A Theory of Multi-Layer Flat Refractive Geometry ∗ Supplementary Materials

In this section, we describe in detail the analytical solutions to compute the layer thicknesses and translation along the axis when refractive indices are unknown. As shown in the paper, the axis can be computed independently of the layer thicknesses and refractive indices. We assume that axis A, rotation R and translation orthogonal to the axis, tA, has been computed as described in Section 3 of the paper. Our goal is to compute the translation tA along the axis, layer thicknesses and refractive indices, using the given 2D-3D correspondences. Let tA = αA, where α is the unknown translation magnitude along the axis. We first apply the computedR and tA⊥ to the 3D points P. Let Pc = RP+ tA⊥ . The plane of refraction is obtained by the estimated axis A and the given camera ray v0. Let [z2, z1] denote an orthogonal coordinate system on the plane of refraction (POR). We choose z1 along the axis. For a given camera ray v0, let z2 = z1 × (z1 × v0) be the orthogonal direction. The projection of Pc on POR is given by u = [u, u], where u = z2 Pc and u = z1 Pc. Similarly, each ray vi on the light-path of v0 can be represented by a 2D vector vpi on POR, whose components are given by z2 vi and z1 vi. Let zp = [0; 1] be a unit 2D vector and ci = vpi zp. On the plane of refraction, the normal n of the refracting layers is given by n = [0;−1]. 1.1. Case 1: Single Refraction In this case, we have three unknowns d0, μ1 and α. When μis are unknown, ray directions cannot be pre-computed and flat refraction constraint needs to be written in terms of camera rays. For Case 1, the flat refraction constraint is given by