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Essential Matrix: Difference between revisions
→Background and Derivation
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* This matrix is skew-symmetric. I.e. <math>[\mathbf{t}]^T_{\times} = -[\mathbf{t}]_{\times}</math> | * This matrix is skew-symmetric. I.e. <math>[\mathbf{t}]^T_{\times} = -[\mathbf{t}]_{\times}</math> | ||
Now if <math>\mathbf{u}'</math> is a feature point from camera 2 matching point <math>\mathbf{u}</math>, then it must lie on this epipolar line. | Now if <math>\mathbf{u}'</math> is a feature point from camera 2 matching point <math>\mathbf{u}</math> from camera 1, then it must lie on this epipolar line.<br> | ||
Thus <math>\mathbf{u}' \in \{(u',v',w') \mid pu' + qv' + rw' = 0\} \implies \mathbf{u}'^T R[T]_{\times} \mathbf{u} = 0</math>. | Thus <math>\mathbf{u}' \in \{(u',v',w') \mid pu' + qv' + rw' = 0\} \implies \mathbf{u}'^T R[T]_{\times} \mathbf{u} = 0</math>. | ||
Now <math>Q = R[T]_{\times}</math> is the essential matrix. | Now <math>Q = R[T]_{\times}</math> is the essential matrix. | ||