An Invariant Theory of Surfaces in the Four-Dimensional Euclidean or Minkowski Space

2010 Mathematics Subject Classification: 53A07, 53A35, 53A10.

Direct Computation of 3D Shape Invariants and the Focus of Expansion

Structure from motion often refers to the computation of 3D structure from a matched sequence of images. However, a depth map of a surface is difficult to compute and may not be a good representation for storage and recognition. Given matched images, I will first show that the sign of the normal curvature in a given direction at a given point in the image can be computed from a simple difference of slopes of line-segments in one image. Using this result, local surface patches can be classified as convex, concave, parabolic (cylindrical), hyperbolic (saddle point) or planar. At the same time the translational component of the optical flow is obtained, from which the focus of expansion can be computed.

Local Influence Under Parameter Constraints

Calculations of local influence curvatures and leverage have been well developed when the parameters are unrestricted. In this article, we discuss the assessment of local influence and leverage under linear equality parameter constraints with extensions to inequality constraints. Using a penalized quadratic function we express the normal curvature of local influence for arbitrary perturbation schemes and the generalized leverage matrix in interpretable forms, which depend on restricted and unrestricted components. The results are quite general and can be applied in various statistical models. In particular, we derive the normal curvature under three useful perturbation schemes for generalized linear models. Four illustrative examples are analyzed by the methodology developed in the article.

Prevalence of bone attrition on knee radiographs and MRI in a community-based cohort

OBJECTIVES: Bone attrition probably constitutes remodeling of the bone, resulting in flattening or depression of the articular surfaces. Defining bone attrition is challenging because it is an accentuation of the normal curvature of the tibial plateaus. We aimed to define bone attrition on magnetic resonance imaging (MRI) of the knee using information from both radiographs and MRIs, and to assess whether bone attrition is common prior to end stage disease osteoarthritis (OA) in the tibio-femoral joint. METHODS: All knees of participants in the community-based sample of the Framingham OA Study were evaluated for bone attrition in radiographs and MRIs. Radiographs were scored based on templates designed to outline the normal contours of the tibio-femoral joint. MRIs were analyzed using the semi-quantitative Whole-Organ Magnetic Resonance Imaging Scoring (WORMS) method. The prevalence of bone attrition was calculated using two different thresholds for MRI scores. RESULTS: Inter-observer agreement for identification of bone attrition was substantial for the radiographs (kappa=0.71, 95% CI 0.67-0.81) and moderate for MRI (kappa=0.56, 95% CI 0.40-0.72). Of 964 knees, 5.7% of the radiographs showed bone attrition. Of these, 91% of MRIs were also read as showing bone attrition. We selected a conservative threshold for bone attrition on MRI scoring (> or = 2 on a 0-3 scale) based on agreement with attrition on the radiograph or when bone attrition on MRI co-occurred with cartilage loss on OA. Using this threshold for bone attrition on MRI, bone attrition was common in knees with OA. For example, in knees with mild OA but no joint space narrowing, 13 of 88 MRIs (14.8%) showed bone attrition. CONCLUSIONS: Using MRI we found that many knees with mild OA without joint narrowing on radiographs had bone attrition, even using conservative definitions. The validity of our definition of bone attrition should be evaluated in further studies. Bone attrition may occur in milder OA and at earlier stages of disease than previously thought.

Spine curvature analysis between participants with obesity and normal weight participants: A biplanar electromagnetic device measurement

To analyse and compare standing thoracolumbar curves in normal weight participants and participants with obesity, using an electromagnetic device, and to analyse the measurement reliability. Material and Methods. Cross-sectional study was carried out. 36 individuals were divided into two groups (normal-weight and participants with obesity) according to their waist circumference. The reference points (T1–T8–L1–L5 and both posterior superior iliac spines) were used to perform a description of thoracolumbar curvature in the sagittal and coronal planes. A transformation from the global coordinate system was performed and thoracolumbar curves were adjusted by fifth-order polynomial equations. The tangents of the first and fifth lumbar vertebrae and the first thoracic vertebra were determined from their derivatives. The reliability of the measurement was assessed according to the internal consistency of the measure and the thoracolumbar curvature angles were compared between groups. Results. Cronbach’s alpha values ranged between 0.824 (95% CI: 0.776–0.847) and 0.918 (95% CI: 0.903–0.949). In the coronal plane, no significant differences were found between groups; however, in sagittal plane, significant differences were observed for thoracic kyphosis. Conclusion. There were significant differences in thoracic kyphosis in the sagittal plane between two groups of young adults grouped according to their waist circumference.

Corneal topography with Scheimpflug imaging and videokeratography : comparative study of normal eyes

PURPOSE: To compare the repeatability within anterior corneal topography measurements and agreement between measurements with the Pentacam HR rotating Scheimpflug camera and with a previously validated Placido disk–based videokeratoscope (Medmont E300). ------ SETTING: Contact Lens and Visual Optics Laboratory, School of Optometry, Queensland University of Technology, Brisbane, Queensland, Australia. ----- METHODS: Normal eyes in 101 young adult subjects had corneal topography measured using the Scheimpflug camera (6 repeated measurements) and videokeratoscope (4 repeated measurements). The best-fitting axial power corneal spherocylinder was calculated and converted into power vectors. Corneal higher-order aberrations (HOAs) (up to the 8th Zernike order) were calculated using the corneal elevation data from each instrument. ----- RESULTS: Both instruments showed excellent repeatability for axial power spherocylinder measurements (repeatability coefficients <0.25 diopter; intraclass correlation coefficients >0.9) and good agreement for all power vectors. Agreement between the 2 instruments was closest when the mean of multiple measurements was used in analysis. For corneal HOAs, both instruments showed reasonable repeatability for most aberration terms and good correlation and agreement for many aberrations (eg, spherical aberration, coma, higher-order root mean square). For other aberrations (eg, trefoil and tetrafoil), the 2 instruments showed relatively poor agreement. ----- CONCLUSIONS: For normal corneas, the Scheimpflug system showed excellent repeatability and reasonable agreement with a previously validated videokeratoscope for the anterior corneal axial curvature best-fitting spherocylinder and several corneal HOAs. However, for certain aberrations with higher azimuthal frequencies, the Scheimpflug system had poor agreement with the videokeratoscope; thus, caution should be used when interpreting these corneal aberrations with the Scheimpflug system.

Miniscleral lens wear influences corneal curvature and optics

Purpose Transient changes in corneal topography associated with soft and conventional or reverse geometry rigid contact lens wear have been well documented; however, only a few studies have examined the influence of scleral contact lens wear upon the cornea. Therefore, in this study, we examined the influence of modern miniscleral contact lenses, which land entirely on the sclera and overlying tissues, upon anterior corneal curvature and optics. Methods Anterior corneal topography and elevation data were acquired using Scheimpflug imaging (Pentacam HR, Oculus) immediately prior to and following 8 hours of miniscleral contact lens wear in 15 young healthy adults (mean age 22 ± 3 years, 8 East Asian, 7 Caucasian) with normal corneae. Corneal diurnal variations were accounted for using data collected on a dedicated measurement day without contact lens wear. Corneal clearance was quantified using an optical coherence tomographer (RS-3000, Nidek) following lens insertion and after 8 hours of lens wear. Results Although corneal clearance was maintained throughout the 8 hour lens wear period, significant corneal flattening (up to 0.08 ± 0.04 mm) was observed, primarily in the superior mid-peripheral cornea, which resulted in a slight increase in against-the-rule corneal astigmatism (mean +0.02/-0.15 x 94 for an 8 mm diameter). Higher order aberration terms of horizontal coma, vertical coma and spherical aberration all underwent significant changes for an 8 mm corneal diameter (p ≤ 0.01), which typically resulted in a decrease in RMS error values (mean change in total higher order RMS -0.035 ± 0.046 µm for an 8 mm diameter). There was no association between the magnitude of change in central or mid-peripheral corneal clearance during lens wear and the observed changes in corneal curvature (p > 0.05). However, Asian participants displayed a significantly greater reduction in corneal clearance (p = 0.04) and greater superior-nasal corneal flattening compared to Caucasians (p = 0.048). Conclusions Miniscleral contact lenses that vault the cornea induce significant changes in anterior corneal surface topography and higher order aberrations following 8 hours of lens wear. The region of greatest corneal flattening was observed in the superior-nasal mid-periphery, more so in Asian participants. Practitioners should be aware that corneal measurements obtained following miniscleral lens removal may mask underlying corneal steepening.

Role of interface curvature on stress distribution under indentation for ZrN/Zr multilayer coating

Contact damage in curved interface nano-layeredmetal/nitride (150 (ZrN)/10 (Zr) nm) multilayer is investigated in order to understand the role of interface morphology on contact damage under indentation. A finite element method (FEM) model was formulated with different wavelengths of 1000 nm, 500 nm, 250 nm and common height of 50 nm, which gives insight on the effect of different curvature on stress field generated under indentation. Elastic-plastic properties were assigned to the metal layer and substrate while the nitride layer was assigned perfectly elastic properties. Curved interface multilayers show delamination along the metal/nitride interface and vertical cracks emanating from the ends of the delamination. FEM revealed the presence of tensile stress normal to the interface even under the contact, along with tensile radial stresses, both present at the valley part of the curve, which leads to vertical cracks associated with interfacial delamination. Stress enhancement was seen to be relatively insensitive to curvature. (C) 2014 Elsevier B.V. All rights reserved.

Metrics of the normal cornea : anterior segment imaging with the Visante OCT

Purpose:  The purpose of the study was to obtain anterior segment biometry for 40 normal eyes and to measure variables that may be useful to design large diameter gas permeable contact lenses that sit outside the region normally viewed by corneal topographers. Also, the distribution of these variables in the normal eye and how well they correlated to each other were determined.

Methods:  This is a cross-sectional study, in which data were collected at a single study visit. Corneal topography and imaging of the anterior segment of the eye were performed using the Orbscan II and Visante OCT. The variables that were collected were horizontal K reading, central corneal/scleral sagittal depth at 15 mm chord, and nasal and temporal angles at the 15 mm chord using the built-in software measurement tools.

Results:  The central horizontal K readings for the 40 eyes were 43 ± 1.73 D (7.85 ± 0.31 mm), with ± 95% confidence interval (CI) of 38.7 (8.7 mm) and 46.6 D (7.24 mm). The mean corneal/scleral sagittal depth at the 15 mm chord was 3.74 ± 0.19 mm and the range was 3.14 to 4.04 mm. The average nasal angle (which was not different from the temporal angle) at the 15 mm chord was 39.32 ± 3.07 degrees and the ± 95%CI was 33.7 and 45.5 degrees. The correlation coefficient comparing the K reading and the corneal/scleral sagittal depth showed the best correlation (0.58, p < 0.001). The corneal/scleral sagittal depth at 15 mm correlated less with the nasal angle (0.44, p = 0.004) and the weakest correlation was for the nasal angle at 15 mm with the horizontal readings (0.32, p = 0.046).

Conclusion:  The Visante OCT is a valuable tool for imaging the anterior segment of the eye. The Visante OCT is especially effective in providing the biometry of the peripheral cornea and sclera and may help in fitting GP lenses with a higher percentage of initial lens success, when the corneal sag and lens sag are better matched.

Effects of varying curvature and width on the electronic states of GaAs quantum rings

Stationary states of an electron in thin GaAs elliptical quantum rings are calculated within the effective-mass approximation. The width of the ring varies smoothly along the centerline, which is an ellipse. The solutions of the Schrödinger equation with Dirichlet boundary conditions are approximated by a product of longitudinal and transversal wave functions. The ground-state probability density shows peaks: (i) where the curvature is larger in a constant-with ring, and (ii) in thicker parts of a circular ring. For rings of typical dimensions, it is shown that the effects of a varying width may be stronger than those of the varying curvature. Also, a width profile which compensates the main localization effects of the varying curvature is obtained.

DETERMINAÇÃO DAS FORMAS DO ARCO DENTÁRIO INFERIOR NA OCLUSÃO NORMAL NATURAL

Com a finalidade de determinar as formas do arco dentário inferior de maior incidência na oclusão normal natural, utilizou-se um método matemático associado ao emprego de uma função polinomial, o qual foi aplicado a 63 modelos de arcadas inferiores selecionados a partir de 6118 adolescentes. Todos os indivíduos eram portadores de dentição permanente, incluindo os segundos molares, e oclusão normal natural. Em cada dente foi fixada uma esfera de vidro, que teve a função de simular o acessório do aparelho ortodôntico, sendo utilizada na medição das distâncias entre o centro da imagem dessas esferas aos eixos x e y. Após a digitalização dos modelos de gesso, as imagens foram plotadas em um programa de computador, a fim de se obterem a função polinomial de sexto grau e o gráfico dessa função para os 126 segmentos de curva, originados das secções das imagens em lado direito e esquerdo. A seguir organizaram-se esses segmentos, de acordo com as características da curvatura anterior dos arcos dentários, em oito grupos diferentes de formas, que receberam as denominações de Forma A, Forma B, Forma C, Forma D, Forma E, Forma F, Forma G, Forma H. Cada grupo foi, então, dividido em três subgrupos, conforme os tamanhos pequeno, médio e grande. Os resultados indicaram 23 formas representativas do arco dentário inferior e uma forma média para a oclusão normal natural.

DETERMINAÇÃO DAS FORMAS DO ARCO DENTÁRIO INFERIOR NA OCLUSÃO NORMAL NATURAL

Com a finalidade de determinar as formas do arco dentário inferior de maior incidência na oclusão normal natural, utilizou-se um método matemático associado ao emprego de uma função polinomial, o qual foi aplicado a 63 modelos de arcadas inferiores selecionados a partir de 6118 adolescentes. Todos os indivíduos eram portadores de dentição permanente, incluindo os segundos molares, e oclusão normal natural. Em cada dente foi fixada uma esfera de vidro, que teve a função de simular o acessório do aparelho ortodôntico, sendo utilizada na medição das distâncias entre o centro da imagem dessas esferas aos eixos x e y. Após a digitalização dos modelos de gesso, as imagens foram plotadas em um programa de computador, a fim de se obterem a função polinomial de sexto grau e o gráfico dessa função para os 126 segmentos de curva, originados das secções das imagens em lado direito e esquerdo. A seguir organizaram-se esses segmentos, de acordo com as características da curvatura anterior dos arcos dentários, em oito grupos diferentes de formas, que receberam as denominações de Forma A, Forma B, Forma C, Forma D, Forma E, Forma F, Forma G, Forma H. Cada grupo foi, então, dividido em três subgrupos, conforme os tamanhos pequeno, médio e grande. Os resultados indicaram 23 formas representativas do arco dentário inferior e uma forma média para a oclusão normal natural.

Ocular residual astigmatism and topographic disparity vector indexes in normal healthy eyes

Purpose: To define a range of normality for the vectorial parameters Ocular Residual Astigmatism (ORA) and topography disparity (TD) and to evaluate their relationship with visual, refractive, anterior and posterior corneal curvature, pachymetric and corneal volume data in normal healthy eyes. Methods: This study comprised a total of 101 consecutive normal healthy eyes of 101 patients ranging in age from 15 to 64 years old. In all cases, a complete corneal analysis was performed using a Scheimpflug photography-based topography system (Pentacam system Oculus Optikgeräte GmbH). Anterior corneal topographic data were imported from the Pentacam system to the iASSORT software (ASSORT Pty. Ltd.), which allowed the calculation of the ocular residual astigmatism (ORA) and topography disparity (TD). Linear regression analysis was used for obtaining a linear expression relating ORA and posterior corneal astigmatism (PCA). Results: Mean magnitude of ORA was 0.79 D (SD: 0.43), with a normality range from 0 to 1.63 D. 90 eyes (89.1%) showed against-the-rule ORA. A weak although statistically significant correlation was found between the magnitudes of posterior corneal astigmatism and ORA (r = 0.34, p < 0.01). Regression analysis showed the presence of a linear relationship between these two variables, although with a very limited predictability (R2: 0.08). Mean magnitude of TD was 0.89 D (SD: 0.50), with a normality range from 0 to 1.87 D. Conclusion: The magnitude of the vector parameters ORA and TD is lower than 1.9 D in the healthy human eye.

Comparative analysis of the relationship between anterior and posterior corneal shape analyzed by Scheimpflug photography in normal and keratoconus eyes

Background To analyze and compare the relationship between anterior and posterior corneal shape evaluated by a tomographic system combining the Scheimpflug photography and Placido-disc in keratoconus and normal healthy eyes, as well as to evaluate its potential diagnostic value. Methods Comparative case series including a sample of 161 eyes of 161 subjects with ages ranging from 7 to 66 years and divided into two groups: normal group including 100 healthy eyes of 100 subjects, and keratoconus group including 61 keratoconus eyes of 61 patients. All eyes received a comprehensive ophthalmologic examination including an anterior segment analysis with the Sirius system (CSO). Antero-posterior ratios for corneal curvature (k ratio) and shape factor (p ratio) were calculated. Logistic regression analysis was used to evaluate if some antero–posterior ratios combined with other clinical parameters were predictors of the presence of keratoconus. Results No statistically significant differences between groups were found in the antero–posterior k ratios for 3-, 5- and 7-mm diameter corneal areas (p ≥ 0.09). The antero–posterior p ratio for 4.5- and 8-mm diameter corneal areas was significantly higher in the normal group than in the keratoconus group (p < 0.01). The k ratio for 3, 5, and 7 mm was significantly higher in the keratoconus grade IV subgroup than in the normal group (p < 0.01). Furthermore, significant differences were found in the p ratio between the normal group and the keratoconus grade II subgroup (p ≤ 0.01). Finally, the logistic regression analysis identified as significant independent predictors of the presence of keratoconus (p < 0.01) the 8-mm anterior shape factor, the anterior chamber depth, and the minimal corneal thickness. Conclusions The antero-posterior k and p ratios are parameters with poor prediction ability for keratoconus, in spite of the trend to the presence of more prolate posterior corneal surfaces compared to the anterior in keratoconus eyes.

DETERMINAÇÃO DAS FORMAS DO ARCO DENTÁRIO INFERIOR NA OCLUSÃO NORMAL NATURAL

Com a finalidade de determinar as formas do arco dentário inferior de maior incidência na oclusão normal natural, utilizou-se um método matemático associado ao emprego de uma função polinomial, o qual foi aplicado a 63 modelos de arcadas inferiores selecionados a partir de 6118 adolescentes. Todos os indivíduos eram portadores de dentição permanente, incluindo os segundos molares, e oclusão normal natural. Em cada dente foi fixada uma esfera de vidro, que teve a função de simular o acessório do aparelho ortodôntico, sendo utilizada na medição das distâncias entre o centro da imagem dessas esferas aos eixos x e y. Após a digitalização dos modelos de gesso, as imagens foram plotadas em um programa de computador, a fim de se obterem a função polinomial de sexto grau e o gráfico dessa função para os 126 segmentos de curva, originados das secções das imagens em lado direito e esquerdo. A seguir organizaram-se esses segmentos, de acordo com as características da curvatura anterior dos arcos dentários, em oito grupos diferentes de formas, que receberam as denominações de Forma A, Forma B, Forma C, Forma D, Forma E, Forma F, Forma G, Forma H. Cada grupo foi, então, dividido em três subgrupos, conforme os tamanhos pequeno, médio e grande. Os resultados indicaram 23 formas representativas do arco dentário inferior e uma forma média para a oclusão normal natural.