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Interactive shape editing is an important field of research in computer graphics, and consequently a variety of different solutions were proposed to solve this problem. Early methods like free-form deformation or space deformations enable high-quality shape modeling by directly manipulating the 3D space where the object is embedded. However, they typically fail to reproduce correct deformation results if only a small number of constraints is used. Some approaches propose to solve the computationally expensive non-linear surface deformation problem directly. Others propose a non-linear differential coordinate setup, while they minimize bending and stretching energies using a coupled shell of prisms; we employ a non-linear version of the volumetric graph Laplacian and present an extension of the non-linear Poisson-based deformation approach applied to mesh sequences.
Alternatively, the article proposes a non-linear handle-aware isoline technique and combines Laplacian-based deformation with skeleton-based inverse kinematics. In general, the main limitation of these nonlinear methods is often that interactive deformation is only feasible on models of reduced complexity. Interactive performance formore complex objects can be achieved by simplifying the inherent non-linear problem. One way to preserve geometric details under global deformations is to use multi-resolution techniques. While these approaches are an effective tool for enhancing fine-scale detail preservation, the generation of the hierarchy can be expensive for complex models. Moreover, it is hard to deal with large deformations in a single step. These limitations are the main reason for differential-based deformation approaches, which represent the model using its local differential coordinates instead of using its spatial coordinates.
Typically, two differential representations can be used: deformation gradients or Laplacian coordinates. Poisson-based methods use the input transformation constraints given by the user to modify the surface gradients of the model. This presents a Poisson-based mesh editing method where the local transformations are propagated based on the geodesic distances. It replaces the geodesic propagation scheme by harmonic field interpolation and shows that this leads to a better estimation of the local transformations while it extends the harmonic field interpolation scheme to deal with different materials. Unfortunately, although these methods work well for rotations, since they are handled explicitly, they are insensitive to translations.
Laplacian-based methods represent vertex coordinates relative to their neighbors in the model. Although the original framework can not correctly deal with rotations, recent improvements allow the methods to work similarly well for translations and rotations. It describes how local rotations can be estimated and incorporated to the original framework, it proposes to use the Laplacian representation combined with implicit transformation optimization; this presents a hybrid scheme combining implicit optimization with a two-step local transformation estimation. In general, although these methods are able to estimate local rotations, the required linearization yields artifacts for large rotations.
Generally, most of these methods suffer from linearization problems: methods which use translational constraints are insensitive to rotations, whereas methods relying on rotational constraints exhibit insensitivity to translations. To solve this problem, recent methods use skeleton-based techniques, multi-step approaches, or iterative approaches. Most linear deformation methods rely on a triangle mesh representation. However, deforming a surface model may cause local self intersections and shrinking effects. To prevent such artifacts, some methods use a volumetric structure as basis for the linear deformation. Another class of approaches is able to manipulate an object while guaranteeing volume preservation by defining deformations based on vector fields. Although it enables the definition of advanced implicit deformation tools, it is still hard to construct vector fields that satisfy the user-defined constraints.