Radially-Distorted Conjugate Translations Supplementary Material

The scene plane is tessellated by a 10x10 square grid of points, denoted {Xi }, with a 1 meter spacing between adjacent points. Suppose that y ↔ y′ is an undistorted point correspondence induced by the conjugate translation Hu = [I3 + ul >] in the imaged scene plane (here we assume that si = 1 since we speak about an individual point correspondence). Points {Xi } are translated by 1 meter on the scene plane in the direction of translation induced by the preimage of the point correspondence y ↔ y′ giving the translated grid {Xi }. The purpose of constructing the grid and it’s translation is to uniformly cover the scene plane that the camera images in its field of view. In this way, the accuracy of the conjugate translation and lens-distortion parameter estimation can be measured across most of the image. The conjugate translation Hu is not used directly because the magnitude of translation may span the extent of the scene plane, so applying it to the tessellation would transform the grid out of the field of view. Let the camera be parameterized by the camera matrix P = (AH)−1 (see Sec. 5.4 for the definition of the camera matrix) that pointwise maps the scene plane Π to the imaged scene plane π and division model parameter λ. The preimages of the undistorted point correspondence y ↔ y′ in the scene-plane coordinate system is, respectively, βY = P−1y and β′Y′ = P−1y′. The translation t of the preimages in the scene plane coordinate system is t = Y −Y′ = ( tx, ty, 0 )> . Then ‖t‖ is the magnitude of translation between the repeated scene elements in the scene-plane coordinate system. Denote the homogeneous translation matrix T(t) to be the matrix constructed from t as T(t) = 1 0 tx 0 1 ty 0 0 1  . (12)