Horn-Schunck Optical Flow Algorithm

Application Background In computer vision, optical flow method is a fundamental technique for detecting moving objects. The principle involves assigning a velocity vector (optical flow) to each pixel to form an optical flow field. When no moving objects are present, the optical flow field remains continuous and uniform. However, moving objects create distinct optical flows that disrupt this continuity, allowing for detection and localization of moving elements. Key Technologies 1.2 Horn-Schunck Model The Horn-Schunck model, proposed in 1981, incorporates an additional constraint based on the continuous and smooth nature of optical flow fields for moving objects. It transforms the global smoothness constraint into a variational problem. The energy equation comprises two components: a data term enforcing brightness constancy and a smoothness term ensuring flow field continuity. In implementation, this typically involves solving a system of equations through iterative methods like Gauss-Seidel relaxation. 1.3 Euler-Lagrange Equation From the Horn-Schunck energy equation, we derive the discrete Euler-Lagrange equations. Here, x represents pixel coordinates, and N denotes the four-connected neighborhood (up, down, left, right) of a pixel (note: boundary pixels have fewer neighbors). Solving these equations is crucial for optical flow computation. Code implementation often involves finite difference methods and iterative solvers to handle the sparse linear systems. Summary While optical flow methods are conceptually straightforward, their implementation involves complex mathematical foundations in computer vision. Understanding the Horn-Schunck model and Euler-Lagrange equations provides essential insights into optical flow algorithms, enabling better optimization and practical applications. Modern implementations often incorporate multi-scale approaches and robust data terms to handle real-world challenges.