Direct Manipulation and Interactive Sculpting of PDE Surfaces

This paper presents an integrated approach and a unified algorithm that combine the benefits of PDE surfaces and powerful physics-based modeling techniques within one single modeling framework, in order to realize the full potential of PDE surfaces. We have developed a novel system that allows direct manipulation and interactive sculpting of PDE surfaces at arbitrary location, hence supporting various interactive techniques beyond the conventional boundary control. Our prototype software affords users to interactively modify point, normal, curvature, and arbitrary region of PDE surfaces in a predictable way. We employ several simple, yet effective numerical techniques including the finite-difference discretization of the PDE surface, the multigrid-like subdivision on the PDE surface, the mass-spring approximation of the elastic PDE surface, etc. to achieve real-time performance. In addition, our dynamic PDE surfaces can also be approximated using standard bivariate B-spline finite elements, which can subsequently be sculpted and deformed directly in real-time subject to intrinsic PDE constraints. Our experiments demonstrate many attractive advantages of our dynamic PDE formulation such as intuitive control, real-time feedback, and usability to the general public.     

Touch-based haptics for interactive editing on point set surfaces

A modeling paradigm for haptics-based editing on point set surfaces exploits implicit surfaces, physics-based modeling, point-sampled surfaces, and haptic. We propose a point-based geometry representation that we initially designed for dynamic physics-based sculpting, but can easily generalize to other relevant applications such as data modeling and human-computer interaction. By extending the idea of the local reference domain in the moving least square (MLS) surface model to the construction of a local and global surface distance field, we naturally incorporate Hua and Qin's dynamic implicit volumetric model into our deformation of the point-based geometry, which not only facilitates topology change but also affords dynamic sculpting and deformation.     

Dynamic PDE-based surface design using geometric and physical constraints

PDE surfaces, which are defined as solutions of partial differential equations (PDEs), offer many modeling advantages in surface blending, free-form surface modeling, and specifying surface’s aesthetic or functional requirements. Despite the earlier advances of PDE surfaces, previous PDE-based techniques exhibit certain difficulties such as lack of interactive sculpting capabilities and restrained topological structure of modeled objects. This paper presents an integrated approach that can incorporate PDE surfaces into the powerful physics-based modeling framework, to realize the full potential of PDE methodology. We have developed a prototype system that allows interactive design of flexible topological surfaces as PDE surfaces and displacements using generalized boundary conditions as well as a variety of geometric and physical constraints, hence supporting various interactive techniques beyond the conventional boundary control. The system offers a set of sculpting toolkits that allow users to interactively modify arbitrary points, curve spans, and/or regions of interest across the entire PDE surfaces and displacements in an intuitive and physically meaningful way. To achieve real-time performance, we employ several simple, yet efficient numerical techniques, including the finite-difference discretization, the multigrid-like subdivision, and the mass-spring approximation of elastic PDE surfaces and displacements. In addition, we present the standard bivariant B-spline finite element approximations of dynamic PDEs, which can subsequently be sculpted and deformed directly in real-time subject to the intrinsic PDE constraints. Our experiments demonstrate many attractive advantages of the physics-based PDE formulation such as intuitive control, real-time feedback, and usability to both professional and common users.     

Snake pedals: compact and versatile geometric models withphysics-based control

We introduce a geometric shape modeling scheme which allows for representation of global and local shape characteristics of an object. Geometric models are well-suited for representing global shapes without local detail, but we propose a scheme which represents global shapes with local detail and permits model shaping as well as topological changes via physics-based control. The scheme represents shapes by pedal curves and surfaces, i.e. the loci of the foot of perpendiculars to the tangents of a fixed curve/surface from a fixed point called the pedal point. By varying the location of the pedal point, one can synthesize a large class of shapes which exhibit both local and global deformations. We introduce physics-based control for shaping these geometric models by letting the pedal point vary and use a snake to represent the position of this varying point. The model, a “snake pedal”, allows for interactive manipulation via forces applied to the snake. We develop a fast numerical iterative algorithm for shape recovery from image data using this scheme. The algorithm involves the Levenberg-Marquardt (LM) method in the outer loop for solving the global parameters and the alternating direction implicit (ADI) method in the inner loop for solving the local parameters of the model. The combination of the global and local scheme leads to an efficient numerical solution to the model fitting problem. We demonstrate the applicability of this modeling scheme via examples of shape synthesis and shape estimation from real image data     

Free-form geometric modeling by integrating parametric and implicit PDEs

Parametric PDE techniques, which use partial differential equations (PDEs) defined over a 2D or 3D parametric domain to model graphical objects and processes, can unify geometric attributes and functional constraints of the models. PDEs can also model implicit shapes defined by level sets of scalar intensity fields. In this paper, we present an approach that integrates parametric and implicit trivariate PDEs to define geometric solid models containing both geometric information and intensity distribution subject to flexible boundary conditions. The integrated formulation of second-order or fourth-order elliptic PDEs permits designers to manipulate PDE objects of complex geometry and/or arbitrary topology through direct sculpting and free-form modeling. We developed a PDE-based geometric modeling system for shape design and manipulation of PDE objects. The integration of implicit PDEs with parametric geometry offers more general and arbitrary shape blending and free-form modeling for objects with intensity attributes than pure geometric models     

D-NURBS: a physics-based framework for geometric design

Presents dynamic non-uniform rational B-splines (D-NURBS), a physics-based generalization of NURBS. NURBS have become a de facto standard in commercial modeling systems. Traditionally, however, NURBS have been viewed as purely geometric primitives, which require the designer to interactively adjust many degrees of freedom-control points and associated weights-to achieve the desired shapes. The conventional shape modification process can often be clumsy and laborious. D-NURBS are physics-based models that incorporate physical quantities into the NURBS geometric substrate. Their dynamic behavior, resulting from the numerical integration of a set of nonlinear differential equations, produces physically meaningful, and hence intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and setting weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. We use Lagrangian mechanics to formulate the equations of motion for D-NURBS curves, tensor-product D-NURBS surfaces, swung D-NURBS surfaces and triangular D-NURBS surfaces. We apply finite element analysis to reduce these equations to efficient numerical algorithms computable at interactive rates on common graphics workstations. We implement a prototype modeling environment based on D-NURBS and demonstrate that D-NURBS can be effective tools in a wide range of computer-aided geometric design (CAGD) applications     

A shape design system using volumetric implicit PDEs

Solid modeling based on partial differential equations (PDEs) can potentially unify both geometric constraints and functional requirements within a single design framework to model real-world objects via its explicit, direct integration with parametric geometry. In contrast, implicit functions indirectly define geometric objects as the level-set of underlying scalar fields. To maximize the modeling potential of PDE-based methodology, in this paper we tightly couple PDEs with volumetric implicit functions in order to achieve interactive, intuitive shape representation, manipulation, and deformation. In particular, the unified approach can reconstruct the PDE geometry of arbitrary topology from scattered data points or a set of sketch curves. We make use of elliptic PDEs for boundary value problems to define the volumetric implicit function. The proposed implicit PDE model has the capability to reconstruct a complete solid model from partial information and facilitates the direct manipulation of underlying volumetric datasets via sketch curves and iso-surface sculpting, deformation of arbitrary interior regions, as well as a set of CSG operations inside the working space. The prototype system that we have developed allows designers to interactively sketch the curve outlines of the object, define intensity values and gradient directions, and specify interpolatory points in the 3D working space. The governing implicit PDE treats these constraints as generalized boundary conditions to determine the unknown scalar intensity values over the entire working space. The implicit shape is reconstructed with specified intensity value accordingly and can be deformed using a set of sculpting toolkits. We use the finite-difference discretization and variational interpolating approach with the localized iterative solver for the numerical integration of our PDEs in order to accommodate the diversity of generalized boundary and additional constraints.     

Compactly-supported smooth interpolators for shape modeling with varying resolution

In applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape itself. Additionally, models for shape representation often need to provide local shape control and they need to be able to reproduce common shape primitives such as ellipsoids, spheres, cylinders, or tori. We present a general framework to construct families of compactly-supported interpolators that are piecewise-exponential polynomial. They can be designed to satisfy regularity constraints of any order and they enable one to build parametric deformable shape models by suitable linear combinations of interpolators. They allow to change the resolution of shapes based on the refinability of B-splines. We illustrate their use on examples to construct shape models that involve curves and surfaces with applications to interactive modeling and character design.     

Boundary Constrained Swept Surfaces for Modelling and Animation

Due to their simplicity and intuitiveness, swept surfaces are widely used in many surface modelling applications. In this paper, we present a versatile swept surface technique called the boundary constrained swept surfaces. The most distinct feature is its ability to satisfy boundary constraints, including the shape and tangent conditions at the boundaries of a swept surface. This permits significantly varying surfaces to be both modelled and smoothly assembled, leading to the construction of complex objects. The representation, similar to an ordinary swept surface, is analytical in nature and thus it is light in storage cost and numerically very stable to compute. We also introduce a number of useful shape manipulation tools, such as sculpting forces, to deform a surface both locally and globally. In addition to being a complementary method to the mainstream surface modelling and deformation techniques, we have found it very effective in automatically rebuilding existing complex models. Model reconstruction is arguably one of the most laborious and expensive tasks in modelling complex animated characters. We demonstrate how our technique can be used to automate this process.     

C-C细分曲面的交互形状修改

为了增强细分曲面的造型功能,讨论了C-C细分曲面的交互形状修改算法。通过实时建立局部坐标系定义C-C细分曲面上点、法向量和局部等参数线等约束并将其转化为对控制顶点的约束,得到全局线性系统,从而可以在满足不同类型的几何约束时修改曲面的形状。基于最小二乘法和能量优化法给出两种修改算法,前者可以保持控制顶点扰动量的总和最小,运行速度快,适合于局部、精确调整;后者利用罚函数法给出了能量极小意义下的最优解,适合于保持光顺性要求的全局修改。两种方法都可以利用广义逆矩阵求得显式解,具有可逆性、可交换性、结合性等优点,提高了曲面形状修改的效率和可控性。     

DigitalSculpture: a subdivision-based approach to interactive implicit surface modeling

This paper presents DigitalSculpture, an interactive sculpting framework founded upon iso-surfaces extracted from recursively subdivided, 3D irregular grids. Our unique implicit surface model arises from an interpolatory, volumetric subdivision scheme that is C¹ continuous across the domains defined by arbitrary 3D irregular grids. We assign scalar coefficients and color to each control vertex and allow these quantities to participate in the volumetric subdivision of irregular grids. In the subdivision limit, a virtual sculpture is obtained by extracting the zero-level from the volumetric, scalar field defined over the irregular grid. This novel shape geometry extends concepts from solid modeling, recursive subdivision, and implicit surfaces; facilitates many techniques for interactive sculpting; permits rapid, local evaluation of iso-surfaces; and affords level-of-detail control of the sculpted surfaces.     

Triangular NURBS and their dynamic generalizations

Triangular B-splines are a new tool for modeling a broad class of objects defined over arbitrary, nonrectangular domains. They provide an elegant and unified representation scheme for all piecewise continuous polynomial surfaces over planar triangulations. To enhance the power of this model, we propose triangular NURBS, the rational generalization of triangular B-splines, with weights as additional degrees of freedom. Fixing the weights to unity reduces triangular NURBS to triangular B-splines. Conventional geometric design with triangular NURBS can be laborious, since the user must manually adjust the many control points and weights. To ameliorate the design process, we develop a new model based on the elegant triangular NURBS geometry and principles of physical dynamics. Our model combines the geometric features of triangular NURBS with the demonstrated conveniences of interaction within a physics-based framework. The dynamic behavior of the model results from the numerical integration of differential equations of motion that govern the temporal evolution of control points and weights in response to applied forces and constraints. This results in physically meaningful hence highly intuitive shape variation. We apply Lagrangian mechanics to formulate the equations of motion of dynamic triangular NURBS and finite element analysis to reduce these equations to efficient numerical algorithms. We demonstrate several applications, including direct manipulation and interactive sculpting through force-based tools, the fitting of unorganized data, and solid rounding with geometric and physical constraints.     

Multi-Level Shape Representation Using Global Deformations and Locally Adaptive Finite Elements

We present a model-based method for the multi-level shape, pose estimation and abstraction of an object's surface from range data. The surface shape is estimated based on the parameters of a superquadric that is subjected to global deformations (tapering and bending) and a varying number of levels of local deformations. Local deformations are implemented using locally adaptive finite elements whose shape functions are piecewise cubic functions with C ¹ continuity. The surface pose is estimated based on the model's translational and rotational degrees of freedom. The algorithm first does a coarse fit, solving for a first approximation to the translation, rotation and global deformation parameters and then does several passes of mesh refinement, by locally subdividing triangles based on the distance between the given datapoints and the model. The adaptive finite element algorithm ensures that during subdivision the desirable finite element mesh generation properties of conformity, non-degeneracy and smoothness are maintained. Each pass of the algorithm uses physics-based modeling techniques to iteratively adjust the global and local parameters of the model in response to forces that are computed from approximation errors between the model and the data. We present results demonstrating the multi-level shape representation for both sparse and dense range data.     

模型显著域上的形状调控和处理

基于3维数字模型的显著性度量和显著域处理技术,提出一种模型显著域上的形状调控和处理方法。该方法首先基于曲面上采样顶点处局部投影高度的Gaussian加权平均双边滤波定义数字模型的表面显著性;然后利用定义在模型显著域上的形状调控函数——显著域低通形状调控函数、显著域高通形状调控函数和显著域增强形状调控函数,使模型的显著特征得到有效抑制、提升和增强,实现了针对模型表面显著特征的形状调控和处理。实验结果表明,该方法能够方便快速地实现3维数字模型的不同形状造型效果。     

Deformable Pedal Curves and Surfaces: Hybrid Geometric Active Models for Shape Recovery

In this paper, we propose significant extensions to the snake pedal model, a powerful geometric shape modeling scheme introduced in (Vemuri and Guo, 1998). The extension allows the model to automatically cope with topological changes and for the first time, introduces the concept of a compact global shape into geometric active models. The ability to characterize global shape of an object using very few parameters facilitates shape learning and recognition. In this new modeling scheme, object shapes are represented using a parameterized function—called the generator—which accounts for the global shape of an object and the pedal curve (surface) of this global shape with respect to a geometric snake to represent any local detail. Traditionally, pedal curves (surfaces) are defined as the loci of the feet of perpendiculars to the tangents of the generator from a fixed point called the pedal point. Local shape control is achieved by introducing a set of pedal points—lying on a snake—for each point on the generator. The model dubbed as a snake pedal allows for interactive manipulation via forces applied to the snake. In this work, we replace the snake by a geometric snake and derive all the necessary mathematics for evolving the geometric snake when the snake pedal is assumed to evolve as a function of its curvature. Automatic topological changes of the model may be achieved by implementing the geometric snake in a level-set framework. We demonstrate the applicability of this modeling scheme via examples of shape recovery from a variety of 2D and 3D image data.     

Image‐based modeling of 3D objects with curved surfaces

This paper addresses an image‐based method for modeling 3D objects with curved surfaces based on the non‐uniform rational B‐splines (NURBS) representation. The user fits the feature curves on a few calibrated images with 2D NURBS curves using the interactive user interface. Then, 3D NURBS curves are constructed by stereo reconstruction of the corresponding feature curves. Using these as building blocks, NURBS surfaces are reconstructed by the known surface building methods including bilinear surfaces, ruled surfaces, generalized cylinders, and surfaces of revolution. In addition to them, we also employ various advanced techniques, including skinned surfaces, swept surfaces, and boundary patches. Based on these surface modeling techniques, it is possible to build various types of 3D shape models with textured curved surfaces without much effort. Copyright © 2007 John Wiley & Sons, Ltd.     

A new 3D-matching method of nonrigid and partially similar models using curve analysis

The 3D-shape matching problem plays a crucial role in many applications, such as indexing or modeling, by example. Here, we present a novel approach to matching 3D objects in the presence of nonrigid transformation and partially similar models. In this paper, we use the representation of surfaces by 3D curves extracted around feature points. Indeed, surfaces are represented with a collection of closed curves, and tools from shape analysis of curves are applied to analyze and to compare curves. The belief functions are used to define a global distance between 3D objects. The experimental results obtained on the TOSCA and the SHREC07 data sets show that the system performs efficiently in retrieving similar 3D models.     

Optimizing BRDF orientations for the manipulation of anisotropic highlights

This paper introduces a system for the direct editing of highlights produced by anisotropic BRDFs, which we call anisotropic highlights. We first provide a comprehensive analysis of the link between the direction of anisotropy and the shape of highlight curves for arbitrary object surfaces. The gained insights provide the required ingredients to infer BRDF orientations from a prescribed highlight tangent field. This amounts to a non‐linear optimization problem, which is solved at interactive framerates during manipulation. Taking inspiration from sculpting software, we provide tools that give the impression of manipulating highlight curves while actually modifying their tangents. Our solver produces desired highlight shapes for a host of lighting environments and anisotropic BRDFs.