Trifocal Tensor - Trilinear Constraints

Trilinear Constraints

One of the most important properties of the trifocal tensor is that it gives rise to linear relationships between lines and points in three images. More specifically, for triplets of corresponding points and any corresponding lines through them, the following trilinear constraints hold:

$({\mathbf l}^{'t} \left {\mathbf l}^{''}) _{\times} = {\mathbf 0}^t$

${\mathbf l}^{'t} \left( \sum_i x_i {\mathbf T}_i \right) {\mathbf l}^{''} = 0$

${\mathbf l}^{'t} \left( \sum_i x_i {\mathbf T}_i \right) _{\times} = {\mathbf 0}^t$

$_{\times} \left( \sum_i x_i {\mathbf T}_i \right) {\mathbf l}^{''} = {\mathbf 0}$

$_{\times} \left( \sum_i x_i {\mathbf T}_i \right) _{\times} = {\mathbf 0}_{3 \times 3}$

where denotes the skew-symmetric cross product matrix.

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