Basic Approaches of Artificial Intelligence and Machine Learning in Thermal Image Processing
U. Snekhalatha, K. Palani Thanaraj, Kurt Ammer in Artificial Intelligence-Based Infrared Thermal Image Processing and Its Applications, 2023
The foundations in mathematics of image processing include linear algebra, probability and statistics, calculus, and formulation of machine-learning algorithms (Pattanayak, 2017). Linear algebra deals with linear equations and their representation by vectors and matrices. Differential calculus focuses on how quickly physical quantities change, e.g., how much intensity of an image varies with respect to spatial domain, while integral calculus provides means to calculate the sum or aggregation of a quantity, e.g., summation of intensity values over the defined area in the image. Probability describes the rate of outcomes of an experiment in the sample space and statistics allow analyzing the distribution of outcomes in experiments with respect to their occurrence by chance.
The non-linear tradition: historical development of complexity
Keiran Sweeney in Complexity in Primary Care, 2017
Parallel developments in mathematics and quantum physics fuelled the development of this non-linear paradigm. Towards the end of the nineteenth century, mathematics had two sets of tools for solving problems, namely deterministic equations and statistical analysis, for simple and complicated systems, respectively. Both shared the key feature of linearity, of which the equation y=x+1 is the simplest example. Geometry, which was the original approach to mathematical solutions originating in Greece, and algebra, which was introduced several hundred years later by the Persians, had been unified by Descartes' analytical geometry, by which technique mathematicians were able to represent linear equations pictorially, using Cartesian coordinates in graphical form. Newton's subsequent contribution was to develop dierential calculus, which allowed mathematicians to represent the motion of a body that was undergoing acceleration. What mathematicians in the early twentieth century found, however, was that the exact solutions provided by the elegant Newtonian mathematics applied to relatively few
Kinetic Theory
Clive R. Bagshaw in Biomolecular Kinetics, 2017
In calculus, the expression d[B]/dt refers to the change in [B] after an infinitesimally small time interval dt. We can approximate the expression by taking a finite but small time interval Δt, such that Δ[B] is defined, from the rearrangement of Equation 2.2, by the expression Δ[B] = k [A] Δt. As a result, [A] is reduced by a small amount Δ[B], and we can repeat the calculation with this new [A] = [A]old − Δ[B]. Therefore, by taking appropriately small time intervals, an exponential function is built up step by step as a series of short linear segments. This is the process of numerical integration. Although this mathematics would be tedious to do by hand, it is trivial by computer and can easily be carried out so that the approximations made are negligible relative to experimental errors [28,29].
Modeling and numerical analysis of fractional order hepatitis B virus model with harmonic mean type incidence rate
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
Modern calculus is the generalization of the integer order calculus having an extra degree of choices for analysis. To check the inside behavior of the dynamics of various problems we can use significantly the idea of fractional calculus. Fractional dynamical systems can be checked on any values lying between two different natural numbers. Therefore fractional order differential equation may model very well the infectious problems under discussions (Baleanu et al. 2021; Khan, Zarin, Humphries et al. 2021; Zarin, Khan, Akgül, et al. 2022; Jajarmi et al. 2022). So many fractional operators have been defined as having a kernel of singularity and non-singularity (Atangana and Baleanu 2016; Atangana and Koca 2016; Zarin, Khan, Inc, et al. 2021; Khan et al. 2022) along with better applicability (Khan and Atangana 2020; Raza et al. 2020; Shah, Arfan, et al. 2020; Jain et al. 2021). Some of the scholars in Bonyah et al. (2020) have taken a problem related to the coupled dynamics of hepatitis and cancer under the fractional operators along with their valuable results.
Intrinsic spherical smoothing method based on generalized Bézier curves and sparsity inducing penalization
Published in Journal of Applied Statistics, 2023
Kwan-Young Bak, Jae-Kyung Shin, Ja-Yong Koo
We first set some notations in the differential calculus. The derivative of x, denoted by x defined through the directional derivative that has the steepest ascending property. Consider now a Riemannian submanifold h is equal to the projection of the gradient of x so that 1] for an overview of the first-order geometry on Riemannian manifolds.
Fractional dynamics and stability analysis of COVID-19 pandemic model under the harmonic mean type incidence rate
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2022
Amir Khan, Rahat Zarin, Saddam Khan, Anwar Saeed, Taza Gul, Usa Wannasingha Humphries
Fractional calculus is the generalization of classical calculus. To get a better insight into a mathematical model and to deeply understand phenomena, non-integer order operators can be used. Moreover, models involving fractional-order derivatives provide a greater degree of accuracy and are able to abduct the fading memory and spanning behavior. Fractional order differential equations models give more understanding about a disease under consideration (Baleanu et al. 2020; Naik et al. 2020; Naik et al. 2020; Yavuz and Sene 2020). Literature has suggested a number of fractional operators with singular and nonsingular kernel (Keten et al. 2019; Yavuz and Bonyah 2019; Yavuz and Özdemir 2020; Zarin et al. 2021) and their applications can be found in some recent studies (Babaei et al. 2019; Babaei et al. 2020; Zarin et al. 2021). In Bonyah et al. (2020), the authors have considered co-dynamics for cancer and hepatitis using a mathematical model with fractional derivative and examined its results.
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