A functional error analysis of differential optical flow methods

We analyze the sources of error in differential optical flow methods using techniques for the analysis of partial differential equations. We first derive an a priori error bound for the estimated optical flow field. We then systematically interpret this error bound and show that the estimation error is primarily bounded by the best-fit approximation error —which quantifies the fidelity with which one can represent the true optical flow field by a chosen or learned set of basis functions—divided by a stability constant —which quantifies one’s ability to infer the optical flow field given the information content of the acquired data. We also show that the estimation error is bounded by effects associated with the finite temporal and spatial resolution of the acquired data. In particular, we show that the main finite resolution effects are related to the finite differencing and time averaging of the measured intensity fields. Finally, we demonstrate the error bound numerically using synthetic three-dimensional data sets based on direct numerical simulations of homogeneous isotropic turbulence and transitional boundary layer flow provided by Johns Hopkins University (Li et al. in J Turbul 9:N31, 2008; Zaki in Flow Turbul Combust in 91(3):451–473, 2013). Graphic abstract