Research Accomplishments in 2000-2001: Mathematics in Multimedia

The meeting was a success beyond my expectation. Not only were the large majority of the talks of extremely high quality, but they complemented themselves very well, and in ways we did not anticipate when putting the program together. As an example, at least four speakers dealt with the phenomenon of a spherical drop of fluid splashing on a solid surface: David Wallace from MicroFab, our first speaker, described how the details of this question are crucial for depositing solder drops onto Flip-Chip semiconductor devices; Our third speaker, Ely Sachs from MIT, showed another beautiful example of this in the context of his three dimensional printing processes: drops depositing on a powder layer create craters, which limits the precision of their engineering. Then we had two, more scientific speakers, who touched on this subject: D. Poulikas from ETH showed his numerical simulations which clearly established the phenomenon as due to capillary waves. And then David Quere, from College de France, showed his beautiful experiments of drops bouncing on hydrophobic surfaces, which exposed the various regimes of bouncing drops. One of David's drops lifted off the solid surface in a shape close to a "baseball bat," which then oscillated through a complex series of shapes. Rather remarkably, another of our speakers, Osman Basaran from Purdue, had just presented a numerical simulation of fluid started in a baseball bat shape, and found a sequence of shapes which were very similar to those of Quere. Basaran's "baseball bat shapes" was motivated by a different experiment, the breaking of a pendant drop near a nozzle, a problem discussed in detail by some of the other speakers.

This is just an example of the types of interactions that occurred at the meeting. In a wide range of technologies, the same issues kept coming up again and again, even though the background and motivation of the speakers was very different. The combination of mathematics, engineering and technology at the same meeting allowed everyone to leave with a different perspective on the problems they have been thinking about previously.

Vicent Caselles (Dept. de Tecnologia, Universitat Pompeu-Fabra)   vicent.caselles@tecn.upf.es

Summary of the work done at the IMA

During my stay at the IMA I wrote essentially two papers: one on Qualitative Properties of the Total Variation Flow (joint paper with F. Andreu, J.I Diaz and J.M. Mazon)  pdf (285KB)   postscript (314KB) and a paper for ICCV written jointly with C. Ballester, J. Verdera, M. Bertalmio and G. Sapiro.

In the first of these papers we study the assymptotic behaviour of the minimizing total variation flow, i.e., of the flow u_t = div(Du/\vert Du\vert) under Dirichlet and Neumann boundary conditions. Essentially, we prove that, if T denotes the extinction time of the solution u(t,x) (corresponding to some initial condition u_0) then v(t,x) = u(t,x)/(T-t) converges as t \to T to a function V which is a solution of the eigenvalue problem - div(DV/\vert DV\vert). Then we study the solutions of this eigenvalue problem and give a complete description of them when the domain is a ball and the solution is radial.

The second of these papers deals with the problem of inpainting or desocclusion in images. We proposed a variational formulation for inpainting which is based in an interpretation of the elastica. Thus, our functional can be considered as a relaxed formulation of the elastica and enables us to interpolate the level lines in a missing part of the image. We display several experiments showing the applications and limits of our model. This paper has been submitted to ICCV and completes a previous paper of the authors on this subject.

General description of my interests:

I am interested in image processing, both in its mathematical and algorithmic aspects. Thus, in my work, those two aspects are considered. Thus, in the last years, my interests focused in

1) the study of the total variation model for restoration, proving the well posedness and the geometric features of this flow. As a consequence, I am working in on new models which try to overcome its drawbacks.

2) The study of data structures for images which try to capture the geometric shapes of the image. This is leading to some applications which I am developping now.

3) The stuty of several interpolation problems in images: namely operators which interpolate images when the dynamic range is poor and variational models for singular interpolation (e.g., inpainting).

I will continue to develop these topics, at least, in the recent future, insisting, in particular in the second topic. In all of them, there is published work, but still there is much to be done.

Marco Antonio Fontelos, Departamentos de Matematica Aplicada Universidad Rey Juan Carlos  fontelos@anacleto.escet.urjc.es   attended the IMA "Hot Topics" Workshop on Analysis and Modeling of Industrial Jetting Processes held on January 10 - 13, 2001. Below is his report.

In my last visit to the I.M.A. I have been working with two visitors of the Institute and completed a personal project.

1) With Avner Friedman, I have been working in the modelling of angiogenic processes. Our work originated the preprint "Mathematical analysis of a model for the initiation of angiogenesis" that has been submitted for publication in a mathematical journal. The model consists of a system of 7 differential equations. 3 of them are nonlinear diffusion equations and describe the densities of three types of cells and the other 4 equations are just kinetic equations modelling the reactions between 4 different chemicals that take part in the process. We prove that there exist global (in time) solutions to the system under certain conditions.

2) With Santiago Betelú, a postdoctoral member of the I.M.A., I have initiated work in a model for thin films of Non-Newtonian (power-law) fluids. Up to now, the work has originated a short note intitled ''Capillarity driven spreading of power-law fluids'' that has been submitted. We hope to continue collaboration on this or related topics since, as far as we know, this is a novel approach to thin film equations.

3) I completed a personal project on the evolution of jets of Non-Newtonian (Oldroyd) fluids. The corresponding preprint is entitled ''Break-up and no break-up in some models for the evolution of Non-Newtonian jets," submitted to a mathematical physics journal.

Dirk Horstmann, Mathematisches Institut der Universitaet zu Koeln  dhorst@MI.Uni-Koeln.DE    pdf (16KB)   postscript (23KB)

W. Christopher Lang, Indiana University Southeast   wclang@ius.edu

I was granted by Indiana University Southeast a one-semester sabbatical leave for the Fall Semester of 2000. My goal for this sabbatical was to become involved in research in interdisciplinary or computational applied mathematics. My primary training and experience as a mathematician is in pure mathematics (I am a functional and abstract harmonic analyst), and it has long been my desire to become involved in more applied work.

To this end, I accepted an invitation for a long-term visitorship to the Fall program on Mathematics in Multimedia scheduled for the 2000/2001 academic year at the Institute for Mathematics and its Applications at the University of Minnesota. I was at the IMA for the months of September, October and November.

My goal during my visit was to learn about the range of applications represented by the workshops in the Fall program, particularly applications of harmonic and wavelet analysis. I found the scientific programs to be extremely interesting. As an outsider, I found the talks to be generally clear and accessible, even those well outside my original specialties. I also benefited from the generous help of various workshop participants. I should thank in particular Sanjeev Khudanpur, Alejandro Murua, Jean-Philippe Vert, Bradley Love, Stu Geman, and Don Geman for questions answered and insights shared.

I now plan on focusing on one of the topics I learned about during my visit, after some reflection to decide where to devote my efforts. (Presently, I am most attracted to lower-level image analysis and processing, but I'm planning on returning in May to the IMA for the short course and workshop on computer graphics.)

I believe that my visit to the IMA was well worth the effort and expense; I learned an tremendous amount in a fast two-and-a-half months, and I am confident that I will be able to realize my goal of doing productive research in an area I learned about at the IMA. I would like to express my gratitude to the Director of the IMA, Professor Willard Miller, and the staff of the IMA (especially Fred Dulles, Judy Sweeny, Michelle Glubke and Steve Bonneville), for making my visit to the IMA so valuable and enjoyable.