To the Graduate Council: I am submitting herewith a dissertation ...

order models entail significantly less computational efforts than second order models.The volume between two surfaces normalized by the surface area was used as an invariantquantitative measure for comparing surface reconstruction results. This measureis invariant with respect to rotations and translations of coordinate systems. The algorithmfor surface reconstruction consists of three steps: an initial reconstruction,partial derivative estimates from the initial reconstruction result, and then a secondreconstruction which uses the estimated derivatives. The estimated derivatives are insertedas constants into an approximately invariant energy functional which make itconvex. In order to estimate the derivatives, the input surface is reconstructed using asimple membrane regularization technique. The final reconstructed results depend onthe reasonable derivative estimates. The stabilizing term is defined as∫∫h =√( 1+zx 2 + zy 2 − 1)dxdy (2.8)where z x and z y are the partial derivatives of the height map. Then (2.8) is approximatedby a convex function. The reason for using (2.8) was not clearly stated, but as shownby the analysis in Section 3.2.1, it is similar to minimizing the surface area of a heightmap. The 3D objects used in the experiments have simple shapes.Because the above methods assume that the surface is a height map, they cannot beapplied to smooth arbitrarily defined surfaces. Another category of surface smoothingmethods, which is known as discrete surface fairing, directly process the surface meshand are able to smooth arbitrary surfaces. Taubin [93] applied a weighted Laplacian14