Discontinuity-Preserving and Viewpoint Invariant Reconstruction of Visible Surfaces Using a First Order Regularization

This paper describes the application of a first order regularization technique to the problem of reconstruction of visible surfaces. Our approach is a computationally efficient first order method that simultaneously achieves approximate invariance and preservation of discontinuities. Our reconstruction method is also robust with respect to the smoothing parameter The robustness property to allows a free choice of the smoothing parameter without struggling to determine an optimal that provides the best reconstruction. A new approximately invariant first order stabilizing function for surface reconstruction is obtained by employing a first order Taylor expansion of a nonconvex invariant stabilizing function that is expanded at the estimated value of the squared gradient instead of at zero. The data compatibility measure used is the squared perpendicular distance between the reconstructed surface and the constraint surface. This combination of stabilizing function and data compatibility measure is necessary to achieve invariance with respect to rotations and translations of the surfaces being reconstructed. Sharp preservation of discontinuities is achieved by a weighted sum of adjacent pixels such that the adjacent pixels that are more likely to be in different regions are less weighted. The results indicate that the proposed methods for surface reconstruction perform well on sparse noisy range data. In addition, the volume between two surfaces normalized by the surface area (interpreted as average distance between two surfaces) is proposed as an invariant measure for the comparison of reconstruction results.