Image Denoising with Convex Optimization

This project implements various image denoising techniques using convex optimization methods, specifically focusing on matrix completion algorithms for image restoration.

Overview

The project explores different approaches to image denoising:

• Quadratic Filter: A basic denoising filter using quadratic regularization
• Total Variation (TV) Filter: Implements TV denoising using primal-dual optimization
• Exact Matrix Completion: The core contribution - uses convex optimization to complete missing/noisy matrix entries

Core Components

Matrix Completion Algorithms

complete_matrix(M, omega)

• Works with any matrix (symmetric or non-symmetric)
• Uses CVXPY for convex optimization
• Minimizes the nuclear norm surrogate (trace) subject to equality constraints

complete_psd_symmetric(M, omega)

• Optimized for positive semi-definite symmetric matrices
• Faster than the general matrix completion algorithm
• Creates a 2n × 2n matrix structure for efficient optimization

Image Processing Functions

Noise Addition

• add_gaussian_noise(im, mean=0, var=0.01): Adds Gaussian noise to images
• add_poisson_noise(im, photons=100): Adds Poisson noise to images
• sp_noise(image): Adds salt-and-pepper noise

Image Utilities

• read_image(filename): Loads images using PIL
• normalize_image(im): Normalizes image pixel values
• PSNR(original_im, cleaned_im): Calculates Peak Signal-to-Noise Ratio for quality assessment

Denoising Filters

quadratic_filter(im, lamb=1)

• Implements quadratic regularization-based denoising
• Uses CVXPY optimization with OSQP solver

TV_filter_pd(im, lamb=1, niter=100)

• Total Variation denoising using primal-dual approach
• Configurable regularization parameter and iteration count

Dependencies

• CVXPY: Convex optimization framework
• NumPy: Numerical computing
• PIL (Pillow): Image processing
• Matplotlib: Visualization and plotting
• OpenCV: Additional image processing capabilities

Usage

The main notebook Exact_matrix_Completion_gaussian_noise.ipynb contains:

1. Noise Generation: Functions to add various types of noise to test images
2. Filter Comparison: Evaluation of different denoising approaches
3. Matrix Completion: Core implementation of exact matrix completion algorithms
4. Performance Analysis: PSNR calculations and result visualization
5. Results Storage: Automatic saving of denoised images and performance metrics

Project Structure

image_denoising/
├── Exact_matrix_Completion_gaussian_noise.ipynb  # Main implementation notebook
├── project.pdf                                   # Project documentation
└── README.md                                     # This file

Key Features

• Multiple Denoising Approaches: Compares traditional filters with matrix completion
• Convex Optimization: Uses CVXPY for robust optimization problems
• Quality Assessment: PSNR-based evaluation of denoising performance
• Flexible Noise Types: Supports Gaussian, Poisson, and salt-and-pepper noise
• Automated Testing: Batch processing of multiple images and noise levels

Implementation Notes

• The matrix completion algorithms are the core contribution of this project
• Uses OSQP solver through CVXPY for efficient optimization
• Results are automatically saved with timestamps and noise parameters
• Supports both grayscale and color image processing

Academic Context

This project appears to be part of ECE602 Introduction to Optimization coursework, focusing on practical applications of convex optimization in image processing.