Degenerate N Point Conngurations of Three Views: Do Critical Surfaces Exist?

When the geometry of 3D space is reconstructed from a pair of views, using the \Fundamental matrix" as the object of analysis, then it is known (as early as the 1940s) that there exists a \critical surface" for which the solution of 3-space is ambiguous. We show that when 3-space is reconstructed from a triplet of views, using the \Trilinear Tensor" as the object of analysis, there are no critical surfaces. In addition to theoretical interest of solving an open problem, this result has profound practical signiicance. The numerical instability associated with Structure from Motion is largely attributed to the existence of \critical volumes" that arise from the existence of critical surfaces coupled with errors in the image measurements. The lack of critical surfaces in the context of three views (provided that the trilinear tensor is used) suggests that better stability in the presence of errors can be gained.