Geometric texture transfer via local geometric descriptors

• Published in: Applied mathematics and computation Vol. 451; p. 128031
• Main Authors: Huska, Martin, Morigi, Serena, Recupero, Giuseppe
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.08.2023
• Subjects: Differential descriptors; Geometric texture transfer; Nonlinear least squares; Variational formulation; Nonlinear least squares; Geometric texture transfer; Differential descriptors; Variational formulation
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2023.128031

•We investigate numerical aspects of both the direct and the inverse representation problem (derive the Euclidean geometry given the descriptors of a shape) using geometric descriptors as Laplacian, normal-controlled and mean value coordinates, highlighting their key properties under transformations as well as the numerical solution to the inverse ill-posed shape-from-operator problem.•Associated with the three considered geometric descriptors, we formulate and analyse three variational models for the solution of the GTT problem which involve two terms: one for the inverse reconstruction problem and one for shape preserving interpolation.•We design a nonlinear numerical optimisation algorithm to efficiently solve the GTT models and we compare the results of transferring geometric textures both qualitatively and quantitatively on a wide range of examples. Geometric Texture Transfer, aimed to add fine grained details to surfaces, can be seen as a realistic advanced geometry modelling technique. At this aim, we investigate and advocate the use of local geometric descriptors as alternative descriptors to the vertex coordinates for surface representation. In particular, we consider the Laplacian coordinates, the normal-controlled coordinates and the mean value encoding, which are well prone to facilitate the transfer of source geometric texture details onto a target surface while preserving the underlying global shape of the target surface. These representations, in general, encode the underlying geometry by describing relative position of a vertex with respect to its local neighborhood, with different levels of invariance to rigid transformations and uniform scaling. We formulate the geometric texture transfer task as a constrained variational nonlinear optimization model that combines an energy term on the shape-from-operator inverse model with constraints aimed to preserve the original underlying surface shape. In contrast to other existing methods, which rely on the strong assumptions of bijectivity, equivalency in local connectivity, and require massive tesselations, we simply map the geometric texture on the base surface, under the only assumption of boundary matching. The proposed geometric texture transfer optimization model is then efficiently solved by nonlinear least squares numerical methods. Experimental results show how the nonlinear texture transfer variational approach based on mean value coordinates overcomes the performance of other alternative descriptors.