Question

What is the essential difference between the mathematical objective of the Lucas-Kanade method and the Farneback method for estimating optical flow?

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The essential difference between the Lucas-Kanade method and the Farneback method lies in the mathematical assumption made about the movement of pixels within an image sequence. Lucas-Kanade is a sparse, local method that assumes the optical flow is constant within a small neighborhood around a specific point. It solves for the velocity of a feature by minimizing the sum of squared differences between image patches, effectively assuming that all pixels within that small window move together as a single rigid block. Because of this local constraint, Lucas-Kanade typically only produces motion vectors for areas with significant intensity gradients, such as corners or edges, rather than every pixel in the frame. In contrast, Farneback is a dense, global-to-local method that approximates the neighborhood of each pixel using a quadratic polynomial. Instead of assuming constant motion, Farneback represents the image intensities in a local window as a second-degree polynomial function. By observing how these polynomial coefficients transform from one frame to the next, it calculates a displacement field for every pixel in the entire image simultaneously. While Lucas-Kanade relies on the direct brightness constancy of a local patch, Farneback effectively models the geometric deformation of the neighborhood to estimate motion, allowing it to generate a dense flow field that provides a motion vector for every single pixel in the image.

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What is the essential difference between the mathematical objective of the Lucas-Kanade method and the Farneback method for estimating optical flow?

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