Focal Flow: Velocity and Depth from Differential Defocus Through Motion

We present the focal flow sensor. It is an unactuated, monocular camera that simultaneously exploits defocus and differential motion to measure a depth map and a 3D scene velocity field. It does this using an optical-flow-like, per-pixel linear constraint that relates image derivatives to depth and velocity. We derive this constraint, prove its invariance to scene texture, and prove that it is exactly satisfied only when the sensor's blur kernels are Gaussian. We analyze the inherent sensitivity of the focal flow cue, and we build and test a prototype. Experiments produce useful depth and velocity information for a broader set of aperture configurations, including a simple lens with a pillbox aperture.