Diffusion-Shock PDEs for Deep Learning on Position-Orientation Space

Area of Science:

• Computer Vision
• Image Processing
• Differential Geometry

Background:

• Regularised diffusion-shock (RDS) filtering is an effective image processing technique.
• Extending RDS filtering to position-orientation space offers potential advantages for handling complex image structures.

Purpose of the Study:

• To extend regularised diffusion-shock (RDS) filtering from Euclidean space to position-orientation space.
• To enhance and inpaint crossing structures in images more effectively.
• To develop a generalized diffusion framework for image processing tasks.

Main Methods:

• Lifting image data to position-orientation space (M2).
• Utilizing gauge frames to mitigate orientation lifting issues.
• Developing generalized diffusion equations not solely reliant on the Laplace-Beltrami operator.
• Integrating RDS filtering into a geometric deep learning framework (PDE-CNN, PDE-G-CNN).

Main Results:

• The extended RDS filtering successfully disentangles and enhances crossing structures in position-orientation space.
• RDS filtering in M2 demonstrates superior performance in denoising images with crossing structures compared to TR-TV flow, NLM, and BM3D.
• M2 RDS inpainting effectively restores crossing structures, unlike its R2 counterpart.
• Theoretical results confirm the well-posedness, smoothing, and analytic properties of the generalized diffusions.
• New RDS filtering PDE layers integrated into deep learning frameworks show benefits in impainting and denoising tasks.

Conclusions:

• Extending RDS filtering to position-orientation space is a significant advancement for image processing, particularly for images with crossing structures.
• The developed gauge frame approach and generalized diffusion framework offer robust solutions.
• The integration with geometric deep learning provides a powerful and versatile tool for various image restoration tasks.