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Allocating Limited Resources in Data Representations

Trying to tease out relevant scientific information from the huge volume of data produced by today’s computers can be like looking for a needle in a high-tech haystack. High-speed computers and the scientific community’s increasing ability to make precise measurements creates more data than anyone has time to sift through.

This is the problem that Dr. Peter Schröder and the team of mathematicians and computer scientists he worked with tackled in their KDI-funded project, Towards Ideal Data Representations. Dr. Schröder, who was a co-project investigator on the project, is a professor of Computer Science and Applied and Computational Mathematics at California Institute of Technology (Caltech).

Dr. Schröder explains, "For example, there are satellites in orbit around the earth that take high resolution imagery—these can be photographs or measurements of topography. If we want to use some of this, say for policy planning or for understanding the environmental impact of various things, we need to analyze these data. And the sheer amount of data often makes it very difficult to do so."

So the group that worked on this project set for themselves the challenge of developing mathematical tools to find new representations of these data and new ways to analyze and manipulate them. They expected that these advances would accelerate progress in many different areas, from medical imaging to particle physics to computer graphic simulations for Hollywood movies.

The group kept in mind that mathematical theorems had to lead to practical applications. According to Dr. Schröder, "To have a partnership between people who have the background to prove the hard theorems in mathematics and people who are more concerned with practical issues on the computer science end of things made it possible for us to develop relevant mathematics that you can actually put to work on real problems."

The team wanted to come up with a method for representing data in a way that would help them allocate resources intelligently. They captured that idea in a mathematically precise manner and then built algorithms to exploit it. "In a digital photograph, for example," says Dr. Schröder, "there’s more detail in some areas and less in others. I want to allocate more resources—computer resources, storage resources—to the region with a lot of detail. In regions where there’s relatively less going on, I can probably get away with allocating fewer resources."

The team focused on representations of different levels of resolution in settings where data aren’t as neatly organized as in an image. "An image is nice and regular," says Dr. Schröder. "But, say, if you wanted to take temperature measurements on planet Earth, you might have measuring stations that are distributed all over the place. They are not nicely arranged in a grid of rows and columns and measurements—they are irregularly placed. We were able to build algorithms that can deal with irregular data and get similar benefits as we did in those settings where everything’s nice and regular."

The team developed algorithms that turned out to be practically relevant and performed far better than previous algorithms that had not used the same mathematical ideas. The team also made prototype implementations of their algorithms to demonstrate how well they perform.

Dr. Schröder credits the KDI grant with making this research possible. He says, "One of the important struggles for academics like myself in terms of taking good research questions is that I have to jump into the future by a significant distance. And the KDI funding was very important for making that possible. It provided support for pursuing ideas that look promising but where lots of detail is yet unknown. It allowed these visions to be pursued more intensely, quicker, deeper, and broader."

Today, Peter Schröder’s work involves computer graphics, with a focus on multi-resolution methods for modeling, simulation, and visualization. To learn more about his research, visit the Web site of the Multi-Res Modeling Group at Caltech,

The Multi-Res Modeling Group has also designed software prototypes, which are available to the public at www.multires.caltech.edu/software/

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