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.

Higher-Order spectra of nonlinear polynomial models for Chua's circuit

Polynomial models are shown to simulate accurately the quadratic and cubic nonlinear interactions (e.g. higher-order spectra) of time series of voltages measured in Chua's circuit. For circuit parameters resulting in a spiral attractor, bispectra and trispectra of the polynomial model are similar to those from the measured time series, suggesting that the individual interactions between triads and quartets of Fourier components that govern the process dynamics are modeled accurately. For parameters that produce the double-scroll attractor, both measured and modeled time series have small bispectra, but nonzero trispectra, consistent with higher-than-second order nonlinearities dominating the chaos.

Decomposition construction for secret sharing schemes with graph access structures in polynomial time

The purpose of this paper is to describe a new decomposition construction for perfect secret sharing schemes with graph access structures. The previous decomposition construction proposed by Stinson is a recursive method that uses small secret sharing schemes as building blocks in the construction of larger schemes. When the Stinson method is applied to the graph access structures, the number of such “small” schemes is typically exponential in the number of the participants, resulting in an exponential algorithm. Our method has the same flavor as the Stinson decomposition construction; however, the linear programming problem involved in the construction is formulated in such a way that the number of “small” schemes is polynomial in the size of the participants, which in turn gives rise to a polynomial time construction. We also show that if we apply the Stinson construction to the “small” schemes arising from our new construction, both have the same information rate.

An accuracy assessment of ASTER stereo images-derived digital elevation model by using rational polynomial coefficient model

The along-track stereo images of Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) sensor with 15 m resolution were used to generate Digital Elevation Model (DEM) on an area with low and near Mean Sea Level (MSL) elevation in Johor, Malaysia. The absolute DEM was generated by using the Rational Polynomial Coefficient (RPC) model which was run on ENVI 4.8 software. In order to generate the absolute DEM, 60 Ground Control Pointes (GCPs) with almost vertical accuracy less than 10 meter extracted from topographic map of the study area. The assessment was carried out on uncorrected and corrected DEM by utilizing dozens of Independent Check Points (ICPs). Consequently, the uncorrected DEM showed the RMSEz of ± 26.43 meter which was decreased to the RMSEz of ± 16.49 meter for the corrected DEM after post-processing. Overall, the corrected DEM of ASTER stereo images met the expectations.

Weather variability, tides, and Barmah Forest Virus Disease in the Gladstone region, Australia

In this study we examined the impact of weather variability and tides on the transmission of Barmah Forest virus (BFV) disease and developed a weather-based forecasting model for BFV disease in the Gladstone region, Australia. We used seasonal autoregressive integrated moving-average (SARIMA) models to determine the contribution of weather variables to BFV transmission after the time-series data of response and explanatory variables were made stationary through seasonal differencing. We obtained data on the monthly counts of BFV cases, weather variables (e.g., mean minimum and maximum temperature, total rainfall, and mean relative humidity), high and low tides, and the population size in the Gladstone region between January 1992 and December 2001 from the Queensland Department of Health, Australian Bureau of Meteorology, Queensland Department of Transport, and Australian Bureau of Statistics, respectively. The SARIMA model shows that the 5-month moving average of minimum temperature (β = 0.15, p-value < 0.001) was statistically significantly and positively associated with BFV disease, whereas high tide in the current month (β = −1.03, p-value = 0.04) was statistically significantly and inversely associated with it. However, no significant association was found for other variables. These results may be applied to forecast the occurrence of BFV disease and to use public health resources in BFV control and prevention.

On the zeros of the Abelian integrals for a class of Liénard systems

A planar polynomial differential system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for the general nonlinear dynamical systems. In this paper, we investigated a class of Liénard systems of the form x'=y, y'=f(x)+y g(x) with deg f=5 and deg g=4. We proved that the related elliptic integrals of the Liénard systems have at most three zeros including multiple zeros, which implies that the number of limit cycles bifurcated from the periodic orbits of the unperturbed system is less than or equal to 3.

Mathematical models for describing the shape of the in vitro unstretched human crystalline lens

We developed orthogonal least-squares techniques for fitting crystalline lens shapes, and used the bootstrap method to determine uncertainties associated with the estimated vertex radii of curvature and asphericities of five different models. Three existing models were investigated including one that uses two separate conics for the anterior and posterior surfaces, and two whole lens models based on a modulated hyperbolic cosine function and on a generalized conic function. Two new models were proposed including one that uses two interdependent conics and a polynomial based whole lens model. The models were used to describe the in vitro shape for a data set of twenty human lenses with ages 7–82 years. The two-conic-surface model (7 mm zone diameter) and the interdependent surfaces model had significantly lower merit functions than the other three models for the data set, indicating that most likely they can describe human lens shape over a wide age range better than the other models (although with the two-conic-surfaces model being unable to describe the lens equatorial region). Considerable differences were found between some models regarding estimates of radii of curvature and surface asphericities. The hyperbolic cosine model and the new polynomial based whole lens model had the best precision in determining the radii of curvature and surface asphericities across the five considered models. Most models found significant increase in anterior, but not posterior, radius of curvature with age. Most models found a wide scatter of asphericities, but with the asphericities usually being positive and not significantly related to age. As the interdependent surfaces model had lower merit function than three whole lens models, there is further scope to develop an accurate model of the complete shape of human lenses of all ages. The results highlight the continued difficulty in selecting an appropriate model for the crystalline lens shape.

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.