Skeleton-based Variational Mesh Deformations

In this paper, a new free-form shape deformation approach is proposed. We combine a skeleton-based mesh deformation technique with discrete differential coordinates in order to create natural-looking global shape deformations. Given a triangle mesh, we first extract a skeletal mesh, a two-sided Voronoibased approximation of the medial axis. Next the skeletal mesh is modified by free-form deformations. Then a desired global shape deformation is obtained by reconstructing the shape corresponding to the deformed skeletal mesh. The reconstruction is based on using discrete differential coordinates. Our method preserves fine geometric details and original shape thickness because of using discrete differential coordinates and skeleton-based deformations. We also develop a new mesh evolution technique which allow us to eliminate possible global and local self-intersections of the deformed mesh while preserving fine geometric details. Finally, we present a multi-resolution version of our approach in order to simplify and accelerate the deformation process. In addition, interesting links between the proposed free-form shape deformation technique and classical and modern results in the differential geometry of sphere congruences are established and discussed.     

Interactive Editing of Lighting and Materials using a Bivariate BRDF Representation

We present a new Precomputed Radiance Transfer (PRT) algorithm based on a two dimensional representation of isotropic BRDFs. Our approach involves precomputing matrices that allow quickly mapping environment lighting, which is represented in the global coordinate system, and the surface BRDFs, which are represented in a bivariate domain, to the local hemisphere at a surface location where the reflection integral is evaluated. When the lighting and BRDFs are represented in a wavelet basis, these rotation matrices are sparse and can be efficiently stored and combined with pre‐computed visibility at run‐time. Compared to prior techniques that also precompute wavelet rotation matrices, our method allows full control over the lighting and materials due to the way the BRDF is represented. Furthermore, this bivariate parameterization preserves sharp specular peaks and grazing effects that are attenuated in conventional parameterizations. We demonstrate a prototype rendering system that achieves real‐time framerates while lighting and materials are edited.     

Global Intrinsic Symmetries of Shapes

Although considerable attention in recent years has been given to the problem of symmetry detection in general shapes, few methods have been developed that aim to detect and quantify the intrinsic symmetry of a shape rather than its extrinsic, or pose‐dependent symmetry. In this paper, we present a novel approach for efficiently computing symmetries of a shape which are invariant up to isometry preserving transformations. We show that the intrinsic symmetries of a shape are transformed into the Euclidean symmetries in the signature space defined by the eigenfunctions of the Laplace‐Beltrami operator. Based on this observation, we devise an algorithm which detects and computes the isometric mappings from the shape onto itself. We show that our approach is both computationally efficient and robust with respect to small non‐isometric deformations, even if they include topological changes.     

Deformable 3D Shape Registration Based on Local Similarity Transforms

In this paper, a new method for deformable 3D shape registration is proposed. The algorithm computes shape transitions based on local similarity transforms which allows to model not only as‐rigid‐as‐possible deformations but also local and global scale. We formulate an ordinary differential equation (ODE) which describes the transition of a source shape towards a target shape. We assume that both shapes are roughly pre‐aligned (e.g., frames of a motion sequence). The ODE consists of two terms. The first one causes the deformation by pulling the source shape points towards corresponding points on the target shape. Initial correspondences are estimated by closest‐point search and then refined by an efficient smoothing scheme. The second term regularizes the deformation by drawing the points towards locally defined rest positions. These are given by the optimal similarity transform which matches the initial (undeformed) neighborhood of a source point to its current (deformed) neighborhood. The proposed ODE allows for a very efficient explicit numerical integration. This avoids the repeated solution of large linear systems usually done when solving the registration problem within general‐purpose non‐linear optimization frameworks. We experimentally validate the proposed method on a variety of real data and perform a comparison with several state‐of‐the‐art approaches.     

Dynamic Multi‐View Exploration of Shape Spaces

Statistical shape modeling is a widely used technique for the representation and analysis of the shapes and shape variations present in a population. A statistical shape model models the distribution in a high dimensional shape space, where each shape is represented by a single point. We present a design study on the intuitive exploration and visualization of shape spaces and shape models. Our approach focuses on the dual‐space nature of these spaces. The high‐dimensional shape space represents the population, whereas object space represents the shape of the 3D object associated with a point in shape space. A 3D object view provides local details for a single shape. The high dimensional points in shape space are visualized using a 2D scatter plot projection, the axes of which can be manipulated interactively. This results in a dynamic scatter plot, with the further extension that each point is visualized as a small version of the object shape that it represents. We further enhance the population‐object duality with a new type of view aimed at shape comparison. This new “shape evolution view” visualizes shape variability along a single trajectory in shape space, and serves as a link between the two spaces described above. Our three‐view exploration concept strongly emphasizes linked interaction between all spaces. Moving the cursor over the scatter plot or evolution views, shapes are dynamically interpolated and shown in the object view. Conversely, camera manipulation in the object view affects the object visualizations in the other views. We present a GPU‐accelerated implementation, and show the effectiveness of the three‐view approach using a number of real‐world cases. In these, we demonstrate how this multi‐view approach can be used to visually explore important aspects of a statistical shape model, including specificity, compactness and reconstruction error.     

Creating and Animating Subject‐Specific Anatomical Models

Creating and animating subject‐specific anatomical models is traditionally a difficult process involving medical image segmentation, geometric corrections and the manual definition of kinematic parameters. In this paper, we introduce a novel template morphing algorithm that facilitates three‐dimensional modelling and parameterization of skeletons. Target data can be either medical images or surfaces of the whole skeleton. We incorporate prior knowledge about bone shape, the feasible skeleton pose and the morphological variability in the population. This allows for noise reduction, bone separation and the transfer, from the template, of anatomical and kinematical information not present in the input data. Our approach treats both local and global deformations in successive regularization steps: smooth elastic deformations are represented by an as‐rigid‐as‐possible displacement field between the reference and current configuration of the template, whereas global and discontinuous displacements are estimated through a projection onto a statistical shape model and a new joint pose optimization scheme with joint limits.     

Animated 3D Line Drawings with Temporal Coherence

Producing traditional animation is a laborious task where the key drawings are first drawn by artists and thereafter inbetween drawings are created, whether it is by hand or computer‐assisted. Auto‐inbetweening of these 2D key drawings by computer is a non‐trivial task as 3D depths are missing. An alternate approach is to generate all the drawings by extracting lines directly from animated 3D models frame by frame, concatenating and rendering them together into an animation. However, animation quality generated using this straightforward method bears two problems. Firstly, the animation contains unsatisfactory visual artifacts such as line flickering and popping. This is especially pronounced when the lines are extracted using high‐order derivatives, such as ridges and valleys, from 3D models represented in triangle meshes. Secondly, there is a lack of temporal continuity as each drawing is generated without taking its neighboring drawings into consideration. In this paper, we propose an improved approach over the straightforward method by transferring extracted 3D line drawings of each frame into individual 3D lines and processing them along the time domain. Our objective is to minimize the visual artifacts and incorporate temporal relationship of individual lines throughout the entire animation sequence. This is achieved by creating correspondent trajectory of each line from each frame and applying global optimization on each trajectory. To realize this target, we present a fully automatic novel approach, which consists of (1) a line matching algorithm, (2) an optimizing algorithm, taking into account both the variations of numbers and lengths of 3D lines in each frame, and (3) a robust tracing method for transferring collections of line segments extracted from the 3D models into individual lines. We evaluate our approach on several animated model sequences to demonstrate its effectiveness in producing line drawing animations with temporal coherence.     

Multi‐Scale Geometry Interpolation

Interpolating vertex positions among triangle meshes with identical vertex‐edge graphs is a fundamental part of many geometric modelling systems. Linear vertex interpolation is robust but fails to preserve local shape. Most recent approaches identify local affine transformations for parts of the mesh, model desired interpolations of the affine transformations, and then optimize vertex positions to conform with the desired transformations. However, the local interpolation of the rotational part is non‐trivial for more than two input configurations and ambiguous if the meshes are deformed significantly. We propose a solution to the vertex interpolation problem that starts from interpolating the local metric (edge lengths) and mean curvature (dihedral angles) and makes consistent choices of local affine transformations using shape matching applied to successively larger parts of the mesh. The local interpolation can be applied to any number of input vertex configurations and due to the hierarchical scheme for generating consolidated vertex positions, the approach is fast and can be applied to very large meshes.     

Reconstructing 3D Shape,Albedo and Illumination from a Single Face Image

The morphable model has been employed to efficiently describe 3D face shape and the associated albedo with a reduced set of basis vectors. The spherical harmonics (SH) model provides a compact basis to well approximate the image appearance of a Lambertian object under different illumination conditions. Recently, the SH and morphable models have been integrated for 3D face shape reconstruction. However, the reconstructed 3D shape is either inconsistent with the SH bases or obtained just from landmarks only. In this work, we propose a geometrically consistent algorithm to reconstruct the 3D face shape and the associated albedo from a single face image iteratively by combining the morphable model and the SH model. The reconstructed 3D face geometry can uniquely determine the SH bases, therefore the optimal 3D face model can be obtained by minimizing the error between the input face image and a linear combination of the associated SH bases. In this way, we are able to preserve the consistency between the 3D geometry and the SH model, thus refining the 3D shape reconstruction recursively. Furthermore, we present a novel approach to recover the illumination condition from the estimated weighting vector for the SH bases in a constrained optimization formulation independent of the 3D geometry. Experimental results show the effectiveness and accuracy of the proposed face reconstruction and illumination estimation algorithm under different face poses and multiple‐light‐source illumination conditions.     

Efficient Packing of Arbitrary Shaped Charts for Automatic Texture Atlas Generation

Texture atlases are commonly used as representations for mesh parameterizations in numerous applications including texture and normal mapping. Therefore, packing is an important post‐processing step that tries to place and orient the single parameterizations in a way that the available space is used as efficiently as possible. However, since packing is NP hard, only heuristics can be used in practice to find near‐optimal solutions. In this publication we introduce the new search space of modulo valid packings. The key idea thereby is to allow the texture charts to wrap around in the atlas. By utilizing this search space we propose a new algorithm that can be used in order to automatically pack texture atlases. In the evaluation section we show that our algorithm achieves solutions with a significantly higher packing efficiency when compared to the state of the art, especially for complex packing problems.     

CubeCover– Parameterization of 3D Volumes

Despite the success of quad‐based 2D surface parameterization methods, effective parameterization algorithms for 3D volumes with cubes, i.e. hexahedral elements, are still missing. Cube Cover is a first approach for generating a hexahedral tessellation of a given volume with boundary aligned cubes which are guided by a frame field. The input of Cube Cover is a tetrahedral volume mesh. First, a frame field is designed with manual input from the designer. It guides the interior and boundary layout of the parameterization. Then, the parameterization and the hexahedral mesh are computed so as to align with the given frame field. Cube Cover has similarities to the Quad Cover algorithm and extends it from 2D surfaces to 3D volumes. The paper also provides theoretical results for 3D hexahedral parameterizations and analyses topological properties of the appropriate function space.     

Discrete Line Congruences for Shading and Lighting

Two‐parameter families of straight lines (line congruences) are implicitly present in graphics and geometry processing in several important ways including lighting and shape analysis. In this paper we make them accessible to optimization and geometric computing, by introducing a general discrete version of congruences based on piecewise‐linear correspondences between triangle meshes. Our applications of congruences are based on the extraction of a so‐called torsion‐free support structure, which is a procedure analogous to remeshing a surface along its principal curvature lines. A particular application of such structures are freeform shading and lighting systems for architecture. We combine interactive design of such systems with global optimization in order to satisfy geometric constraints. In this way we explore a new area where architecture can greatly benefit from graphics.     

ProcDef: Local‐to‐global Deformation for Skeleton‐free Character Animation

Animations of characters with flexible bodies such as jellyfish, snails, and, hearts are difficult to design using traditional skeleton‐based approaches. A standard approach is keyframing, but adjusting the shape of the flexible body for each key frame is tedious. In addition, the character cannot dynamically adjust its motion to respond to the environment or user input. This paper introduces a new procedural deformation framework (ProcDef) for designing and driving animations of such flexible objects. Our approach is to synthesize global motions procedurally by integrating local deformations. ProcDef provides an efficient design scheme for local deformation patterns; the user can control the orientation and magnitude of local deformations as well as the propagation of deformation signals by specifying line charts and volumetric fields. We also present a fast and robust deformation algorithm based on shape‐matching dynamics and show some example animations to illustrate the feasibility of our framework.     

Conformal Flattening by Curvature Prescription and Metric Scaling

We present an efficient method to conformally parameterize 3D mesh data sets to the plane. The idea behind our method is to concentrate all the 3D curvature at a small number of select mesh vertices, called cone singularities, and then cut the mesh through those singular vertices to obtain disk topology. The singular vertices are chosen automatically. As opposed to most previous methods, our flattening process involves only the solution of linear systems of Poisson equations, thus is very efficient. Our method is shown to be faster than existing methods, yet generates parameterizations having comparable quasi‐conformal distortion.     

Automatic Registration for Articulated Shapes

We present an unsupervised algorithm for aligning a pair of shapes in the presence of significant articulated motion and missing data, while assuming no knowledge of a template, user‐placed markers, segmentation, or the skeletal structure of the shape. We explicitly sample the motion, which gives a priori the set of possible rigid transformations between parts of the shapes. This transforms the problem into a discrete labeling problem, where the goal is to find an optimal assignment of transformations for aligning the shapes. We then apply graph cuts to optimize a novel cost function, which encodes a preference for a consistent motion assignment from both source to target and target to source. We demonstrate the robustness of our method by aligning several synthetic and real‐world datasets.     

View and Time Interpolation in Image Space

The ability to interpolate between images taken at different time and viewpoints directly in image space opens up new possiblities. The goal of our work is to create plausible in‐between images in real time without the need for an intermediate 3D reconstruction. This enables us to also interpolate between images recorded with uncalibrated and unsynchronized cameras. In our approach we use a novel discontiniuity preserving image deformation model to robustly estimate dense correspondences based on local homographies. Once correspondences have been computed we are able to render plausible in‐between images in real time while properly handling occlusions. We discuss the relation of our approach to human motion perception and other image interpolation techniques.     

Generalized Discrete Ricci Flow

Surface Ricci flow is a powerful tool to design Riemannian metrics by user defined curvatures. Discrete surface Ricci flow has been broadly applied for surface parameterization, shape analysis, and computational topology. Conventional discrete Ricci flow has limitations. For meshes with low quality triangulations, if high conformality is required, the flow may get stuck at the local optimum of the Ricci energy. If convergence to the global optimum is enforced, the conformality may be sacrificed. This work introduces a novel method to generalize the traditional discrete Ricci flow. The generalized Ricci flow is more flexible, more robust and conformal for meshes with low quality triangulations. Conventional method is based on circle packing, which requires two circles on an edge intersect each other at an acute angle. Generalized method allows the two circles either intersect or separate from each other. This greatly improves the flexibility and robustness of the method. Furthermore, the generalized Ricci flow preserves the convexity of the Ricci energy, this ensures the uniqueness of the global optimum. Therefore the algorithm won't get stuck at the local optimum. Generalized discrete Ricci flow algorithms are explained in details for triangle meshes with both Euclidean and hyperbolic background geometries. Its advantages are demonstrated by theoretic proofs and practical applications in graphics, especially surface parameterization.     

Energy-Based Image Deformation

We present a general approach to shape deformation based on energy minimization, and applications of this approach to the problems of image resizing and 2D shape deformation. Our deformation energy generalizes that found in the prior art, while still admitting an efficient algorithm for its optimization. The key advantage of our energy function is the flexibility with which the set of "legal transformations" may be expressed; these transformations are the ones which are not considered to be distorting. This flexibility allows us to pose the problems of image resizing and 2D shape deformation in a natural way and generate minimally distorted results. It also allows us to strongly reduce undesirable foldovers or self-intersections. Results of both algorithms demonstrate the effectiveness of our approach.     

A Vectorial Framework for Ray Traced Diffusion Curves

Diffusion curves allow creating complex, smoothly shaded images by diffusing colours defined at curves. These methods typically require the solution of a global optimization problem (over either the pixel grid or an intermediate tessellated representation) to produce the final image, making fully parallel implementation challenging. An alternative approach, inspired by global illumination, uses 2D ray tracing to independently compute each pixel value. This formulation allows trivial parallelism, but it densely computes values even in smooth regions and sacrifices support for instancing and layering. We describe a sparse, ray traced, multi‐layer framework that incorporates many complementary benefits of these existing approaches. Our solution avoids the need for a global solve and trivially allows parallel GPU implementation. We leverage an intermediate triangular representation with cubic patches to synthesize smooth images faithful to the per‐pixel solution. The triangle mesh provides a resolution–independent, vectorial representation and naturally maps diffusion curve images to a form natively supported by standard vector graphics and triangle rasterization pipelines. Our approach supports many features which were previously difficult to incorporate into a single system, including instancing, layering, alpha blending, texturing, local blurring, continuity control and parallel computation. We also show how global diffusion curves can be combined with local painted strokes in one coherent system.     

Improved Stochastic Progressive Photon Mapping with Metropolis Sampling

This paper presents an improvement to the stochastic progressive photon mapping (SPPM), a method for robustly simulating complex global illumination with distributed ray tracing effects. Normally, similar to photon mapping and other particle tracing algorithms, SPPM would become inefficient when the photons are poorly distributed. An inordinate amount of photons are required to reduce the error caused by noise and bias to acceptable levels. In order to optimize the distribution of photons, we propose an extension of SPPM with a Metropolis‐Hastings algorithm, effectively exploiting local coherence among the light paths that contribute to the rendered image. A well‐designed scalar contribution function is introduced as our Metropolis sampling strategy, targeting at specific parts of image areas with large error to improve the efficiency of the radiance estimator. Experimental results demonstrate that the new Metropolis sampling based approach maintains the robustness of the standard SPPM method, while significantly improving the rendering efficiency for a wide range of scenes with complex lighting.