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Notes:

  • Need 6 points to find transform matrix P in the equation x = P * X. 11 unknowns, each point gives 2 variables, so 11 / 2 = 5.5 ~= 6

  • P = [H | h]

  • X0 = -H^-1 * h

  • To decompose H = KR into the intrinsic and rotation matrices, use QR-decomposition.

    • In QR-decomposition, Q is a rotation matrix, R is a triangular matrix.
    • H^-1 = (K * R)^-1 = R^-1 * K^-1 = R^T * K^-1
      • Q = R^T
      • R = K^-1
    • Need to normalize K, e.g. K = 1/K33 * K
    • Need to do a coordinate tranform by a rotation of 180 deg
      • K = K * R(z, 180)
      • R = R(z, 180) * R

  • DLT in a nutshell

    1. Build M for the linear system: M is (2 * i, 12) and p is (12, 1). M * p = 0. For every point we measure, we add 2 rows to the matrix M (minimum of 6 points which is 12 rows).

    2. Solve by SVD M = U S V^T, solution is the last column of V, which are the values of p which gives us P.

    3. Solve for K, R, X0. Let P = [H | h]

      • X0 = -H^-1 * h
      • QR(H^-1) = R^T * K^-1
      • R = R(z, 180) * R
      • K = (1/K33) * K * R(z, 180)

  • What were the innovations of Zhang calibration over the prior state of the art?

    • Previous calibration techniques required more expensive or procedures: specially made 3D calibration targets, or targets that are moved in a precise way.
    • Zhang's method requires only a 2D planar target (cheap to print) and requires no special movements

  • Under what conditions will this calibration method fail?

    • If the calibration target undergoes pure, unknown translation, Zhang's method will not work.
    • This is because additional views on the same model plane do not add additional constraints.
    • But if the translation of the target is precisely known, then calibration is possible if we impose those constraints.

  • At a high level, what are the steps to the Zhang calibration algorithm?

    • Collect feature points (2D / 3D point associations) from several images (assumed to be done)
    • Estimate the intrinsic and extrinsic parameters using the closed form solution
    • Estimate the radial distortion parameters
    • Refine all parameters by minimizing

  • What is SVD, DLT, and QR, and how do they relate to Zhang calibration?

    • In the DLT case, QR-decomposition is used to decouple the intrinsics (K) and the rotation matrix (R) from the full projection matrix P.

      • But in Zhang's method, we cannot use QR-decomposition because the product contains the intrinsic matrix and a matrix which is not orthogonal (r1, r2, t)
        • x = P * X, P = [H | h], H = K * R
        • QR-decomposition separates H into its two products: an orthogonal matrix (the rotation matrix, R) and an upper-diagonal matrix (the intrinsic matrix, K)
      • Instead, we will drop the z terms (every 3rd col) of the linear system and solve for the 3x3 homography H

    • Still need to estimate K from H: H = K * [r1 r2 t]. So we need a custom solution to exploit properties we know about K, r1, and r2

      1. Exploit constraints on K, r1, r2
      2. Define a matrix B = K^-T * K^-1
      3. Compute B by solving another homogeneous linear system
      4. Decompose matrix B to get K