Travelling salesman-based compressive sampling

The issue of random sampling in compressive sensing was made more clear, in my view, by the work of Ben Adcock and Anders Hansen ( see A Q&A with Ben Adcock and Anders Hansen: Infinite Dimensional Compressive Sensing, Generalized Sampling, Wavelet Crimes, Safe Zones and the Incoherence Barrier. ). In short, the whole random sampling story has some problems at low frequencies. In MRI, the field at the leading edge of compressive sensing, several sampling techniques have been evaluated to corner this issue down. What is most interesting in this whole story is the connection with the hardware: We are here seeing a direct connection between actual hardware constraints (sampling authorized by the hardware) and the mathematics of sampling. One needs to realize that this connection between those two fields is rare. Here is another example of the connection between mathematics of sampling and hardware constraints:

Travelling salesman-based compressive sampling by Nicolas Chauffert, Philippe CIUCIU, Jonas Kahn, Pierre Weiss. The abstract reads:

Compressed sensing theory indicates that selecting a few measurements independently at random is a near optimal strategy to sense sparse or compressible signals. This is infeasible in practice for many acquisition devices that acquire samples along \textit{continuous} trajectories. Examples include magnetic resonance imaging (MRI), radio-interferometry, mobile-robot sampling, ... In this paper, we propose to generate continuous sampling trajectories by drawing a small set of measurements independently and joining them using a travelling salesman problem solver. Our contribution lies in the theoretical derivation of the appropriate probability density of the initial drawings. Preliminary simulation results show that this strategy is as efficient as independent drawings while being implementable on real acquisition systems.

If somebody were to explain to me why they have pi^2 as opposed to pi^(1/2), I'd really appreciate it. 

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