Advanced PDE-Based Image Processing

A complete course on edge-preserving anisotropic image filtering. The course takes you from the elementary heat equation all the way to a publishable-quality filter library grounded in Finsler geometry, following

H. Kilic, S. Ceyhan & O. N. Gerek, "A novel family of edge preserving anisotropic filters", Digital Signal Processing 128 (2022) 103623. https://doi.org/10.1016/j.dsp.2022.103623

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Why this course?

Classical filters (Gaussian, mean, median) remove noise but blur edges. Anisotropic, geometry-driven filters remove noise while preserving edges, which is critical when filtering is a pre-processing step before edge detection, segmentation, or measurement — for example in biomedical or satellite imaging. This course explains why these filters work, derives them from first principles, and has you build your own.

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Course outline

Week	Notebook	Topic	Role of diffusion
1	01_foundations.ipynb	Foundations	Anatomy of diffusion, heat equation, FDM, Perona–Malik, vectorisation
2	02_variational_theory.ipynb	Variational approach	Energy functionals (Dirichlet energy) and minimisation
3	03_polyakov_action.ipynb	Polyakov action	Deriving the energy integral; modelling the image surface
4	04_laplace_beltrami.ipynb	Laplace–Beltrami flow	Diffusion as a geometric operator
5	05_finsler_geometry.ipynb	Finsler geometry	From Riemann to Finsler; defining metrics
6	06_new_randers.ipynb	Randers metrics	The New Randers metric in the flow
7	07_miron_normalized.ipynb	Miron metrics	New Normalized Miron and edge analysis
8	08_numerical_schemes.ipynb	Advanced PDE solving	Stability (CFL), boundary conditions, implicit solvers
9	09_cuda_acceleration.ipynb	GPU & parallelisation	Numba/CUDA hardware acceleration
10	10_multichannel_flow.ipynb	Colour (RGB) flow	Vector diffusion, colour-space coupling
11	11_benchmarking.ipynb	Quality & validation	PSNR, SSIM, IF, MI and the Edge-Preservation Index (EPI)
12	12_final_thesis.ipynb	Capstone project	Build your own Custom Anisotropic Filter library

A printable syllabus with weekly objectives and the English language-focus map is in SYLLABUS.md.

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The anisofilt package

All notebooks import a small, tested reference library that implements the paper's algorithms:

import anisofilt as af
from anisofilt import utils, metrics

img   = utils.sample_image("camera", size=256)
noisy = utils.add_gaussian_noise(img, sigma=0.08)

# Paper's best-performing filters (beta=1, dt=0.05, 20 iterations):
denoised = af.new_metric(noisy)            # New Metric  (Synge-Beil, c = 1 - 1/g)
denoised = af.new_normalized_miron(noisy)  # New Normalized Miron, c = (1 - 1/g)/V
denoised = af.new_randers(noisy)           # New Randers metric

print("PSNR:", metrics.psnr(img, denoised))
print("EPI :", metrics.edge_preservation_index(img, denoised))

Module	Contents
anisofilt.core	finite-difference derivatives, metric tensor g_{μν}, determinant, inverse, isotropic Laplace–Beltrami
anisofilt.tangents	3×3 maximum-contrast tangent-vector estimation
anisofilt.finsler	Synge–Beil family: Normalized Miron, New Normalized Miron, New Metric
anisofilt.randers	New Randers metric filter
anisofilt.classical	isotropic Beltrami flow + Gaussian / mean / median / Perona–Malik baselines
anisofilt.metrics	MSE, PSNR, IF, SSIM, MI, Edge-Preservation Index
anisofilt.utils	test images, noise models, synthetic edge scenes

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Installation

git clone https://github.com/HAYDARKILIC/adv_pde_based_image_processing.git
cd adv_pde_based_image_processing
python -m venv .venv && source .venv/bin/activate    # optional
pip install -r requirements.txt
pip install -e .          # installs the `anisofilt` package in editable mode
jupyter lab               # open the notebooks/ folder

GPU note. Week 9 auto-detects your hardware. The Numba CPU path runs everywhere; the CUDA path executes only if an NVIDIA GPU is present and otherwise prints the kernel for study — no error either way.

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Running the tests

pip install pytest
pytest -q

The test suite checks that every filter improves PSNR on a noisy image and that the quality metrics behave correctly (e.g. PSNR of an image against itself is infinite).

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How to study

1. Read the markdown of a week top-to-bottom before running the code.
2. Run each code cell and tie its output back to the equation it implements.
3. Do the end-of-notebook Exercises — they include a writing task each week.
4. By Week 12 you assemble everything into your own filter and write a short report.

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License & citation

Course code released under the MIT License (see LICENSE).

If you use this material, please cite the original paper:

@article{kilic2022anisotropic,
  title   = {A novel family of edge preserving anisotropic filters},
  author  = {Kilic, Haydar and Ceyhan, Salim and Gerek, Omer Nezih},
  journal = {Digital Signal Processing},
  volume  = {128},
  pages   = {103623},
  year    = {2022},
  doi     = {10.1016/j.dsp.2022.103623}
}