A major benefit of multitemporal, remotely sensed images is their applicability to change detection over time. Because of concerns about global and environmental change, these data are becoming increasingly more important. However, to maximize the usefulness of the data from a multitemporal point of view, an easy-to-use, cost-effective, and accurate radiometric calibration and correction procedure is needed. The atmosphere effects the radiance received at the satellite by scattering, absorbing, and refracting light; corrections for these effects, as well as for sensor gains and offsets, solar irradiance, and solar zenith angles, must be included in radiometric correction procedures that are used to convert satellite-recorded digital counts to ground reflectances. To generate acceptable radiometric correction results, a model is required that typically uses in-situ atmospheric measurements and radiative transfer code (RTC) to correct for atmospheric effects. The main disadvantage of this type of correction procedure is that it requires in-situ field measurements during each satellite overflight. This is unacceptable for many applications and is often impossible, as when using historical data or when working in very remote locations. The optimum radiometric correction procedure is one based solely on the digital image and requiring no in-situ field measurements during the satellite overflight. The darkobject subtraction (DOS) method, a strictly image-based technique, is an attempt to achieve this ideal procedure. However, the accuracy is not acceptable for many applications, mostly because it corrects only for the additive scattering effect and not for the multiplicative transmittance effect. This paper presents an entirely image-based procedure that expands on the ~10s model by including a simple multiplicative correction for the effect of atmospheric transmittance. Two straightforward methods to derive the multiplicative transmittance-correction coefficient are presented. The COSITZ) or COST method uses the cosine of the solar zenith angle, which, to a first order, is a good approximation of the atmospheric transmittance for the dates und sites used in this study. The default TAUS method uses the average of the transmittance values computed by using in-situ atmospheric field measurements made during seven different satellite overflights. Published and unpublished data made available for this study by Moran et al. (1992) are used, and my model results are compared with their results. The corrections generated by the entirely image-based COST model are as accurate as those generated by the models that used in-situ atmospheric field measurements and RTC software.

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Image-Based Atmospheric Corrections - Revisited and Improved

Leakage of Gd-DTPA through a defective blood-brain barrier is measured quantitatively using dynamic MRI scanning, in which repeated scans are made after a bolus injection. Image registration artifacts are minimized; a dose of 0.1 mM/kg and an IR sequence enable enhancement to be measured quantitatively. The triexponential enhancement curve is fitted to a theoretical model based on compartmental analysis. The transfer constant, or permeability surface area product per unit volume of tissue (k), and leakage space per unit volume of tissue (v1) are measured. Estimates for a quickly enhancing multiple sclerosis lesion are k = 0.050 min-1, v1 = 21%; for a slow one k = 0.013 min-1, v1 = 49%. This implies permeability in the range 4-17 x 10(-6) cm s-1, in broad agreement with other physiological methods. The method is noninvasive and can be used to make serial measurements in patients and in experimental animal models. The time course of pathological aspects of diseases with blood-brain barrier breakdown, such as multiple sclerosis, tumors, and infections (e.g., HIV) can be studied, along with their response to therapy. The measurements are of physiological variables and are therefore independent of imaging equipment and field.

/pdf/measurement-of-the-blood-brain-barrier-permeability-and-5es5b3nnn6.pdf

Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging. 1. Fundamental concepts.

Three major models (from Tofts, Larsson, and Brix) for collecting and analyzing dynamic MRI gadolinium-diethylene-triamine penta-acetic acid (Gd-DTPA) data are examined. All models use compartments representing the blood plasma and the abnormal extravascular extracellular space (EES), and they are intercompatible. All measure combinations of three parameters; (1) kPSp is the influx volume transfer constant (min-1), or permeability surface area product per unit volume of tissue, between plasma and EES; (2) ve is the volume of EES space per unit volume of tissue (0 < ve < 1); and (3) K(ep), the efflux rate constant (min-1), is the ratio of the first two parameters (k(ep) = kPSp/ve). The ratio K(ep) is the simplest to measure, requiring only signal linearity with Gd tracer concentration or, alternatively, a measurement of T1 before injection of Gd (T10). To measure the physiologic parameters kPSp and ve separately requires knowledge of T10 and of the tissue relaxivity R1 (approximately in vitro value).

Modeling tracer kinetics in dynamic Gd-DTPA MR imaging

Consider the function defined for $\alpha _{1},\ \alpha _{2} \in \mathbb{R}$ (α 1 2 +α 2 2 ≠ 0) and $\beta _{1},\beta _{2} \in \mathbb{C}$ by the series 

$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$

(6.1.1)

Such a function with positive α 1 > 0, α 2 > 0 and real $\beta _{1},\beta _{2} \in \mathbb{R}$ was introduced by Dzherbashian [Dzh60].

Multi-index Mittag-Leffler Functions

https://pure.mpg.de/pubman/item/item_719624_8/component/file_3025682/Hutton_Draganski_2009.pdf

A comparison between voxel-based cortical thickness and voxel-based morphometry in normal aging

Chapter 1: Infinite Series, Power Series.The Geometric Series.Definitions and Notation.Applications of Series.Convergent and Divergent Series.Convergence Tests.Convergence Tests for Series of Positive Terms.Alternating Series.Conditionally Convergent Series.Useful Facts about Series.Power Series Interval of Convergence.Theorems about Power Series.Expanding Functions in Power Series.Expansion Techniques.Accuracy of Series Approximations.Some Uses of Series.Chapter 2: Complex Numbers.Introduction.Real and Imaginary Parts of a Complex Number.The Complex Plane.Terminology and Notation.Complex Algebra.Complex Infinite Series.Complex Power Series Disk of Convergence.Elementary Functions of Complex Numbers.Euler's Formula.Powers and Roots of Complex Numbers.The Exponential and Trigonometric Functions.Hyperbolic Functions.Logarithms.Complex Roots and Powers.Inverse Trigonometric and Hyperbolic Functions.Some Applications.Chapter 3: Linear Algebra.Introduction.Matrices Row Reduction.Determinants Cramer's Rule.Vectors.Lines and Planes.Matrix Operations.Linear Combinations, Functions, Operators.Linear Dependence and Independence.Special Matrices and Formulas.Linear Vector Spaces.Eigenvalues and Eigenvectors.Applications of Diagonalization.A Brief Introduction to Groups.General Vector Spaces.Chapter 4: Partial Differentiation.Introduction and Notation.Power Series in Two Variables.Total Differentials.Approximations using Differentials.Chain Rule.Implicit Differentiation.More Chain Rule.Maximum and Minimum Problems.Constraints Lagrange Multipliers.Endpoint or Boundary Point Problems.Change of Variables.Differentiation of Integrals.Chapter 5: Multiple Integrals.Introduction.Double and Triple Integrals.Applications of Integration.Change of Variables in Integrals Jacobians.Surface Integrals.Chapter 6: Vector Analysis.Introduction.Applications of Vector Multiplication.Triple Products.Differentiation of Vectors.Fields.Directional Derivative Gradient.Some Other Expressions Involving V.Line Integrals.Green's Theorems in the Plane.The Divergence and the Divergence Theorem.The Curl and Stokes' Theorem.Chapter 7: Fourier Series and Transforms.Introduction.Simple Harmonic Motion and Wave Motion Periodic Functions.Applications of Fourier Series.Average Value of a Function.Fourier Coefficients.Complex Form of Fourier Series.Other Intervals.Even and Odd Functions.An Application to Sound.Parseval's Theorem.Fourier Transforms.Chapter 8: Ordinary Differential Equations.Introduction.Separable Equations.Linear First-Order Equations.Other Methods for First-Order Equations.Linear Equations (Zero Right-Hand Side).Linear Equations (Nonzero Right-Hand Side).Other Second-Order Equations.The Laplace Transform.Laplace Transform Solutions.Convolution.The Dirac Delta Function.A Brief Introduction to Green's Functions.Chapter 9: Calculus of Variations.Introduction.The Euler Equation.Using the Euler Equation.The Brachistochrone Problem Cycloids.Several Dependent Variables Lagrange's Equations.Isoperimetric Problems.Variational Notation.Chapter 10: Tensor Analysis.Introduction.Cartesian Tensors.Tensor Notation and Operations.Inertia Tensor.Kronecker Delta and Levi-Civita Symbol.Pseudovectors and Pseudotensors.More about Applications.Curvilinear Coordinates.Vector Operators.Non-Cartesian Tensors.Chapter 11: Special Functions.Introduction.The Factorial Function.Gamma Function Recursion Relation.The Gamma Function of Negative Numbers.Formulas Involving Gamma Functions.Beta Functions.Beta Functions in Terms of Gamma Functions.The Simple Pendulum.The Error Function.Asymptotic Series.Stirling's Formula.Elliptic Integrals and Functions.Chapter 12: Legendre, Bessel, Hermite, and Laguerre functions.Introduction.Legendre's Equation.Leibniz' Rule for Differentiating Products.Rodrigues' Formula.Generating Function for Legendre Polynomials.Complete Sets of Orthogonal Functions.Orthogonality of Legendre Polynomials.Normalization of Legendre Polynomials.Legendre Series.The Associated Legendre Polynomials.Generalized Power Series or the Method of Frobenius.Bessel's Equation.The Second Solutions of Bessel's Equation.Graphs and Zeros of Bessel Functions.Recursion Relations.Differential Equations with Bessel Function Solutions.Other Kinds of Bessel Functions.The Lengthening Pendulum.Orthogonality of Bessel Functions.Approximate Formulas of Bessel Functions.Series Solutions Fuch's Theorem.Hermite and Laguerre Functions Ladder Operators.Chapter 13: Partial Differential Equations.Introduction.Laplace's Equation Steady-State Temperature.The Diffusion of Heat Flow Equation the Schrodinger Equation.The Wave Equation the Vibrating String.Steady-State Temperature in a Cylinder.Vibration of a Circular Membrane.Steady-State Temperature in a Sphere.Poisson's Equation.Integral Transform Solutions of Partial Differential Equations.Chapter 14: Functions of a Complex Variable.Introduction.Analytic Functions.Contour Integrals.Laurent Series.The Residue Theorem.Methods of Finding Residues.Evaluation of Definite Integrals.The Point at Infinity Residues of Infinity.Mapping.Some Applications of Conformal Mapping.Chapter 15: Probability and Statistics.Introduction.Sample Space.Probability Theorems.Methods of Counting.Random Variables.Continuous Distributions.Binomial Distribution.The Normal or Gaussian Distribution.The Poisson Distribution.Statistics and Experimental Measurements.

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Mathematical Methods in the Physical Sciences

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Mathematical Methods in the Physical Sciences.

Mathematical Methods in the Physical Sciences, 2nd ed.

It has been recently recognized that there is a difference between the measured Lorentz contracted shape of an object moving at relativistic speed and the shape as seen by a single observer. The case of an object which subtends a small solid angle at the observer has been discussed by several authors. This paper discusses objects so large or so near that the subtended solid angle cannot be considered small, and gives simple proofs that spheres always present a circular outline and that straight lines may appear curved. These results are applied to revise Gamow's well-known picture of the bicyclist seen by Mr. Tompkins.

Mary L. Boas

Papers

Mathematical Methods in the Physical Sciences

Mathematical Methods in the Physical Sciences

Mathematical Methods in the Physical Sciences.

Mathematical Methods in the Physical Sciences, 2nd ed.

Apparent Shape of Large Objects at Relativistic Speeds