Given a rectified stereo pair, what constraint can you put on the homogeneous coordinates of all epipolar lines \(\ell_p = [a, b, c]\)?
In a rectified stereo pair, give the homogeneous coordinates of the right epipole (expressed in the right camera’s image coordinates).
Write down the residuals for the reprojection error given a camera matrix \(P\), a 3D world point \(p_{world}=(X_i, Y_i, Z_i, W_i)\), and its image space coordinates \(p_{img} = (x_j, y_j)\). Note that we’ll need to use the same “multiply by the denominator” trick we used when solving for Homographies, meaning these residuals don’t perfectly correspond to the reprojection error.
Give the first two rows of the \(A\) matrix in the least squares system \(Ax = 0\) that you’d solve to find the elements of the camera matrix \[ P = \begin{bmatrix} p_{00} & p_{01} & p_{02} & p_{03} \\ p_{10} & p_{11} & p_{12} & p_{13} \\ p_{20} & p_{21} & p_{22} & p_{23} \\ \end{bmatrix}. \] given a (known) set of \(n\) 3D points \(\{(X_i, Y_i, Z_i, 1) : 0 < i < n\}\) and their (known) corresponding 2D projections \(\{(x_i, y_i) : 0 < i < n\}\). Note that we’ll assume here that the 3D points are normalized, i.e., \(W_i=1\).
How many 3D-2D point correspondences do you need to compute the entries of \(P\)?
Give the first two rows of the \(A\) matrix in the least squares system \(Ax = 0\) that you’d solve to find the location of a 3D point \([X, Y, Z, W]\) given a set of \(m\) camera matrices \(P_{1\ldots m}\) and corresponding observed 2D locations \(\{(x_i, y_i) : 0 < i < m\}\). Note here that we’re not assuming \(W = 1\); this is not requried, but should help with numerical stability.
How many cameras (and corresponding 2D point locations) do you need to compute the location of \(X, Y, Z, W\)?