Edge Enhancing Diffusion with Perona-Malik Filtering | Mathematical Image Processing | Exercise 08
Channel: Fabian Gabel
Date: 2022-10-29

This video is on the #peronamalik equations, a #nonlinear #diffusion equation used in #imageprocessing for #edgepreserving #denoising of images.
Our goal is to understand how the Perona-Malik equation works and how it accomplishes to control diffusion based on the amount of details that are inside the image.
In particular, we want to get a feel for the parameters of this equation.

This is the live recording of Exercise 08 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/ex08.pdf

Further Resources to books, lecture notes, and software are linked below. Let me know if anything is unclear.

Chapters
0:00 Intro
1:36 Notions of differential calculus: #divergence, #gradient, #laplacian
4:20 Interplay of divergence of gradient in the Perona-Malik equation
6:26 Controlling diffusion with the Perona-Malik function
8:36 Perona-Malik and #edges: equation in the one-dimensional case and reformulation using the chain rule to decompose the equation into a flux-corrector and a Laplace-diffusion term
13:26 Notion/definition of edges in the one-dimensional case: certain types of inflection points
16:58 How do solutions to the Perona-Malik behave in terms of edge improvement and preservation? Which edges become steeper and which become more flat as time proceeds
28:23 Special case of functions g in the Perona-Malik diffusion matrix. How does the parameter kappa influence the evolution of an edge over time?
37:04 Octave / MATLAB  example: parameter choice for kappa in Perona-Malik and steepening edges, flattening, and staircasing effects
44:32 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 adding Gaussian noise to the original image and denoising it with an own implementation of a explicit Euler stepping scheme to solve the Peronal-Malik equation).
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 #diffusion #peronamalik #staircasing #edgedetection #denoising