Image Denoising

In this experiment we test different image denoising techniques through the lens of convex optimization. Specifically, we test a quadratic filter, a total-variation filter, the non-local means approach, and the non-local weighted nuclear norm minimization approach. We benchmark these approaches against 5 images from the University of Southern California's image database (specifically aerial, boat, bridge, clock, and couple), when these images have been corrupted by additive zero-mean Gaussian noise and Poisson noise. Our final report explains in further detail our approach, the math behind it, and the references used.

Installation

Clone repo

git clone https://github.com/Federico-PizarroBejarano/image_denoising.git
cd image_denoising

Create a Conda Environment

Create and access a Python 3.8 environment using conda

conda create -n denoise python=3.8.10
conda activate denoise

Install the image_denoising repository

pip install --upgrade pip
pip install -r requirements.txt

Install Mosek

Obtain MOSEK's license (free for academia). Once you have received (via e-mail) and downloaded the license to your own ~/Downloads folder, install it by executing

$ mkdir ~/mosek                                                    # Create MOSEK license folder in your home '~'
$ mv ~/Downloads/mosek.lic ~/mosek/                                # Copy the downloaded MOSEK license to '~/mosek/'

Denoising Techniques

Consider a greyscale image Y. We are looking for a denoised image X. We have implemented the following techniques:

• Quadratic Filtering: Solve the optimization

  min_X ||X - Y||_sq + λ ||∇X||_sq

  where ||X||_sq is the sum-of-squares norm, and ∇X is the gradient of the image X.

• Total Variation (TV) Filtering: Solve the optimization

  min_X ||X - Y||₁ + λ ||X||_TV

  where ||X||_TV is the total-variation operator.

• Non-Local Means (NLM) Filtering: Denoising every pixel by making it a weighted average of every other pixel in the image, weighted by the similarity between the patches around the two pixels.

• Non-Local Weighted Nuclear Norm Minimization (WNNM): Finding similar patches to every patch in the image and using a weighted nuclear norm minimization to remove noisy singular values using a quadratic program.

Optimization Techniques

Quadratic filtering and TV filtering are bicriteria optimizations since they have two different objectives. To solve this, we have implemented the primal-dual algorithm, and compared this approach with the out-of-the-box convex optimization provided by cvxpy.

Testing

We compare the various approaches with additive white Gaussian noise (zero-mean but known variance) and Poisson noise (varying photon availability). We compare the efficacy of the approaches both qualitatively, and using the peak signal-to-noise ratio (PSNR). The original test images can be found in images and the pickled results and images can be found in results.

The final results for additive white Gaussian noise are:

The final results for Poisson noise are: