The concepts of Multiple View Geometry & 3D reconstruction from multiple images

To recover the projection matrices from the trifocal tensor  T , we can use the relationship between the trifocal tensor and the projection matrices of the three views. Here’s the process: 1. Relationship Between the Trifocal Tensor and Projection Matrices The trifocal tensor T encodes the relationship among three views: The first projection matrix is typically chosen as P 1 = [ I   ∣   0 ], where  I is the  3 × 3 3 \times 3  identity matrix and 0 0  is a 3 × 1 3 \times 1  vector. The second and third projection matrices P 2 = [ A   ∣   a 4 ] P_2 = [A \, | \, a_4]  and P 3 = [ B   ∣   b 4 ] P_3 = [B \, | \, b_4]  can be derived from the trifocal tensor. For canonical projective geometry: Each slice T i T_i ​ of T T  is given by: T i = a i b 4 ⊤ − a 4 b i ⊤ T_i = a_i b_4^\top - a_4 b_i^\top where a i a_i ​ and b i b_i ​ are the columns of A A  and B B , respectively. 2. Recovering a 4 a_4 ​ , b 4 b_4  (Epipoles) The epipoles e 21...

The trilinearity equation describes the geometric relationship between corresponding points in three views using the trifocal tensor . It encapsulates how points in the first two views can be used to predict the corresponding point in the third view. The Trilinearity Equation For three corresponding points x 1 \mathbf{x}_1 , x 2 \mathbf{x}_2 , and x 3 \mathbf{x}_3 in three images: [ x 2 ] × ( ∑ i = 1 3 x 1 i T i ) [ x 3 ] × = 0 [\mathbf{x}_2]_\times \left( \sum_{i=1}^3 x_{1i} T_i \right) [\mathbf{x}_3]_\times = 0 Where: [ x 2 ] × [\mathbf{x}_2]_\times : Skew-symmetric matrix derived from the coordinates of x 2 \mathbf{x}_2 (view 2). [ x 3 ] × [\mathbf{x}_3]_\times : Skew-symmetric matrix derived from the coordinates of x 3 \mathbf{x}_3 (view 3). ∑ i = 1 3 x 1 i T i \sum_{i=1}^3 x_{1i} T_i : A linear combination of the slices T 1 , T 2 , T 3 T_1, T_2, T_3 of the trifocal tensor T T , weighted by the coordinates x 11 , x 12 , x 13 x_{11}, x_{12}, x_{13} of the point x 1 ...