberrations, the vergence map is a function of meridian only and the power vectors from all three methods are identical. The differences in the methods begin to appear as we include higher order aberrations, in which case the wavefront vergence map is more complicated. Finally, we discuss the advantages and limitations of vergence map representation of ocular aberrations.

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.

Distributed orthogonal space-time block codes with adaptive diversity gain

In this paper we present a novel distributed coding protocol for multi-user cooperative networks. The proposed distributed coding protocol exploits the existing orthogonal space-time block codes to achieve higher diversity gain by repeating the code across time and space (available relay nodes). The achievable diversity gain depends on the number of relay nodes that can fully decode the signal from the source. These relay nodes then form space-time codes to cooperatively relay to the destination using number of time slots. However, the improved diversity gain is archived at the expense of the transmission rate. The design principles of the proposed space-time distributed code and the issues related to transmission rate and diversity trade off is discussed in detail. We show that the proposed distributed space-time coding protocol out performs existing distributed codes with a variable transmission rate.

Model-order selection in Zernike polynomial expansion of corneal surfaces using the efficient detection criterion

Corneal-height data are typically measured with videokeratoscopes and modeled using a set of orthogonal Zernike polynomials. We address the estimation of the number of Zernike polynomials, which is formalized as a model-order selection problem in linear regression. Classical information-theoretic criteria tend to overestimate the corneal surface due to the weakness of their penalty functions, while bootstrap-based techniques tend to underestimate the surface or require extensive processing. In this paper, we propose to use the efficient detection criterion (EDC), which has the same general form of information-theoretic-based criteria, as an alternative to estimating the optimal number of Zernike polynomials. We first show, via simulations, that the EDC outperforms a large number of information-theoretic criteria and resampling-based techniques. We then illustrate that using the EDC for real corneas results in models that are in closer agreement with clinical expectations and provides means for distinguishing normal corneal surfaces from astigmatic and keratoconic surfaces.

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.

Multiphase experiments with at least one later laboratory phase. I. orthogonal designs

The paper provides a systematic approach to designing the laboratory phase of a multiphase experiment, taking into account previous phases. General principles are outlined for experiments in which orthogonal designs can be employed. Multiphase experiments occur widely, although their multiphase nature is often not recognized. The need to randomize the material produced from the first phase in the laboratory phase is emphasized. Factor-allocation diagrams are used to depict the randomizations in a design and the use of skeleton analysis-of-variance (ANOVA) tables to evaluate their properties discussed. The methods are illustrated using a scenario and a case study. A basis for categorizing designs is suggested. This article has supplementary material online.

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.

Ideal threshold schemes from orthogonal arrays

The work investigates the design of ideal threshold secret sharing in the context of cheating prevention. We showed that each orthogonal array is exactly a defining matrix of an ideal threshold scheme. To prevent cheating, defining matrices should be nonlinear so both the cheaters and honest participants have the same chance of guessing of the valid secret. The last part of the work shows how to construct nonlinear secret sharing based on orthogonal arrays.

λ-orthogonal pericyclic macromolecular photoligation

A photochemical strategy enabling λ-orthogonal reactions is introduced to construct macromolecular architectures and to encode variable functional groups with site-selective precision into a single molecule by the choice of wavelength. λ-Orthogonal pericyclic reactions proceed independently of one another by the selection of functional groups that absorb light of specific wavelengths. The power of the new concept is shown by a one-pot reaction of equimolar quantities of maleimide with two polymers carrying different maleimide-reactive endgroups, that is, a photoactive diene (photoenol) and a nitrile imine (tetrazole). Under selective irradiation at λ=310–350 nm, any maleimide (or activated ene) end-capped compound reacts exclusively with the photoenol functional polymer. After complete conversion of the photoenol, subsequent irradiation at λ=270–310 nm activates the reaction of the tetrazole group with functional enes. The versatility of the approach is shown by λ-orthogonal click reactions of complex maleimides, functional enes, and polymers to the central polymer scaffold.

Populations of models, experimental designs and coverage of parameter space by Latin Hypercube and Orthogonal Sampling

In this paper we have used simulations to make a conjecture about the coverage of a t-dimensional subspace of a d-dimensional parameter space of size n when performing k trials of Latin Hypercube sampling. This takes the form P(k,n,d,t) = 1 - e^(-k/n^(t-1)). We suggest that this coverage formula is independent of d and this allows us to make connections between building Populations of Models and Experimental Designs. We also show that Orthogonal sampling is superior to Latin Hypercube sampling in terms of allowing a more uniform coverage of the t-dimensional subspace at the sub-block size level. These ideas have particular relevance when attempting to perform uncertainty quantification and sensitivity analyses.

Difference covering arrays and pseudo-orthogonal Latin squares

A pair of Latin squares, A and B, of order n, is said to be pseudo-orthogonal if each symbol in A is paired with every symbol in B precisely once, except for one symbol with which it is paired twice and one symbol with which it is not paired at all. A set of t Latin squares, of order n, are said to be mutually pseudo-orthogonal if they are pairwise pseudo-orthogonal. A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007. In this paper we develop row complete MNOLS from difference covering arrays. We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146.