Geometry-aware direction field processing

Direction fields and vector fields play an increasingly important role in computer graphics and geometry processing. The synthesis of directional fields on surfaces, or other spatial domains, is a fundamental step in numerous applications, such as mesh generation, deformation, texture mapping, and many more. The wide range of applications resulted in definitions for many types of directional fields: from vector and tensor fields, over line and cross fields, to frame and vector-set fields. Depending on the application at hand, researchers have used various notions of objectives and constraints to synthesize such fields. These notions are defined in terms of fairness, feature alignment, symmetry, or field topology, to mention just a few. To facilitate these objectives, various representations, discretizations, and optimization strategies have been developed. These choices come with varying strengths and weaknesses. This course provides a systematic overview of directional field synthesis for graphics applications, the challenges it poses, and the methods developed in recent years to address these challenges.

Combinatorial topology, also known as rubber sheet geometry, has extensive applications in geometry and analysis, many of which result from connections with the theory of differential equations. A link between topology and differential equations is vector fields. Recent developments in scientific visualization have shown that vector fields also play an important role in the analysis of second-order tensor fields. A second-order tensor field can be transformed into its eigensystem, namely, eigenvalues and their associated eigenvectors without loss of information content. Eigenvectors behave in a similar fashion to ordinary vectors with even simpler topological structures due to their sign indeterminacy. Incorporating information about eigenvectors and eigenvalues in a display technique known as hyperstreamlines reveals the structure of a tensor field. The simplify and often complex tensor field and to capture its important features, the tensor is decomposed into an isotopic tensor and a deviator. A tensor field and its deviator share the same set of eigenvectors, and therefore they have a similar topological structure. A a deviator determines the properties of a tensor field, while the isotopic part provides a uniform bias. Degenerate points are basic constituents of tensor fields. In 2-D tensor fields, there are only two types of degenerate points; while in 3-D, the degenerate points can be characterized in a Q'-R' plane. Compressible and incompressible flows share similar topological feature due to the similarity of their deviators. In the case of the deformation tensor, the singularities of its deviator represent the area of vortex core in the field. In turbulent flows, the similarities and differences of the topology of the deformation and the Reynolds stress tensors reveal that the basic addie-viscosity assuptions have their validity in turbulence modeling under certain conditions.

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We give an easily computable description of these vector fields using valuations along the critical locus. We apply this to answer the question when the liftable derivations can be defined by a tangency condition along the discriminant. 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We also generalise in positive characteristic Seidenberg’s theorem that all tangent vector fields can be lifted to the normalisation, assuming tame ramification.

Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and non-photorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of user-provided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges (i.e., discrete 1-forms), we obtain an intrinsic, coordinate-free formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient pre-factorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.

Many natural and man-made surface patterns are created by interactions between texture elements and surface geometry. We believe that the best way to create such patterns is to synthesize a texture directly on the surface of the model. Given a texture sample in the form of an image, we create a similar texture over an irregular mesh hierarchy that has been placed on a given surface.Our method draws upon texture synthesis methods that use image pyramids, and we use a mesh hierarchy to serve in place of such pyramids. First, we create a hierarchy of points from low to high density over a given surface, and we connect these points to form a hierarchy of meshes. Next, the user specifies a vector field over the surface that indicates the orientation of the texture. The mesh vertices on the surface are then sorted in such a way that visiting the points in order will follow the vector field and will sweep across the surface from one end to the other. Each point is then visited in turn to determine its color. The color of a particular point is found by examining the color of neighboring points and finding the best match to a similar pixel neighborhood in the given texture sample. The color assignment is done in a coarse-to-fine manner using the mesh hierarchy. A texture created this way fits the surface naturally and seamlessly.

Explicit flows associated to any tangent vector fields on any isospectral manifold for the AKNS operator acting in L2 × L2 on the unit interval are written down. The manifolds are of infinite dimension (and infinite codimension). The flows are called isospectral and also are Hamiltonian flows. It is proven that they may be explicitly expressed in terms of regularized determinants of infinite matrix-valued functions with entries depending only on the spectral data at the starting point of the flow. The tangent vector fields are decomposed as ∑ξkTk where ξ ∈ ℓ2 and the Tk ∈ L2 × L2 form a particular basis of the tangent vector spaces of the infinite dimensional manifold. The paper here is a continuation of Amour [“Explicit isospectral flows for the AKNS operator on the unit interval,” Inverse Probl. 25, 095008 (2009)]10.1088/0266-5611/25/9/095008 where, except for a finite number, all the components of the sequence ξ are zero in order to obtain an explicit expression for the isospectral flows. The regularized determinants induce counter-terms allowing for the consideration of finite quantities when the sequences ξ run all over ℓ2.

We present a wavelet‐based approach for selecting patches in patch‐based texture synthesis. We randomly select the first block that satisfies a minimum error criterion, computed from the wavelet coefficients (using 1D or 2D wavelets) for the overlapping region. We show that our wavelet‐based approach improves texture synthesis for samples where previous work fails, mainly textures with prominent aligned features. Also, it generates similar quality textures when compared against texture synthesis using feature maps with the advantage that our proposed method uses implicit edge information (since it is embedded in the wavelet coefficients) whereas feature maps rely explicitly on edge features. In previous work, the best patches are selected among all possible using a L2 norm on the RGB or grayscale pixel values of boundary zones. The L2 metric provides the raw pixel‐to‐pixel difference, disregarding relevant image structures — such as edges — that are relevant in the human visual system and therefore on synthesis of new textures.

Research into texture synthesis is normally concerned with generation of 2D images of texture. However, real-world surface textures comprise rough surface geometry and various reflectance properties. These surface textures are different from 2D still texture as their images can therefore vary dramatically with illumination directions. This paper presents a simple framework for 3D surface texture synthesis. Firstly, we propose a novel 2D texture synthesis algorithm based on wavelet transform that can be efficiently extended to synthesise surface representations in multi-dimensional space. Then, photometric stereo (PS) is used to generate surface gradient and albedo maps from three synthesized surface texture images. The surface gradient maps can be further integrated to produce a surface height map (surface profile). With the albedo and height or gradient maps, new images of a Lambertian surface under arbitrary illuminant directions can be generated. Experiments show that the proposed approach can not only produce 3D surface textures under arbitrary illumination directions, but also has the ability to retain the surface geometry structure.

A vector wavelet approach to iono- and magnetospheric geomagnetic satellite data

It is well known that the sphere S6(1) admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector field ξ=−JN is said to be characteristic or the Reeb vector field. The Jacobi operator with respect to ξ is called the structure Jacobi operator, and is denoted by l=R(·,ξ)ξ, where R is the curvature tensor on M. The study of Riemannian submanifolds in different ambient spaces by means of their Jacobi operators has been highly active in recent years. In particular, many recent results deal with questions around the existence of hypersurfaces with a structure Jacobi operator that satisfies conditions related to their parallelism. In the present paper, we study the parallelism of the structure Jacobi operator of real hypersurfaces in the nearly Kähler sphere S6(1). More precisely, we prove that such real hypersurfaces do not exist.

Designing rotational symmetries on surfaces is a necessary task for a wide variety of graphics applications, such as surface parameterization and remeshing, painterly rendering and pen-and-ink sketching, and texture synthesis. In these applications, the topology of a rotational symmetry field such as singularities and separatrices can have a direct impact on the quality of the results. In this paper, we present a design system that provides control over the topology of rotational symmetry fields on surfaces. As the foundation of our system, we provide comprehensive analysis for rotational symmetry fields on surfaces and present efficient algorithms to identify singularities and separatrices. We also describe design operations that allow a rotational symmetry field to be created and modified in an intuitive fashion by using the idea of basis fields and relaxation. In particular, we provide control over the topology of a rotational symmetry field by allowing the user to remove singularities from the field or to move them to more desirable locations. At the core of our analysis and design implementations is the observations that N-way rotational symmetries can be described by symmetric N-th order tensors, which allows an efficient vector-based representation that not only supports coherent definitions of arithmetic operations on rotational symmetries but also enables many analysis and design operations for vector fields to be adapted to rotational symmetry fields. To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing.

In this paper, we make use of the theory of Clifford algebras for anisotropic smoothing of vector-valued data. It provides a common framework to smooth functions, tangent vector fields and mappings taking values in $\mathfrak{so}(m)$, the Lie algebra of SO(m), defined on surfaces and more generally on Riemannian manifolds. Smoothing process arises from a convolution with a kernel associated to a second order differential operator: the Hodge Laplacian. It generalizes the Beltrami flow in the sense that the Laplace-Beltrami operator is the restriction to functions of minus the Hodge operator. We obtain a common framework for anisotropic smoothing of images, vector fields and oriented orthonormal frame fields defined on the charts.

CHAPTER VI - Geometry of Surfaces in E3

Digital inpainting provides a means for reconstruction of damaged portions of an image. Although the inpainting basics are straightforward, most inpainting techniques published in the literature are only suitable for remarkable small portion or smooth color image. In order to avoid such shortcomings, we present a new algorithm for digital reconstruction based on combination of wavelet decomposition, Surface-based/PDE-based inpainting and texture synthesis. In which, wavelet transform at first decomposes the image into high frequency and low frequency level parts. Subsequently, CSRBF which is generally used for surface interpolation or PDE-based inpainting is employed for low frequency level and texture synthesis is used for high frequency level. It results in that not only slightly portion but also the common blotched image can be reconstructed with high quality. Especially, our algorithm makes many difficult cases for other methods possible.

Studies the topology of 2nd-order symmetric tensor fields. Degenerate points are basic constituents of tensor fields. From the set of degenerate points, an experienced researcher can reconstruct a whole tensor field. We address the conditions for the existence of degenerate points and, based on these conditions, we predict the distribution of degenerate points inside the field. Every tensor can be decomposed into a deviator and an isotropic tensor. A deviator determines the properties of a tensor field, while the isotropic part provides a uniform bias. Deviators can be 3D or locally 2D. The triple-degenerate points of a tensor field are associated with the singular points of its deviator and the double-degenerate points of a tensor field have singular local 2D deviators. This provides insights into the similarity of topological structure between 1st-order (or vectors) and 2nd-order tensors. Control functions are in charge of the occurrences of a singularity of a deviator. These singularities can further be linked to important physical properties of the underlying physical phenomena. For a deformation tensor in a stationary flow, the singularities of its deviator actually represent the area of the vortex core in the field; for a stress tensor, the singularities represent the area with no stress; for a Newtonian flow, compressible flow and incompressible flow as well as stress and deformation tensors share similar topological features due to the similarity of their deviators; for a viscous flow, removing the large, isotropic pressure contribution dramatically enhances the anisotropy due to viscosity.