In this paper we reconsider the sub-Riemannian cortical model of image completion introduced in [G. Citti and A. Sarti, J. Math. Imaging Vision, 24 (2006), pp. 307--326]. This model combines two mechanisms, the sub-Riemannian diffusion and the concentration, giving rise to a diffusion driven motion by curvature. In this paper we give a formal proof of the existence of viscosity solutions of the sub-Riemannian motion by curvature. Furthermore we illustrate the sub-Riemannian finite difference scheme used to implement the model and we discuss some properties of the algorithm. Finally results of completion and enhancement on a number of natural images are shown and compared with other models.

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton--Jacobi--Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case $\mathcal{C}=1$. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.

We designed a strategy for extracting the shapes of cell membranes and nuclei from time lapse confocal images taken throughout early zebrafish embryogenesis using a partial-differential-equation-based segmentation. This segmentation step is a prerequisite for an accurate quantitative analysis of cell morphodynamics during embryogenesis and it is the basis for an integrated understanding of biological processes. The segmentation of embryonic cells requires live zebrafish embryos fluorescently labeled to highlight sub-cellular structures and designing specific algorithms by adapting classical methods to image features. Our strategy includes the following steps: the signal-to-noise ratio is first improved by an edge-preserving filtering, then the cell shape is reconstructed applying a fully automated algorithm based on a generalized version of the Subjective Surfaces technique. Finally we present a procedure for the algorithm validation either from the accuracy and the robustness perspective.

Automatic ultrasound (US) image segmentation is a difficult task due to the quantity of noise present in the images and the lack of information in several zones produced by the acquisition conditions. In this paper, we propose a method that combines shape priors and image information to achieve this task. In particular, we introduce knowledge about the rib-eye shape using a set of images manually segmented by experts. A method is proposed for the automatic segmentation of new samples in which a closed curve is fitted taking into account both the US image information and the geodesic distance between the evolving curve and the estimated mean rib-eye shape in a shape space. This method can be used to solve similar problems that arise when dealing with US images in other fields. The method was successfully tested over a database composed of 610 US images, for which we have the manual segmentations of two experts.