Horn–Schunck method

Horn–Schunck method

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 Lucas–Kanade method for further description).

A global energy function is sought to be minimized, this function is given as:: f=int (( abla I cdot vec{V} + I_t)^2 + alpha (| abla V_x|^2+| abla V_y|^2+| abla V_z|^2)) m d}x{ m d}y{ m d}z} where abla I=egin{bmatrix}I_x\I_y\I_zend{bmatrix} are the derivatives of the image intensity values along the x,y, and z dimensions, I_t is the derivative in the t (time-) direction, vec{V} is the optical flow vector with the components V_x,V_y,V_z. The parameter alpha is a regularization constant. Larger values of alpha lead to a smoother flow. This function can be solved by calculating the Euler–Lagrange equations corresponding to the solution of the above equation. These are given as follows:: I_x(I_xV_x+I_yV_y+I_zV_z+I_t) - alpha Delta V_x = 0: I_y(I_xV_x+I_yV_y+I_zV_z+I_t) - alpha Delta V_y = 0: I_z(I_xV_x+I_yV_y+I_zV_z+I_t) - alpha Delta V_z = 0

where Delta denotes the Laplace operator so that Delta = frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2} + frac{partial^2}{partial z^2} .

Solving these equations with Gauss–Seidel for the flow components V_x,V_y,V_z gives an iterative scheme::V_x^{k+1}=frac{Delta V_x^k - frac{1}{alpha}I_x(I_yV_y^k+I_zV_z^k+I_t)}{frac{1}{alpha}I_x^2}:V_y^{k+1}=frac{Delta V_y^k - frac{1}{alpha}I_y(I_xV_x^k+I_zV_z^k+I_t)}{frac{1}{alpha}I_y^2}:V_z^{k+1}=frac{Delta V_z^k - frac{1}{alpha}I_z(I_xV_x^k+I_yV_y^k+I_t)}{frac{1}{alpha}I_z^2}where the superscript "k+1" denotes the next iteration, which is to be calculated and "k" is the last calculated result.

One simple way to obtain Delta V_i is::Delta V_i(p) = frac{1}{4} sum_{N(p)} V_i(N(p)) - V_i(p)where "N(p)" are the four direct neighbors of the pixel "p".

An alternative algorithmic implementation based upon the Jacobi method is given as::V_x^{k+1}=overline{V_x^k} - frac{I_x(I_xoverline{V_x^k}+I_yoverline{V_y^k}+I_zoverline{V_z^k}+I_t)}{alpha ^2+I_x^2+I_y^2+I_z^2}:V_y^{k+1}=overline{V_y^k} - frac{I_y(I_xoverline{V_x^k}+I_yoverline{V_y^k}+I_zoverline{V_z^k}+I_t)}{alpha ^2+I_x^2+I_y^2+I_z^2}:V_z^{k+1}=overline{V_z^k} - frac{I_z(I_xoverline{V_x^k}+I_yoverline{V_y^k}+I_zoverline{V_z^k}+I_t)}{alpha ^2+I_x^2+I_y^2+I_z^2}where overline{V_i^k} refers to the average of V_i^k in the neighborhood of the current pixel position.

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.

References

* B.K.P. Horn and B.G. Schunck, [http://www.caam.rice.edu/~zhang/caam699/opt-flow/horn81.pdf "Determining optical flow."] "Artificial Intelligence", vol 17, pp 185-203, 1981.


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