## Section: New Results

### Algebraic Geometric Modeling

#### Elimination and nonlinear equations of Rees Algebra

Participant : Laurent Busé.

A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is everywhere defined this analysis provides free resolutions of graded parts of the Rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra. A surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion. An additional point is that this contribution – which of course corresponds to non linear equations of the Rees algebra – can be described in these degrees in terms of non Koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by J. Herzog, A. Simis and W. Vasconcelos. As a measure of the reach of this torsion analysis we could say that, in the case of a general everywhere defined map, half of the degrees where the torsion does not vanish are understood.

This work is in collaboration with Marc Chardin, Univ. Paris VI, and Aron Simis, Univ. Recife. A preprint version of it, submitted for publication, is available at http://hal.inria.fr/inria-00431783/en/ .

#### Multihomogeneous resultant formulae for systems with scaled support

Participant : Angelos Mantzaflaris.

Constructive methods for matrices of multihomogeneous resultants for
unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels,
Dickenstein and Emiris. We generalize these constructions to *mixed* systems, whose Newton polytopes are scaled copies of one
polytope, thus taking a step towards systems with arbitrary
supports. First, we specify matrices whose determinant equals the
resultant and characterize the systems that admit such formulae.
Bézout-type determinantal formulae do not exist, but we describe
all possible Sylvester-type and hybrid formulae. We establish tight
bounds for the corresponding degree vectors, as well as precise
domains where these concentrate; the latter are new even for the
unmixed case. Second, to specify resultant matrices explicitly in
the general case we make use of multiplication tables and
strong duality theory. The encountered matrices are classified; these
include a new type of Sylvester-type matrix as well as Bézout-type
matrices, which we call partial Bezoutians. Our public-domain
Maple implementation includes efficient storage of complexes in
memory, and construction of resultant matrices.

It is done in collaboration with Ioannis Emiris (NKUA) and is published in [23] .

#### Convolution surfaces based on polygonal curve skeletons

Participant : Evelyne Hubert.

For its application to the reconstruction of tree branches within the project PlantScan3D, we reviewed and generalized Convolution Surfaces. The technique is used in Computer Graphics to generate smooth 3D models around polygonal line serving as skeletons. Convolution surfaces are defined as level set of a function obtained by integrating a kernel function along this skeleton. To allow interactive modeling, the technique has relied on closed form formulae for integration obtained through symbolic computation software.

We provided new qualitative results and generalizations on the topic. On the one hand we provide the relationship between the level set and the thickness around the skeleton elements. On the other hand we obtained the closed form formulae for all the kernels in the most commonly used families - power inverse and Cauchy - through recurrences. We also exhibit recurrences to include polynomial weights on the skeleton. This allows to have a varying shape along the skeleton, the coefficients of the polynomial weight acting as controls.

Other types of skeletons are under study such as polygons and arcs of circles. Helices and super-helices appeared in the modelisation of hair and could also be of use for modeling tree branches as those can present torsion. We also plan to investigate kernels with compact supports, which are given by piecewise polynomials. Those allow locality and provide piecewise polynomial representations of the shape to be approximated.

This work is done in collaboration with Marie-Paul Cani, INRIA Grenoble. A preprint version of this work is available at http://hal.inria.fr/inria-00429358/en/ .

#### Detail-preserving axial deformation using curve pairs

Participant : Gang Xu.

Traditional axial deformation is simple and intuitive for users to modify the shape of objects. However, unexpected twist of the object may be obtained. The use of a curve-pair allows the local coordinate frame to be controlled intuitively. However, some important geometric details may be lost and changed in the deformation process. In this work, we present a detail-preserving axial deformation algorithm based on Laplacian coordinates. Instead of embedding the absolute coordinates into deformation space in the traditional axial deformation, we transform the Laplacian coordinates at each vertex according to the transformation of local frames at the closest points on the axial curve. Then the deformed mesh is reconstructed by solving a linear system that describes the reconstruction of the local details in least squares sense. By associating a complex 3D object to a curve-pair, the object can be stretched, bend, twisted intuitively through manipulating the curve-pair, and can also be edited by means of view-dependent sketching. This method combines the advantages of axial deformation and Laplacian mesh editing. Experimental results are presented to show the effectiveness of the proposed method.

This work was done in collaboration with Wenbing Ge, Peking University, Kin-Chuen Hui, The Chinese University of Hong Kong, Guoping Wang, Peking University, and the results are published in [25] .

#### Detail-preserving sculpting deformation

Participant : Gang Xu.

Sculpting deformation is a powerful tool to modify the shape of objects intuitively. However, the detail preserving problem has not been considered in sculpting deformation. In the deformation of a source object by pressing a primitive object against it, the source object is deformed while geometric details of the object should be maintained. In order to address this problem, we present a detail preserving sculpting deformation algorithm by using Laplacian coordinates. Based on the property of Laplacian coordinate, we propose two feature invariants to encode the Laplacian coordinate. Instead of mapping the source mesh to the primitive mesh, we map the smooth version of source mesh to the primitive mesh and use the Laplacian coordinates to encode the geometric details. When the smooth version of the source mesh is deformed, the Laplacian coordinates of the deformed mesh are computed for each vertex firstly and then the deformed mesh is reconstructed by solving a linear system that satisfies the reconstruction of the local details in least squares sense. Several examples are presented to show the effectiveness of the proposed approach.

This work was done in collaboration with Wenbing Ge, Peking University, Kin-Chuen Hui, The Chinese University of Hong Kong, Guoping Wang, Peking University, and the results are published in [26] .

#### Tree reconstructions from scanner point clouds

Participant : Bernard Mourrain.

In this work, we present a reconstruction pipeline for recovering branching structure of trees from laser scanned data points. The process is made up of two main blocks: segmentation and reconstruction. Based on a variational k -means clustering algorithm, cylindrical components and ramified regions of data points are identified and located. An adjacency graph is then built from neighborhood information of components. Simple heuristics allow us to extract a skeleton structure and identify branches from the graph. Finally, a B-spline model is computed to give a compact and accurate reconstruction of the branching system.

This work was done in collaboration with Fréderic Boudon, Virtual plants, Christophe Godin, Virtual plants, Julien Wintz, Wenping Wang, Hong Kong Univ., Donming Yuan, Hong Kong Univ. and the results are published in [29] .