Horn–Schunck method

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The Horn–Schunck method of estimating optical flow is a global method which introduces a global constraint of smoothness to solve the aperture problem (see Optical Flow for further description).

Mathematical details[edit]

The Horn-Schunck algorithm assumes smoothness in the flow over the whole image. Thus, it tries to minimize distortions in flow and prefers solutions which show more smoothness.

The flow is formulated as a global energy functional which is then sought to be minimized. This function is given for two-dimensional image streams as:

 E=\iint \left[(I_xu + I_yv + I_t)^2 + \alpha^2(\lVert\nabla u\rVert^2+\lVert\nabla v\rVert^2)\right]{{\rm d}x{\rm d}y}

where I_x, I_y and I_t are the derivatives of the image intensity values along the x, y and time dimensions respectively, \vec{V} = [u(x,y),v(x,y)]^\top is the optical flow vector, and the parameter \alpha is a regularization constant. Larger values of \alpha lead to a smoother flow. This functional can be minimized by solving the associated Euler–Lagrange equations. These are

 \frac{\partial L}{\partial u} - \frac{\partial}{\partial x}\frac{\partial L}{\partial u_x} - \frac{\partial}{\partial y}\frac{\partial L}{\partial u_y} = 0
 \frac{\partial L}{\partial v} - \frac{\partial}{\partial x}\frac{\partial L}{\partial v_x} - \frac{\partial}{\partial y}\frac{\partial L}{\partial v_y} = 0

where L is the integrand of the energy expression, giving

 I_x(I_xu+I_yv+I_t) - \alpha^2 \Delta u = 0
 I_y(I_xu+I_yv+I_t) - \alpha^2 \Delta v = 0

where subscripts again denote partial differentiation and \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} denotes the Laplace operator. In practice the Laplacian is approximated numerically using finite differences, and may be written \Delta u(x,y) = \overline{u}(x,y) - u(x,y) where \overline{u}(x,y) is a weighted average of u calculated in a neighborhood around the pixel at location (x,y). Using this notation the above equation system may be written

(I_x^2 + \alpha^2)u + I_xI_yv = \alpha^2\overline{u}-I_xI_t
 I_xI_yu + (I_y^2 + \alpha^2)v = \alpha^2\overline{v}-I_yI_t

which is linear in u and v and may be solved for each pixel in the image. However, since the solution depends on the neighboring values of the flow field, it must be repeated once the neighbors have been updated. The following iterative scheme is derived:

u^{k+1}=\overline{u}^k - \frac{I_x(I_x\overline{u}^k+I_y\overline{v}^k+I_t)}{\alpha^2+I_x^2+I_y^2}
v^{k+1}=\overline{v}^k - \frac{I_y(I_x\overline{u}^k+I_y\overline{v}^k+I_t)}{\alpha^2+I_x^2+I_y^2}

where the superscript k+1 denotes the next iteration, which is to be calculated and k is the last calculated result. This is in essence the Jacobi method applied to the large, sparse system arising when solving for all pixels simultaneously.

Properties[edit]

Advantages of the Horn–Schunck algorithm include that it yields a high density of flow vectors, i.e. the flow information missing in inner parts of homogeneous objects is filled in from the motion boundaries. On the negative side, it is more sensitive to noise than local methods.

See also[edit]

References[edit]

  • B.K.P. Horn and B.G. Schunck, "Determining optical flow." Artificial Intelligence, vol 17, pp 185–203, 1981. Manuscript available on MIT server.

External links[edit]