Optimal transport applied to infographics (color transportation problem)

Optimal transport applied to color transportation in the field of infographics / image processing.

Abstract

The shortest path principle guides most decisions in life and sciences and therefore, optimization problems have came to the fore. The goal of Optimal Transport as a mathematical gem at the interface between probability, analysis and optimization is to find the least costly transport. This work reviews this field with a bias toward numerical methods and their applications in computer graphics, and sheds lights on the impact of the given distance on the final result.

Article

A short article has been written (in french) to sum up the ideas behind and the key results.

Installation

Clone this repository :

bash git clone https://github.com/AdrienC21/optimal-transport-color-transportation.git

Make sure the following packages are installed. If not, type in a python console :

```python pip install --upgrade pip pip install setuptools pip install --upgrade setuptools --ignore-installed

pip install numpy pip install matplotlib pip install scipy pip install cython pip install POT pip install colour-science pip install colour-science[optional] pip install colour-science[plotting] pip install colour-science[tests] pip install colour-science[docs] pip install colour-science[development]

pip install pymanopt autograd ```

How to use

Edit in parameters.py the following lines :

```python

name of the source image (the one that will change color)

imageSourceName = "bluebutterfly.jpg"

name of the target image (colors of this one will be transported

onto the source image)

imageTargetName = "pinkfield.jpeg" nbpixels = 1000 # number of pixels that will be randomly chosen ```

Run runoptimaltransport.py to apply transport optimal algorithms.

WARNING : The complexity of our methods are O(nm1 + n*4) where n is equal to nbpixels and m1 is the number of pixels in the source image, which means that nbpixels is a really sensitive parameter regarding the running time.

Bibliography

[1] Cohen Scott : Finding color and shape patterns in images, Thèse, InfoLab Stanford, Chapitre 4, Mai 1999

[2] Gabriel Peyré : Le transport optimal: de Gaspard Monge à la science des données,Conférence, 2018

[3] Ferradans, S., Papadakis, N., Peyre, G., & Aujol, J. F. : (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882

[4] M. Perrot, N. Courty, R. Flamary, A. Habrard : "Mapping estimation for discrete optimal transport", Neural Information Processing Systems (NIPS), 2016

[5] Gabriel Peyré : Convex Optimization, note de cours

[6] Lindbloom Bruce : RGB/XYZ Matrices : http://www.brucelindbloom.com/index.html?EqnRGBXYZ_Matrix.html, consultation : Nov 2018

[7] Gabriel Peyré : Computational Optimal Transport, Mars 2018

[8] Yann Brenier, Thierry Viéville : «La brouette de Monge ou le transport optimal » - Images des Mathématiques, CNRS, 2012

License

MIT