Projection Matrices from Trifocal Tensor

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 $\mathrm{T\; encodes\; the\; relationship\; among\; three\; views:}$

• The first projection matrix is typically chosen as $P_1=[I\mathrm{\mid }0],\; where$$\mathrm{I\; is\; the }$$3 \times 3$ identity matrix and $0$ is a $3 \times 1$ vector.
• The second and third projection matrices $P_2 = [A \, | \, a_4]$ and $P_3 = [B \, | \, b_4]$ can be derived from the trifocal tensor.

For canonical projective geometry:

• Each slice $T_i$ of $T$ is given by: $$T_i = a_i b_4^\top - a_4 b_i^\top$$where $a_i$ and $b_i$ are the columns of $A$ and $B$, respectively.

2. Recovering $a_4$, $b_4$ (Epipoles)

The epipoles $e_{21}$ and $e_{31}$, corresponding to $a_4$ and $b_4$, can be extracted as follows:

• $e_{31}$ (third image epipole in the first image) is the common null space of the matrices $T_1$, $T_2$, and $T_3$.
• $e_{21}$ (second image epipole in the first image) is similarly obtained from the slices $T_1^\top$, $T_2^\top$, $T_3^\top$.

3. Recovering $A$ and $B$ (Camera Matrices)

Given the epipoles $a_4 = e_{21}$ and $b_4 = e_{31}$:

• The slices of $T$ can be used to form a linear system to recover $A$ and $B$.

Python Code to Recover Projection Matrices

import numpy as np

def recover_projection_matrices(T):
    """
    Recover the projection matrices P1, P2, and P3 from the trifocal tensor T.

    Parameters:
        T: numpy array of shape (3, 3, 3) - The trifocal tensor.

    Returns:
        P1, P2, P3: Projection matrices of shape (3, 4).
    """
    # Canonical choice for P1
    P1 = np.hstack((np.eye(3), np.zeros((3, 1))))  # [I | 0]

    # Recover epipoles (a4 and b4)
    # Epipole e31: Null space of [T1, T2, T3]
    _, _, Vt = np.linalg.svd(np.stack([T[0].flatten(), T[1].flatten(), T[2].flatten()]))
    e31 = Vt[-1].reshape(-1)  # Last row of Vt gives the null space
    e31 /= e31[-1]  # Normalize to homogeneous coordinates

    # Epipole e21: Null space of [T1.T, T2.T, T3.T]
    _, _, Vt = np.linalg.svd(np.stack([T[0].T.flatten(), T[1].T.flatten(), T[2].T.flatten()]))
    e21 = Vt[-1].reshape(-1)
    e21 /= e21[-1]

    # Recover matrices A and B from T and epipoles
    A = np.zeros((3, 3))
    B = np.zeros((3, 3))
    for i in range(3):
        A[:, i] = T[i] @ e31
        B[:, i] = T[i].T @ e21

    # Construct P2 and P3
    P2 = np.hstack((A, e21.reshape(-1, 1)))
    P3 = np.hstack((B, e31.reshape(-1, 1)))

    return P1, P2, P3

# Example usage:
# Replace T with the actual trifocal tensor
T = np.random.rand(3, 3, 3)  # Replace with your trifocal tensor
P1, P2, P3 = recover_projection_matrices(T)

print("P1:\n", P1)
print("P2:\n", P2)
print("P3:\n", P3)

Explanation:

1. Canonical Initialization: $P_1$ is set as $[I \, | \, 0]$.
2. Epipole Recovery: The epipoles $e_{21}$ and $e_{31}$ are extracted as the null spaces of the appropriate slices of $T$.
3. Matrix Recovery: Using the epipoles and slices of $T$, the projection matrices $A$ and $B$ are computed.
4. Projection Matrices: $P_2$ and $P_3$ are assembled from $A, e_{21}$ and $B, e_{31}$.