GESIDICS

Generalized Sampling and Infinite-Dimensional Compressed Sensing

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 Obiettivo del progetto (Objective)

'Sampling theory is one of the mainstays of modern signal processing and encompasses mathematical theory from harmonic analysis, functional analysis, operator theory, approximation theory and computational mathematics. It is a field with a vast amount of applications ranging from medical imaging (Magnetic Resonance Imaging and X-ray Computed Tomography) to sound engineering and image processing.

The key to a successful mathematical sampling theory for use in applications is to have a model that fits the real world scenarios. And the main focus of this proposal is to emphasize the following: Due to the physical models that are the foundation of modern science one can heuristically say (from a signal processing point of view) that the world is analog (continuous-time or infinite-dimensional) whereas computer science is discrete (finite-dimensional). The gap between how we actually model the world and how we can carry out computations on a computer is a fundamental hurdle.

We will in this proposal display and suggest new techniques in sampling theory that will help bridging the gap between the true model and the model used in computations. These techniques stem from recent developments in functional analysis and will ultimately provide tools that allow for improved reconstruction techniques for use in medical imaging, sound engineering and in signal processing in general.'

Introduzione (Teaser)

One of the fundamental issues in signal analysis is ensuring that the sample adequately represents the real thing. Scientists developed improved techniques for optimal reconstruction with immediate application to medical imaging.

Descrizione progetto (Article)

Real processes are continuous (analogue) yet computers are discrete (digital). Obtaining an accurate reconstruction of a signal is critically related to the sampling technique applied. The key is ensuring adequate representation without expending extra time and computational effort in collecting and analysing redundant data.

Sampling theory is at the heart of signal processing with virtually limitless applications ranging from medical imaging to sound engineering to global positioning systems. EU-funded scientists initiated the project 'Generalized sampling and infinite-dimensional compressed sensing' (GESIDICS) to develop innovative sampling techniques that enhance reconstruction of the true signal from the model.

The researchers dealt with the theory of generalised sampling that improves signal reconstruction by not imposing restrictions on the sampling or reconstruction space. While it is a powerful technique, it is known to fail in certain cases. GESIDICS scientists introduced a stable sampling rate to deliver a stable and convergent solution in cases of previous failure. This enabled the stable and accurate recovery of wavelet coefficients in a linear manner from Fourier samples in signals up to a certain constant value. These results clearly demonstrated that their algorithm is an optimal stable reconstruction scheme.

GESIDICS has made a significant contribution to the mathematical field of sampling theory with important implications for enhanced signal processing. In particular, more accurate and consistent signal reconstruction from magnetic resonance images should have important impact on medical diagnostics and signal processing in general.