README.md

Variational Methods for Image Processing

In this project, we implement the variational method to do the image denoising and contrast enhancement. Since the minimum problems are NP-hard, we use the Split Bregman method to approximate the solution. It's efficient and has a convergence guarantee. This work is programmed on MATLAB, so if you want to reproduce our result, please install the MATLAB at the first.

Methodology

Let $f$ be the original image and $u$ be the recover image. We do the image processing by solving the following variational problem.

ROF model (Physica D, 1992)

$\displaystyle \min_{u \in BV(\Omega)}\left(|u|_{TV(\Omega)}+\frac{\lambda}{2}\int _{\Omega}(u-f)^2 dx\right)$

Hsieh-Shao-Yang model (SIIMS 2020)

$\displaystyle \min_{u} \left(\frac{1}{2} \int _{\Omega} |\nabla u - \nabla h|^2 dx + \frac{\lambda}{2} \int _\Omega (u-g)^2 dx + \chi _{[0, 255]}(u) \right)$

Getting the code

You can download a copy of all the files in this repository by cloning this repository:

https://github.com/Jia-Wei-Liao/Variational_Methods_for_Image_Processing.git

Experiments and results

1. Image Denoising
The PSNR of the recover image is 29.702 (dB).

2. Image Contrast Enhancement

Reference

[1] T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), pp. 323-343.
[2] P.-W. Hsieh, P.-C. Shao, and S.-Y. Yang, Adaptive variational model for contrast enhancement of low-light images, SIAM Journal on Imaging Sciences, 13 (2020), pp. 1-28.
[3] L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), pp. 259-268.

Citation

If you find our work useful in your project, please cite:

@misc{
    title  = {variational_methods_for_image_processing},
    author = {Jia-Wei Liao},
    url    = {https://github.com/Jia-Wei-Liao/Variational_Methods_for_Image_Processing},
    year   = {2022}
}