Image Restoration usingGraphical Models

Margaret ThorenAnna Grim

Problem: Image Restoration

https://www.smapip.is.tohoku.ac.jp/~kazu/SMAPIP-KazuKazu/Summary/index-e.html

Introduction

https://blog.statsbot.co/probabilistic-graphical-models-tutorial-and-solutions-e4f1d72af189

Def: A graphical model consists of a collection of random variables corresponding to some graph.

Example:(1) Random Variables

(2) Graph

Converting an Image into a Graph

Ising Model

https://www.cs.ubc.ca/~murphyk/Papers/intro_gm.pdf

Local Energy:

Pairwise Energy:

● We associate a random variable with each pixel;

Ising Model

● The joint pdf corresponding to the random variables is energy based and defined to be

● Maximizing the probability is equivalent to minimizing the energy

Overview

https://slideplayer.com/slide/8088915https://www.slideshare.net/zukun/02-probabilistic-inference-in-graphical-models

Gibbs Sampling

https://slideplayer.com/slide/8141891/

Main Idea: Restoring pixels to values close to original observation and close to neighboring observations

Formulas for Gibbs Sampling

Restore image by drawing samples from the following distribution

Goal:

● Using properties of Markov Random Fields, we can simplify this distribution such that

Gibbs Sampling Algorithm(1) Initialization

(2) Sampling

(3) Check for Convergence

Coding in MATLAB

Belief Propagation

Main Idea: Message passing algorithm that approximates the marginal probability distribution where pixels take on values based on messages they receive from neighbors

http://ssg.mit.edu/nbp/

Formulas for Belief Propagation

Messages from node i to node j:

Belief at node j:

Intuition of Belief Propagation Formulas

Belief Propagation Algorithm(1) Initialization

(2) Update Messages

(3) Check for convergence

Messages to Restoration

•Once the messages have converged, we restore the

image by computing

Convergence

https://en.wikipedia.org/wiki/Tree_(graph_theory)

https://slideplayer.com/slide/5090395/

Belief Propagation Results

Conclusions

Belief Propagation Gibbs Sampling