基于Poisson方程的曲线形状渐变方法

以定义在分段线性曲线上的离散Poisson方程为理论基础,提出了一种同时适用于平面和空间曲线形状渐变的方法.通过在源曲线和目标曲线上定义局部标架,给出了一种非线性梯度场插值算法,使得源曲线的梯度场逐步过渡到目标曲线的梯度场,所得到的中间梯度场与用户指定的关键节点路径一起输入离散Poisson方程求解得到渐变序列 .该方法不直接插值顶点坐标,而是将源曲线与目标曲线视为定义在公共定义域上的标量场,并在梯度域进行梯度场操纵.对中间帧曲线周长以及平面曲线所包围的内部面积变化的统计表明:该算法尽可能地保持了几何形状的刚性,在中间帧求解的稳定性方面该算法优于同类其他方法.     

A Linear Variational System for Modelling From Curves

We present a linear system for modelling 3D surfaces from curves. Our system offers better performance, stability and precision in control than previous non‐linear systems. By exploring the direct relationship between a standard higher‐order Laplacian editing framework and Hermite spline curves, we introduce a new form of Cauchy constraint that makes our system easy to both implement and control. We introduce novel workflows that simplify the construction of 3D models from sketches. We show how to convert existing 3D meshes into our curve‐based representation for subsequent editing and modelling, allowing our technique to be applied to a wide range of existing 3D content.     

Flow Curves: an Intuitive Interface for Coherent Scene Deformation

Effective composition in visual arts relies on the principle of movement, where the viewer's eye is directed along subjective curves to a center of interest. We call these curves subjective because they may span the edges and/or center‐lines of multiple objects, as well as contain missing portions which are automatically filled by our visual system. By carefully coordinating the shape of objects in a scene, skilled artists direct the viewer's attention via strong subjective curves. While traditional 2D sketching is a natural fit for this task, current 3D tools are object‐centric and do not accommodate coherent deformation of multiple shapes into smooth flows. We address this shortcoming with a new sketch‐based interface called Flow Curves which allows coordinating deformation across multiple objects. Core components of our method include an understanding of the principle of flow, algorithms to automatically identify subjective curve elements that may span multiple disconnected objects, and a deformation representation tailored to the view‐dependent nature of scene movement. As demonstrated in our video, sketching flow curves requires significantly less time than using traditional 3D editing workflows.     

Sketch-based design for green geometry and image deformation

User interfaces have traditionally followed the WIMP (window, icon, menu, pointer) paradigm. Though functional and powerful, they are usually cumbersome for a novice user to design a complex model, requiring considerable expertise and effort. This paper presents a system for designing geometric models and image deformation with sketching curves, with the use of Green coordinates. In 3D modeling, the user first creates a 3D model by using a sketching interface, where a given 2D curve is interpreted as the projection of the 3D curve. The user can add, remove, and deform these control curves easily, as if working with a 2D line drawing. For a given set of curves, the system automatically identifies the topology and face embedding by applying graph rotation system. Green coordinates are then used to deform the generated models with detail-preserving property. Also, we have developed a sketch-based image-editing interface to deform image regions using Green coordinates. Hardware-assisted schemes are provided for both control shape deformation and the subsequent surface optimization, the experimental results demonstrate that 3D/2D deformations can be achieved in realtime.     

3D Face Similarity Measure by Fréchet Distances of Geodesics

3D face similarity is a critical issue in computer vision, computer graphics and face recognition and so on. Since Fréchet distance is an effective metric for measuring curve similarity, a novel 3D face similarity measure method based on Fréchet distances of geodesics is proposed in this paper. In our method, the surface similarity between two 3D faces is measured by the similarity between two sets of 3D curves on them. Due to the intrinsic property of geodesics, we select geodesics as the comparison curves. Firstly, the geodesics on each 3D facial model emanating from the nose tip point are extracted in the same initial direction with equal angular increment. Secondly, the Fréchet distances between the two sets of geodesics on the two compared facial models are computed. At last, the similarity between the two facial models is computed based on the Fréchet distances of the geodesics obtained in the second step. We verify our method both theoretically and practically. In theory, we prove that the similarity of our method satisfies three properties: reflexivity, symmetry, and triangle inequality. And in practice, experiments are conducted on the open 3D face database GavaDB, Texas 3D Face Recognition database, and our 3D face database. After the comparison with iso-geodesic and Hausdorff distance method, the results illustrate that our method has good discrimination ability and can not only identify the facial models of the same person, but also distinguish the facial models of any two different persons.     

一种三角网格模型的轮廓生成方法

从三维网格模型中提取轮廓信息是一个具有挑战性的过程。现有的方法一般是基于局部形状特征 分析,如曲面的曲率和相邻面法向之间的夹角,但其特性通常对模型中的局部特征变化敏感。为了解决这个问题, 提出一种基于三维形状几何近似的轮廓提取方法。利用完善的变分几何分割算法来得到一套完整的描述性特征曲 线,首先基于变分几何近似方法划分模型为若干分片;其次提取所有分片的内部特征曲线,并过滤较短的特征曲 线;然后将分片的边界曲线平滑化;最后合并分片边界曲线与特征曲线,并延伸曲线得到闭合的轮廓。该方法的 优点是：在几何近似的基础上结合特征提取方法,使轮廓能够体现三维形状的全局结构。通过在各类网格模型上 进行实验和比较表明,该方法在提取模型轮廓的正确性和完整性方面优于现有方法。     

基于Mumford-Shah模型和开样条曲线的边界检测

受Cremers方法启发, 本文提出了一种新的开边界自动检测算法, 如图像中海岸线和天际线的检测. 这一算法的设计主要是基于样条函数、曲线演化理论和Mumford-Shah图像分割泛函模型. 由于所要检测的目标为图像区域中开曲线, 在一般Mumford-Shah模型中引入了两个约束条件. 这就将开边界的检测问题转化为一般的曲线最小分割问题. 通过样条曲线控制点所满足的微分方程和约束条件, 曲线将演化至所要求的边界. 如果图像中有一条开曲线将图像分为两个明显不同质区域, 这一算法将能有效地自动检测出该边界曲线, 且不需要边界的梯度信息. 即使在图像中有大量噪声情况下, 该算法同样有效. 此外, 通过两条曲线演化方程, 该算法可推广到图像中带状区域的(如河流、道路等)自动检测.     

Efficient 3D face recognition handling facial expression and hair occlusion

This paper presents an efficient 3D face recognition method to handle facial expression and hair occlusion. The proposed method uses facial curves to form a rejection classifier and produce a facial deformation mapping and then adaptively selects regions for matching. When a new 3D face with an arbitrary pose and expression is queried, the pose is normalized based on the automatically detected nose tip and the principal component analysis (PCA) follows. Then, the facial curve in the nose region is extracted and used to form the rejection classifier which quickly eliminates dissimilar faces in the gallery for efficient recognition. Next, six facial regions which cover the face are segmented and curves in these regions are used to map facial deformation. Regions used for matching are automatically selected based on the deformation mapping. In the end, results of all the matching engines are fused by weighted sum rule. The approach is applied on the FRGC v2.0 dataset and a verification rate of 96.0% for ROC III is achieved as a false acceptance rate (FAR) of 0.1%. In the identification scenario, a rank-one accuracy of 97.8% is achieved.     

3D cartoon face generation by local deformation mapping

We present a data-driven method for automatically generating a 3D cartoon of a real 3D face. Given a sparse set of 3D real faces and their corresponding cartoon faces modeled by an artist, our method models the face in each subspace as the deformation of its nearby exemplars and learn a mapping between the deformations defined by the real faces and their cartoon counterparts. To reduce the exemplars needed for learning, we regress a collection of linear mappings defined locally in both face geometry and identity spaces and develop a progressive scheme for users to gradually add new exemplars for training. At runtime, our method first finds the nearby exemplars of an input real face and then constructs the result cartoon face from the corresponding cartoon faces of the nearby real face exemplars and the local deformations mapped from the real face subspace. Our method greatly simplifies the cartoon generation process by learning artistic styles from a sparse set of exemplars. We validate the efficiency and effectiveness of our method by applying it to faces of different facial features. Results demonstrate that our method not only preserves the artistic style of the exemplars, but also keeps the unique facial geometric features of different identities.     

Surface Patches from Unorganized Space Curves

Recent 3D sketch tools produce networks of three‐space curves that suggest the contours of shapes. The shapes may be non‐manifold, closed three‐dimensional, open two‐dimensional, or mixed. We describe a system that automatically generates intuitively appealing piecewise‐smooth surfaces from such a curve network, and an intelligent user interface for modifying the automatically chosen surface patches. Both the automatic and the semi‐automatic parts of the system use a linear algebra representation of the set of surface patches to track the topology. On complicated inputs from ILoveSketch [ [BBS08] ], our system allows the user to build the desired surface with just a few mouse‐clicks.     

Linear blending of curvature profiles

A technique is presented for blending curvature profiles and creating a set of intermediate curves which gradually change their shape from that of one boundary curve to that of a second boundary curve. The use of this technique for calculating a 3D shape and its extension to blending both curvature and torsion profiles are discussed.     

Template Assembly for Detailed Urban Reconstruction

We propose a new framework to reconstruct building details by automatically assembling 3D templates on coarse textured building models. In a preprocessing step, we generate an initial coarse model to approximate a point cloud computed using Structure from Motion and Multi View Stereo, and we model a set of 3D templates of facade details. Next, we optimize the initial coarse model to enforce consistency between geometry and appearance (texture images). Then, building details are reconstructed by assembling templates on the textured faces of the coarse model. The 3D templates are automatically chosen and located by our optimization‐based template assembly algorithm that balances image matching and structural regularity. In the results, we demonstrate how our framework can enrich the details of coarse models using various data sets.     

Cartoon-style simulation of water coupled with objects

This paper proposes a novel method for cartoon-style modelling and simulation of water coupled with static or dynamic objects. For the coupling with static objects, we specially design a main wave splitting algorithm, making the simulation results more natural. By deformation of elliptic cylindrical surfaces, we introduce a novel 3D splashed water model for the coupling with dynamic objects. Finally, the boundary curves around objects are automatically extracted for the combination of waves and objects, which reduces user interventions to as few as possible. Various experiment results demonstrate the validity of our method.     

线框设计中轮廓线的自动生成

通常的二维布尔运算只适用于封闭的轮廓线，无法应用于任意类型的曲线．本文在分析了二维封闭轮廓线布尔运算的基础上，给出了基于曲线交点处局部邻域的曲线分类方法，并由此得到任意曲线间的布尔运算，来自动生成所需的轮廓线，这种方法还可以推广到三维设计．     

Hierarchical part-type segmentation using voxel-based curve skeletons

We present an effective framework for segmenting 3D shapes into meaningful components using the curve skeleton. Our algorithm identifies a number of critical points on the efficiently computed curve skeleton, either fully automatically as the junctions of the curve skeleton, or based on user input. We use these points to construct a partitioning of the object surface using geodesics. Because the segmentation is based on the curve skeleton, it intrinsically reflects the shape symmetry and articulation, and can handle shapes with tunnels. We describe a voxel-based implementation of our method which is robust and noise resistant, able to handle shapes of complex articulation and topology, produces smooth segment borders, and delivers hierarchical level-of-detail segmentations. We demonstrate the framework on various real-world 3D shapes. Additionally, we discuss the use of both curve and surface skeletons to produce part-type and patch-type, respectively, segmentations of 3D shapes.     

Feature Curve Co‐Completion in Noisy Data

Feature curves on 3D shapes provide important hints about significant parts of the geometry and reveal their underlying structure. However, when we process real world data, automatically detected feature curves are affected by measurement uncertainty, missing data, and sampling resolution, leading to noisy, fragmented, and incomplete feature curve networks. These artifacts make further processing unreliable. In this paper we analyze the global co‐occurrence information in noisy feature curve networks to fill in missing data and suppress weakly supported feature curves. For this we propose an unsupervised approach to find meaningful structure within the incomplete data by detecting multiple occurrences of feature curve configurations (co‐occurrence analysis). We cluster and merge these into feature curve templates, which we leverage to identify strongly supported feature curve segments as well as to complete missing data in the feature curve network. In the presence of significant noise, previous approaches had to resort to user input, while our method performs fully automatic feature curve co‐completion. Finding feature reoccurrences however, is challenging since naïve feature curve comparison fails in this setting due to fragmentation and partial overlaps of curve segments. To tackle this problem we propose a robust method for partial curve matching. This provides us with the means to apply symmetry detection methods to identify co‐occurring configurations. Finally, Bayesian model selection enables us to detect and group re‐occurrences that describe the data well and with low redundancy.     

Geometry curves: A compact representation for 3D shapes

We propose a novel compact surface representation, namely geometry curves, which record the essence of shape geometry and topology. The geometry curves mainly contain two parts: the interior and boundary lines. The interior lines, which correspond to the feature lines, record the geometry information of the 3D shapes; the boundary lines, which correspond to the boundary or fundamental polygons, record the topology information of the 3D shapes. As a vector representation, geometry curves can depict highly complex geometry details. The concept of geometry curves can be utilized in many potential applications, e.g., mesh compression, shape modeling and editing, animation, and level of details. Furthermore, we develop a procedure for automatically constructing geometry curves which obtain an excellent approximation to the original mesh.     

Progressive surface modelling scheme from unorganised curves

This paper presents a novel surface modelling scheme to construct a freeform surface progressively from unorganised curves representing the boundary and interior characteristic curves. The approach can construct a base surface model from four ordinary or composite boundary curves and support incremental surface updating from interior characteristic curves, some of which may not be on the final surface. The base surface is first constructed as a regular Coons surface and upon receiving an interior curve sketch, it is updated. With this progressive modelling scheme, a final surface with multiple sub-surfaces can be obtained from a set of unorganised curves and transferred to commercial surface modelling software for detailed modification. The approach has been tested with examples based on 3D motion sketches; it is capable of dealing with unorganised design curves for surface modelling in conceptual design. Its limitations have been discussed.     

A new subdivision based approach for piecewise smooth approximation of 3D polygonal curves

This paper presents an algorithm dealing with the data reduction and the approximation of 3D polygonal curves. Our method is able to approximate efficiently a set of straight 3D segments or points with a piecewise smooth subdivision curve, in a near optimal way in terms of control point number. Our algorithm is a generalization for subdivision rules, including sharp vertex processing, of the Active B-Spline Curve developed by Pottmann et al. We have also developed a theoretically demonstrated approach, analysing curvature properties of B-Splines, which computes a near optimal evaluation of the initial number and positions of control points. Moreover, our original Active Footpoint Parameterization method prevents wrong matching problems occurring particularly for self-intersecting curves. Thus, the stability of the algorithm is highly increased. Our method was tested on different sets of curves and gives satisfying results regarding to approximation error, convergence speed and compression rate. This method is in line with a larger 3D CAD object compression scheme by piecewise subdivision surface approximation. The objective is to fit a subdivision surface on a target patch by first fitting its boundary with a subdivision curve whose control polygon will represent the boundary of the surface control polyhedron.     

Variational Dynamics of Free Triple Junctions

We propose a level set framework for representing triple junctions either with or without free endpoints. For triple junctions without free endpoints, our method uses two level set functions to represent the three segments that constitute the structure. For free triple junctions, we extend our method using the free curve work of Schaeffer and Vese (J Math Imaging Vis, 1–17, 2013), Smereka (Phys D Nonlinear Phenom 138(3–4):282–301, 2000). For curves moving under length minimizing flows, it is well known that the endpoints either intersect perpendicularly to the boundary, do not intersect the boundary of the domain or the curve itself (free endpoints), or meet at triple junctions. Although many of these cases can be formulated within the level set framework, the case of free triple junctions does not appear in the literature. Therefore, the proposed free triple junction formulation completes the important curve structure representations within the level set framework. We derive an evolution equation for the dynamics of the triple junction under length and area minimizing flow. The resulting system of partial differential equations are both coupled and highly non-linear, so the system is solved numerically using the Sobolev preconditioned descent. Qualitative numerical experiments are presented on various triple junction and free triple junction configurations, as well as an example with a quadruple junction instability. Quantitative results show convergence of the preconditioned algorithm to the correct solutions.