Imaging and Math Scientists See a Future of Continued Close Interaction

May 3, 2002

A highlight of a session on the Visible Human Project's "Adam" and "Eve," organized by Fred Bookstein, was the demonstration of a new program for the dynamic navigation of continuously tumbling sections of the images of the two cadavers. Shown here is a series of planes perpendicular to Eve's corpus callosum.
Tony Chan, dean of physical science in the College of Letters and Science at UCLA and a member of the organizing committee for the First SIAM Conference on Imaging Science, wrote the following personal perspective as an introduction to a series of "snapshots" from several of the conference speakers (see pages 4-8 in this issue of SIAM News). David Wilson of the University of Florida, who chaired the organizing committee,* helped out with recollections of the early days of the SIAM Activity Group on Imaging Science.

The first SIAM Conference on Imaging Science, held in Boston, March 4-6, came at a time of unusual prominence for imaging science. Object tracking in videos and facial recognition, long-standing problems for imaging scientists, had suddenly taken on new meaning and heightened urgency.

Long before 9/11, however, it was clear to many SIAM members that imaging science had matured and developed to a stage that there was sufficient interest to support an activity group and also to hold a national conference. Researchers were already talking about an activity group in imaging in the early 1990s. Things really got under way in March 1998, when then SIAM president Gilbert Strang, in an open forum at the SIAM-SEAS meeting in Knoxville, Tennessee, invited the audience to suggest new areas where SIAM should be putting its energy.

Gerhard Ritter immediately proposed that SIAM form an activity group in imaging science. Strang's response, also immediate, was positive. Not long afterward, with the bylaws of the recently created SIAM Activity Group on Life Sciences as a model, the SIAG on Imaging Science was created. The group's first officers were Gerhard Ritter, chair; Robert J. Plemmons, vice chair; Bernard Mair, secretary; and David Wilson, program director.

Of course, imaging science is a huge enterprise and has long been an established subject (in, for example, the medical, computer science, and electrical engineering communities). What got SIAM interested was the increasing realization that the kinds of mathematical and computational techniques being employed in imaging science have much in common with those used in other more traditional SIAM areas and that bringing imaging scientists together with computational and applied mathematicians would benefit both groups. Imaging science also has a wide variety of important applications that would be unwise for SIAM to overlook.

Judging from the Boston meeting, this thinking was valid. The meeting was extremely well attended (with 229 registered attendees) and offered a remarkable variety of mathematical topics and application areas. The meeting obviously benefited from the large number of academic and medical institutions in the Boston area, as participation of local attendees was significant. A particularly interesting and successful feature was that the last day of the three-day conference was held in conjunction with the first SIAM Conference on Life Sciences.

The meeting was organized around five plenary talks---Margaret Cheney, on synthetic aperture radar; Stan Osher, on PDEs and level sets; Stéphane Mallat, on geometric wavelets; Ed Dougherty, on morphology and genomics; and David Mumford, on stochastic image models-each associated with a number of related minisymposia. These topics represent some of the major recent mathematical developments in imaging science. With a total of 27 minisymposia, seven contributed sessions, and 14 poster presentations, the conference covered a broad spectrum of the main areas of imaging science, including compression, restoration, segmentation, tracking, recognition. . . . Also featured were a wide variety of applications, many in the medical and life sciences, others in computer vision and computer graphics, and in security and defense applications.

Invited speaker Stanley Osher used a simulation of a Grand Canyon fly-through to illustrate dynamic visibility, one of several imaging applications of level set methods described in the talk. Invited speaker Stanley Osher used a simulation of a Grand Canyon fly-through to illustrate dynamic visibility, one of several imaging applications of level set methods described in the talk.

A number of unifying themes can be identified. Different modes of imaging, from SAR to tomography (e.g., PET, MRI), were discussed. Speakers presented a wide variety of tools for image representation: pixels, continuous functions, multiresolution techniques, level sets, Markov random fields, mathematical morphology.

One very prominent emerging set of techniques revolves around the use of variational principles, and PDE and level set methods. This theme was beautifully summarized by Osher, and then elaborated in a series of four very well-attended minisymposia organized by Andrea Bertozzi (Duke University) and Luminita Vese (University of California, Los Angeles). Particularly noteworthy about these methods is their natural incorporation of geometric features of the images (e.g., curvature of level curves, shapes of objects), along with their ability to preserve discontinuities (edges) in the images. Interesting and surprising connections arise with models and techniques in computational fluid dynamics. Bertozzi, for example, talked about the similarity between the image inpainting PDEs of Sapiro et al. and the stream function-vorticity form of the Navier-Stokes equations. Vese's talk touched on the use of multiple level set representations of images to exploit the four-color theorem.

Wavelets have played an increasingly central role in image compression (the JPEG 2000 standard, for instance, was designed with wavelets in mind). Today, geometric aspects have become a focus for many researchers in wavelets. Mallat discussed bandelets, a way of providing a sparse orthogonal basis for approximating images with geometric regularity. Emmanuel Candès (California Institute of Technology) described curvelets, which are designed for the same purpose. Candès was a speaker in one of two minisymposia organized by Michael Unser (Ecole Polytechnique Fédérale de Lausanne) to complement Mallat's talk.

Stochastic models with a Bayesian statistical framework represent a third way of representing, analyzing, and thinking about images. These techniques have been very successful and have been widely adopted in practical applications. But a deeper look from this stochastic viewpoint also reveals fundamental issues, major challenges, and big surprises.

Natural images, Mumford explained, appear to be samples from a highly non-Gaussian but scale-invariant distribution; the decomposition of the world into objects emerges from the empirical study of image statistics. The Bayesian theme was also taken up in two minisymposia, one organized by Anuj Srivastava (Florida State University) and the other by Hamid Krim (North Carolina State University). An interesting discussion arose between Mumford and Osher during the former's talk as to whether images are bounded variation (BV) functions (and, correspondingly, whether the total variation method is the right regularization). A recent comment of Yves Meyer seems to apply here: Images can be decomposed into two parts, one part in BV functions and the other consisting of textures (and more appropriately modeled stochastically). The truth, as in many other contexts, may be that all three ways of looking at images---stochastic models, wavelets, and PDEs---are needed to deal with the infinite variety of images in the real world.

Complementing the technical sessions were other conference activities, the most important, of course, being the informal and impromptu meeting of people from different fields who would probably never get together otherwise. I personally made quite a few new acquaintances and learned many new things. The book exhibits were another very useful feature of the conference; for a cross-disciplinary field like imaging, I found it particularly useful to have many books from different disciplines and publishers brought together for easy browsing. I bought five or six-all at a discount!

One especially informative special session, a panel chaired by Gerhard Ritter, gave meeting participants the chance to hear from funding agency representatives. Wen Masters (Office of Naval Research), Arje Nachman (Air Force Office of Scientific Research), and Carey Schwartz (Defense Advanced Research Projects Agency) discussed application-driven research areas of importance to their agencies, including inverse problems, classification, target recognition, compression, and steganography.

Funding panel (left to right): Wen Masters (Office of Naval Research), Carey Schwartz (Defense Advanced Research Projects Agency), Arje Nachman (Air Force Office of Scientific Research), and moderator Gerhard Ritter (University of Florida).

Rounding out the activities was a well-attended business meeting of the SIAG on Imaging Science, convened by David Wilson. Consensus was reached that the next SIAM conference in imaging science should be held in May or September of 2004. The SIAG will elect new officers by the end of 2002; the members of the nominating committee are the current officers, with Gil Strang and Wen Masters.


*The members of the organizing committee, in addition to Wilson and Chan, were Akram Aldroubi, Vanderbilt University; Fred Bookstein, University of Michigan; Longin Jan Latecki, University of Hamburg; Chris Johnson, University of Utah; Bernard Mair, University of Florida; Robert J. Plemmons, Wake Forest University; Gerhard Ritter, University of Florida; Guillermo Sapiro, University of Minnesota; and Michael Unser, Ecole Polytechnique Fédérale de Lausanne.

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