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    <p>Actually the methods of Horn, Schunk and Lucas, Kanade deal in different ways with the equation:</p> <pre><code>Fx*U + Fy*V = -Ft </code></pre> <p>As you see this equation is an underdetermined system of equations. So Horn and Schunk proposed to integrate a secound assumption. The smoothness constrain that the deviation of <code>U</code> and <code>V</code> should be small. This is integrated into a least square framework where you have:</p> <pre><code>(Fx*U + Fy*V + Ft)² + lambda * (gradient(U)² + gradient(V)² = E E -&gt; min </code></pre> <p>with that equation it is possible to solve <code>U</code> and <code>V</code> by setting the deviation of <code>E</code> to <code>0</code>. Consequently the solutions of the motion vectors are connected via the gradient operator of <code>U</code> and <code>V</code>.</p> <p>Lucas and Kanade proposed to that in a defined region the Lucas Kanade window only one motion vector is computed (or a region has the only one motion / motion constancy constrain) and put it into a least square framework:</p> <pre><code>sum(Fx*U + Fy*V + Ft)² = E E-&gt;min </code></pre> <p>The summation is done for each pixel in the defined region. And <code>U</code> and <code>V</code> could also be easily computed be the deviation of <code>E</code> set to <code>0</code>.</p> <p>With these two equations you see that the pixel correspondences are found by using the temporal (<code>Ft</code>) and spatial image gradients (<code>Fx</code>, <code>Fy</code>). There is a nice picture in the origin Lucas and Kanade paper, that shows this correlation graphically. However there are some points to consider:</p> <ul> <li>these kind of methods are only able to compute motion vectors, if the image contains texture (aperture problem)</li> <li><code>Fx*U + Fy*V + Ft</code> is a first order Taylor approximation of <code>F(x, y, t) = F(x + U, y + V, t + 1)</code>. That means your image signal needs to be linear. In consequence you are just able to compute motions up to a few pixels. That's why image pyramids are used to deliver linearity. </li> <li>Motion constancy or smoothness constrains prevent sharp motion boundaries. This could be important in some application.</li> <li>The framework does not prevent you from classical correspondence problem. </li> </ul>
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