Diameter | EPFL Graph Search

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length d of a diameter is also called the diameter. In this sense one speaks of diameter rather than diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the radius r. :d = 2r \qquad\text{or equivalently}\qquad r = \frac{d}{2}. For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers. For a curve of constant width such as the Reuleaux t

Optical spectral probing for epithelial tissue characterisation

Philippe Thueler

In this thesis, a new method has been developed, which determines the optical coefficients spectra (absorption and scattering) of small volumes of tissue (1 mm3) by the measurement of spatially resolved reflectance between 480 and 950 nm. Firstly, an algorithm has been developed to determine the optical coefficients form reflectance measurements: It is based on the optical propagation model introduced by Bevilacqua [Bevilacqua 99 JOSA]. This model takes the first two moments of the phase function into account through the γ parameter, allowing to accurately simulate the propagation of photons close to the source by the Monte Carlo method. These simulations are then fitted to the reflectance to obtain the optical coefficients. To obtain any value of absorption, scattering or γ parameter, the simulations are interpolated by 3D B-splines. A complete robustness to noise study of the algorithm has been presented. A theoretical study of the γ parameter in tissue over broad wavelengths has been reported. Tissue was simulated with Mie theory for a fractal distribution of sphere diameters. For such a distribution, the γ spectrum is almost flat, instead of being wavy like in the case of mono-disperse sphere diameter. The correlation of the mean value of γ with the fractal power (or dimension) of the distribution was made and showed an analytical form of hyperbolic tangent. It allowed deepening the interpretation of the γ parameter as being an indicator of the relative number of small scatterers (much smaller than the wavelength, e.g. of Rayleigh type) in comparison of bigger ones (their size being comparable to the wavelength). The assessment of the set-up and the algorithm was made by measuring the reflectance of a solution containing 5 different diameters of polystyrene spheres simulating the scattering and Nigrosin colorant as absorber. It showed a good agreement between measurements and prediction. In order to built a better comprehension of absorption and scattering at the tissue scale, a first part of in vivo investigations has been made on an animal model. The skin of the mouse back was altered with Freund's adjuvant and TPA and the optical coefficients were measured on treated and control sites. The effect of alteration (epidermis and dermis thickening, oil vesicle appearing, inflammatory cells infiltration, etc.) was clearly correlated with optical coefficients variations. The scattering spectrum always showed the better contrast. The absorption spectrum, especially in the haemoglobin peak region (between 550 and 600 nm) also showed significant information, in terms of oxy and deoxy-haemoglobin concentration and saturation. Sensitivity and specificity of the "diagnosis" was evaluated. The study of stomach epithelium in vivo on 29 human subjects was made using a small optical fibre probe (

EPFL2001

Non-Exponential Variations for Classical Results in First Passage Percolation

Jacques Saliba

We study in this thesis the asymptotic behavior of optimal paths on a random graph model, the configuration model, for which we assign continuous random positive weights on its edges. We start by describing the asymptotic behavior of the diameter and the flooding time on the graph for a set of light-tailed edge-weights, and we prove that it is the largest class of densities for which these precise asymptotic expressions for the diameter/flooding time hold. We then show how the weighted optimal path can be constructed using a particular positive recurrent Markov chain. We finally study, in the last chapter, the noise sensitivity of the model by replacing every edge-weight independently by a new realization of the same distribution, with probability epsilon. We show that the optimal paths between two vertices before and after this modification are asymptotically ''independent'' conditioning on the graph, as the number of vertices tends to infinity.

EPFL2020

Vieillissement et fiabilité des parcs de poteaux bois des réseaux de lignes aériennes

Yann Benoit

The reliability of new overhead electric and telecommunication lines depends principally on the quality of their support structures. These structures are generally made of wood, metal or concrete. The complexity of a natural substance such as wood requires a thorough analysis of the various factors that influence its overall quality. In the case of wood poles, such factors include initial forest growth pattern, the species of wood and its preservative treatment, ageing characteristics, and its various mechanical defects such as knots, cracks etc. The accumulation of knowledge on the effect of the various variables that contribute to the overall quality of a wood support structure permits an optimum use of such a resource. For example, less variability and higher strength of wood support structures permits optimum loading and spacing between structures, thus reducing the number needed in a specific length of an overhead line. If one assumes that in Western Europe 1 wood pole is employed for every 2 inhabitants, and that this proportion increases in less densely populated countries such as the US and Scandinavia, the economics of optimum use of wood as a resource soon become apparent. In less developed countries, the proportions and the economics vary depending on the natural resources such as wood that they employ. The goal of this research is to establish, thanks to non destructive evaluations, a general ageing probabilistic law of the wooden pole based on two distinguished laws: one on the new pole in studying the influence of a grading of the bad elements based on a normal law: "left-truncation of a normal distribution", point 1; and another one based on the in-field wooden pole in exploiting the different parameters such as: the age of the pole, its chemical treatment, its species, its knots etc. in order to define the pole's damage law, point 2. Statistical distribution law of the new wooden pole after grading by non destructive sorting (ultrasounds) of the high mechanical performances supports: This new distribution law is a Gaussian law or evolves to a Log or Weibull's law with 3 parameters according to the inspected species. This grading allows a revalorization of the properties of the new poles and of the design values while guaranteeing an index of reliability required by the design standards, or in improving directly this nominal reliability (economic gain and reliability gain). Statistical distribution law of an aged in-field population (20-50 years old) approached by a bi-modal law which depends on: The distribution law of the new component (see point 1) and its minimal extreme law, which is asymmetrical, for an observation on 50 years. The statistical distribution at the time t of the residual mechanical performances of a group of supports making a local net, evaluated by non destructive methods. The non destructive evaluation is based on the measurements of physical variables (density, biological moisture content) and some descriptive variables from natural origins (diameter, knots, cracks...) and from accidental origins (diameter reduction, lightning cracks...). The statistical distribution at the time t is then obtained on the basis of a model of multivariate non destructive evaluation, generalized to the whole of species and treatments. This model is the other concrete goal to reach in this thesis. As a conclusion, the research demonstrates the influence and the interaction of the new pole grading (distribution at t0) on the modelisation of the distribution at ti (multivariate non destructive model). The data used for the mentioned modelisations come from a significant international database with a large amount of inspected wood poles and with studied cases. This database is the synthesis of about 15 years of research and development leaded by IBOIS-EPFL and its international partners. The probabilistic approaches are then validated by a huge database allowing thus to be directly exploitable. On this basis, all the standards dealing with the new poles and dealing with the controls and maintenances of a wooden pole networks, could be re-examined for a double gain: Concerning the economy: by increasing the capacity of the new poles profiting of an objective quality assurance, and by increasing the life time of the in-field pole, in knowing how to purge only the ones which are under the critical threshold of damage Concerning the reliability: by increasing the reliability of the network from the stage "new pole", by eliminating the weakest components, and by maintaining this reliability during all the life time of the network thanks to a cyclic preventive maintenance (every 5 to 8 years) and the replacement of only the weakened poles.

EPFL2004