Yiheng Zhang and Dana H. Brooks
Diffusion Optical Tomography (DOT) imaging is a relatively new technique for imaging inside biological tissues by using near infrared light. The DOT imaging problem is typically ill-posedness due to the large attenuation and scattering of the diffuse photon density wave. Moreover, locations where we can put sensors and detectors are generally quite restricted, which results in problems that are typically highly underdetermined. All these lead to poor quality reconstructions of optical absorption and ultimately of blood volume and oxygenation.
To address these problems, standard regularized solutions employ an a priori constraint to achieve reliability, trading off between fidelity to the constraint and fidelity to the data plus propagation model. Choosing a good constraint, as well as the regularization parameter which governs this tradeoff, is difficult to do, with no clearly optimal choices. Recently we reported our use of an admissible solution approach; we reformulate the problem to search for a solution that is a member of an admissible set. Admissibility is defined in terms of as many available constraints as we wish to employ. (We restrict ourselves to constraints which define convex sets). Each constraint defines a particular region in the solution space, and the intersection of all of these constraint regions defines the admissible solution domain. The problem, then, is to find any solution in this intersection; such a solution satisfies all constraints and thus is admissible.
In this work, we address the reconstruction of the optical absorption coefficient. We use three constraints: a constraint on the residual between the measured data and what is predicted by a candidate solution and our forward (propagation) model, a min/max constraint on the total excursion of absorption coefficient from the background, and a constraint on the total variation over the reconstruction (the sum of absolute value of variation both in x and y directions). We employ the Deep-Cut Ellipsoid algorithm, an iterative convex optimization algorithm, to find a point in the admissible solution region. With proper constraint values, we found that simulation experiments give promising result.
However, the computational complexity is quite high, and increases rapidly as the dimension of the problem (i.e. the resolution of the reconstruction) increases. Here we introduce a new multi-resolution method which first uses a coarse mesh to get an initial result, then uses this initial result to selectively refine the mesh. The idea is to achieve finer resolution only in the regions indicated as likely to contain an anomaly. We increase the resolution by focusing only on these selected domains, while combining other pixels into one element. Thus we can always keep the problem dimensionality to a reasonable level, and at the same time obtain as fine a resolution as possible in the anomalous regions. Simulations experiments show this approach reduces the computation time by more than 95% while results are equivalent or even slightly improved.