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In this paper, a numerical method optimizing the coefficients of the semi empirical mass formula or those of similar mass formulas is presented. The optimization is based on the least-squares adjustments method and leads to the resolution of a linear system which is solved by iterations according to the Gauss–Seidel scheme. The steps of the algorithm are given in detail. In practice, the method is very simple to implement and is able to treat large data in a very fast way. In fact, although this method has been illustrated here by specific examples, it can be applied without difficulty to any experimental or statistical data of the same type, i.e. those leading to linear system characterized by symmetric and positive-definite matrices.

The toughness data obtained from Charpy impact tests are presented by a temperature-energy relationship. Data fitting for quantitative evaluation of the transition temperature, upper shelf energy and ductile-brittle transition, in other words, the representation with general mathematical model equation for energy variation according to the temperature is necessary. In this study, the Charpy impact tests to two representative steels were carried out as a research work for the material property standardization technique development. The fitting procedure of the scattering in data according to materials and temperature in the transition region was described. The data fitting procedure using the tangent hyperbolic function was established through variances treatment in the transition region.

A feature reduction technique is proposed for the hyperspectral (HS) data classification problem. The new features have been developed through a curve fitting step which fits specific rational function approximations to every spectral response curve (SRC) of HS image pixels. Then, the coefficients of the numerator and denominator polynomials of these fitted functions are considered as new extracted features. The method concentrates on the geometrical nature of SRCs and is utilizing the information that exists in sequence discipline — ordinance of reflectance coefficients in SRC — which has not been addressed by many other statistical analysis based methods. Maximum likelihood (ML) classification results show that the proposed method provides better classification accuracies compared to some basic and state-of-the-art feature extraction methods. Moreover, the proposed algorithm has the capability of being applied individually and simultaneously to all pixels of image.

In real-time rendering transparency is an important multi-fragment effect to visualize the structure of three-dimensional models. The per-pixel transmittance implicitly describes how the light is attenuated by traveling through several transparent fragments. We present a hybrid approach to fit the transmittance using the Heaviside step function and the trigonometric function. The k fragments with the largest contribution are exactly composited and the remaining ones are accurately compressed in an unified formulation. With a single geometry pass, fragments are sorted into a fixed-size array and overflowing ones are expanded by a truncated Fourier series. Then the transmittance is reconstructed on the fly to modulate the surface color in another geometry pass. Our approach favors high scene complexity but operates in bounded memory without losing noticeable high-frequency detail. We demonstrate that it is able to closely match the image quality at competitive frame rate, comparing to a realtime A-buffer implementation and other approximate transparency techniques.

Current state-of-the-art techniques in 2D chart analysis primarily emphasize the recognition of textual information as a means of comprehending and summarizing chart contents. However, the effective analysis and understanding of information embedded in chart images depends on accurate reverse-engineering of the behavior of depicted variables. In this paper, we propose a methodology, named Abax, as an initial study for recognizing and approximating the mathematical functions that describe the behavior of variables illustrated in chart images, particularly those containing curves. Abax is focused on approximating the values of function parameters using spatial pixel information derived from the identified keypoints of each curve. Qualitative results of the described method are presented as a proof of concept, demonstrating accurate extraction of information from fives types of functions: linear, polynomial, asymptotic, sinusoidal and arbitrary.

A popular approach to training feed-forward nets is to treat the problem of adaptation as a function approximation and to use curve fitting techniques. We discuss here the problems which the use of pure curve fitting techniques entail for the generalization capability and robustness of the net. These problems are in general inherently associated with the use of pure supervised learning techniques. We argue that a better approach to the training of feed-forward nets is to use adaptive techniques that combine properties of both supervised and unsupervised learning. A new formulation of the training problem is presented here. According to this formulation the net is viewed as two coupled sub-nets the first of which is trained by an unsupervised learning technique and the second by a supervised one. The same formulation gives rise to analytic expressions of the goals of the adaptation and leads to a new method for the adaptation of feed-forward nets.

The reconstruction of an unknown function providing a set of Lagrange data can be approached by means of fractal interpolation. The power of that methodology allows us to generalize any other interpolant, both smooth and nonsmooth, but the important fact is that this technique provides one of the few methods of nondifferentiable interpolation. In this way, it constitutes a functional model for chaotic processes. This paper studies a generalization of an approximation formula proposed by Dunham Jackson, where a wider range of values of an exponent of the basic trigonometric functions is considered. The trigonometric polynomials are then transformed in close fractal functions that, in general, are not smooth. For suitable election of this parameter, one obtains better conditions of convergence than in the classical case: the hypothesis of continuity alone is enough to ensure the convergence when the sampling frequency is increased. Finally, bounds of discrete fractal Jackson operators and their classical counterparts are proposed.

In this paper, we show that the spaces of some types of fractal interpolation functions are reproducing kernel Hilbert spaces with two different types of inner products. Then we apply these results to curve fitting problems. We establish the fractal interpolation functions that are in reproducing kernel Hilbert spaces and that minimize the regularized empirical error.

The paper is devoted to ℓ₁ and ℓ_∞ approximation with parametric spline curves. We discuss the questions of existence and uniqueness of solutions. With the help of a suitable linearization of the Euclidean norm, we derive a method for computing the approximating spline curves. The method uses linear and quadratic programming in order to find the solution.

The long-range dependence of Internet traffic has been experimentally observed. One issue in handling long-range dependent traffic is how to simulate random traffic data with long-range dependence. The authors discuss a correlation-based simulator with a white noise input for generating long-range dependent traffic data. With the real TCP traffic traces, a simulation model of TCP arrival traffic is empirically developed and the experimental results are satisfactory.

A new multipurpose curve technique has been introduced which is meant to automatically provide a fit to any ordered data in a plane. The technique is particularly economical for designing purposes as well as for the visualization of a large amount of data sets. A more flexible class of cubic functions is the basis of this technique. This class of functions involves two control parameters, to produce more flexible shapes than ordinary Bézier cubics or Hermite cubics, in each segment. These functions, together with the control parameters, are utilized to fit a design curve in an interactive way. These functions are also utilized in an optimal way to fit a design curve to the data arose from any image or a scientific phenomenon. The design curve method is highly useful to capture the outlines of images. It differs, in its methodology, from the existing techniques in the literature using Bézier cubics. The curve technique has used various ideas in its construction. These ideas include end-point interpolation, detection of characteristic points, least squares approximation. The final shape is achieved by stitching the generalized Bézier cubic pieces with GC¹ smoothness. Finally, three algorithms have been proposed for various applications.

This paper presents a new method for inverse geometric reconstruction of conics in 3D space using a ray–surface intersection. The perspective views of the conic in both the image planes are used as the input of the reconstruction algorithm. Least-square curve fitting is used in one of the 2D image planes to obtain the algebraic equation of the projected conic. The ray–surface intersection is performed using a second-order method, where a new criterion is given to provide the unique intersection. A plane is fitted through the evolved intersection points. The constructed plane cuts the conical surface to the desired conic. The proposed method does not require to establish correspondence between the two perspective views. Moreover, it requires only three intersection points. Various experiments are presented to support the validity of the proposed algorithm. Simulation studies are also performed to observe the effect of noise on errors of reconstruction. Effect of quantization errors are also considered in the final reconstruction.

Curve and surface-fitting are classic problems of approximation that find use in many fields, including computer vision. There are two broad approaches to the problem — interpolation, which seeks to fit points exactly, and regression, which seeks a rougher approximation which is more robust to noise. This survey looks at several techniques of both kinds, with a particular focus on applications in computer vision. We make use of an empirical first-level evaluation approach which scores the techniques on multiple features based on how important they are to users of the technique and developers. This provides a quick summary of the broad applicability of the technique to most situations, rather than a deep evaluation of the performance and accuracy of the technique obtained by running it on several datasets.

Five different physically motivated analytic isotherm models are fit to experimental $(P,V)$ data from seven different sources reporting studies of the adsorption of CO₂ by activated carbon. The model behavior upon parameter optimization suggests that multi-layer adsorption does not play a dominant role in CO₂ uptake by activated carbon. Only by explicitly modeling two distinct types of binding sites in the first adsorption layer does the model fully capture the nuances of the data. The values of the best-fit parameters provide good support for a widely used structural model of activated carbon: that it may be represented by nanoscopic flakes of hexagonally bonded carbon, the edges of which are terminated by functional groups. This conclusion is confirmed by comparison of the fitting parameter values to published results of first-principles calculations of the interaction of CO₂ with systems having chemical features representative of this structural model.

In this paper, we propose a framework to estimate the yield curve in the illiquid market. Within this framework, seven different curve-fitting models are compared from four aspects with the trading data of government bonds listed in the Shanghai Stock Exchange (SSE) of China. We find that the exponential spline model is optimal for this market. The characteristics and reasons underlying SSE interest rate fluctuations in the past two years are also analyzed.

In this paper, a method for analyzing transversal plane images obtained by computer tomography (CT) scans is presented. A mathematical model that describes the ribs-bounded contour was created and the problem of approximation is solved by finding out the optimal parameters of the model in the least-squares sense. The paper discloses the problems that appear in building the proper model. Such a model would be useful in the registration of images independently on the patient position on the bed and of the radiocontrast agent injection. We consider the slices where ribs are visible because many important internal organs are located here: liver, heart, stomach, pancreas, lungs, etc. The model is flexible and describes the ribs-bounded contour independently on the patient age, sex and disease. The only exception is patients with the bone fracture. This makes the basis for the proper registration of slices.

For video indexing, the problem of video cut detection remains largely an open problem because of the wide nature of transitions that occur in a digital video. This paper describes a shot boundary detection technique, which is an amalgamation of few statistical methods and measures, and robustly detects camera breaks in a full-motion video clip. The proposed algorithm incorporates a weighted histogram, an error-propagation technique for increased robustness, and a curve-fitting technique to extract partitions from the similarity curve for avoiding heuristically chosen threshold value. The algorithm has been validated on many video clips and is shown to give improved results, including for videos with rapid scanning, changes in illumination, fade-ins and fade-outs, and with special effects like dissolve and filters.

The Global Innovation Index (GII) aims to rank countries using different innovation factors. This ranking list enables countries to observe their potential status according to the rankings of other countries. The countries are classified under four groups according to the World Bank Income Group Classification on the GII list. The groups are named as; low income (LI), lower-middle income (LM), upper-middle income (UM) and high income (HI). Also, every country has a score in this ranking list. In this study, the ranking scores of 128 countries are estimated using the artificial neural network (ANN). We chose the relevant 27 features on GII 2016 Report, as input data. The significance of this paper is that; it is the first curve fitting and estimation of the score processes on GII 2016 dataset. The low root mean square error (RMSE) value which is obtained in an experimental study shows that the fitting structure is good enough to determine the approximate score of the countries in GII list. The results also show that the selected 27 features are sufficient for obtaining the income score of the countries. Increasing the number of features would lower the RMSE value and enable better approximation in the curve fitting process. The final results can assist the countries in achieving long-term output growth and improving their innovation capabilities.

T-wave alternans (TWA) in surface electrocardiograph (ECG) signal is considered a marker of abnormal ventricular function which may be associated with ventricular tachycardia. Several methods have been developed in recent years to evaluate the important feature. One such method is known as modified moving average (MMA) analysis, which performs well for different levels of TWA, but it is sensitive to the noise in T-waves. In this paper we propose an improved MMA algorithm, which adds a stage of T-wave curve fitting for the MMA method before intermediate averaging. The curve fitting is performed by means of least square method technique. Our assessment study demonstrates the improved performance.

The COVID-19 is a global disease that occurred at the end of 2019 and it has shown its effects all over the world in a very short time. World Health Organization has mobilized all the countries of the world to survive with minimal damage from this outbreak. The situation in some countries was under control as their health infrastructure is robust enough. On the other hand, many countries suffered significant damage from the outbreak. The countries that have already taken their precautions have suffered less, Turkey is one of the leading countries. Besides taking precautions in advance, countries are guiding each other throughout the outbreak. Therefore, the countries leading the fight against the outbreak should be analyzed and each country should update its precautions to fight the outbreak. In this study, COVID-19 deaths are taken into account and similar countries to Turkey are identified by K-means clustering. Later, by comparing the various characteristics of Turkey with these similar countries, Turkey’s status in fighting the outbreak is revealed. The precautions Turkey took before the outbreak showed that Turkey can fight the COVID-19 outbreak successfully.