Non-Local Morphological PDEs for Image Processing

This repository contains the complete code implementation and results for our B.Tech Major Project titled "Non-Local Morphological PDEs for Image Processing", carried out by Adarsh Roy and Abhinna Agarwal under the supervision of Dr. Mani Mehra, Department of Mathematics, IIT Delhi.

---

Project Overview

This project explores the use of non-local morphological Partial Differential Equations (PDEs) and p-Laplacian operators on graph structures for advanced image denoising and smoothing.

We extend traditional local models with graph-based non-local operators and also introduce fractional order anisotropic diffusion models for improved preservation of image features.

---

📖 Full Report

A complete PDF report of the project including theory, methodology, and results is available here:

main.pdf

---

Folder Structure

.
├── barbara.png                 # Sample grayscale image used in experiments
├── main.pdf                    # Final project report
├── fpde.ipynb                  # Notebook for fractional PDE-based image denoising
├── pde_big_graph.ipynb         # PDE implementation on large graph (non-local)
├── pde_mid_graph.ipynb         # PDE on mid-sized grid graph
├── pde_small_graph.ipynb       # PDE on small 4/8-adjacency grid
├── pLaplacianDiffusion.py      # Graph-based p-Laplacian diffusion using torch_pdegraph
├── psnr.ipynb                  # Notebook to compute PSNR for denoising results
└── README.md

---

Installation

Make sure Python ≥ 3.8 is installed. Then install dependencies:

pip install numpy scipy matplotlib opencv-python scikit-image tqdm
pip install torch

Then check the instructions for installing torch_pdegraph locally here.

---

Running the Code

1. Clone the repository:

   git clone https://github.com/Adarsh-Roy/pde-fpde-image-processing.git
   cd pde-fpde-image-processing

2. For fractional PDE model:

   jupyter notebook fpde.ipynb

3. For non-local PDEs on graphs:

   jupyter notebook pde_big_graph.ipynb
   # or try pde_small_graph.ipynb or pde_mid_graph.ipynb

4. To compute PSNR of results:

   jupyter notebook psnr.ipynb

5. For direct PyTorch-based graph PDE experiments:

   python pLaplacianDiffusion.py

---

Dependencies and Libraries

This project heavily uses:

• torch_pdegraph: For pre-built morphological and diffusion operators on graphs.
• numpy, matplotlib, scikit-image: For image manipulation and visualization.

Special thanks to the developers of torch_pdegraph for enabling easy-to-use GPU-compatible graph PDE modeling.

---

Sample Results

Graph PDE Denoising with different α and β values:

• α = 1.0, β = 0.0 → ∞-Laplacian
• α = 0.5, β = 0.5 → Normalized p-Laplacian
• α = 0.0, β = 1.0 → Laplacian

Fractional PDE Results with α ∈ [1.0, 2.2] showed improved edge preservation and reduced noise for intermediate values.

Visual comparisons and figures are included in the main.pdf.

---

Authors

• Adarsh Roy - GitHub
• Abhinna Agarwal - GitHub

Supervised by: Dr. Mani Mehra, IIT Delhi

---

References

• Bougleux et al., Nonlocal Regularization on Weighted Graphs (2009)
• Elmoataz et al., Morphological PDEs on Graphs (2012)
• Lindqvist, Notes on the p-Laplace Equation (2006)
• Aronsson, Infinity Laplacian Theory (1967)
• torch_pdegraph: https://github.com/aGIToz/Pytorch_pdegraph

---