Sparse vs dense optical flow and Lucas-Kanade

WHAT THIS TESTS Whether you understand that optical flow estimates apparent pixel motion between frames and that sparse and dense differ in coverage, not merely quality.

A GOOD ANSWER COVERS Optical flow describes the apparent motion of brightness patterns between consecutive frames. Sparse optical flow estimates motion only at a set of selected points, typically good-to-track features like corners detected by Shi-Tomasi or Harris, producing a few reliable vectors cheaply. Dense optical flow estimates a displacement vector for every pixel, giving a full motion field useful for segmentation or warping but far more expensive. The Lucas-Kanade method is a classic sparse approach. It rests on two assumptions: brightness constancy, that a pixel's intensity is unchanged as it moves, and small motion, letting a first-order Taylor expansion linearize the constraint. A single pixel gives one equation in two unknowns, the aperture problem, so Lucas-Kanade assumes every pixel in a small neighborhood moves identically, yielding an overdetermined system solved by least squares.

COMMON WRONG ANSWERS Saying sparse flow is just low-quality dense flow; they target different point sets. Forgetting the aperture problem and why a window is needed. Claiming Lucas-Kanade handles large motions directly without a coarse-to-fine pyramid.

LIKELY FOLLOW-UPS What is the aperture problem? Why use an image pyramid with Lucas-Kanade for larger motion? How does Farneback or a deep network like RAFT do dense flow? Where does the least-squares system become singular?

ONE CONCRETE EXAMPLE Tracking a moving car, detect 200 corners on it, then for each corner take a 5 by 5 window, build the least-squares system from spatial and temporal gradients, and solve for the per-corner motion vector, getting sparse arrows on the car rather than a per-pixel field.