As design is an inverse problem both approaches share similar problematic properties as e.g. indeterminate in-plane location of surface discretization or necessary regularization and filtering of sensitivity and other data.
In practice this target functional has to be augmented by terms which penalize deviation from the reference surface and by a regularization term (for example, one which penalizes deviation of 2nd order differences from their previous values).
We therefore believe the SV in CHAOS-6 is satisfactory out at least to degree 16, and possibly even as far as degree 18. Turning to the SA spectrum, in CHAOS-6 this converges at high degree at the core surface due to the applied regularization.
Based on the integral equation method and regularization theory, the regularized conjugate gradient method (RCGM) is presented for studying wave scattering from a one-dimensional rough surface with tapered wave incidence.
Furthermore, the behavior of the contacting phases in multipoint junctions and their contact angles could depend on the regularization term, as the surface energy density also consists of an additional term.
A priori information (surface topography and the optical properties) and regularization approaches (mainly Tikhonov method [75]) are usually required to solve the ill-posed inverse problem [70, 76].
The L-surface framework is employed to select optimal regularization parameters for the inverse problem.
Our results show that the TPS approach is effective, as it is able to create a globally smooth surface whose properties are easily controlled by the regularization parameter and the number of control points used.
We describe the framework and present examples in computer graphics and image processing applications, including texture synthesis, flow field visualization, and image and vector field intrinsic regularization for data defined on 3D surfaces.
The acoustic pressure and normal velocity on the source surface are reconstructed using singular value decomposition, modified Tikhonov regularization and generalized cross-validation method.
