Large Scale Geometric Deformation

As an important research topic in the field of computer grahics, editing 3D modelsby deformation and simulating the deformation of elastic objects are intensively involvedin many applications, such as geometric modeling, computer animation, film industry,computer game, virtual reality, and so on. Its main focus lies in the performance, con-trollability and deformation quality of the algorithm. However, previous methods cannotmeet all of the requirements of the above applications.The widely-applied Skeletal-Subspace Deformation (SSD), Free Form Deforma-tion(FFD), and multiresolution deformation technique all follow interpolation and for-ward reconstruction schemes. Despite their low computational costs, these methods maylead to serious artifacts, such as unnatural shrinkage and local self-intersection, whenapplied to large deformation,.In recent years, some energy optimization based methods have been proposed. Theyachieve better deformation quality by minimizing some kinds of quadratic deformationenergy which measure the change of geometrical differential properties through linearleast square, but it is still hard to avoid the artifacts mentioned above. Furthermore, thesemethods can describe few geometric features and need many user specified parameters ormanual intervention, which limit the application of these techniques greatly. The goal ofphysical based elastic simulation methods is to simulate the motion of the elastic objects inthe real world within reasonable computational cost. The property of deformation energyhas great impact on the simulation algorithm. It's easy to solve the motion equationwith quadratic elastic deformation energy, but large distortion still occurs when the objectdeforms greatly. To solve the problem of large deformation, previous methods adopt non-quadratic one, but fail to efficiently solve the motion equation in the end.This thesis focuses on the formulation of deformation energies and numerical meth-ods for solving deformation equation and motion equation, and presents several tech-niques to solve the above problems. The main contributions include:●Extend the linear least squares based deformation energy from surface only to thevolume space. By minimizing the distortion of volume graph which is easy tobe constructed, the quality of the deformation result can be greatly improved. Anovel curve based deformation user interface is also proposed for transfering the animation in 2D cartoon to 3D models.●Propose a quasi-linear deformation energy to handle many important deformationconstraints. Our method provides many deformation effects and reduces the re-quirement of user interaction. We also proposed a subspace method to accelerateand stabilize tbe energy optimiztion procedure.●To simulate large deformation of elastic objects efficiently and stablely, the abovegeometrically based quasi-linear deformation energy is used as the elastic deforma-tion energy in the motion equation. Dimensionality reduction and adaptive schemewithout the requirement of hierarchical structure are applied to improving the per-formance.●The non-linear elastic deformation energy coming from continuity mechanics ishighly non-linear in global, but it can be linearized without large error after the ob-ject is divided into multiple parts. Based on the above observation, we accerlate thesimulation greatly by Domain Decomposition method (DDM) with precomputedimpulse-response at the interface between the parts.The exploration about deformation energy in large deformation applications not onlyprovides useful techniques, but also solid theoretical analysis about the relationship he-tween the energy property and corresponding numerical methods.