This paper describes the application of a first order regularization technique to the problem of reconstruction of curves and surfaces from sparse data. The reconstruction methods achieve approximate invariance, sharp preservation of discontinuities and are robust to the smoothing parameter A. The robustness property to X allows a free choice of the smoothing parameter X without struggling to determine an optimal X that provides the best reconstrilction. 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 ideas employed for surface reconstruction are also applied to curve reconstruction. The results indicate that the proposed methods for curve and surface reconstruction perform well on sparse noisy range data. Curved surfaces (or curved sections in the case of curve reconstruction) are well reconstructed even though a first order model is employed. In addition, the volume between two surfaces normalized by the surface area is proposed as an invariant measure for the comparison of reconstruction results. Similarly, the area between two curves normalized by the arc length is proposed for comparing curve reconstruction results.
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