Reconstruction from Range Data: A Radial Basis Function for Multiple Orders of Smoothness

In [Turk 99], the functional that was minimized is the thin-plate energy in 3D. When we applied thin-plate regularization to a real space-carved data set, we found that the resulting reconstruction was too blobby. Consequently, we began using a basis that minimizes a combination of first, second, and third order energy based upon derivations in [Chen 96] for the family of Laplacian splines, of which the first, second, and third order energy-minimizing splines are members.

Thin-plate energy is equivalent to second order energy, and membrane to first order energy. For the first three dimensions, the basis are comprised of r^k, r^klog|r|, exponential, and Bessel function terms. r is the distance from the center of the radially symmetric basis. Turk and O'Brien use b(r) = |r|²log|r| for two dimensional thin-plate interpolation, and b(r) = |r|³ for three dimensional thin-plate interpolation.

Surprisingly, a more complex radial basis function yields a better conditioned system matrix. Chen and Suter do not separate orders of smoothness in their derivation of the basis function. Their resulting formulations are combinations of first, second, third, and fourth order smoothness basis for the first three dimensions. The basis function for obtaining a combination of first, second, and third order smoothness in three dimensions is as follows:

r is the distance from the center of the radial basis function. delta controls the amount of first order smoothness, and tau controls the amount of third order smoothness. delta and tau are the only free parameters in defining the basis function. The system matrix formed by the above equations is diagonally dominant and is especially amenable the biconjugate gradient method of solving linear equations. Timing results show that the system matrix used to find the unknown weights for the implicit function was solved in 1.7 minutes using the multi-order basis function with delta = 0.01 and tau = 10, while the system matrix generated for the same set of 3300 constraints using the thin-plate radial basis function required 36.7 minutes. The results also show that reconstructions using the multi-order basis function retain global smoothness, while enhancing local features. The figure below is a reconstruction of the toy dinosaur using the multi-order basis with lambda = 0.001, delta = 30, and tau = 0.01. The same constraints were used in this reconstruction as in the previous reconstructions. The feet and tail are no longer fused. In addition, the tail, feet, and turn key are much more well-defined.