If you read this Sunday Morning Insight entry two years ago entitled "The Linear Boltzmann Equation and Co-Sparsity" you may remember that the analysis approach as used in compressive sensing (the TV norm minimization falls in that category) is nothing less than an attempt at putting some of the physics back into a signal processing problem.

Co-sparsity is really a statement of how a specific operator describes field equations in homogenous media. Boundary conditions between homogenous media and sources within homogenous media provide additional constraints within this field approach. The measurement equations eventually further constrain the problem so that only one solution can be physically inferred.

Up until now, I do not think we had many publications that tried to discretize the forward operator of the underlying physics and put it in an analysis approach. Here is a new path in that direction: Brain source localization using a physics-driven structured cosparse representation of EEG signals by Laurent Albera Srđan Kitić Nancy Bertin, Gilles Puy, Rémi Gribonval

Localizing several potentially synchronous brain activities with low signal-to-noise ratio from ElectroEncephaloGraphic (EEG) recordings is a challenging problem. In this paper we propose a novel source localization method, named CoRE, which uses a Cosparse Representation of EEG signals. The underlying analysis operator is derived from physical laws satisfied by EEG signals, and more particularly from Poisson's equation. In addition, we show how physiological constraints on sources, leading to a given space support and fixed orientations for current dipoles, can be taken into account in the optimization scheme. Computer results, aiming at showing the feasability of the CoRE technique, illustrate its superiority in terms of estimation accuracy over dictionary-based sparse methods and subspace approaches.

We previously mentioned a Data Driven or Zero Knowledge Sensor Design approach. Here though, we now have a whole new world of Physics Driven Sensor Design where measurement ensembles, instead of being RIP admissible or gaussian, are required to fit Physics based set of constraints (instantiated, in part, through the appearance of sharp phase transitions). Back in 1996, this work pretty much convinced many people that dictionary learning was physics based/biologically relevant. In the same way, I am sure both approaches (Data Driven and Physics based designs), instead of looking diametrically opposed, will eventually converge. More on that later.

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