Reconstructing gas distribution maps via an adaptive sparse regularization algorithm

In this paper, we present an algorithm to be used by an inspection robot to produce a gas distribution map and localize gas sources in a large complex environment. The robot, equipped with a remote gas sensor, measures the total absorption of a tuned laser beam and returns integral gas concentrations. A mathematical formulation of such measurement facility is a sequence of Radon transforms, which is a typical ill-posed problem. To tackle the ill-posedness, we develop a new regularization method based on the sparse representation property of gas sources and the adaptive finite-element method. In practice, only a discrete model can be applied, and the quality of the gas distribution map depends on a detailed 3-D world model that allows us to accurately localize the robot and estimate the paths of the laser beam. In this work, using the positivity of measurements and the process of concentration, we estimate the lower and upper bounds of measurements and the exact continuous model (mapping from gas distribution to measurements), and then create a more accurate discrete model of the continuous tomography problem. Based on adaptive sparse regularization, we introduce a new algorithm that gives us not only a solution map but also a mesh map. The solution map more accurately locates gas sources, and the mesh map provides the real gas distribution map. Moreover, the error estimation of the proposed model is discussed. Numerical tests for both the synthetic problem and practical problem are given to show the efficiency and feasibility of the proposed algorithm.