Constrained models for optical absorption tomography

We consider the inverse problem of concentration imaging in optical absorption tomography with limited data sets. The measurement setup involves simultaneous acquisition of near infrared wavelength-modulated spectroscopic measurements from a small number of pencil beams equally distributed among six projection angles surrounding the plume. We develop an approach for image reconstruction that involves constraining the value of the image to the conventional concentration bounds and a projection into low-dimensional subspaces to reduce the degrees of freedom in the inverse problem. Effectively, by re-parameterising the forward model we impose simultaneously spatial smoothness and a choice between three types of inequality constraints, namely positivity, boundedness and logarithmic boundedness in a simple way that yields an unconstrained optimisation problem in a new set of surrogate parameters. Testing this numerical scheme with simulated and experimental phantom data indicates that the combination of affine inequality constraints and subspace projection leads to images that are qualitatively and quantitatively superior to unconstrained regularised reconstructions. This improvement is more profound in targeting concentration profiles of small spatial variation. We present images and convergence graphs from solving these inverse problems using Gauss-Newton's algorithm to demonstrate the performance and convergence of our method.