In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function. In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance y from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance r from the center. Abel transform is limited to applications with axially symmetric geometries. For more general asymmetrical cases, more general-oriented reconstruction algorithms such as algebraic reconstruction technique (ART), maximum likelihood expectation maximization (MLEM), filtered back-projection (FBP) algorithms should be employed. In recent years, the inverse Abel transform (and its variants) has become the cornerstone of data analysis in photofragment-ion imaging and photoelectron imaging. Among recent most notable extensions of inverse Abel transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis. (Wikipedia).

Introduction to the z-Transform

http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Introduces the definition of the z-transform, the complex plane, and the relationship between the z-transform and the discrete-time Fourier transfor

From playlist The z-Transform

The Two-Dimensional Discrete Fourier Transform

The two-dimensional discrete Fourier transform (DFT) is the natural extension of the one-dimensional DFT and describes two-dimensional signals like images as a weighted sum of two dimensional sinusoids. Two-dimensional sinusoids have a horizontal frequency component and a vertical frequen

From playlist Fourier

Introduction to additive combinatorics lecture 7.3 -- dual groups and the discrete Fourier transform

The discrete Fourier transform is a fundamental tool in additive combinatorics that makes it possible to prove many interesting results that would be very hard or even impossible to prove otherwise. Here I discuss the characters on a finite Abelian group G, prove that they are orthogonal a

From playlist Introduction to Additive Combinatorics (Cambridge Part III course)

The Fourier Transform and Derivatives

This video describes how the Fourier Transform can be used to accurately and efficiently compute derivatives, with implications for the numerical solution of differential equations. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf These lectures follow

From playlist Fourier

The Laplace Transform: A Generalized Fourier Transform

This video is about the Laplace Transform, a powerful generalization of the Fourier transform. It is one of the most important transformations in all of science and engineering. @eigensteve on Twitter Brunton Website: eigensteve.com Book Website: http://databookuw.com Book PDF: http:/

From playlist Data-Driven Science and Engineering

Electrical Engineering: Ch 19: Fourier Transform (2 of 45) What is a Fourier Transform? Math Def

Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the mathematical definition and equation of a Fourier transform. Next video in this series can be seen at: https://youtu.be/yl6RtWp7y4k

From playlist ELECTRICAL ENGINEERING 18: THE FOURIER TRANSFORM

Intro to Fourier transforms: how to calculate them

Free ebook https://bookboon.com/en/partial-differential-equations-ebook A basic introduction to Fourier transforms. The transforms is motivated and defined. Two examples are discussed and solved. Quick correction to the final answer in the second example: replace "4a" with "4a^2".

From playlist Partial differential equations

Abel Award ceremony 2017 - Yves Meyer

The Abel Prize Award Ceremony, May 23, 2017. Place: The University Aula, Oslo, Norway 0:00 Procession accompanied by the “Abel Fanfare” (Klaus Sandvik) Performed by musicians from The Staff Band of the Norwegian Armed Forces 0:39 His Majesty King Harald enters the University Aula 1:37 Ri

From playlist Abel Prize Ceremonies

The Abel Prize announcement 2017 - Yves Meyer

0:45 The Abel Prize announced by Ole M. Sejersted, President of The Norwegian Academy of Science and Letters 2:06 Citation by John Rognes, Chair of the Abel committee 6:49 Popular presentation of the prize winners work by professor Terrence Tao 25:19 Phone interview with Yves Meyer 37:12 C

From playlist The Abel Prize announcements

Abel Award Ceremony 2019 - Karen Uhlenbeck

The Abel Prize Award Ceremony, May 21, 2019. Place: The University Aula, Oslo, Norway Programme: 0:08 Procession accompanied by the “Abel Fanfare” (Klaus Sandvik). Performed by musicians from The Staff Band of the Norwegian Armed Forces 1:00 His Majesty King Harald V enters the University

From playlist Karen K. Uhlenbeck

Electrical Engineering: Ch 19: Fourier Transform (1 of 45) What is a Fourier Transform?

Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is a Fourier transform and how is it different from the Fourier series. Next video in this series can be seen at: https://youtu.be/fMHk6_1ZYEA

From playlist ELECTRICAL ENGINEERING 18: THE FOURIER TRANSFORM

Stephanos Venakides: Rigorous semiclassical asymptotics for integrable systems

The title of the lecture is shortened to comply with Youtubes' title policy. The original title of this lecture is "Rigorous semiclassical asymptotics for integrable systems:The KdV and focusing NLS cases". Programme for the Abel Lectures 2005: 1. "Abstract Phragmen-Lindelöf theorem & Sa

From playlist Abel Lectures

Emmanuel Candès: Wavelets, sparsity and its consequences

Abstract: Soon after they were introduced, it was realized that wavelets offered representations of signals and images of interest that are far more sparse than those offered by more classical representations; for instance, Fourier series. Owing to their increased spatial localization at f

From playlist Abel Lectures

Stéphane Mallat: A Wavelet Zoom to Analyze a Multiscale World

Abstract: Complex physical phenomena, signals and images involve structures of very different scales. A wavelet transform operates as a zoom, which simplifies the analysis by separating local variations at different scales. Yves Meyer found wavelet orthonormal bases having better propertie

From playlist Abel Lectures

Henri Darmon: Andrew Wiles' marvelous proof

Abstract: Pierre de Fermat famously claimed to have discovered “a truly marvelous proof” of his last theorem, which the margin in his copy of Diophantus' Arithmetica was too narrow to contain. Fermat's proof (if it ever existed!) is probably lost to posterity forever, while Andrew Wiles' p

From playlist Abel Lectures

Compositional Structure of Classical Integral Transforms

The recently implemented fractional order integro-differentiation operator, FractionalD, is a particular case of more general integral transforms. The majority of classical integral transforms are representable as compositions of only two transforms: the modified direct and inverse Laplace

From playlist Wolfram Technology Conference 2022

Abel Award Ceremony 2018 - Robert Langlands

0:02 Procession accompanied by the “Abel Fanfare” (Klaus Sandvik). Performed by musicians from The Staff Band of the Norwegian Armed Forces 0:52 His Majesty King Harald enters the University Aula 2:09 Dance of the Drums. Performed by Teodor Berg og Arild Torvik | Music: Gene Koshinski 7:19

From playlist Abel Prize Ceremonies

Michael Hopkins: Bernoulli numbers, homotopy groups, and Milnor

Abstract: In his address at the 1958 International Congress of Mathematicians Milnor described his joint work with Kervaire, relating Bernoulli numbers, homotopy groups, and the theory of manifolds. These ideas soon led them to one of the most remarkable formulas in mathematics, relating f

From playlist Abel Lectures

Ingrid Daubechies: Wavelet bases: roots, surprises and applications

This lecture was held by Ingrid Daubechies at The University of Oslo, May 24, 2017 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations. Ingrid Daubechies is a Belgian physicist and mathematician. She is best known for her work with wavelets in imag

From playlist Abel Lectures

Laplace transform of e^(at). We will use the definition of Laplace transform to determine L{e^(at)}. Laplace transform of the exponential function. Laplace Transformation (ultimate study guide) 👉 https://youtu.be/ftnpM_RO0Jc Get a Laplace Transform For You t-shirt 👉 https://bit.ly/lapla

From playlist Laplace Transform (Nagle Sect7.2)