1 / 25

Understanding Inverse Problems and Regularization Theory: A Comprehensive Overview

This presentation, delivered by Chuck Groetsch at Texas A&M University, dives into the fascinating world of inverse problems (I.P.s) and regularization theory. It covers the fundamentals, historical context, and key issues of well-posedness in inverse problems. The talk discusses important concepts such as the Moore-Penrose inverse, compact operators, and singular value decomposition (SVD). Real-world applications of inverse problems are explored, including imaging techniques and the mathematics behind effective deblurring. Join us for an academic journey through models, frameworks, and the essence of regularization.

harva
Télécharger la présentation

Understanding Inverse Problems and Regularization Theory: A Comprehensive Overview

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. INVERSE PROBLEMS and REGULARIZATION THEORY – Part I AIP 2011 Texas A&M University MAY 21, 2011 CHUCK GROETSCH

  2. OUTLINE What are I.P.s? - Some History Some Model I.P.s A Framework for I.P.s Key Issue: Well-posedness The Moore-Penrose Inverse Compact Operators and the SVD What is ‘Regularization’?

  3. WHAT ARE INVERSE PROBLEMS? PLATO’S CAVE

  4. Dürer: Man drawing a lute A Renaissance Inverse Problem

  5. Renaissance Ballistics I knew that a cannon could strike in the same place with two different elevations or aimings, I found a way of bringing this about, a thing not heard of and not thought by any other, ancient or modern. NicolòTartaglia, 1537

  6. The Grand Academy of Lagado “He had been Eight Years upon a Project for extracting Sun-Beams out of Cucumbers …” J. Swift 1726

  7. Add some low amplitude noise : Another way to look at it:

  8. Direct: Super Smooth

  9. DEBLURRING AS AN I.P. IMAGE OBJECT The Perfect Imager:

  10. Imaging as Reverse Diffusion

  11. Axial Attraction

  12. Ion Channel Distribution in Olfactory Cilia

  13. Framework for Inverse Problems MODEL CAUSE EFFECT K PHENOMENON OBSERVATION PROCESS

  14. WELL-POSEDNESS: Jacques Hadamard 1902

  15. The Moore-Penrose Inverse

  16. Compact Operators Linear Measurement Theory Object Observation

  17. Weak Convergence Finite Rank Operator F.R. Operators honor weak convergence: Compact Operators: (Uniform) Limits of F.R. Operators

  18. SVD: SINGULAR VALUE DECOMPOSITION

  19. SVD & M-P Inverse

  20. A SIMPLE EXAMPLE

  21. Instability

  22. REGULARIZATION

More Related