Essential Matrix: Difference between revisions

Essential Matrix (view source)

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 ===Planar Images===
+Copied from section 5 of Hartley<ref name="hartley"/>.
+Here we will focus on calculating the essential matrix given 8 or more points.
+It is possible to calculate the essential matrix using 7 points using a non-linear equation if
+your correspondences are very accurate and are not linearly dependent.
+For this, see section 5.1 of Hartley<ref name="hartley"/>.
+For each correspondence <math>\mathbf{u}_i</math> and <math>\mathbf{u}'_i</math>.,
+we get the equation <math>\mathbf{u}_i'^T Q \mathbf{u}_i = 0</math>.
+This system of equations is linear in the entries of <math>Q</math> and can be
+rewritten as <math>A\mathbf{x} = 0</math> where <math>\mathbf{x}</math> contains the entries of <math>Q</math>.<br>
+Here,
+<math>\mathbf{x} = \begin{pmatrix}
+q_{11} \\ q_{12} \\ q_{13} \\
+q_{21} \\ q_{22} \\ q_{23} \\
+q_{31} \\ q_{32} \\ q_{33} \\
+\end{pmatrix}</math>
+and each row of <math>A</math> is
+<math>\mathbf{a_i} = \begin{pmatrix}
+u'_1 u_1 \\ u'_1 u_2 \\ u'_1 \\
+u'_2 u_1 \\ u'_2 u_2 \\ u'_2 \\
+u'_3 u_1 \\ u'_3 u_2 \\ 1 \\
+\end{pmatrix}
+</math>
+The goal is to minimize <math>\Vert A\mathbf{x} \Vert </math> such that <math>\Vert \mathbf{x} \Vert = 1</math>