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In recent years, interactive shape deformation and editing techniques have become an active field of research in computer graphics. Commonly, the input to such techniques is a triangle mesh to be deformed, denoted by Mtri = (Vtri,Ttri), which consists of n vertices and m triangles. The goal is the development of algorithms to efficiently edit and manipulate Mtri as naturally as possible under the influence of a set of constraints specified by the user. Physical simulation and non-linear deformation methods are able to deliver accurate and physically-correct deformation results. However, unfortunately thesemethods require the minimization of complex non-linear energies,which often makes them difficult to implement and computationally too expensive to be used in an interactive environment, where different constraints are used to update and correct the shape of a model on-the-fly.
In general, in order to be interactive, editing methods need to be based on easyto- compute linear deformations that still generate physically plausible and aesthetically pleasing deformation results, i.e. deformations should be smooth or piecewise smooth and the result should preserve the local appearance of the surface under deformation. Recently, linear deformation methods based on differential representations have gained more popularity because they are fast to compute, robust, and easy to implement, as the associated linear system is sparse. Instead of directly modify the spatial location of each vertex in the model, they use a local differential representation of the shape, which encodes information about its local shape and the size and orientation of the local details, to obtain a detail-preserving deformation result. Deformation is performed by constructing a differential representation of the shape, manipulating it according to the given constraints, and finally reconstructing the shape from the modified differential representation.While sharing the same general framework, the two main categories of differential techniques differ by the particular representation they use. In general, when applying these methods, the resulting deformation is dependent on the particular embedding of the surface in space. During model manipulation, its local representation is not updated, which may lead to unnatural deformations.
This happens since the surface deformation problem is inherently non-linear, as it requires the estimation of local rotations to be applied to the local differential representations. To correct this limitation in the linearization process, many approaches were developed in the last years attacking this problem from different directions. After reviewing the most relevant related work on interactive shape deformation techniques, we first present two surface-based techniques used. Thereafter, an iterative volumetric approach is described, As can be seen later, the mesh deformation techniques presented are key components for the advanced methods proposed here.