xDDx: A Numerical Toolbox for Ultrasound Transducer Characterization and Design With Acoustic Holography

Transient acoustic holography is a useful technique for characterization of ultrasound transducers. It involves hydrophone measurements of the 2-D distribution of acoustic pressure waveforms in a transverse plane in front of the transducer-a hologram-and subsequent numerical forward projection (FP) or backward projection of the ultrasound field. This approach enables full spatiotemporal reconstruction of the acoustic field, including the vibrational velocity at the transducer surface. This allows identification of transducer defects as well as structural details of the radiated acoustic field such as sidelobes and hot spots. However, numerical projections may be time-consuming ( $10^{{10}}$ - $10^{{11}}$ operations with complex exponents). Moreover, backprojection from the measurement plane to the transducer surface is sensitive to misalignment between the axes of the positioning system and the axes associated with the transducer. This article presents an open-access transducer characterization toolbox for use in MATLAB or Octave on Windows computers (https://github.com/pavrosni/xDDx/releases). The core algorithm is based on the Rayleigh integral implemented in C++ executables for graphics and central processing units (GPUs and CPUs). The toolbox includes an automated procedure for correcting axes misalignments to optimize the visualization of transducer surface vibrations. Beyond using measured holograms, the toolbox can also simulate the fields radiated by user-defined transducers. Measurements from two focused 1.25-MHz 12-element sector transducers (apertures of 87 mm and focal distances of 65 and 87 mm) were used with the toolbox for demonstration purposes. Simulation speed tests for different computational devices showed a range of 0.2 s-3 min for GPUs and 1.6 s-57 min for CPUs.