Convex analysis fundamentals | Mathematical Image Processing | Ex. 13
Channel: Fabian Gabel
Date: 14/03/2023

In this video, we peek into #convexanalysis. We will give the definition of a convex functional and cover some classical examples as well as results on the decomposition of convex functionals with #affine functions

This is a prerecorded video for Exercise 13 of the course "Mathematical Image Processing" held at #tuhh in 2021/2022.

Watch the full series at
https://www.youtube.com/playlist?list=PLqB7maHdRHhNdx1QX7kUNediGlGqo8NGw

You find the exercise sheet on the webpage
https://math.fabian-gabel.de/teaching/improc/ex13.pdf

Further resources, such as books, lecture notes, and software, are linked below. Let me know if anything is unclear.

Chapters
0:00 Intro
0:32 #Convexity of sets
6:10 Examples of convex sets: vector spaces
9:07 Convexity in denoising: variational approach
13:05 Convexity in #inpainting / #restauration
23:06 Definition of convex functionals 
26:40 Example: #norms on vector spaces are convex
30:08 Example: x maps to x*x is a convex function on the reals
33:30 Example: concatenation of affine linear function with convex function is convex
42:32 Outro

===References===

==Book==
 "Mathematical Image Processing" by K. Bredies and D. Lorenz:
https://doi.org/10.1007/978-3-030-01458-2

==Lecture Notes==
https://bit.ly/math-improc-script

==Software==
GNU Octave is free software licensed under the terms of the GNU General Public License (GPL).
Get Octave:  https://octave.org/
License: https://www.gnu.org/licenses/gpl-3.0.html

===Credits===
Thumbnail uses a photo by Jannet Corgi: https://bit.ly/3feoSEn (also converted to grayscale and further processed by applying Laplace sharpening to the original image).
Outro music titled "Motion Blur" by "http://www.nihilore.com/synthwave" is licensed under a Creative Commons Attribution 4.0 license (https://creativecommons.org/licenses/by/4.0/legalcode).


#math #imageprocessing #signalprocessing #octave #matlab #compact #banachspace #illposed #wellposed #inverseproblem #deconvolution #convolution #sequences #multiplication #hilbertspace