High Quality Real-time Volumetric Deformation

As a fundamental task in computer graphics and geometry processing,shape deformation is widely used in many applications such as animation,physical simulation and geometric modeling.Shape deformation is a mapping which best satisfies some user specified constraints.Current research focuses on the shape preservation,smoothness,injectivity and performance.However,there is no existing method that has all of these properties.Linear blend skinning and methods which are based on barycentric coordinates use precomputed basis functions and formulate the deformation as a linear combination of these basis functions with some coefficients.Despite these techniques usually achieve real-time performance,they are usually not able to produce maps with good distortion control and these methods cannot preserve geometry in details under large deformation.Most spatial 3D methods discretize the interior of the shape using a tetrahedral mesh.The map is assumed to be continuous and linear on each tetrahedron.The deformation results are obtained by optimizing the geometric energy function to measure the mapping quality.Such a mesh-based schemes are able to control distortion,although the deformation is C0,hence nonsmooth.Furthermore,the execution time grows dramatically when fine tetrahedral meshes are used.This article focuses on the meshless deformation method.Meshless maps are generated from a set of smooth basis functions,so they are also smooth maps.In meshless deformation,the embedding fine shape can be smoothly deformed by using cage and the generalized barycentric coordinates.The quality of cage determines the fitting of thebasis function to the original mesh,and there are different requirements for the quality of cage in different applications.Nevertheless,cages are mostly constructed manually in practice,which can be a tedious and time-consuming process.The existing automatic generation cage methods are not robust enough for complex input shape.Therefore,in this article,we present an efficient and robust method for the automatic construction of high quality cage.By leveraging the versatility of SDF,we propose a simple modification of the SDF,so that the resulting isosurface will be homeomorphic to the given shape and better capture the details of the shape.Then,we construct a cage by simplifying the isosurface with the quadric error metric while making sure the cage stays enclosing and does not self-intersect.We propose to further optimize various qualities of the cage for different applications.Through extensive experiments,we demonstrate that our method is robust and efficient for a wide variety of shapes with complex geometry and topology.A general optimization framework based on meshless deformation is proposed to generate locally injective maps.Unlike mesh-based methods in which local injectivity is enforced on tetrahedral elements,our method enforces injectivity on a sparse set of domain samples.The main difficulty is then to certify the map as locally injective throughout the entire domain.This is done by utilizing the Lipschitz continuity property of the smooth basis functions.One of key contributions of our work is providing a method to evaluate the Lipschitz constants for the singular value without ever evaluating the singular value.This allows us to develop a sufficient condition for the injectivity certification.In the framework of meshless deformation,we use shape-aware harmonic basis functions to improve the deformation quality.Firstly,we compare the theoretical difference between 2D and 3D harmonic mapping in bounded distortion theory.We found that the number of samples in 3D case will be cubic times than the number in the 2D case for the injectivity certification.The Lipschitz constant determines the change of the distortion of the surrounding area relative to the sampling points.Hence,in order to reduce the sampling density as much as possible to achieve the purpose of real-time deformation while ensuring the deformation quality,we carefully derive a Lipschitz constant for the singular value of Jacobian of maps.In addition,by utilizing the special structure of the harmonic basis functions and combining with a novel regularization term that pushes the Lipschitz constants further down.As a result,the injectivity analysis can be performed on a relatively sparse set of samples.Combined with a parallel GPU-based implementation,our method preserves the quality of deformations while guarantees a real-time rate which were only possible in 2D case so far.