A Complex View of Barycentric Mappings

Barycentric coordinates are very popular for interpolating data values on polyhedral domains. It has been recently shown that expressing them as complex functions has various advantages when interpolating two‐dimensional data in the plane, and in particular for holomorphic maps. We extend and generalize these results by investigating the complex representation of real‐valued barycentric coordinates, when applied to planar domains. We show how the construction for generating real‐valued barycentric coordinates from a given weight function can be applied to generating complex‐valued coordinates, thus deriving complex expressions for the classical barycentric coordinates: Wachspress, mean value, and discrete harmonic. Furthermore, we show that a complex barycentric map admits the intuitive interpretation as a complex‐weighted combination of edge‐to‐edge similarity transformations, allowing the design of “home‐made” barycentric maps with desirable properties. Thus, using the tools of complex analysis, we provide a methodology for analyzing existing barycentric mappings, as well as designing new ones.     

General Spectral Camera Lens Simulation

We present a camera lens simulation model capable of producing advanced photographic phenomena in a general spectral Monte Carlo image rendering system. Our approach incorporates insights from geometrical diffraction theory, from optical engineering and from glass science. We show how to efficiently simulate all five monochromatic aberrations, spherical and coma aberration, astigmatism, field curvature and distortion. We also consider chromatic aberration, lateral colour and aperture diffraction. The inclusion of Fresnel reflection generates correct lens flares and we present an optimized sampling method for path generation.     

Exploring Non‐Linear Relationship of Blendshape Facial Animation

Human face is a complex biomechanical system and non‐linearity is a remarkable feature of facial expressions. However, in blendshape animation, facial expression space is linearized by regarding linear relationship between blending weights and deformed face geometry. This results in the loss of reality in facial animation. To synthesize more realistic facial animation, aforementioned relationship should be non‐linear to allow the greatest generality and fidelity of facial expressions. Unfortunately, few existing works pay attention to the topic about how to measure the non‐linear relationship. In this paper, we propose an optimization scheme that automatically explores the non‐linear relationship of blendshape facial animation from captured facial expressions. Experiments show that the explored non‐linear relationship is consistent with the non‐linearity of facial expressions soundly and is able to synthesize more realistic facial animation than the linear one.     

Least Squares Vertex Baking

We investigate the representation of signals defined on triangle meshes using linearly interpolated vertex attributes. Compared to texture mapping, storing data only at vertices yields significantly lower memory overhead and less expensive runtime reconstruction. However, standard approaches to determine vertex values such as point sampling or averaging triangle samples lead to suboptimal approximations. We discuss how an optimal solution can be efficiently calculated using continuous least‐squares. In addition, we propose a regularization term that allows us to minimize gradient discontinuities and mach banding artifacts while staying close to the optimum. Our method has been integrated in a game production lighting tool and we present examples of representing signals such as ambient occlusion and precomputed radiance transfer in real game scenes, where vertex baking was used to free up resources for other game components.     

SSD: Smooth Signed Distance Surface Reconstruction

We introduce a new variational formulation for the problem of reconstructing a watertight surface defined by an implicit equation, from a finite set of oriented points; a problem which has attracted a lot of attention for more than two decades. As in the Poisson Surface Reconstruction approach, discretizations of the continuous formulation reduce to the solution of sparse linear systems of equations. But rather than forcing the implicit function to approximate the indicator function of the volume bounded by the implicit surface, in our formulation the implicit function is forced to be a smooth approximation of the signed distance function to the surface. Since an indicator function is discontinuous, its gradient does not exist exactly where it needs to be compared with the normal vector data. The smooth signed distance has approximate unit slope in the neighborhood of the data points. As a result, the normal vector data can be incorporated directly into the energy function without implicit function smoothing. In addition, rather than first extending the oriented points to a vector field within the bounding volume, and then approximating the vector field by a gradient field in the least squares sense, here the vector field is constrained to be the gradient of the implicit function, and a single variational problem is solved directly in one step. The formulation allows for a number of different efficient discretizations, reduces to a finite least squares problem for all linearly parameterized families of functions, and does not require boundary conditions. The resulting algorithms are significantly simpler and easier to implement, and produce results of quality comparable with state‐of‐the‐art algorithms. An efficient implementation based on a primal‐graph octree‐based hybrid finite element‐finite difference discretization, and the Dual Marching Cubes isosurface extraction algorithm, is shown to produce high quality crack‐free adaptive manifold polygon meshes.     

A Graph‐based Approach to Continuous Line Illustrations with Variable Levels of Detail

This paper introduces a method for automatically generating continuous line illustrations, drawings consisting of a single line, from a given input image. Our approach begins by inferring a graph from a set of edges extracted from the image in question and obtaining a path that traverses through all edges of the said graph. The resulting path is then subjected to a series of post‐processing operations to transform it into a continuous line drawing. Moreover, our approach allows us to manipulate the amount of detail portrayed in our line illustrations, which is particularly useful for simplifying the overall illustration while still retaining its most significant features. We also present several experimental results to demonstrate that our approach can automatically synthesize continuous line illustrations comparable to those of some contemporary artists.     

Skeleton computation of orthogonal polyhedra

Skeletons are powerful geometric abstractions that provide useful representations for a number of geometric operations. The straight skeleton has a lower combinatorial complexity compared with the medial axis. Moreover, while the medial axis of a polyhedron is composed of quadric surfaces the straight skeleton just consist of planar faces. Although there exist several methods to compute the straight skeleton of a polygon, the straight skeleton of polyhedra has been paid much less attention. We require to compute the skeleton of very large datasets storing orthogonal polyhedra. Furthermore, we need to treat geometric degeneracies that usually arise when dealing with orthogonal polyhedra. We present a new approach so as to robustly compute the straight skeleton of orthogonal polyhedra. We follow a geometric technique that works directly with the boundary of an orthogonal polyhedron. Our approach is output sensitive with respect to the number of vertices of the skeleton and solves geometric degeneracies. Unlike the existing straight skeleton algorithms that shrink the object boundary to obtain the skeleton, our algorithm relies on the plane sweep paradigm. The resulting skeleton is only composed of axis‐aligned and 45° rotated planar faces and edges.     

A Multiscale Approach to Optimal Transport

In this paper, we propose an improvement of an algorithm of Aurenhammer, Hoffmann and Aronov to find a least square matching between a probability density and finite set of sites with mass constraints, in the Euclidean plane. Our algorithm exploits the multiscale nature of this optimal transport problem. We iteratively simplify the target using Lloyd's algorithm, and use the solution of the simplified problem as a rough initial solution to the more complex one. This approach allows for fast estimation of distances between measures related to optimal transport (known as Earth‐mover or Wasserstein distances). We also discuss the implementation of these algorithms, and compare the original one to its multiscale counterpart.     

Direct Ray Tracing of Phong Tessellation

There are two major ways of calculating ray and parametric surface intersections in rendering. The first is through the use of tessellated triangles, and the second is to use parametric surfaces together with numerical methods such as Newton's method. Both methods are computationally expensive and complicated to implement. In this paper, we focus on Phong Tessellation and introduce a simple direct ray tracing method for Phong Tessellation. Our method enables rendering smooth surfaces in a computationally inexpensive yet robust way.     

Interactive Ray Tracing of Large Models Using Voxel Hierarchies

We propose an efficient approach for interactive visualization of massive models with CPU ray tracing. A voxel‐based hierarchical level‐of‐detail (LOD) framework is employed to minimize rendering time and required system memory. In a pre‐processing phase, a compressed out‐of‐core data structure is constructed, which contains the original primitives of the model and the LOD voxels, organized into a kd‐tree. During rendering, data is loaded asynchronously to ensure a smooth inspection of the model regardless of the available I/O bandwidth. With our technique, we are able to explore data sets consisting of hundreds of millions of triangles in real‐time on a desktop PC with a quad‐core CPU.