Approximating a surface up to curvature signs using the depth data alone

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Abstract
Gaussian curvature K and mean curvature H are stringent properties of surfaces. Surfaces can be segmented based on the (K,H) sign pair, which classifies surfaces into eight types. To obtain a good KH sign image from raw depth data, it requires a surface approximation up to the degree of curvature signs. The conventional Hermite's interpolation, which can produce an approximation version up to given derivatives at certain surface nodes, is no use for this task because 1) there is only data for surface depth and nothing for derivatives at all; 2) noise in raw data should be reduced rather than be retained in approximation. These suggest that using depth data alone to approximate a surface up to the degree of curvature signs is an underconditioned fitting problem. In practice, a feasible approximation can be obtained by using a recursive piecewise surface fitting with a set of selected low order basis functions, where the surface fitting is decomposed into 1-D polygonal fittings associated with parabolic corrections. In this paper five lemmas, a theorem and two corollaries are given to discuss the feasibility of such a solution.

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