Mean value coordinates–based caricature and expression synthesis

We present a novel method for caricature synthesis based on mean value coordinates (MVC). Our method can be applied to any single frontal face image to learn a specified caricature face pair for frontal and 3D caricature synthesis. This technique only requires one or a small number of exemplar pairs and a natural frontal face image training set, while the system can transfer the style of the exemplar pair across individuals. Further exaggeration can be fulfilled in a controllable way. Our method is further applied to facial expression transfer, interpolation, and exaggeration, which are applications of expression editing. Additionally, we have extended our approach to 3D caricature synthesis based on the 3D version of MVC. With experiments we demonstrate that the transferred expressions are credible and the resulting caricatures can be characterized and recognized

Caricature Synthesis Based on Mean Value Coordinates

In this paper, a novel method for caricature synthesis is developed based on mean value coordinates (MVC). Our method can be applied to any single frontal face image to learn a specified caricature face exemplar pair for frontal and side view caricature synthesis. The technique only requires one or a small number of caricature face pairs and a natural frontal face training set, while the system can transfer the style of the exemplar pair across individuals. Further exaggeration can be fulfilled in a controllable way. Our method is further extended to facial expression transfer, interpolation and exaggeration, which are applications of expression editing. Moreover, the deformation equation of MVC is modified to handle the case of polygon intersections and applied to lateral view caricature synthesis from a single frontal view image. Using experiments we demonstrate that the transferred expressions are credible and the resulting caricatures can be characterized and recognized

Quadratic Mean Value Barycentric Coordinates for Image Deformation

随着计算几何这个领域的逐渐发展，重心坐标作为计算几何中的一个重要工具也在逐渐进步。最开始的重心坐标是定义在三角形上的，它具有仿射不变性、lagrange性质、正性、归一性等很多很好的性质。然而在实际应用中碰到更多的是多边形，甚至是多面体，所以在此基础上，将重心坐标推广到了广义的重心坐标。根据不同的广义重心坐标的用途，广义重心坐标的种类也很多。但是，在本文之前所提出的广义重心坐标大多数都是一次的重心坐标，很少涉及到高阶的重心坐标，即使有部分文章研究了高阶的重心坐标，也没有给出明确的计算方法，这是不利于实际使用的。然而在实际应用中，相比于一次的重心坐标，高阶的重心坐标不仅可以提高自由度，加速收敛，...Barycentric coordinates is a powerful and yet simple way to interpolate data values on polyhedral domains . At first , barycentric coordinates is defined in the triangles . It has a large variety of properties like lagrange , affine invariant , positive , normalization and so on . However, in reality we often meet polygons, even polyhedrons , So we need to extend the barycentric coordinates . Befo...学位：理学硕士院系专业：数学科学学院_计算数学学号：1902013115265

Construction of smooth maps with mean value coordinates

Bernstein polynomials are a classical tool in Computer Aided Design to create smooth maps with a high degree of local control. They are used for the construction of B\'ezier surfaces, free-form deformations, and many other applications. However, classical Bernstein polynomials are only defined for simplices and parallelepipeds. These can in general not directly capture the shape of arbitrary objects. Instead, a tessellation of the desired domain has to be done first. We construct smooth maps on arbitrary sets of polytopes such that the restriction to each of the polytopes is a Bernstein polynomial in mean value coordinates (or any other generalized barycentric coordinates). In particular, we show how smooth transitions between different domain polytopes can be ensured

A mean value formula for elliptic curves

It is proved in this paper that for any point on an elliptic curve, the mean value of x-coordinates of its n-division points is the same as its x-coordinate and that of y-coordinates of its n-division points is n times of its y-coordinate

Derivation of Mean Value Coordinates Using Interior Distance and Their Application on Mesh Deformation

The deformation methods based on cage controls became a subject of considerable interest due its simplicity and intuitive results. In this technique, the model is enclosed within a simpler mesh (the cage) and its points are expressed as function of the cage elements. Then, by manipulating the cage, the respective deformation is obtained on the model in its interior.In this direction, in the last years, extensions of barycentric coordinates, such as Mean Value coordinates, Positive Mean Value Coordinates, Harmonic coordinates and Green's coordinates, have been proposed to write the points of the model as a function of the cage elements.The Mean Value coordinates, proposed by Floater in two dimensions and extended later to three dimensions by Ju et al. and also by Floater, stands out from the other coordinates because of their simple derivation. However the existence of negative coordinates in regions bounded by non-convex cage control results in a unexpected behavior of the deformation in some regions of the model.In this work, we propose a modification in the derivation of Mean Value Coordinates proposed by Floater. Our derivation maintains the simplicity of the construction of the coordinates and eliminates the undesired behavior in the deformation by diminishing the negative influence of a control vertex on regions ofthe model not related to it. We also compare the deformation generated with our coordinates and the deformations obtained with the original Mean Value coordinates and Harmonic coordinates

Geometrical Well Posed Systems for the Einstein Equations

We show that, given an arbitrary shift, the lapse $N$ can be chosen so that the extrinsic curvature $K$ of the space slices with metric $\overline g$ in arbitrary coordinates of a solution of Einstein's equations satisfies a quasi-linear wave equation. We give a geometric first order symmetric hyperbolic system verified in vacuum by $\overline g$, $K$ and $N$. We show that one can also obtain a quasi-linear wave equation for $K$ by requiring $N$ to satisfy at each time an elliptic equation which fixes the value of the mean extrinsic curvature of the space slices.Comment: 13 pages, latex, no figure