SDP-based approach to monocular reconstruction of inextensible surfaces
This study addresses the problem of monocular reconstruction of surfaces that deform isometrically, using points tracked in a single image. To tackle this problem, a flat three-dimensional (3D) shape of the surface and its image are used as template. Such deformations are characterised by certain geometric constraints and to reconstruct the surfaces, these constraints have to be properly exploited. Therefore, the authors propose an algebraic formula that aims at the joint expression of the geometric constraints, namely those based on the differential properties and also those based on the upper-bound model. This expression is, in fact, a unique formulation that results from integrating these two types of constraints, and leads to the intended reconstructions, even when the surface is not strictly isometric. The template shape is used to set the parameters of the expression, which is then optimised (along with the projection equations) by means of a semi-definite programming (SDP) problem. This optimisation enables the estimation of 3D positions of the points on the surface. However, and for implementation purposes, this optimisation is applied separately to patches which, together, make up the whole surface. The experimental results show that the proposed approach improves the results from other methods in terms of accuracy.