Projective Group Representations in Quaternionic Hilbert Space

We extend the discussion of projective group representations in quaternionic Hilbert space which was given in our recent book. The associativity condition for quaternionic projective representations is formulated in terms of unitary operators and then analyzed in terms of their generator structure. The multi--centrality and centrality assumptions are also analyzed in generator terms, and implications of this analysis are discussed.Comment: 16 pages, no figures, plain Te

Optimizing land use decision-making to sustain Brazilian agricultural profits, biodiversity and ecosystem services

AbstractDesigning landscapes that can meet human needs, while maintaining functioning ecosystems, is essential for long-term sustainability. To achieve this goal, we must better understand the trade-offs and thresholds in the provision of ecosystem services and economic returns. To this end, we integrate spatially explicit economic and biophysical models to jointly optimize agricultural profit (sugarcane production and cattle ranching), biodiversity (bird and mammal species), and freshwater quality (nitrogen, phosphorus, and sediment retention) in the Brazilian Cerrado. We generate efficiency frontiers to evaluate the economic and environmental trade-offs and map efficient combinations of agricultural land and natural habitat under varying service importance. To assess the potential impact of the Brazilian Forest Code (FC), a federal policy that aims to promote biodiversity and ecosystem services on private lands, we compare the frontiers with optimizations that mimic the habitat requirements in the region. We find significant opportunities to improve both economic and environmental outcomes relative to the current landscape. Substantial trade-offs between biodiversity and water quality exist when land use planning targets a single service, but these trade-offs can be minimized through multi-objective planning. We also detect non-linear profit-ecosystem services relationships that result in land use thresholds that coincide with the FC requirements. Further, we demonstrate that landscape-level planning can greatly improve the performance of the FC relative to traditional farm-level planning. These findings suggest that through joint planning for economic and environmental goals at a landscape-scale, Brazil's agricultural sector can expand production and meet regulatory requirements, while maintaining biodiversity and ecosystem service provision

Developing the Miami-Dade County Injury Surveillance System: using surveillance to build community capacity for injury prevention

The Miami-Dade County Injury Surveillance System was created as part of a hospital-based injury prevention program associated with the Injury Free Coalition for Kids. Initially the program utilized trauma center and mortality data to describe injury. However, as community programming and coalition-building developed, so did the demands on the surveillance system. Coalition partners and potential partners desired a more comprehensive and population-based system. As a result of the county-wide approach and open access to results, the surveillance system has engaged new partners and leveraged additional resources to injury prevention

Refractive index measurement of the isolated crystalline lens using optical coherence tomography

AbstractAn optical coherence tomography system has been developed that was designed specifically for imaging the isolated crystalline lens. Cross-sectional OCT images were recorded on 40 lenses from 32 human donors with an age range of 6–82 years. A method has been developed to measure the axial thickness and average refractive index of the lens from a single recorded image. The measured average group refractive index at the measurement wavelength of 825nm was converted to the average phase refractive index at 589nm using lens dispersion data from the literature. The average refractive index for all lenses measured was 1.408±0.005 which agrees well with recent MRI measurements of the lens index gradient. A linear regression of the data resulted in a statistically significant decrease in the average refractive index with age, but a simple linear model was insufficient to explain the age dependence. The results presented here suggest that the peak refractive index in the nucleus is closer to 1.420, rather than the previously accepted value of 1.406

Noncontact optical measurement of lens capsule thickness ex vivo

Purpose: To design a non-contact optical system to measure lens capsule thickness in cadaver eyes. Methods: The optical system uses a 670nm laser beam delivered to a single-mode fiber coupler. The output of the fiber coupler is focused onto the tissue using an aspheric lens (NA=0.68) mounted on a motorized translation stage. Light reflected from the sample is collected by the fiber coupler and sent to a silicon photodiode connected to a power meter. Peaks in the power signal are detected when the focal point of the aspheric lens coincides with the capsule boundaries. The capsule thickness is proportional to the distance between successive peaks. Anterior and posterior lens capsule thickness measurements were performed on 13 human, 10 monkey, and 34 New Zealand white rabbit lenses. The cadaver eyes were prepared for optical measurements by bonding a PMMA ring on the sclera. The posterior pole was sectioned, excess vitreous was removed, and the eye was placed on a Teflon slide. The cornea and iris were then sectioned. After the experiments, the lenses were excised, placed in 10% buffered formalin, and prepared for histology. Results: Central anterior lens capsule thickness was 9.4±2.9 m (human), 11.2±6.6 m (monkey), and 10.3±3.6 m (rabbit) optically and 14.9±1.6 m (human), 17.7±4.9 m (monkey), and 12.6±2.3 m (rabbit) histologically. The values for the central posterior capsule were 9.4±2.9 m (human), 6.6±2.5 m (monkey), and 7.9±2.3 m (rabbit) optically and 4.6±1.4 m (human), 4.5±1.2 m (monkey), and 5.7±1.7 m (rabbit) histologically. Conclusions: This study demonstrates that a non-contact optical system can successfully measure lens capsule thickness in cadaver eyes

Automated analysis of OCT images of the crystalline lens

Presbyopia is the age related, gradual loss of accommodation, mainly due to changes in the crystalline lens. As part of research efforts to understand and cure this condition, ex vivo, cross-sectional OCT images of crystalline lenses were obtained and analyzed to extract their physical and optical properties. The raw OCT images are distorted, as the probing beam passing through media of different refractive indices and refraction on curved surfaces. In a first step, various filters, edge detection and pattern matching methods are applied to isolate the edge contour. An ellipse is fitted to the lens outline to obtain central reference point for transforming the pixel data into the analysis coordinate system. This allows for the fitting of high order equation to obtain a mathematical description of the edge contour, which obeys constraints of continuity as well as zero to infinite surface slopes from apex to equator. Robustness of these algorithms are tested by analyzing the images at various contrast levels. Gradient refractive index of the lens is determined and the physical shape is reconstructed. In a further refinement, the refraction on the curved anterior surface is compensated to obtain the actual shape of the posterior surface. Once the physical shape is fully reconstructed, the optical properties are determined by fitting conic sections to both surfaces and calculating the power profile across the lens. The relative contribution of each of these refinement steps is investigated by comparing their influence on the effective power of the lens

Distortion correction of OCT images of the crystalline lens: Gradient index approach

PURPOSE.: To propose a method to correct optical coherence tomography (OCT) images of posterior surface of the crystalline lens incorporating its gradient index (GRIN) distribution and explore its possibilities for posterior surface shape reconstruction in comparison to existing methods of correction. METHODS.: Two-dimensional images of nine human lenses were obtained with a time-domain OCT system. The shape of the posterior lens surface was corrected using the proposed iterative correction method. The parameters defining the GRIN distribution used for the correction were taken from a previous publication. The results of correction were evaluated relative to the nominal surface shape (accessible in vitro) and compared with the performance of two other existing methods (simple division, refraction correction: assuming a homogeneous index). Comparisons were made in terms of posterior surface radius, conic constant, root mean square, peak to valley, and lens thickness shifts from the nominal data. RESULTS.: Differences in the retrieved radius and conic constant were not statistically significant across methods. However, GRIN distortion correction with optimal shape GRIN parameters provided more accurate estimates of the posterior lens surface in terms of root mean square and peak values, with errors <6 and 13 μm, respectively, on average. Thickness was also more accurately estimated with the new method, with a mean discrepancy of 8 μm. CONCLUSIONS.: The posterior surface of the crystalline lens and lens thickness can be accurately reconstructed from OCT images, with the accuracy improving with an accurate model of the GRIN distribution. The algorithm can be used to improve quantitative knowledge of the crystalline lens from OCT imaging in vivo. Although the improvements over other methods are modest in two dimension, it is expected that three-dimensional imaging will fully exploit the potential of the technique. The method will also benefit from increasing experimental data of GRIN distribution in the lens of larger populations. Copyright © 2012 American Academy of Optometry.Peer Reviewe

Noncontact optical measurement of lens capsule thickness

PURPOSE. To measure interspecies thickness differences in the central anterior and posterior capsules of postmortem crystalline lenses, by a technique that maintains the anatomic integrity of the lens. METHODS. Central capsule thickness was measured with a custom-built, noncontact optical system, using a focus detection technique. Anterior and posterior lens capsule thickness measurements were performed on 22 human, 29 monkey, and 34 New Zealand White rabbit intact postmortem lenses in situ. Eyes were prepared for optical measurements by bonding a PMMA ring to the sclera in the region of the ciliary body after the conjunctiva, adipose, and muscle tissues were removed. The posterior pole was removed by making a circumferential incision through the sclera approximately 7 mm posterior to the limbus. Excess vitreous was removed to expose the posterior capsule surface, and the eye assembly was placed on a Teflon slide. The cornea and iris were sectioned to expose the anterior capsule surface. After the experiments, the lenses were excised, placed in 10% buffered formalin, and prepared for histology. Lens capsule thickness was measured from the histologic slides and compared to the optical results. RESULTS. Central anterior lens capsule thickness was 8.2 Ϯ 5.5 (human), 7.5 Ϯ 4.4 (monkey), and 10.7 Ϯ 4.2 (rabbit) m optically and 12.4 Ϯ 2.5 (human), 10.7 Ϯ 3.7 (monkey), and 10.4 Ϯ 2.0 (rabbit) m histologically. Central posterior capsule thickness was 6.3 Ϯ 2.2 (human), 5.9 Ϯ 1.7 (monkey), and 7.8 Ϯ 2.3 (rabbit) m optically and 4.1 Ϯ 1.5 (human), 3.5 Ϯ 1.6 (monkey), and 4.7 Ϯ 2.5 (rabbit) m histologically. CONCLUSIONS. The central anterior and posterior lens capsule thicknesses do not appear to vary considerably among human, rabbit, and monkey eyes. There were significant differences between optical in situ measurements and histology, which indicates that histologic preparation may affect lens capsule thickness. (Invest Ophthalmol Vis Sci