# Alternating Direction Implicit (ADI) schemes for a PDE-based image osmosis model

## TL;DR

This paper introduces ADI splitting schemes for efficiently solving a PDE-based image osmosis model, preserving key properties and enabling large time steps, with applications demonstrated in shadow removal.

## Contribution

It presents a novel ADI splitting approach for the PDE osmosis model that improves computational efficiency and preserves model properties.

## Key findings

- ADI schemes efficiently solve the PDE model.
- The method preserves properties of the continuous model.
- Applications successfully demonstrate shadow removal.

## Abstract

We consider \emph{Alternating Direction Implicit} (ADI) splitting schemes to compute efficiently the numerical solution of the PDE osmosis model considered by Weickert et al. for several imaging applications. The discretised scheme is shown to preserve analogous properties to the continuous model. The dimensional splitting strategy traduces numerically into the solution of simple tridiagonal systems for which standard matrix factorisation techniques can be used to improve upon the performance of classical implicit methods, even for large time steps. Applications to the shadow removal problem are presented.