キーワード：標準正則化理論、多価関数、透明視、テンソル積、最適化問題、表面復元
An extension of the standard regularization theory is proposed for data approximations by multi-valued functions which are essential for such as the transparency problems in computational vision. Conventional standard regularization theory can approximate scattered data by a single-valued function which is smooth everywhere in the domain. However, to incorporate discontinuities of the functions, we need to introduce the line process or equivalent techniques for breaking the coherence or smoothness of the approximating functions. Recently, multi-layer representations have been utilized for multiple overlapping surface reconstruction. However, it should incorporate auxiliary fields for segmenting the given data. Further, these two different approaches share the difficulty in implementing optimizations of their energy functionals, since they become non-quadric, non-convex minimization problem with respect to unknown surface and auxiliary field parameters. In this paper, by using a direct representation for multi-fold surfaces based on tensor product, we show that the data approximation by a multi-valued function can be reduced to minimization of a single quadric functional. Therefore, since the Euler-Lagrange equation of the functional becomes linear, we can get benefit from simple relaxation techniques of guaranteed convergence to the optimal solution.
Keywords: Standard regularization theory, Multi-valued functions, Transparency, Tensor product, Optimization, Surface reconstruction.