  We introduce Doppler time-of-flight (D-ToF) rendering, an extension of ToF
rendering for dynamic scenes, with applications in simulating D-ToF cameras.
D-ToF cameras use high-frequency modulation of illumination and exposure, and
measure the Doppler frequency shift to compute the radial velocity of dynamic
objects. The time-varying scene geometry and high-frequency modulation
functions used in such cameras make it challenging to accurately and
efficiently simulate their measurements with existing ToF rendering algorithms.
We overcome these challenges in a twofold manner: To achieve accuracy, we
derive path integral expressions for D-ToF measurements under global
illumination and form unbiased Monte Carlo estimates of these integrals. To
achieve efficiency, we develop a tailored time-path sampling technique that
combines antithetic time sampling with correlated path sampling. We show
experimentally that our sampling technique achieves up to two orders of
magnitude lower variance compared to naive time-path sampling. We provide an
open-source simulator that serves as a digital twin for D-ToF imaging systems,
allowing imaging researchers, for the first time, to investigate the impact of
modulation functions, material properties, and global illumination on D-ToF
imaging performance.
