In this paper, we propose a novel method to solve the forward and inverse problems in diffuse optical tomography.
Our forward solution is based on the diffusion approximation equation and is constructed using the Feynman-Kac
formula with an interacting particle method. It can be implemented using Monte-Carlo (MC) method and thus
provides great flexibility in modeling complex geometries. But different from conventional MC approaches, it
uses excursions of the photons' random walks and produces a transfer kernel so that only one round of MC-based
forward simulation (using an arbitrarily known optical distribution) is required in order to get observations
associated with different optical distributions. Based on these properties, we develop a perturbation-based
method to solve the inverse problem in a discretized parameter space. We validate our methods using simulated
2D examples. We compare our forward solutions with those obtained using the finite element method and find
good consistency. We solve the inverse problem using the maximum likelihood method with a greedy optimization
approach. Numerical results show that if we start from multiple initial points in a constrained searching space,
our method can locate the abnormality correctly.
Early detection and estimation of the spread of a biochemical contaminant are major issues for homeland security applications.
We present an integrated approach combining the measurements given by an array of biochemical sensors with a physical model of the dispersion and statistical analysis to solve these problems and provide system performance measures. We approximate the dispersion model of the contaminant in a realistic environment through numerical simulations of reflected stochastic diffusions describing the microscopic transport phenomena due to wind and chemical diffusion using the Feynman-Kac formula. We consider arbitrary complex geometries and account for wind turbulence. Localizing the
dispersive sources is useful for decontamination purposes and estimation of the cloud evolution. To solve the associated inverse problem, we propose a Bayesian framework based on a random field that is particularly powerful for localizing multiple sources with small amounts of measurements. We also develop a sequential detector using the numerical transport model we propose. Sequential detection allows on-line analysis and detecting wether a change has occurred. We first focus on the formulation of a suitable sequential detector that overcomes the presence of unknown parameters (e.g. release time, intensity and location). We compute a bound on the expected delay before false detection in order to decide the threshold of the test. For a fixed false-alarm rate, we obtain the detection probability of a substance release as a function of its location and initial concentration. Numerical examples are presented for two real-world scenarios: an urban area and an indoor ventilation duct.