Gradient Domain Mesh Deformation — A Survey

Abstract

This survey reviews the recent development of gradient domain mesh deformation method. Different to other deformation methods, the gradient domain deformation method is a surface-based, variational optimization method. It directly encodes the geometric details in differential coordinates, which are also called Laplacian coordinates in literature. By preserving the Laplacian coordinates, the mesh details can be well preserved during deformation. Due to the locality of the Laplacian coordinates, the variational optimization problem can be casted into a sparse linear system. Fast sparse linear solver can be adopted to generate deformation result interactively, or even in real-time. The nonlinear nature of gradient domain mesh deformation leads to the development of two categories of deformation methods: linearization methods and nonlinear optimization methods. Basically, the linearization methods only need to solve the linear least-squares system once. They are fast, easy to understand and control, while the deformation result might be suboptimal. Nonlinear optimization methods can reach optimal solution of deformation energy function by iterative updating. Since the computation of nonlinear methods is expensive, reduced deformable models should be adopted to achieve interactive performance. The nonlinear optimization methods avoid the user burden to input transformation at deformation handles, and they can be extended to incorporate various nonlinear constraints, like volume constraint, skeleton constraint, and so on. We review representative methods and related approaches of each category comparatively and hope to help the user understand the motivation behind the algorithms. Finally, we discuss the relation between physical simulation and gradient domain mesh deformation to reveal why it can achieve physically plausible deformation result.

A Gauss-Newton Method for the Integration of Spatial Normal Fields in Shape Space

“…Another contribution of ours concerns the relationship with gradient domain mesh editing, for which, in Sect. 4.3.4, we will supply a more theoretical justification than previous works [22,23]. The advent of level-set methods [24] enabled a unified theoretical and numerical treatment of shape functions and their analysis.…”

“…Obviously, its (weak) Jacobian D τ is constant on the faces of h and somehow encodes the orientation of their images under τ . A technique called gradient domain mesh editing, closely related to our approach, exploits this by attaching to the faces of h a finite collection F of frames or handles, which can be rigidly transformed by the user into any desired configuration [22,23]. The edited mesh τ ( h ) is recovered as the minimizer of…”

Fast Deformation for Modelling of Musculoskeletal System

“…A good survey of gradient domain deformation techniques is given in (Xu and Zhou, 2009). For a surface mesh, these techniques define its energy as a function that contains terms for detail and volume preservation, position and other constraints.…”

As-Rigid-As-Possible Surface Morphing

“…This method produces good results in many cases. More detailed introduction to the gradient method can be found in [25].…”