New data sampling theory offer flexibility, speed in processing

Nonuniform model improves any digital image

A new theory developed by mathematicians at Vanderbilt University and the University of Connecticut could lead to methods for quickly generating digital images that are more robust and accurate than those created using conventional methods. The theory sets guidelines that could be applied when data sets acquired during medical imaging procedures are sampled.

Any digital imaging modality could benefit, according to the two developers, Akram Aldroubi of Vanderbilt and Karlheinz Grochenig of UConn. Two prime candidates are spiral sampling in MR and polar sampling in CT. Ultrasound may also benefit.

"Our theory, which is based on beautiful new mathematics, can produce more accurate digital representations of all kinds of samples, including those that classical methods handle poorly or cannot handle at all," Aldroubi said. "It generates algorithms that are fast, efficient, stable, and robust."

Conventional algorithms for image reconstruction depend on regularly sampling signals. In medical imaging, these signals correspond to a volume of tissue. Conventional methods also constrain signal sampling to a limited band.

"Computational aspects using the band limited space can be tricky," he said. "They can induce errors and, in some applications, slow down the reconstruction."

Errors specific to a certain part of the data set tend to ripple throughout the set. Correcting them, therefore, takes more time than if the errors were limited.

The nonuniform sampling model created by Aldroubi and Grochenig promises to accelerate reconstruction and increase accuracy. The mathematical model may also be more widely applicable than the theory upon which modern digital reconstruction is based, allowing improvements in signal sampling used to record and enhance music as well as television pictures.

Modern information theory is based on work done by Bell Labs mathematician Claude Shannon. Aldroubi and Grochenig extended this uniform sampling theory to allow nonuniform sampling.

"In Shannon's sampling theory, function must be band limited," Aldroubi said. "That is not bad. It's just not the best assumption in some cases."

The two researchers used MR images to illustrate the effect of their nonuniform sampling theory on what they call the "missing-data problem." In this case, what may have started as a regular, uniform sample becomes irregular when data are lost; during transmission, for instance. Algorithms created on the basis of the nonuniform sampling theory were shown to more accurately reconstruct an MR image with missing data.

Another issue involving image processing arises when details are lost due to image compression. Using algorithms based on their new theory, Aldroubi and Grochenig shrank an MR image and then blew it up again. They then completed the process using conventional algorithms. Less information was lost during the process when they used the new data sampling method than with conventional methods.

"If you do reconstruction in the right way, you get a little error, but it is much less visible," Aldroubi said.

The work so far has been largely theoretical, he said. Practical benefits will come only when the mathematical framework is used in specific applications by engineers and computer scientists.

"The new theory could have a lot of applications," Aldroubi said. "But these have yet to be worked out.