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Critical Parameter Values and Reconstruction Properties of Discrete Tomography : Application to Experimental Fluid Dynamics

Petra S., Schnörr C., Schröder A.

Fundamenta Informaticae

2013

Vol. 125, nr 3/4

285--312

We analyze representative ill-posed scenarios of tomographic PIV (particle image velocimetry) with a focus on conditions for unique volume reconstruction. Based on sparse random seedings of a region of interest with small particles, the corresponding systems of linear projection equations are probabilistically analyzed in order to determine: (i) the ability of unique reconstruction in terms of the imaging geometry and the critical sparsity parameter, and (ii) sharpness of the transition to non-unique reconstruction with ghost particles when choosing the sparsity parameter improperly. The sparsity parameter directly relates to the seeding density used for PIV in experimental fluids dynamics that is chosen empirically to date. Our results provide a basic mathematical characterization of the PIV volume reconstruction problem that is an essential prerequisite for any algorithm used to actually compute the reconstruction. Moreover, we connect the sparse volume function reconstruction problem from few tomographic projections to major developments in compressed sensing.

Asymptotic results for exit probabilities of stochastic processes governed by an integral type rate function

Abundo M., Macci C., Stabile G.

Probability and Mathematical Statistics

2012

Vol. 32, Fasc. 1

25--39

In this paper we present asymptotic results for exit probabilities of stochastic processes in the fashion of large deviations. The main result concerns stochastic processes which satisfy the large deviation principle with an integral type rate function. We also present results for exit probabilities of linear diffusions and particular growth processes, and we give two examples.

An Erdös-Rényi law for mixing processes

Denker M., Kabluchko Z.

Probability and Mathematical Statistics

2007

Vol. 27, Fasc. 1

139--149

We prove a large deviation type result for ψ-mixing processes and derive an ergodie version of the Erdøs-Rényi law. The result applies to expanding and Gibbs-Markov dynamical systems, including Gibbs measures and continued fractions.