Acoustic pressure field estimation methods for synthetic schlieren tomography

Synthetic schlieren tomography is a recently proposed three-dimensional (3D) optical imaging technique for studying ultrasound fields. The imaging setup is composed of an imaged target, a water tank, a camera, and a pulsed light source, which is stroboscopically synchronized with an ultrasound transducer to achieve tomographically stationary imaging of an ultrasound field. In this technique, ultrasound waves change the propagation of light rays by inducing a change in refractive index via the acousto-optic effect. The change manifests as optical flow in the imaged target. By performing the imaging in a tomographic fashion, the two-dimensional tomographic dataset of the optical flow can be transformed into a 3D ultrasound field. In this work, two approaches for acoustic pressure field estimation are introduced. The approaches are based on optical and potential flow regularized least square optimizations where regularization based on the Helmholtz equation is introduced. The methods are validated via simulations in a telecentric setup and are compared quantitatively and qualitatively to a previously introduced method. Cases of a focused, an obliquely propagating, and a standing wave ultrasound field are considered. The simulations demonstrate the efficiency of the introduced methods also in situations in which the previously applied method has weaknesses.