Our aim is to make a methodological contribution to the analysis of three applications in Computer Vision. The methodologies being linked with what we are looking for, our second aim will be to explicitely write and justify the chosen algorithms and to show obtained results.
Applications are :
* Restore one image.
Among the great variety of nonlinear PDE based techniques, we begin with a comparative study. We propose a common formalism for this method and we proceed to a benchmark to test their capabilities. Then we establish a convergent method for noise and blur removal. This approach combines parabolic and hyperbolic operators. We will justify this model using the theory of viscosity solutions and we will give a suitable numerical scheme which take into account the specificity of each term.
* Know the movement.
For a given sequence of images, our aim is to estimate the velocity field, usually called the optical flow. We propose a variational method based on the optical flow constraint. We study this problem on the space of bounded variations. Two situations are considered, depending on the regularity of the data :
- If the data is Lipschitz, we prove that the problem on BV is well posed and we construct via arguments from Gamma-convergence and duality, a convergent algorithm.
- If the data is only BV, we establish the integral representation of the relaxed problem for a suitable topology.
* Analyse sequence of images.
We propose an original variational approach to deal with sequences of noisy images with static background. The specificity of this method is to estimate simultaneously the background and the movement detection. We justify theoretically this model and give a stable and convergent algorithm. Numerous examples will illustrate the capabilities of this original approach.