Optimal Transport

The optimal mass transport problem was first introduced 200 years ago by the mathematician  Gaspard Monge. Monge wanted to solve the problem of transporting construction materials from mining sites to construction sites with a minimum cost. The cost depends on the mass of the construction materials and the distance traveled. Despite its apparent simplicity, the problem attracted the attention of the research community since that time, resulting a large body of work.
In the first half of the 20th century, another mathematician, and economist, Leonid Kantorovich made important contributions to the field and proposed the first practical method to solve the problem using linear programming. Kantorovich received the Nobel prize in economics for related work in 1975.
Much later, in 1997,  McCann introduced a time-dependent version of the problem, where the two densities are assumed to lie on the same space, and the goal is to find a time-continuous transport plan, also called a displacement interpolation, that minimizes the transport cost. An important contribution to field came sooner after this, in 2001, when Benamou and Brenier showed that time-dependent optimal transport with the squared Euclidean distance as cost admits a computational fluid dynamics formulation. It is shown equivalent to finding a flow of minimum kinetic energy that transports the initial density to the final one. This allowed the development of an iterative numerical scheme to solve the time-dependent transport problem with the squared Euclidean distance as cost.

Since its introduction, and especially in the last two decades, optimal transport came to be applied and connected to various fields in mathematics, science and engineering, including geometry, partial differential equations (Cedric Villani received the Fields Medal in 2001 for his work in optimal transport and the kinetic theory of the Boltzmann equation), probability theory, fluid dynamics, optics, geophysics and oceanography, meteorology, cosmology, crowd movement modeling, antenna design, image processing and computer vision.

My work on optimal transport is concerned with numerically solving the optimal transport problem under soft constraints (additional terms in the cost function) and hard constraints, as well as the potential applications of constrained optimal transport, for instance in the registration of medical images.

For further details, see the list of publications.

Image Interpolation and Registration

Interpolation between images, also known as image morphing, is the process of fnding a continuous deformation of one image to another. It is a basic image processing task with many applications, especially in computer graphics, where it is frequently used for special effects in cinematography. Another application of image interpolation is in computer vision, where it is applied for view morphing.
Image registration is the process of putting in correspondence two images of the same scene or object taken at different times, from different positions or  using different imaging modalities. The correspondence needs to be meaningful, in the sense that similar parts in the two images are mapped to each other.
Image interpolation is tightly linked to image registration, since many techniques for registration achieve the desired transformation by gradually deforming one image to the other. Image registration is an important step for many image processing and computer vision tasks such as medical image analysis, remote sensing, video surveillance and segmentation.
The application of optimal transport to image registration was pioneered by Tannenbaum and his group at Georgia tech. My work in this area consists in applying constrained optimal transport to obtain a better registration, especially in the case of medical images.

For further details, see the list of publications.

A Parallel Solver for Optimal Transport: Optrans

Solving the optimal transport problem, especially the time-dependent version and its fluid dynamics formulation is a computationally demanding task. In addition to this, there are no available softwares dedicated to optimal transport.
To remedy to this shortcoming, I developed Optrans, a parallel library for solving the time-dependent optimal transport problem. The library handles constrained as well as unconstrained problems. The time-dependent problem is computationally complex but lends itself to parallelism, which motivates the choice of a parallel implementation.
The figure bellow shows the solution of a three dimensional transport problem computed by Optrans.

For further details, see the list of publications.