Fall 2005

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New perspectives on imaging

The workshops in the fall semester of the IMA program Imaging—Imaging from Wave Propagation, Frontiers in Imaging, and Integration of Sensing and Processing—brought together an eclectic mix of mathematicians, physicists, physicians, and engineers from academia, industry, and government research programs to share ideas, results, and wish lists. In spite of the participants’ diverse backgrounds and goals, several common themes and concerns emerged. For example, target recognition methodologies developed by the military can be used in medical imaging, where the hidden targets are tumors, rather than tanks. A spectacular assortment of new mathematics and imaging applications were discussed at the workshops. In this article we highlight an innovative approach to image acquisition: integrated sensing and processing.

The concept of Integrated Sensing and Processing (ISP) suggests that a sensor should collect data in a manner that is consistent with the end objective. Thus ISP seeks to minimize the collection of redundant data, reduce processing time and improve overall performance.
—Robert Muise, Lockheed Martin
Lenslet array, less
than one square mm.
(Image courtesy David Brady.)
ISP offers tremendous potential benefits in situations in which conventional high resolution imaging techniques are unsafe, unwieldy, or prohibitively expensive, extracting essentially all of the relevant information captured in a limited amount of data. It also promises to yield cost-effective imaging techniques in situations in which only a limited amount of image data will be transmitted or stored—rather than capturing millions of pixels, then compressing them to kilobytes of data, the Brownies of the not-so-distant future may use tiny, very affordable arrays of lenslets that collect relatively small amounts of data but yield high quality images. As James Fienup (University of Rochester) explains, “One can trade off complexity in imaging system hardware and strict system tolerances for complexity in post-detection data processing.” This shift in complexity from hardware to analysis poses mathematical challenges—one revolutionary new approach to some of these challenges that has captured the attention of the ISP community is compressive sampling.

Left: image captured by lenslet 4,4. Right: reconstructed image. (Images courtesy David Brady.)

Emmanuel Candes (CalTech), one of the leaders in compressive sampling, describes the situation as follows: “Conventional wisdom and common practice in acquisition and reconstruction of images or signals from frequency data follows the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image/signal, the number of Fourier samples we need to acquire must match the desired resolution of the image/signal, e.g. the number of pixels in the image. A newly emerged sampling theory shows that this conventional wisdom is inaccurate.... perhaps surprisingly, images or signals of scientific interest can be recovered accurately and sometimes even exactly from a limited number of nonadaptive random measurements... By collecting a comparably small number of measurements rather than pixel values, one could in principle reconstruct an image with essentially the same resolution as that one would obtain by measuring all the pixels.”

Compressive sampling is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction.
—Marco Duarte, Rice University