Toward Shape from a Single Specular Flow

In “Shape From Specular Flow: Is One Flow Enough?” (Vasilyev, et al., 2011 [5]) we show that mirror shape can often be reconstructed from the observation of a single specular flow. In this technical report we provide additional details that, due to space constraints, could not be included in the paper. First we provide a derivation of the linear system for the reflection field derivative in the direction orthogonal to the flow, r̂y. Second, we derive an expression for the determinant of this system which is independent of coordinate system. Third, we show that the sphere is reconstructable whenever the scene rotation is neither on the equator nor parallel to the view direction. Finally we provide additional details for the outline of the proof that reconstructability is a generic property and for our numerical investigation of the dimensionality of the variety described by the “bad” conditions. 1 Linear system for r̂y 1.1 Derivation of equation (6) in [5] Let û = u ‖u‖ = (û1, û2) . Starting with equation (5) in [5], ∇y (∇xr̂) = ∇y (Jx) = ∇y (r̂xû1 + r̂yû2) = ∇y ( 1 ‖u‖ [ω]×r̂ ) (1) and expanding using product rule we get: ∇y (r̂xû1 + r̂yû2) = ∇y(r̂x)û1 + r̂x∇y(û1) +∇y(r̂y)û2 + r̂y∇y(û2) (2) and ∇y ( 1 ‖u‖ [ω]×r̂ ) = ( 1 ‖u‖ ) y [ω]×r̂ + 1 ‖u‖ [ω]×r̂y (3)