In this paper we consider the problem of creating and controlling volume deformations used to articulate characters for use in high-end applications such as computer generated feature films. We ...introduce a method we call harmonic coordinates that significantly improves upon existing volume deformation techniques. Our deformations are controlled using a topologically flexible structure, called a cage, that consists of a closed three dimensional mesh. The cage can optionally be augmented with additional interior vertices, edges, and faces to more precisely control the interior behavior of the deformation. We show that harmonic coordinates are generalized barycentric coordinates that can be extended to any dimension. Moreover, they are the first system of generalized barycentric coordinates that are non-negative even in strongly concave situations, and their magnitude falls off with distance as measured within the cage.
We introduce Probabilistic Coordinate Fields (PCFs), a novel geometric-invariant coordinate representation for image correspondence problems. In contrast to standard Cartesian coordinates, PCFs ...encode coordinates in correspondence-specific barycentric coordinate systems (BCS) with affine invariance. To know when and where to trust the encoded coordinates, we implement PCFs in a probabilistic network termed PCF-Net, which parameterizes the distribution of coordinate fields as Gaussian mixture models. By jointly optimizing coordinate fields and their confidence conditioned on dense flows, PCF-Net can work with various feature descriptors when quantifying the reliability of PCFs by confidence maps. An interesting observation of this work is that the learned confidence map converges to geometrically coherent and semantically consistent regions, which facilitates robust coordinate representation. By delivering the confident coordinates to keypoint/feature descriptors, we show that PCF-Net can be used as a plug-in to existing correspondence-dependent approaches. Extensive experiments on both indoor and outdoor datasets suggest that accurate geometric invariant coordinates help to achieve the state of the art in several correspondence problems, such as sparse feature matching, dense image registration, camera pose estimation, and consistency filtering. Further, the interpretable confidence map predicted by PCF-Net can also be leveraged to other novel applications from texture transfer to multi-homography classification.
We introduce an extremely scalable and efficient yet simple palette-based image decomposition algorithm. Given an RGB image and set of palette colors, our algorithm decomposes the image into a set of ...additive mixing layers, each of which corresponds to a palette color applied with varying weight. Our approach is based on the geometry of images in RGBXY-space. This new geometric approach is orders of magnitude more efficient than previous work and requires no numerical optimization. We provide an implementation of the algorithm in 48 lines of Python code. We demonstrate a real-time layer decomposition tool in which users can interactively edit the palette to adjust the layers. After preprocessing, our algorithm can decompose 6 MP images into layers in 20 milliseconds.
Fifth-generation (5G) networks providing much higher bandwidth and faster data rates will allow connecting vast number of stationary and mobile devices, sensors, agents, users, machines, and ...vehicles, supporting Internet-of-Things (IoT), real-time dynamic networks of mobile things. Positioning and location awareness will become increasingly important, enabling deployment of new services and contributing to significantly improving the overall performance of the 5G system. Many of the currently talked about solutions to positioning in 5G are centralized, mostly requiring direct communication to the access nodes (or anchors, i.e., nodes with known locations), which in turn requires a high density of anchors. But such centralized positioning solutions may become unwieldy as the number of users and devices continues to grow without limit in sight. As an alternative to the centralized solutions, this paper discusses distributed localization in a 5G-enabled IoT environment where many low power devices, users, or agents are to locate themselves without a direct access to anchors. Even though positioning is essentially a nonlinear problem (solving circle equations by trilateration or triangulation), we discuss a cooperative linear distributed iterative solution with only local measurements, local communication, and local computation needed at each agent. Linearity is obtained by reparametrization of the agent location through barycentric coordinate representations based on local neighborhood geometry that may be computed in terms of certain Cayley-Menger determinants involving relative local inter-agent distance measurements. After a brief introduction to the localization problem, and other available distributed solutions primarily based on directly addressing the nonlinear formulation, we present the distributed linear solution for stationary agent networks and study its convergence, its robustness to noise, and extensions to mobile scenarios, in which agents, users, and (possibly) anchors are dynamic.
Object deformation with linear blending dominates practical use as the fastest approach for transforming raster images, vector graphics, geometric models and animated characters. Unfortunately, ...linear blending schemes for skeletons or cages are not always easy to use because they may require manual weight painting or modeling closed polyhedral envelopes around objects. Our goal is to make the design and control of deformations simpler by allowing the user to work freely with the most convenient combination of handle types. We develop linear blending weights that produce smooth and intuitive deformations for points, bones and cages of arbitrary topology. Our weights, called bounded biharmonic weights, minimize the Laplacian energy subject to bound constraints. Doing so spreads the influences of the controls in a shape-aware and localized manner, even for objects with complex and concave boundaries. The variational weight optimization also makes it possible to customize the weights so that they preserve the shape of specified essential object features. We demonstrate successful use of our blending weights for real-time deformation of 2D and 3D shapes.
Flame particles (FP) are massless, virtual particles which follow material points on the flame surface. This work presents a tracking algorithm for FPs which utilizes barycentric coordinates. The ...methodology can be used with any cell shape in the computational mesh and allows computationally fast spatial interpolation as well as efficient determination of the intersection of FP trajectories with iso-surfaces. In contrast to previous flame particle tracking (FPT) approaches, the code is fully parallelized and can therefore be used in-situ during the simulation. It also includes fully parallelized computation of flame consumption speed by integrating reaction rates along a line normal to the flame surface at each FP position. Direct numerical simulations of laminar pulsating premixed hydrogen–air Bunsen flames serve as validation cases and showcase the added value of tracking material points for studying local flame dynamics. Exciting the inlet flow harmonically with frequencies equal to the inverse flame time scale leads to a pulsating mode where the flame front is corrugated. Ten times higher frequencies nearly resemble the steady state solution. The FPs are seeded along the flame surface and are used to track the unsteady diffusive, convective and chemical contributions at arbitrary points on the flame front over time. Their trajectories reveal a phase shift between the unsteady flame stretch rate and local flame speed of the order of 0.1 flame time scales for rich hydrogen flames. This is caused by a time delay between straining and stretch due to curvature. The reason is that diffusive processes follow the time signal of curvature while chemical processes are most strongly affected by the straining rate, which dominates the high Lewis number hydrogen flames investigated. This time history effect may help to explain the large scattering in the correlation of local flame speed with flame stretch found in turbulent flames.
•Non-linear objective functions are considered.•Intuitionistic fuzzy TOPSIS approach is proposed.•Linear membership and non-membership functions are used.•Defuzzification based upon barycentric ...coordinates of a triangle is proposed.•Two different types of problems are solved with the proposed methodology.
The given study intends to propose a new approach, named intuitionistic fuzzy TOPSIS, to solve a non-linear multi-objective intuitionistic fuzzy problem, whose application is not confined to only one type but has a wider scope. Considering the distance of positive and negative ideal solutions from the optimal values directly in terms of triangular intuitionistic fuzzy numbers makes the study unique. Along with this, a new technique, based upon barycentric coordinates of a triangle, to defuzzify triangular intuitionistic fuzzy numbers has been proposed. Two different types of substantive illustrations viz., non-linear multi-objective intuitionistic fuzzy transportation problem and non-linear multi-objective intuitionistic fuzzy manufacturing problem, are solved and compared with the paradigmatic approaches existing in the literature.
Constructing a function that interpolates a set of values defined at vertices of a mesh is a fundamental operation in computer graphics. Such an interpolant has many uses in applications such as ...shading, parameterization and deformation. For closed polygons, mean value coordinates have been proven to be an excellent method for constructing such an interpolant. In this paper, we generalize mean value coordinates from closed 2D polygons to closed triangular meshes. Given such a mesh
P
, we show that these coordinates are continuous everywhere and smooth on the interior of
P
. The coordinates are linear on the triangles of
P
and can reproduce linear functions on the interior of
P
. To illustrate their usefulness, we conclude by considering several interesting applications including constructing volumetric textures and surface deformation.
•Use polynomials of Wachspress GBCs to construct smooth vertex splines over convex quadrilaterals.•The derivation of the construction is aided by MATHEMATICA and MATLAB to ensure the ...correctness.•These smooth vertex splines are locally supported which can be used to adjust and mend a surface.•Quasi-interpolatory operators are constructed and their approximation properties are shown.•These vertex splines are used to construct smooth surfaces such as suitcase corners.
In this paper we construct smooth bivariate spline functions over a polygonal partition, e.g. a convex quadrilateral partition by using vertex spline techniques. Vertex splines, introduced in Chui and Lai (1985) are smooth piecewise polynomial functions supported over a collection of triangles sharing a vertex. In this paper, we extend the concept of vertex splines to the partition of polygons and describe how to construct C1 vertex polygonal splines over a collection of quadrilaterals. We begin with our construction of C1 vertex splines over a collection of parallelograms, although they may not be axis-orientated. Then the construction is generalized to the setting of general convex quadrilaterals. We will use various monomials of Wachspress GBC functions of degrees 5 and 7 to explain how to construct C1 vertex splines together with additional special splines called edge and face splines. With these splines at hand, we construct quasi-interpolatory formulas, whose approximation properties will be shown. Numerical interpolation and approximation results will be presented. Finally, three applications of these splines are explained: the first one is to form smooth locally supported GBC functions, the second one is to construct smooth suitcase corners, and the third one to construct C1 surfaces over quadrilateral partitions with extraordinary points (EP). Several examples will be demonstrated to show the convenience of using these splines.
PDF
