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Essential Matrix: Difference between revisions
→Determining rotation \mathbf{R} and translation \mathbf{t}
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A <math>3 \times 3</math> real matrix can be factored into a product of a rotation matrix <math>R</math> and a non-zero skew symmetric matrix <math>S</math> iff <math>Q</math> is two equal non-zero singular values and one zero singular value. | A <math>3 \times 3</math> real matrix can be factored into a product of a rotation matrix <math>R</math> and a non-zero skew symmetric matrix <math>S</math> iff <math>Q</math> is two equal non-zero singular values and one zero singular value. | ||
Let the singular value decomposition of our essential matrix <math>Q</math> be <math>U D V^T</math> where <math>D = \operatorname{diag}(k, k, 0)</math>. | Let the singular value decomposition of our essential matrix <math>Q</math> be <math>U D V^T</math> where <math>D = \operatorname{diag}(k, k, 0)</math>.<br> | ||
Let <math>E = \begin{pmatrix} | Let <math>E = \begin{pmatrix} | ||
0 & 1 & 0\\ | 0 & 1 & 0\\ | ||
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1 & 0 & 0 \\ | 1 & 0 & 0 \\ | ||
0 & 0 & 0 | 0 & 0 & 0 | ||
\end{pmatrix}</math> | \end{pmatrix}</math><br> | ||
Then we have the following: | Then we have the following: | ||
* <math>S = V Z V^T</math> | * <math>S = V Z V^T</math> | ||