EPFL | Biomedical Imaging Group | Impact of Mathematics

Advanced Bio-Imaging: The Impact of Mathematics and Signal Processing

M. Unser

Plenary talk, Fifth Caesarium, Advances in Biomedicine, Internationale Tagung des Forschungszentrums Caesar (CAESAR'04), Bonn, Federal Republic of Germany, September 6-8, 2004.

In this presentation, we emphasize the key role that is played by mathematics and signal processing in modern bio-imaging. The main reason for this is that developers and engineers have been taking full advantage of the increasing power of computers; they are applying more and more sophisticated algorithms for extracting structural and functional volumetric information from raw measurements (computed imaging), and for processing, visualizing, and analyzing the image data. Most observers will agree that the algorithmic part has become an essential component of the imaging process and that its importance is likely to grow even further in the future. We will make our point by focusing on one particular class of techniques called image transforms, the principle of which is to decompose the signal (or image) of interest into a sum of elementary components. First, we will discuss the Fourier transform and show that it provides the mathematical and algorithmic foundation for a number of prominent imaging modalities; these include x-ray and emission tomographies (CT, PET, SPECT), cryo-electron tomography, several types of optical microscopy (including fluorescence), and magnetic resonance imaging (MRI). Second, we will present the wavelet transform, a more recent development that appears to be ideally suited for signal and image processing. We will explain the basic principles of the approach and illustrate its use in advanced biomedical image processing. In particular, we will consider the problem of detecting neuronal activity patterns in functional MRI.

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