Project : evasion
Section: New Results
Multi-resolution representation of shapes
Subdivision Surfaces: Hierarchical surface modelling
The purpose of this research is to define a surface model that has all the advantages of subdivision surfaces, while still having a low degree polynomial parameterization (See figure 2). This research is the result of a collaboration with Stefanie Hahmann from the LMC/IMAG.
In previous works a polynomial interpolation method for triangular meshes has been introduced. This interpolant can be used to design smooth surfaces of arbitrary topological type. In a design process, it is very useful to be able to locate the deformation made on a geometric model. The previously introduced interpolant has the so-called strict locality property: when a mesh vertex is moved, only the surface patches containing this vertex are changed. This enables to locate the deformation at the size of the input triangles. Unfortunately this is not sufficient if the designer wants to add some detail at a smaller level than that of the input triangles. In this research work, we propose a modification of our interpolant, that enables to arbitrarily refine the input triangulation, without changing the resulting surface. We call this property the subdivision invariance. After the refinement of the input triangulation, the modification of one of the vertices will change the shape of the interpolant at the scale of the refined triangulation. In this way, it is possible to add details at an arbitrary fine scale.
Multiresolution geometric modelling with constraints
The purpose of this research is to allow complex non-linear geometric constraints in a multiresolution geometric modelling environment. Two kinds of constraints have been investigated: constraint of constant area inside a planar curve, and constraint of constant length of a curve.
Constant Area: A method for multiresolution deformation of closed planar curves that keeps the enclosed area constant has been developed (see figure 3). We use a wavelet based multiresolution representation of the curves. The key contribution of this work is the efficient computation of the area in the wavelet decomposition form: the area is expressed through all levels of resolution as a bilinear form of the coarse and detail coefficients.
Constant Length: A multiresolution editing tool for planar curves which allows satisfying the non-linear constraint of length preserving has been developed (See figure 4). The focus is on the generation of folds when the curve is compressed. Since the deformation is based solely on a geometric model, the computing time is improved compared to physical models. Also the multiresolution framework allows to easily and precisely control the scales of the folds.
Our work on multi-resolution implicit modelling introduces multi-resolution representations to skeleton-based implicit surfaces. This is done through the use of subdivision curves and surfaces as skeleton (see figure 1). Convolution kernels admitting a closed form solution for the integral are used for generating a smooth surface around them. Our first contributions in the area are now available in a journal paper . This work will be further pursued in Olivier Palombi's PhD thesis which just started this fall and will examine the use of this representation for a specific anatomical study (see section 6.6.1).
In the scope of the Vertigo collaboration (RIAM funding, cf 7.1), we are extending our past work on realistic, real-time volumetric textures, the target being the management of forest covered scenery in virtual reality applications (see figure 5).