Iterative implicit constraints for controlling gradient collapse and thickness variations.

@inproceedings{Laurent3GM2014,
 abstract = { Implicit modelling approaches consist of interpolating 3D scalar fields whose iso-values implicitly represent structural surfaces. The discrete implicit approach implements this interpolation by defining the scalar field as a piece-wise linear field based on a tetrahedron mesh, which entails solving a linear system. Data points are introduced into the system as local constraints that can enforce: (1) the value of the scalar field, (2) both the norm and direction of the gradient of the scalar field, or (3) the direction of this gradient. Constraining the norm of the gradient without specifying the direction raises mathematical complications as it produces quadratic terms that do not fit in the linear system. This poses two kinds of problems. First, gradient norm is related to the thickness of the layers located between two given iso-surfaces. A gradient norm constraint would be helpful to control the variation of thickness, and produce more consistent models. The second issue appears when the scalar field is interpolated with a relatively high number of gradient direction constraints and little control on the norm or scalar values. In that case, the interpolator accommodates a part of the gradient direction variations by decreasing the norm of the gradient, which produces unrealistic thickening. When these variations are too strong or too poorly constrained, it may cause the gradient norms to collapse, resulting in scalar fields with dramatically high ranges of values and showing “bubbles” around the data points. In this paper, we propose an original approach to overcome this problem by iteratively adjusting the scalar field and bringing the gradient norms back into acceptable ranges while conforming to gradient orientations. This adjustment is enforced by a constraint that specifies the variation of the scalar field between two points taken in the current direction of the gradient, which yields a linear constraint. This process generates a scalar field honouring both the classical orientation constraints and the gradient norm constraints. Even if there is no theoretical proof of convergence, we illustrate how this technique effectively allows controlling thickness variation and gradient collapse. },
 author = { Laurent, Gautier AND Ailleres, Laurent AND Caumon, Guillaume },
 booktitle = { Proc. 34th Gocad Meeting, Nancy },
 title = { Iterative implicit constraints for controlling gradient collapse and thickness variations. },
 year = { 2014 }
}