Local regularization method applied to estimating oxygen consumption during muscle activities

In this paper, we consider a one-dimensional inverse problem of estimating an input signal that a priori is known to be a smooth function with a few jump discontinuities. A popular method of regularizing the inverse problem with such prior information is to use a total variation (TV) penalty as a regularizing functional. In this paper, we adopt a different approach, considering the problem from the Bayesian statistics viewpoint and using a hierarchical Gaussian smoothness prior. We demonstrate that the approach allows us to construct a local regularization scheme that is computationally effective and reproduces well the jump discontinuities. The approach avoids the non-differentiability problems encountered by TV methods. Once discretized, the algorithm is completely data driven, the parameter selections requiring no user intervention. The method is applied to an inverse problem in metabolic modelling, where the objective is to estimate the mitochondrial oxygen consumption during muscle activities based on noisy observations of the time decay of the oxygen concentration on the muscle surface.