This paper presents a method for task-oriented path planning and collision avoidance for a group of heterogeneous holonomic mobile sensors. It is a generalization of the authors' prior work on diffusion-based path planning. The proposed variant allows one to plan paths in environments cluttered with obstacles. The agents follow flow dynamics, i.e., the negative gradient of a function that is the sum of two functions: the first minimizes the distance from desired target regions and the second captures distance from other agents within a field of view. When it becomes necessary to steer around an obstacle, this function is augmented by a projection term that is carefully designed in terms of obstacle boundaries. More importantly, a diffusion term is added intermittently so that agents can exit local minima. In addition, the new approach skips the offline planning phase in the prior approach to improve computational performance and handle collision avoidance with a completely decentralized method. This approach also provably finds collision-free paths under certain conditions. Numerical simulations of three deployment missions further support the performance of ID-based diffusion.

Photoacoustic tomography (PAT) is a hybrid imaging modality that intends to construct high-resolution images of optical properties of heterogeneous media from measured acoustic data generated by the photoacoustic effect. To date, most of the model-based quantitative image reconstructions in PAT are performed with either the radiative transport equation or its classical diffusion approximation as the model of light propagation. In this work, we study quantitative image reconstructions in PAT using the simplified $P_2$ equations as the light propagation model. We provide numerical evidences on the feasibility of this approach and derive some stability results as theoretical justifications.