Planar Ego-Motion Without Correspondences

Motion estimation as Hough

• Camera viewing points P_i vectors relative to the camera's reference system.
• P_i's are projected to p_i = P_i/||P_i || by a spherical projective projection
• The camera undergoes a rigid motion in the X-Y plane
  
• This motion is given by (R,t) where R = R_z(α) and t, a vector in theX-Y plane
• t = R_z(θ)·e_1
• From the camera's new coordinate frame, the world points are given as Q_i = RP_i + t and the world points Q_i map to image points q_i = Q_i/||Q_i ||
  
• We infer that Rp_i, q_i and t lie on the same plane X-Y. This is expressed as

(Rz(α)pi × qi)'Rz(θ) e1 = 0.

• Extension to any plane given by the angles β and γ.
• The vector Rz(γ)Ry(β)e_3 is orthogonal to all vectors lying in the plane.
• We can define any plane of motions with rotations.
• Motions on the plane (β, γ) are simply a coordinate frame rotation from motions in the X-Y plane.
• If the point pairs (p_i, q_i) satisfy a motion in the plane (β, γ), then the point pairs (p'_i, q'_i ), where {p'_i, q'_i } = Ry(−β)Rz(−γ){p_i, q_i} satisfie the same motion in the X-Y plane.

Epipolar constraint for general planar motions can be expressed as

Rz((Rz(α)(Rz(γ)Ry(β)' p_i × (Rz(γ)Ry(β))' q_i)'Rz(θ) e1 = 0