Sparse Delta Integration method for the calculation of spatiotemporal pressure fields of arbitrary ultrasound transducer geometries
Accurate and efficient simulation of ultrasound pressure fields and pulse-echo responses is essential for transducer design, beamforming optimization, and model-based imaging research. Conventional Spatial Impulse Response methods compute acoustic fields by explicitly sampling trapezoidal impulse responses for each rectangular aperture subdivision, which leads to high computational cost for large apertures, dense spatial grids, and high sampling frequencies. We introduce Sparse Delta Integration, a mathematical framework for the far-field SIR of rectangular apertures that expresses the trapezoidal spatial impulse response as the double integration of a sparse set of Dirac delta distributions. This formulation yields compact expressions for time-domain and frequency-domain SIRs and enables vectorized implementations whose computational cost is independent of the trapezoid duration. We further derive a spectral formulation for pulse-echo simulation that removes part of the conventional Fourier-domain convolution pipeline. Implemented in the open-source Python package eSDIva, these methods achieve speedups of up to 180x for temporal SIR computation and up to 20x for pulse-echo RF simulation relative to Field II, while maintaining high numerical accuracy with mean squared errors below 1e-5 and correlation coefficients close to unity.