The aperture problem for refractive motion

When viewed through a small aperture, a moving image provides incomplete information about the local motion. Only the component of motion along the local image gradient is constrained. In an essential part of optical flow algorithms, information must be aggregated from nearby image locations in order to estimate all components of motion. This limitation of local evidence for estimating optical flow is called “the aperture problem”. We pose and solve a generalization of the aperture problem for moving refractive elements. We consider a common setup in air flow imaging or telescope observation: a camera is viewing a static background, and an unknown refractive elements undergoing unknown motion between them. Then we are addressing this fundamental question: what does the local image motion tell us about the motion of refractive elements? We show that the information gleaned through a local aperture for this case is very different than that for optical flow. In optical flow, the movement of 1D structure already constrains the motion in a certain direction. However, we cannot infer any information about the refractive motion from the movement of 1D structure in the observed sequence, and can only recover one component of the motion from 2D structure. Results on both simulated and real sequences are shown to illustrate our theory.