Planar Shape Interpolation Based on Local Injective Mapping

在只给出用简单多边形表示的两输入形状的情况下,实现一种简单易用、自然高效的形状插值方法.首先利用基于形状感知的特征匹配算法生成源形状和目标形状之间的匹配;之后在源形状上构造三角剖分,并通过求解映射到目标形状上的尽量刚体的局部单射得到同构三角剖分;最后利用扭曲有界的插值方法得到中间序列.实验结果表明,该方法构造的形变结果能较好地体现源形状和目标形状的特征对应信息,形变过程自然,扭曲较小.This paper presents an efficient and easy-to-use planar shape interpolation method, given two input shapes represented by simple polygons. We firstly used a perception-based feature matching algorithm to match the feature points in the source shape with the target shape, then built compatible triangulations by constructing a locally injective mapping between the source and target shapes. Finally, an interpolation method with bounded distortion was adopted to get intermediate frames. Experimental results show that the interpolation results by our method can well reflect the feature correspondences between the source and the target shapes, and the resultant deformation is visually pleasing with less distortion.国家自然科学基金(61472332);; 中央高校基本科研业务费专项基金(20720140520

Repurpose 2D Animations for a VR Environment using BDH Shape Interpolation

Virtual Reality technology has spread rapidly in recent years. However, its growth risks ending soon due to the absence of quality content, except for few exceptions. We present an original framework that allows artists to use 2D characters and animations in a 3D Virtual Reality environment, in order to give an easier access to the production of content for the platform. In traditional platforms, 2D animation represents a more economic and immediate alternative to 3D. The challenge in adapting 2D characters to a 3D environment is to interpret the missing depth information. A 2D character is actually flat, so there is not any depth information, and every body part is at the same level of the others. We exploit mesh interpolation, billboarding and parallax scrolling to simulate the depth between each body segment of the character. We have developed a prototype of the system, and extensive tests with a 2D animation production show the effectiveness of our framework

Repurpose 2D Character Animations for a VR Environment Using BDH Shape Interpolation.

Virtual Reality technology has spread rapidly in recent years. However, its growth risks ending soon due to the absence of quality content, except for few exceptions. We present an original framework that allows artists to use 2D characters and animations in a 3D Virtual Reality environment, in order to give an easier access to the production of content for the platform. In traditional platforms, 2D animation represents a more economic and immediate alternative to 3D. The challenge in adapting 2D characters to a 3D environment is to interpret the missing depth information. A 2D character is actually flat, so there is not any depth information, and every body part is at the same level of the others. We exploit mesh interpolation, billboarding and parallax scrolling to simulate the depth between each body segment of the character. We have developed a prototype of the system, and extensive tests with a 2D animation production show the effectiveness of our framework

Non-isometric 3D shape registration.

3D shape registration is an important task in computer graphics and computer vision. It has been widely used in the area of film industry, 3D animation, video games and AR/VR assets creation. Manually creating the 3D model of a character from scratch is tedious and time consuming, and it can only be completed by professional trained artists. With the development of 3D geometry acquisition technology, it becomes easier and cheaper to capture high-resolution and highly detailed 3D geometries. However, the scanned data are often incomplete or noisy and therefore cannot be employed directly. To deal with the above two problems, one typical and efficient solution is to deform an existing high-quality model (template) to fit the scanned data (target). Shape registration as an essential technique to do so has been arousing intensive attention. In last decades, various shape registration approaches have been proposed for accurate template fitting. However, there are still some remaining challenges. It is well known that the template can be largely different with the target in respect of size and pose. With the large (usually non-isometric) deformation between them, the shear distortion can easily occur, which may lead to poor results, such as degenerated triangles, fold-overs. Before deforming the template towards the target, reliable correspondences between them should be found first. Incorrect correspondences give the wrong deformation guidance, which can also easily produce fold-overs. As mentioned before, the target always comes with noise. This is the part we want to filter out and try not to fit the template on it. Hence, non-isometric shape registration robust to noise is highly desirable in the scene of geometry modelling from the scanned data. In this PhD research, we address existing challenges in shape registration, including how to prevent the deformation distortion, how to reduce the foldover occurrence and how to deal with the noise in the target. Novel methods including consistent as-similar as-possible surface deformation and robust Huber-L1 surface registration are proposed, which are validated through experimental comparison with state-of-the-arts. The deformation technique plays an important role in shape registration. In this research, a consistent as similar-as-possible (CASAP) surface deformation approach is proposed. Starting from investigating the continuous deformation energy, we analyse the existing term to make the discrete energy converge to the continuous one, whose property we called as energy consistency. Based on the deformation method, a novel CASAP non-isometric surface registration method is proposed. The proposed registration method well preserves the angles of triangles in the template surface so that least distortion is introduced during the surface deformation and thus reduce the risk of fold-over and self-intersection. To reduce the noise influence, a Huber-L1 based non-isometric surface registration is proposed, where a Huber-L1 regularized model constrained on the transformation variation and position difference. The proposed method is robust to noise and produces piecewise smooth results while still preserving fine details on the target. We evaluate and validate our methods through extensive experiments, whose results have demonstrated that the proposed methods in this thesis are more accurate and robust to noise in comparison of the state-of-the arts and enable us to produce high quality models with little efforts

Theory and applications of bijective barycentric mappings

Barycentric coordinates provide a convenient way to represent a point inside a triangle as a convex combination of the triangle's vertices, and to linearly interpolate data given at these vertices. Due to their favourable properties, they are commonly applied in geometric modelling, finite element methods, computer graphics, and many other fields. In some of these applications it is desirable to extend the concept of barycentric coordinates from triangles to polygons. Several variants of such generalized barycentric coordinates have been proposed in recent years. An important application of barycentric coordinates consists of barycentric mappings, which allow to naturally warp a source polygon to a corresponding target polygon, or more generally, to create mappings between closed curves or polyhedra. The principal practical application is image warping, which takes as input a control polygon drawn around an image and smoothly warps the image by moving the polygon vertices. A required property of image warping is to avoid fold-overs in the resulting image. The problem of fold-overs is a manifestation of a larger problem related to the lack of bijectivity of the barycentric mapping. Unfortunately, bijectivity of such barycentric mappings can only be guaranteed for the special case of warping between convex polygons or by triangulating the domain and hence renouncing smoothness. In fact, for any barycentric coordinates, it is always possible to construct a pair of polygons such that the barycentric mapping is not bijective. In the first part of this thesis we illustrate three methods to achieve bijective mappings. The first method is based on the intuition that, if two polygons are sufficiently close, then the mapping is close to the identity and hence bijective. This suggests to ``split'' the mapping into several intermediate mappings and to create a composite barycentric mapping which is guaranteed to be bijective between arbitrary polygons, polyhedra, or closed planar curves. We provide theoretical bounds on the bijectivity of the composite mapping related to the norm of the gradient of the coordinates. The fact that the bound depends on the gradient implies that these bounds exist only if the gradient of the coordinates is bounded. We focus on mean value coordinates and analyse the behaviour of their directional derivatives and gradient at the vertices of a polygon. The composition of barycentric mappings for closed planar curves leads to the problem of blending between two planar curves. We suggest to solve it by linearly interpolating the signed curvature and then reconstructing the intermediate curve from the interpolated curvature values. However, when both input curves are closed, this strategy can lead to open intermediate curves. We present a new algorithm for solving this problem, which finds the closed curve whose curvature is closest to the interpolated values. Our method relies on the definition of a suitable metric for measuring the distance between two planar curves and an appropriate discretization of the signed curvature functions. The second method to construct smooth bijective mappings with prescribed behaviour along the domain boundary exploits the properties of harmonic maps. These maps can be approximated in different ways, and we discuss their respective advantages and disadvantages. We further present a simple procedure for reducing their distortion and demonstrate the effectiveness of our approach by providing examples. The last method relies on a reformulation of complex barycentric mappings, which allows us to modify the ``speed'' along the edges to create complex bijective mappings. We provide some initial results and an optimization procedure which creates complex bijective maps. In the second part we provide two main applications of bijective mapping. The first one is in the context of finite elements simulations, where the discretization of the computational domain plays a central role. In the standard discretization, the domain is triangulated with a mesh and its boundary is approximated by a polygon. We present an approach which combines parametric finite elements with smooth bijective mappings, leaving the choice of approximation spaces free. This approach allows to represent arbitrarily complex geometries on coarse meshes with curved edges, regardless of the domain boundary complexity. The main idea is to use a bijective mapping for automatically warping the volume of a simple parametrization domain to the complex computational domain, thus creating a curved mesh of the latter. The second application addresses the meshing problem and the possibility to solve finite element simulations on polygonal meshes. In this context we present several methods to discretize the bijective mapping to create polygonal and piece-wise polynomial meshes

Bijective Parameterization with Free Boundaries

When displaying 3D surfaces onto computer screens, additional information is often mapped onto the surface to enhance the quality of the rendering. Surface parameterization generates a correspondence, or mapping, between the 3D surface and 2D parameterization space. This mapping has many applications in computer graphics, but in most cases cannot be performed without introducing large distortions in the 2D parameterization. Along with problems of distortion, the mapping of the 2D space to 3D for many applications can be invalidated if the property of bijectivity is violated. While there is previous research guaranteeing bijectivity, these methods must constrain or modify the boundary of the 2D parameterization. This dissertation, describes a fully automatic method for generating guaranteed bijective surface parameterizations from triangulated 3D surfaces. In particular, a new isometric distortion energy metric is introduced preventing local folds of triangles in the parameterization as well as a barrier function that prevents intersection of the 2D boundaries. By using a computationally efficient isometric metric energy, the dissertation achieves fast and comparable optimization times to previous methods. The boundary of the parameterization is free to change shape during the optimization to minimize distortion. A new optimization approach is introduced called singularity aware optimization and in conjunction with an interior point approach and barrier energy functions guarantee bijectivity. This optimization framework is then modified to allow for an importance weighting allowing for customizable and more efficient texel usage

Transformations quasi-conformes de maillages volumiques et applications en infographie

La modélisation géométrique est importante autant en infographie qu'en ingénierie. Notre capacité à représenter l'information géométrique fixe les limites et la facilité avec laquelle on manipule les objets 3D. Une de ces représentations géométriques est le maillage volumique, formé de polyèdres assemblés de sorte à approcher une forme désirée. Certaines applications, tels que le placage de textures et le remaillage, ont avantage à déformer le maillage vers un domaine plus régulier pour faciliter le traitement. On dit qu'une déformation est \emph{quasi-conforme} si elle borne la distorsion. Cette thèse porte sur l’étude et le développement d'algorithmes de déformation quasi-conforme de maillages volumiques. Nous étudions ces types de déformations parce qu’elles offrent de bonnes propriétés de préservation de l’aspect local d’un solide et qu’elles ont été peu étudiées dans le contexte de l’informatique graphique, contrairement à leurs pendants 2D. Cette recherche tente de généraliser aux volumes des concepts bien maitrisés pour la déformation de surfaces. Premièrement, nous présentons une approche linéaire de la quasi-conformité. Nous développons une méthode déformant l’objet vers son domaine paramétrique par une méthode des moindres carrés linéaires. Cette méthode est simple d'implémentation et rapide d'exécution, mais n'est qu'une approximation de la quasi-conformité car elle ne borne pas la distorsion. Deuxièmement, nous remédions à ce problème par une approche non linéaire basée sur les positions des sommets. Nous développons une technique déformant le domaine paramétrique vers le solide par une méthode des moindres carrés non linéaires. La non-linéarité permet l’inclusion de contraintes garantissant l’injectivité de la déformation. De plus, la déformation du domaine paramétrique au lieu de l’objet lui-même permet l’utilisation de domaines plus généraux. Troisièmement, nous présentons une approche non linéaire basée sur les angles dièdres. Cette méthode définit la déformation du solide par les angles dièdres au lieu des positions des sommets du maillage. Ce changement de variables permet une expression naturelle des bornes de distorsion de la déformation. Nous présentons quelques applications de cette nouvelle approche dont la paramétrisation, l'interpolation, l'optimisation et la compression de maillages tétraédriques.Geometric modeling is important for both computer graphics and engineering. Our ability to represent geometric information sets the limits and the ease with which we manipulate 3D objects. One such representation is the volume mesh, that is composed of polyhedra assembled to approximate a desired shape. Some applications, such as texturing and remeshing, benefit from deforming the mesh to a more regular domain in order to perform some operations. We say that a deformation is \emph{quasi-conformal} if its distortion is bounded. In this thesis, we propose algorithms for quasi-conformal deformations of volume meshes. We study these deformations because of their good local shape preservation properties and because they are still relatively unknown to the graphics community, as opposed to their 2D counterparts. This research attempts to generalize \gilles{some well-known surface deformation concepts to volumes}. First, we present a linear approach to quasi-conformality. We develop a method that deforms a solid to a parameterization domain using a linear least squares method. This method is fast and simple to implement, but the result is an approximation to quasi-conformality because distortion is not bounded. Second, we solve this latter limitation with a nonlinear approach based on vertex positions. We develop a technique to deform the parameterization domain to a solid shape using a nonlinear least squares method. Nonlinearity lets us include constraints that guarantee the injectivity of the deformation. Moreover, deforming the parameterization domain instead of the shape itself lets us use more general domains. Third, we present a nonlinear approach based on dihedral angles. Our method defines the deformation of the volume mesh using its dihedral angles instead of its vertex positions. This change of variables permits a natural expression of the bounds of the deformation distortion. We present some applications of this new approach that include volume parameterization, shape interpolation, mesh optimization, and mesh compression of tetrahedral meshes

Generation of 3D characters from existing cartoons and a unified pipeline for animation and video games.

Despite the remarkable growth of 3D animation in the last twenty years, 2D is still popular today and often employed for both films and video games. In fact, 2D offers important economic and artistic advantages to production. In this thesis has been introduced an innovative system to generate 3D character from 2D cartoons, while maintaining important 2D features in 3D as well. However, handling 2D characters and animation in a 3D environment is not a trivial task, as they do not possess any depth information. Three different solutions have been proposed in this thesis. A 2.5D modelling method, which exploits billboarding, parallax scrolling and 2D shape interpolation to simulate the depth between the different body parts of the characters. Two additional full 3D solution have been presented. One based on inflation and supported by a surface registration method, and one that produces more accurate approximations by using information from the side views to solve an optimization problem. These methods have been introduced into a new unified pipeline that involves a game engine, and that could be used for animation and video games production. A unified pipeline introduces several benefits to animation production for either 2D and 3D content. On one hand, assets can be shared for different productions and media. On the other hand, real-time rendering for animated films allows immediate previews of the scenes and offers artists a way to experiment more during the making of a scene