Lucas-Kanade Algorithm - A Classical Optical Flow Field Method

The Lucas-Kanade (LK) algorithm is a classical optical flow field technique widely used in computer vision for motion estimation between consecutive image frames. Optical flow fields capture the direction and velocity of pixel movements over time, making them essential for applications like object tracking, video stabilization, and 3D reconstruction. In practice, the algorithm operates by processing grayscale image pairs through localized gradient analysis.

The core principle of the LK algorithm relies on the local motion smoothness assumption, where pixels within a small window (typically 3x3 or 5x5) are assumed to share identical motion vectors. It minimizes motion errors within these local windows through a least-squares solution. Key implementation steps involve: 1) enforcing brightness constancy (pixel intensity conservation between frames), 2) applying Taylor expansion for linear approximation of the optical flow equation, and 3) solving the system using matrix operations (often via numpy.linalg.pinv() in Python for pseudoinverse calculations). The final displacement vector (u,v) for each pixel is derived from the equation: G*d = b, where G is the spatial gradient matrix and b represents temporal intensity differences.

The algorithm's advantages include computational efficiency suitable for real-time systems and robustness to small motions. However, it exhibits sensitivity to illumination changes and large displacements. Practical implementations often incorporate pyramid-based coarse-to-fine strategies (using cv2.buildOpticalFlowPyramid() in OpenCV) to handle larger motions. As a foundational work, the LK algorithm underpins many modern optical flow techniques and remains a vital tool in computer vision pipelines.