05.10.15 - Efficient compressive sampling strategies and novel reconstruction methods with applications in MRI, EPFL thesis n° 6036 (2014)
Thesis director : Prof. P. Vandergheynst

“For his remarkable contributions to the theoretical analysis and the applications of sparse representations for inverse problems and compressive sensing, pioneering accelerated spread spectrum sensing techniques with applications ranging from medical imaging to radio‐astronomy.”

Compressive sampling is a theory which proves that sparse signals can be sampled at a much lower rate than the Nyquist rate. The main focus of this thesis is to develop new compressive sampling methods to efficiently capture and reconstruct signals that have a small number of degrees of freedom.

First, we propose a new compressive sampling strategy based on a spread spectrum principle. We prove that this technique is optimal in the sense that the number of measurements needed to guarantee exact recovery of sparse signals is reduced to its minimum.

Second, we propose to use this compressive sampling strategy to accelerate magnetic resonance imaging. We implement this technique on a real scanner and test its efficiency with numerical simulations as well as in real-life conditions. We show that these techniques can reduce the amount of acquired measurements by 75 percent, and consequently accelerate the MR acquisition by a factor of 4, without impairing image quality.

Finally, we propose a novel algorithm which advantageously combines multiple measurements of a moving object, acquired at different positions, to improve the accuracy of the reconstructed image of this object. We demonstrate the efficiency of our method in multiple settings including compressive sampling, super-resolution, or cardiac MRI.