Reconstruction of multiple overlapping surfaces via standard regularization techniques

A fundamental extension of the standard regularization technique is proposed for making data approximations using multivalued functions which are essential for solving the transparency problems in computational vision. Conventional standard regularization techniques can approximate scattered data by using a single-valued function which is smooth everywhere in the domain. However, to incorporate discontinuities of the functions, it is necessary to introduce the line process or an equivalent technique to break the coherence or smoothness of the approximating functions. Multilayer representations have been used in reconstruction of multiple overlapping surfaces. However this technique should incorporate auxiliary fields for segmenting given data. Furthermore, these two different approaches both have the difficulty implementing optimizations of their energy functionals since they always become nonquadratic, nonconvex minimization problems with respect to an unknown surface and auxiliary field parameters. This paper shows that by using a direct representation of multivalued functions, data approximation made using a multivalued function can be reduced to minimizations of a single quadratic convex functional. Therefore, since the Euler-Lagrange equation of the functional becomes linear in this case, it is possible to benefit from simple relaxation techniques of guaranteed convergence to the optimal solution.