Solving ill-posed Image Processing problems using Data Assimilation

Data Assimilation is a mathematical framework used in environmental sciences to improve forecasts performed by meteorological, oceanographic or air quality simulation models. It aims to solve an evolution equation, describing the temporal dynamics, and an observation equation, linking the state vector and observations. In this article we use this framework to study a class of ill-posed Image Processing problems, usually solved by spatial and temporal regularization techniques. An approach is proposed to convert an ill-posed Image Processing problem in terms of a Data Assimilation system, solved by a 4D-Var method. This is illustrated by the estimation of optical flow from a noisy image sequence, with the dynamic model ensuring the temporal regularity of the result. The innovation of the paper concerns first, the extensive description of the tasks to be achieved for going from an image processing problem to a data assimilation description; second, the theoretical analysis of the covariance matrices involved in the algorithm; and third a specific discretisation scheme ensuring the stability of computation for the application on optical flow estimation.