Quadric Metric Research Articles

Fiber-Optic Hydrophone Based on Michelson's Interferometer with Active Stabilization for Liquid Volume Measurement.

Sensing technologies using optical fibers have been studied and applied since the 1970s in oil and gas, industrial, medical, aerospace, and civil areas. Detecting ultrasound acoustic waves through fiber-optic hydrophone (FOH) sensors can be one solution for continuous measurement of volumes inside production tanks used by these industries. This work presents an FOH system composed of two optical fiber coils made with commercial single mode fiber (SMF) working in the sensor head of a Michelson’s interferometer (MI) supported by an active stabilization mechanism that drives another optical coil wound around a piezoelectric actuator (PZT) in the reference arm to mitigate external mechanical and thermal noise from the environment. A 1000 mL glass graduated cylinder filled with water is used as a test tank, inside which the sensor head and an ultrasound source are placed. For detection, amplitudes and phases are measured, and machine learning algorithms predict their respective liquid volumes. The acoustic waves create patterns electronically detected with resolution of 1 mL and sensitivity of 340 mrad/mL and 70 mvolts/mL. The nonlinear behavior of both measurands requires classification, distance metrics, and regression algorithms to define an adequate model. The results show the system can determine liquid volumes with an accuracy of 99.4% using a k-nearest neighbors (k-NN) classification with one neighbor and Manhattan’s distance. Moreover, Gaussian process regression using rational quadratic metrics presented a root mean squared error (RMSE) of 0.211 mL.

Least-squares and maximum-likelihood in computed tomography.

Purpose: Existing maximum-likelihood (ML) methods in computed tomography usually require significant computing resources to implement, and/or are limited to particular measurement noise models that are representative of the simplest theoretical archetypes. There is an absence of general procedures to produce rapid ML methods that account precisely for the noise model of a given experiment. We investigate a mathematical-computational procedure of producing constrained quadratic optimization reconstruction algorithms that fill this niche, requiring less computing resources than the exact (expectation-maximization) procedures and having comparable performance with least-squares iterative methods. This allows high-fidelity reconstructions to be practically achievable for largely arbitrary noise models. Approach: We identify a systematic mathematical procedure to produce constrained quadratic optimization methods that maximize tomogram likelihood under arbitrary noise models, which are tunable to specific characteristics of the experiment. This procedure is applied to a general theory of mixed Poisson-Gaussian noise in transmission tomography, and to a theory of invertible linear transformations of measurement intensity subject to Poisson noise. We perform tomographic reconstructions of a very highly attenuating two-dimensional object phantom and compare the speed and fidelity of reconstruction with alternative quadratic metrics ( -minimization among others). Results: Quantitative metrics reveal that reconstructions under our systematically produced quadratic methods achieved significantly greater reconstruction fidelity with less computation than the optimized conventional, untuned quadratic metrics with a comparable procedure. Conclusion: Constrained quadratic optimization methods appear to apply sufficiently good approximations to achieve a high reconstruction fidelity with a simple quadratic metric amenable to a broad class of minimization methods. These preliminary simulation-based results are very promising and suggest that such methods may be used to produce high-fidelity reconstructions with less computation than many other statistical methods. By design, these quadratic methods are also explicit and quantitative in their description, allowing fine-tuning according to the specific uncertainties and noise model of the experiment. Further research is required to ascertain the full practical potential of these methods.

Flat left-invariant pseudo-Riemannian metrics on quadratic Lie groups

A flat quadratic Lie algebra (g,〈,〉,k) is a Lie algebra g endowed with a flat pseudo-Euclidean metric 〈,〉 and a quadratic structure k. In geometrical terms, it is a Lie algebra of a Lie group endowed with both a flat left-invariant pseudo-Riemannian metric and a bi-invariant metric. It is known that if a Lie algebra admits a symplectic form and a quadratic structure then it is a flat quadratic Lie algebra. In this paper, we show that there exist non-abelian Lie algebras which admit a flat Lorentzian metric and a quadratic structure. If g is such Lie algebra, then we show that g is a trivial central extension of R⋊H2k+1 and [g,g]≃H2k+1, where H2k+1 is the Heisenberg Lie algebra and k≥2. In particular, we prove that g is 3-step solvable, not nilpotent and hence admits no symplectic structure. We show that there exists only one non-abelian Lie algebra of dimension n≤6 which admits a flat Lorentzian metric and a quadratic structure. Using a variant of the double extension method, which conserves both the flat and the quadratic metrics, we prove that any Lie algebra which admits a flat Lorentzian metric and a quadratic structure, is a double extension of an abelian Lie algebra and we construct a large class of such (non-abelian) Lie algebras in higher dimensions. Oscillator Lie algebras denoted by gλ constitute an important class of quadratic Lie algebras which are of the form R⋊H2k+1. It is known that gλ admits no flat Lorentzian metric. We show in this paper that gλ admits a Ricci-flat Lorentzian metric. The paper also study the class of quadratic 2-step nilpotent Lie algebras (g,k) which constitute examples of flat quadratic Lie algebras since k is also flat in this case.

The consumer’s demand functions defined to study contingent consumption plans

Given two probability distributions expressing returns on two single risky assets of a portfolio, we innovatively define two consumer’s demand functions connected with two contingent consumption plans. This thing is possible whenever we coherently summarize every probability distribution being chosen by the consumer. Since prevision choices are consumption choices being made by the consumer inside of a metric space, we show that prevision choices can be studied by means of the standard economic model of consumer behavior. Such a model implies that we consider all coherent previsions of a joint distribution. They are decomposed inside of a metric space. Such a space coincides with the consumer’s consumption space. In this paper, we do not consider a joint distribution only. It follows that we innovatively define a stand-alone and double risky asset. Different summary measures of it characterizing consumption choices being made by the consumer can then be studied inside of a linear space over {mathbb {R}}. We show that it is possible to obtain different summary measures of probability distributions by using two different quadratic metrics. In this paper, our results are based on a particular approach to the origin of the variability of probability distributions. We realize that it is not standardized, but it always depends on the state of information and knowledge of the consumer.

A Generalized Asymmetric Dual-Front Model for Active Contours and Image Segmentation.

The Voronoi diagram-based dual-front scheme is known as a powerful and efficient technique for addressing the image segmentation and domain partitioning problems. In the basic formulation of existing dual-front approaches, the evolving contour can be considered as the interfaces of adjacent Voronoi regions. Among these dual-front models, a crucial ingredient is regarded as the geodesic metrics by which the geodesic distances and the corresponding Voronoi diagram can be estimated. In this paper, we introduce a new dual-front model based on asymmetric quadratic metrics. These metrics considered are built by the integration of the image features and a vector field derived from the evolving contour. The use of the asymmetry enhancement can reduce the risk for the segmentation contours being stuck at false positions, especially when the initial curves are far away from the target boundaries or the images have complicated intensity distributions. Moreover, the proposed dual-front model can be applied for image segmentation in conjunction with various region-based homogeneity terms. The numerical experiments on both synthetic and real images show that the proposed dual-front model indeed achieves encouraging results.

Non-Parametric Probability Distributions Embedded Inside of a Linear Space Provided with a Quadratic Metric

There exist uncertain situations in which a random event is not a measurable set, but it is a point of a linear space inside of which it is possible to study different random quantities characterized by non-parametric probability distributions. We show that if an event is not a measurable set then it is contained in a closed structure which is not a σ-algebra but a linear space over R. We think of probability as being a mass. It is really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space. It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within this context. It always depends on the state of information and knowledge of an individual. We study different intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing them. We define the notion of α-distance between two non-parametric probability distributions.

A Portfolio of Risky Assets and Its Intrinsic Properties

We show a canonical expression of a univariate risky asset. We find out a canonical expression of the product of two univariate risky assets when they are jointly considered. We find out a canonical expression of a portfolio of two univariate risky assets when it is viewed as a stand-alone entity. We prove that a univariate risky asset is an isometry. We define different distributions of probability on R inside of metric spaces having di erent dimensions. We use the geometric property of collinearity in order to obtain this thing. We obtain the expected return on a portfolio of two univariate risky assets when it is viewed as a stand-alone entity. We also obtain its variance. We show that it is possible to use two di erent quadratic metrics in order to analyze a portfolio of two univariate risky assets. We consider two intrinsic properties of it. If a portfolio of two univariate risky assets is viewed as a stand-alone entity then it is an antisymmetric tensor of order 2. What we say can be extended to a portfolio of more than two univariate risky assets.

Графические алгоритмы построения квадрики, заданной девятью точками

The fundamental issue of constructing a nine-point quadric was frequently discussed by mathematicians in the 19th century. They failed to find a simple linear geometric dependence that would join ten points of a quadric (similar to Pascal's theorem, which joins six points of a conic section). Nevertheless, they found different algorithms for a geometrically accurate construction (using straightedge and compass or even using straightedge alone) of any number of points of a quadric that passes through nine given points. While the algorithms are quite complex, they can be implemented only with the help of computer graphics. The paper proposes a simplified computer-based realization of J.H. Engel’s well-known algorithm, which makes it possible to define the ninepoint quadric metric. The proposed graphics algorithm can be considered an alternative to the algebraic solution of the stated problem. The article discusses two well-known graphical algorithms for constructing a quadric (the Rohn — Papperitz algorithm and the J.H. Engel algorithm) and proposes a simplified version of the J.H. algorithm. For its constructive implementation using computer graphics. All algorithms allow you to determine the set of points and the set of flat sections of the surface of the second order, given by nine points. The Rohn — Papperitz algorithm, based on the spatial configuration of Desargues, is best suited for its implementation on an axonometric drawing using 3D computer graphics. Algorithm J.H. Engel allows you to solve a problem on the plane. The proposed simplified constructive version of the algorithm J.H. Engel is supplemented with an algorithm for constructing the principal axes and symmetry planes of a quadric, given by nine points. The construction cannot be performed with a compass and a ruler, since this task reduces to finding the intersection points of two second-order curves with one known general point (third degree task). For its constructive solution, a computer program is used that performs the drawing of a second order curve defined by an arbitrarily specified set of five points and tangents (both real and imaginary). The proposed graphic algorithm can be considered as an alternative to the algebraic solution of the problem.

Best Member Selection for convective-scale ensembles

This paper addresses the question how to identify the best members of a convection-permitting numerical weather prediction ensemble. Two different metrics, a classical quadratic approach using conventional observations and a spatial approach based on radar derived precipitation estimates, are employed to identify the ensemble members closest to the observations and to separate good and bad ensemble members. The characteristics of the best member selections and their performance in different weather regimes are investigated and evaluated in the context of three potential applications. The results show clear differences between the different best member selections and based on their characteristics their use in different applications is suggested. The classical quadratic metric has a higher persistence and is thus well suited for synoptic-scale applications, while the spatial metric shows good correlations with a reference measure for precipitation for short lead times and is thus better suited for very short-range applications.

Image resizing via non-homogeneous warping

Image resizing aims to adapt images to displays with different sizes and aspect ratios. In this paper, we provide a new image resizing approach for efficiently determining the non-homogeneous warp that better preserves the global image configuration and concentrates the distortion in regions of the image where they are least-likely to be noticed. Considering the different properties of large displays and small displays, we design different strategies for upsizing and downsizing. We define a variety of quadratic metrics to measure image distortion. We introduce a patch-linking scheme that can better preserve the global image configuration. We formulate image resizing as a quadratic minimization problem, which can be efficiently solved. We experiment with our method on a variety category of images and compare our results to the state of the art.

Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators, and quadratic metrics

Optical correlators used in pattern recognition should have their operation optimized according to physically measurable behavior related to the correlator's task. The optimization is best done with explicit consideration of the limited set of complex values available to the filter. We earlier developed a theoretical filter computing strategy, MEDOF (minimum Euclidean distance optimal filter), that optimizes a wide variety of correlation metrics on physically realistic modulators. We use a single reference image and add noise according to a stated model. We show laboratory results for filters optimizing a variety of metrics on a highly coupled SLM. The results show improvements in the metrics by using MEDOF when compared with some other filter computing strategies such as matching phase. Optimizing SNR appears preferable to the other metrics when detecting an object in clutter of known power spectral density. © 1999 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(99)01802-4)

On optimum binning for spectral distributions

The paper describes a method for optimum selection of parameters of a histogram bin for a very large class of regressions those of spectral distributions. It is shown that for a regression s( x) the minimum statistical errors are given by bins of equal weights: they are intervals X k such that the integrals of expectations of s( x) over them are equal; whereas the minimum errors of the representation of a function by a histogram are obtained when the abscissa of any histogram ordinate is assigned the value where the function is equal to its integral mean (when using quadratic metrics) or to the amplitude mean (when using uniform metrics). In both cases this abscissa may differ from the middle of the bin. The described method has been applied to the most frequently used models of spectral functions — exponential, Gaussian and Lorentzian.

The Problem of Quality for Nonlinear Self-Regulating Systems with Quadratic Metric

The systems considered in this paper are characterized by differential equations of the form \dot{x}_{k} = \sum_{\alpha} b_{k\alpha} x_{\alpha} + f_{k}(\alpha_1,\cdots,x_n, t) (k= 1, \cdots, n) which are defined over a region N of Euclidean space E_n with metric R^2 = \sum_{i=1} x_i^2 , and with t ranging over some interval T . The f_k (k+1,\cdots, n) are assumed to be such that 1) f(0,\cdots,0,t) \equiv 0 and 2) there exist positive constants L_k such that |f_k(x_1,\cdots,x_n,t) | \leq L_k R for all points in N and all t in T . The problem of quality means the problem of determining the values of m adjustable parameters p_i,\cdots,p_m in the b_{k\alpha} , and f_k in such a way as to result in a rapid return to equilibrium of the representative point in phase-space subject to a limitation on the amount of overshoot. This problem is formulated in precise terms, and a method of solution for it is indicated. As an illustration, the method is applied to a problem in regulation which was formulated by Bulgakov.

Experimental demonstration of the Euclidean-Pythagorean structure and the quadratic metrics in the perceptual manifold of the cutaneous tactile sense.

Zusammenfassung.Der Zusammenhang zwischen dem bei voluntärer Aktion eines Muskels erbrachten Kraftimpuls und der integrierten, elektrischen Aktivität im Elektromyogramm wurde untersucht. Die Versuchs‐ergebnisse zeigen, dass bei der nicht‐isometrischen Zeigefingerab‐duktion die integrierte, elektrische EMG‐Aktivität bei stärkeren Aktionen relativ schneller zuwachst als bei schwächeren.Summary.The relation between the mechanical impulse produced by a muscle in voluntary action and the integrated electric. activity in the electromyogram was investigated. The experimental results indicated that in non‐isometric abduction of the forefinger the integrated electric activity increased relatively more rapidly in connection with stronger than weaker actions.

Relation of the amount of motility induced by voluntary muscular action & the electrical activity of muscles

Summary.It is demonstrated experimentally that quadratic metrics apply in the component structures with intensity and surface area dimensions of the haptic type of perception.