Brightness constancy and small-motion assumptions

WHAT THIS TESTS Whether you can connect the two classical assumptions to the math of the optical flow constraint and reason about real-world breakdowns.

A GOOD ANSWER COVERS Brightness constancy assumes that as a point moves between frames its image intensity stays the same. Writing intensity as a function of position and time and setting the moved intensity equal to the original gives the optical flow constraint equation, linking spatial and temporal image gradients to the unknown velocity. The small-motion assumption justifies a first-order Taylor expansion of that equation; without small displacements the linear approximation is invalid. Together they reduce flow to a tractable linear problem.

VIOLATIONS Brightness constancy fails under changing illumination, shadows, specular highlights, or auto-exposure shifts, since a pixel's intensity changes without motion. Small motion fails for fast objects, low frame rates, or large displacements, where the Taylor expansion no longer holds and naive solving aliases.

ROBUSTNESS Use coarse-to-fine image pyramids so large motions appear small at coarse scales, then refine. Replace raw intensity with illumination-invariant cues such as gradient constancy or normalized features. Swap quadratic least squares for robust penalty functions like the Charbonnier or L1 norm to downweight outliers. Modern learned methods such as RAFT sidestep the linearization with iterative correlation-based updates.

LIKELY FOLLOW-UPS Derive the optical flow constraint equation. Why does a pyramid help large motion? How does gradient constancy survive lighting changes? Why are robust norms better than L2 at occlusion boundaries?

ONE CONCRETE EXAMPLE A car speeds across a frame at 40 pixels per frame under flickering streetlights. Plain Lucas-Kanade fails: motion is too large and intensity flickers. Running it in a 4-level pyramid makes the top-level motion only a few pixels, and using gradient constancy makes the estimate ignore the lighting flicker, recovering accurate flow.