Section: New Results

Geometry of vision and sub-Riemannian geometry: new results

Let us list here our new results in the geometry of vision axis and, more generally, on hypoelliptic diffusion and sub-Riemannian geometry.

• In [12] we propose a variational model for joint image reconstruction and motion estimation applicable to spatiotemporal imaging. This model consists of two parts, one that conducts image reconstruction in a static setting and another that estimates the motion by solving a sequence of coupled indirect image registration problems, each formulated within the large deformation diffeomorphic metric mapping framework. The proposed model is compared against alternative approaches (optical flow based model and diffeomorphic motion models). Next, we derive efficient algorithms for a time-discretized setting and show that the optimal solution of the time-discretized formulation is consistent with that of the time-continuous one. The complexity of the algorithm is characterized and we conclude by giving some numerical examples in 2D space + time tomography with very sparse and/or highly noisy data.

• The article [16] presents a method to incorporate a deformation prior in image reconstruction via the formalism of deformation modules. The framework of deformation modules allows to build diffeomorphic deformations that satisfy a given structure. The idea is to register a template image against the indirectly observed data via a modular deformation, incorporating this way the deformation prior in the reconstruction method. We show that this is a well-defined regularization method (proving existence, stability and convergence) and present numerical examples of reconstruction from 2-D tomographic simulations and partially-observed images.

• The article [28] adapts the framework of metamorphosis to the resolution of inverse problems with shape prior. The metamorphosis framework allows to transform an image via a balance between geometrical deformations and changes in intensities (that can for instance correspond to the appearance of a new structure). The idea developed here is to reconstruct an image from noisy and indirect observations by registering, via metamorphosis, a template to the observed data. Unlike a registration with only geometrical changes, this framework gives good results when intensities of the template are poorly chosen. We show that this method is a well-defined regularization method (proving existence, stability and convergence) and present several numerical examples.

• In [8] we prove the $C^1$ regularity for a class of abnormal length-minimizers in rank 2 sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank 2 sub-Riemannian structures of step up to 4 are of class $C^1$

• In [33] we show that, for a sub-Laplacian $\Delta$ on a 3-dimensional manifold $M$, no point interaction centered at a point $q_0\in M$ exists.

• In [39] we consider a one-parameter family of Grushin-type singularities on surfaces, and discuss the possible diffusions that extend Brownian motion to the singularity. This gives a quick proof and clear intuition for the fact that heat can only cross the singularity for an intermediate range of the parameter. When crossing is possible and the singularity consists of one point, we give a complete description of these diffusions, and we describe a “best" extension, which respects the isometry group of the surface and also realizes the unique symmetric one-point extension of the Brownian motion, in the sense of Chen-Fukushima. This extension, however, does not correspond to the bridging extension, which was introduced by Boscain-Prandi, when they previously considered self-adjoint extensions of the Laplace-Beltrami operator on the Riemannian part for these surfaces. We clarify that several of the extensions they considered induce diffusions that are carried by the Marin compactification at the singularity, which is much larger than the (one-point) metric completion. In the case when the singularity is more than one-point, a complete classification of diffusions extending Brownian motion would be unwieldy. Nonetheless, we again describe a “best" extension which respects the isometry group, and in this case, this diffusion corresponds to the bridging extension. A prominent role is played by Bessel processes (of every real dimension) and the classical theory of one-dimensional diffusions and their boundary conditions.

• In [50] we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc length in the Heisenberg group, that is the simplest sub-Riemannian structure. Our goal is to give a metric interpretation of this notion of geodesic curvature as the first corrective term in the Taylor expansion of the distance between two close points of the curve.

We would also like to mention the monograph [30] and the PhD thesis of Mathieu Kohli [3].