In my research, I like to combine computational methods with analysis in order to provide more efficient and simpler algorithms. One aspect of my research looks at the computations that can be done on implicitly defined or non parametric interfaces. The other aspect of my research looks at segmentation algorithms for phase-contrast images of bacterial populations. Below I describe some of the work I have done, together with ongoing research and projects I am interested in working on in the future.

Theme 1: Computing with implicit and non parametric interfaces

Algorithms for area preserving flows

With Selim Esedoglu and Jeffrey Fessler, I developed efficient algorithms for computing certain area preserving geometric motions of curves in the plane, such as area preserving motion by curvature, when the curve is described implicitly by a signed distance function. The signed distance function is a specific level set function. Our schemes are based on a class of diffusion generated motion algorithms that simulate the evolution of an interface by alternately diffusing (solving the heat equation) and constructing the signed distance function to the interface. These algorithms are very fast and very accurate. Below I show an example of the computations that can be done with these algorithms. The algorithm in these figures computes the area preserving curvature motion on a large number of interfaces (an example of large scale computations and of how accurate it is even when the interface is resolved with few pixels).

Area preserving motion by curvature on initial data with 40% area fraction. The very good accuracy of our algorithms allows us to carry out simulations with an initial number of droplets as large 25000 on a 4097^2 grid with droplets as small as 15 pixels in diameter. The computations were performed using an adaptive time step regulated by the average size of the droplets, since the average size of the droplets increases during the evolution, thus slowing down the dynamics.

Boundary integrals on implicit and non parametric interfaces

The overall theme of this research pertains to the efficient and seamless computation of boundary integrals in the context of level set methods (implicit interfaces) and non parametric interfaces. With Richard Tsai and Nicolay Tanushev, I obtained an integral formulation for computing boundary integrals when the boundary is a closed manifold of codimension 1. This integral formulation rewrites the boundary integral exactly as an integral over the ambiant space R^n.  The advantage of this result is that integrals over R^n are discretized easily using simple Riemann sums. Richard Tsai and I later generalized this integral formulation to open curves and surfaces of codimension 1, and curves in R^3 (codimension 2). In addition, we showed that the Jacobian involved in these formulations can be expressed as the product of the nonzero singular values of the closest point mapping. The closest point mapping (or orthogonal projection) is a map that projects a point x in R^n onto the manifold, in other words that associates to any point x in R^n its closest point x* on the manifold.
These integral formulations are particularly useful when using boundary integral methods to solve PDEs. We have used these formulations to compute the solution of Poisson's equation and to compute the motion of interfaces evolving under Mullins-Sekerka dynamics on unbounded domains. Below I show the solution of Poisson's equation with Dirichlet boundary conditions.

                                                                                            Solution of Poisson's equation with Dirichlet boundary conditions on a flower domain.

One of the current projects looks at extending the integral formulation to curves and surfaces with singularities (corner and cusp in 2D, corner, cusp, edges in 3D).
Another project involves the computation of the Laplace-Beltrame operator on surfaces using our integral formulation.
I am also interested in extending the integral formulation to any level set function and to investigate if this generalization is practical and if it has computational advantages. One large advantage is that one would not need to construct the signed distance function after each time step, as is currently needed.

I have recently become interested in the closest point mapping and some of its general properties. Can we prove a general statement relating the number of singular values of its Jacobian matrix to the dimension of the manifold? Can this be applied to manifold learning, namely can we use this property to compute the dimension of a set of unorganized point clouds sampled from a manifold?

Finally I am interested in understanding an area called shape optimization and in learning whether we can combine their technique with our formulation to obtain new computational algorithms for generating interfacial motions.

Theme 2: Image processing

Coarsening in high order, ill-posed diffusion equations

Motivated by questions raised by Selim Esedoglu, I studied the coarsening of a discrete set of equations generalizing the Perona-Malik and You-Kaveh PDE models used to denies and sharpen images. The idea behind these models was to devise a method for gradually simplifying (or coarsening) an image by diffusing out its details starting with the smallest scales. An important property was to keep object boundaries in the image sharp during the diffusion, until their disappearance at some point in the process. 

I studied a generalized discrete set of equations that were generalizing the Perona-Malik and You-Kaveh models, and obtained upper bounds for the coarsening laws. I also obtained a rigorous result directly stated in terms of the number of edges in the image, which is more relevant in applications.

Segmentation of phase-contrast images

With Alan Veliz-Cuba (University of Dayton), I am working on an algorithm for segmentation phase-contrast images of bacterial population. These segmentations are currently done by hand and take a long time. The goal is to automate the segmentation with minimal manual intervention. Below I show an example of a phase-contrast image of bacterial population.

In this work, we use simple geometric information on the shape of the cells, to segment most of the easy cells in the image. The complicated situations arise when cells are very close to each other: this is when the simple criteria based on geometry cannot separate them. We deal with the complicated situations by using a multi scale decomposition of these union of cells and then building a graph the vertices of which are the pieces of cells in the subdivided images that are convex. In addition, we plan to add weights to this graph based on how the convexity changes from one scale to the next. The goal is to build a weighted graph such that its graph cuts match the separation between the cells in the image. In essence, we are trying to represent the information given in the image as a graph, and this graph needs to have the right information such that its graph cuts coincide with the boundaries of the cells in the image.