Integral transforms | Integral geometry
In mathematics, the X-ray transform (also called John transform) is an integral transform introduced by Fritz John in 1938 that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over lines rather than over hyperplanes as in the Radon transform. The X-ray transform derives its name from X-ray tomography (used in CT scans) because the X-ray transform of a function ƒ represents the attenuation data of a tomographic scan through an inhomogeneous medium whose density is represented by the function ƒ. Inversion of the X-ray transform is therefore of practical importance because it allows one to reconstruct an unknown density ƒ from its known attenuation data. In detail, if ƒ is a compactly supported continuous function on the Euclidean space Rn, then the X-ray transform of ƒ is the function Xƒ defined on the set of all lines in Rn by where x0 is an initial point on the line and θ is a unit vector giving the direction of the line L. The latter integral is not regarded in the oriented sense: it is the integral with respect to the 1-dimensional Lebesgue measure on the Euclidean line L. The X-ray transform satisfies an ultrahyperbolic wave equation called John's equation. The Gauss hypergeometric function can be written as an X-ray transform . (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
Inversion of the z-Transform: Power Series Expansion
http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Finding inverse z-tranforms by writing the z-transform as a power series expansion. Includes long division and inverting transcendental functions.
From playlist The z-Transform
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
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
Lecture 9: Computational imaging: a survey of medical and scientific applications
MIT MAS.531 Computational Camera and Photography, Fall 2009 Instructor: Douglas Lanman (guest lecturer from Brown University) View the complete course: https://ocw.mit.edu/courses/mas-531-computational-camera-and-photography-fall-2009/ YouTube Playlist: https://www.youtube.com/playlist?li
From playlist MIT MAS.531 Computational Camera and Photography, Fall 2009
Light and Beyond (Lecture 3) by Rajaram Nityananda
SUMMER COURSES : LIGHT AND BEYOND SPEAKER : Rajaram Nityananda (Azim Premji University) DATE : 31 May 2020 to 28 June 2020 VENUE : Online Lectures and Tutorials This short and intensive advanced undergraduate level course starts with the understanding of light as an electromagnetic wave
From playlist Summer Course 2020: Light And Beyond
Inverse problems for transport equations (Lecture 1) by Alexandre Jollivet
DISCUSSION MEETING WORKSHOP ON INVERSE PROBLEMS AND RELATED TOPICS (ONLINE) ORGANIZERS: Rakesh (University of Delaware, USA) and Venkateswaran P Krishnan (TIFR-CAM, India) DATE: 25 October 2021 to 29 October 2021 VENUE: Online This week-long program will consist of several lectures by
From playlist Workshop on Inverse Problems and Related Topics (Online)
Inverse problems for transport equations (Lecture 2) by Alexandre Jollivet
DISCUSSION MEETING WORKSHOP ON INVERSE PROBLEMS AND RELATED TOPICS (ONLINE) ORGANIZERS: Rakesh (University of Delaware, USA) and Venkateswaran P Krishnan (TIFR-CAM, India) DATE: 25 October 2021 to 29 October 2021 VENUE: Online This week-long program will consist of several lectures by
From playlist Workshop on Inverse Problems and Related Topics (Online)
Baggage X-Ray machine teardown Part 1
What's inside a baggage x-ray machine..?. Part 2 : http://youtu.be/Qjw0NDeP-0Q Part 3 : http://youtu.be/M9V3kt0bXNM Part 4 : http://www.youtube.com/watch?v=5OebkJu48Vw&list=UUcs0ZkP_as4PpHDhFcmCHyA&index=7 Part 5 : http://www.youtube.com/watch?v=X-gdUVBbTjE Part 6 : http://youtu.be/Q4ffx5e
From playlist Teardowns
Lecture 16 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood continues his lecture on diffraction and connects it to his next topic, sampling and interpolation. The Fourier transform is a tool for solving
From playlist Lecture Collection | The Fourier Transforms and Its Applications
Introduction to Solid State Physics, Lecture 9: Scattering Experiments (X-ray Diffraction)
Upper-level undergraduate course taught at the University of Pittsburgh in the Fall 2015 semester by Sergey Frolov. The course is based on Steven Simon's "Oxford Solid State Basics" textbook. Lectures recorded using Panopto, to see them in Panopto viewer follow this link: https://pitt.host
From playlist Introduction to Solid State Physics
Lecture 29 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood continues to lecture on the general stretch theorem and begins covering medical imaging. The Fourier transform is a tool for solving physical p
From playlist Lecture Collection | The Fourier Transforms and Its Applications
X Ray Head Part 1: Equipment Autopsy #4
Chris Boden performs an educational exploration though an actual X-Ray head unit. As part of our series of Equipment Autopsies we take interesting things apart to show people how they're made and what makes things tick. Learn more at thegeekgroup.org
From playlist Equipment Autopsies
Second Imaging & Inverse Problems (IMAGINE) OneWorld SIAM-IS Virtual Seminar Series Talk
Date: Wednesday, October 21, 10:00am EDT Speaker: Chang-Ock Lee, Computational Mathematics and Imaging Lab, Department of Mathematical Sciences, KAIST Title: Artifact suppression in X-ray CT images Abstract: X-ray Computed Tomography (CT) is one of the most powerful techniques for visua
From playlist Imaging & Inverse Problems (IMAGINE) OneWorld SIAM-IS Virtual Seminar Series