Update 7 files

- /images/DOT_3_simplified_curve.jpg
- /images/DOT_3_simplified.jpg
- /README.md
- /requirements.txt
- /ot-image-restoration.ipynb
- /model_fashion_mnist.pt
- /Deep_Optimal_Transport_Presentation.pdf

README.md

+# README
+
+### Team : Fatma Zahra Zaghouani, Islem Chaari, Léo Rouppert
+
+## Description
+The paper "Deep Optimal Transport: A Practical Algorithm for Photo-realistic Image Restoration" presents an image restoration algorithm that allows control over the perceptual quality and/or mean square error (MSE) of any pre-trained model, enabling a trade-off between these aspects during testing. This few-shot algorithm requires only about a dozen images restored by the model to significantly enhance the perceptual quality or MSE for new images without additional training.
+
+The approach is based on theoretical findings that connect the minimum MSE predictor with the predictor that minimizes MSE under a perfect perceptual quality constraint, achieved by optimally transporting the output of the former to match the source data distribution. This optimal transport is approximated by a linear transformation in the latent space of a variational autoencoder, computed in closed-form using empirical means and covariances.
+
+## Installation
+Make sure Python is installed on your machine. Then, install the required dependencies with:
+
+```sh
+pip install -r requirements.txt
+```
+
+## Usage
+Run the notebook with the parameters you want to test. A pretrained VAE model on the Fashion-MNIST dataset is provided.
+
+## Difference Between the Original and Simplified Transport Operator
+
+The original optimal transport operator based on the paper has been modified for computational simplification while retaining its fundamental properties.
+
+In the original version, the transport operator is defined as:
+
+$$
+T_{p_x \longrightarrow p_y}(x) = \Sigma_x^{-\frac{1}{2}} \left(\Sigma_x^{\frac{1}{2}} \Sigma_{y} \Sigma_x^{\frac{1}{2}}\right)^{\frac{1}{2}} \Sigma_x^{-\frac{1}{2}} (x - \mu_{x}) + \mu_y
+$$
+
+where $x \sim \mathcal{N}(\mu_{x}, \Sigma_x)$ and $y \sim \mathcal{N}(\mu_{y}, \Sigma_y)$.
+
+In the simplified version, the transport operator is rewritten by replacing the successive multiplications with simpler transform :
+
+$$
+T_{p_{x} \longrightarrow p_{y}}(x) = \Sigma_y^{\frac{1}{2}}  \Sigma_x^{-\frac{1}{2}}  (x - \mu_x) + \mu_y
+$$
+
+This reformulation reduces the number of matrix multiplications and facilitates the inversion of matrices, improving the numerical stability of the model while maintaining a transformation close to the original. This approach led to some successful results whereas the original transport operator failed to improve perceptual quality.
+
+## Example Results
+
+![Results](images/DOT_3_simplified.jpg)
+![Curve](images/DOT_3_simplified_curve.jpg)
+
+## References
+- [Deep Optimal Transport: A Practical Algorithm for Photo-realistic Image Restoration](https://arxiv.org/pdf/2306.02342).
+
+## Author
+This code was developed by Léo Rouppert.
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requirements.txt

+torch
+torchvision
+numpy
+pandas
+matplotlib
+pillow
+scipy
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