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Essential Matrix: Difference between revisions
→Calculating the Essential Matrix from two images
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u'_3 u_1 \\ u'_3 u_2 \\ 1 \\ | u'_3 u_1 \\ u'_3 u_2 \\ 1 \\ | ||
\end{pmatrix} | \end{pmatrix} | ||
</math> | </math><br> | ||
Here <math>A</math> is an <math>n\ times 9</math> matrix (where <math>n=8</math> if using 8 points). | |||
The goal is to minimize <math>\Vert A\mathbf{x} \Vert </math> such that <math>\Vert \mathbf{x} \Vert = 1</math> | The goal is to minimize <math>\Vert A\mathbf{x} \Vert </math> such that <math>\Vert \mathbf{x} \Vert = 1</math> | ||
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;Solution | ;Solution | ||
* First take the SVD of A: <math>A = UDV^T</math> | * First take the SVD of A: <math>A = UDV^T</math> | ||
** <math>U</math> is <math>8 \times 8</math>, <math>D</math> is <math>8 \times 9</math> diagonal matrix, and <math>V^T</math> is a <math>9 \times 9</math> matrix. | |||
* Now <math>x = V_j</math>, the <math>j</math>-th column of <math>V</math>. Reshape this to get <math>Q_{est}</math>. | * Now <math>x = V_j</math>, the <math>j</math>-th column of <math>V</math>. Reshape this to get <math>Q_{est}</math>. | ||
* In practice, this may not be rank 2 so we take the another SVD <math>Q_{est}=U diag(r,s,t) V^T</math> | * In practice, this may not be rank 2 so we take the another SVD <math>Q_{est}=U diag(r,s,t) V^T</math> | ||