Sampling is a core component of many applications including: imaging, rendering, geometry processing and visualization. Prior research has primarily focused on blue noise sampling with a single class of samples. Limited research has been done on multi-class sampling. Multi-class blue noise sampling is still a challenging problem when the density distribution of every class is non-uniform and different from each other. In this paper, we present a Wasserstein blue noise sampling algorithm for multi-class sampling by throwing samples as the Wasserstein barycenter of multiple density distributions. We employ a more general representation of the optimal transport problem to break up the partition necessary in other optimal sampling. Moreover, an adaptive blue noise distribution property for every individual class is guaranteed, as well as their combined class. The sampling efficiency is also improved by applying the Wasserstein distance with entropic constraints. Our method can be applied to multi-class sampling on the point set surface. We also demonstrate applications in object distribution and color stippling.
We present a fast, novel image-based technique, for reverse engineering the Bidirectional Texture Function (BTF) of woven fabrics. In order to recover our models, we estimate a depth map and a set of yarn parameters (yarn width, yarn crossovers and so on) from spatial and frequency domain cues. We solve for the woven fabric pattern, and from this build a volumetric data set. We use a combination of image space analysis, frequency domain analysis and in challenging cases match image statistics with those from previously captured known patterns. Our method determines, from a single digital image, the woven cloth structure, depth and albedo, thus removing the need for separately measured depth data. The focus of this work is on the rapid acquisition of woven cloth structure and therefore we use standard approaches to render the results. Our pipeline first estimates the weave pattern, yarn characteristics, albedo and noise statistics using a novel combination of low level image processing and Fourier Analysis. Next, we estimate a depth map for the fabric sample using a first order Markov chain and our estimated noise model as input. A volumetric BTF model is constructed from the recovered depth and albedo maps.
Many graphics and vision problems are naturally expressed as optimizations with either linear or non-linear least squares objective functions over visual data, such as images and meshes. The mathematical descriptions of these functions are extremely concise, but their implementation in real code is tedious, especially when optimized for real-time performance in interactive applications. We propose a new language, Opt in which a user simply writes energy functions over image- or graph-structured unknowns, and a compiler automatically generates state-of-the-art GPU optimization kernels. The end result is a system in which real-world energy functions in graphics and vision applications are expressible in tens of lines of code. They compile directly into highly-optimized GPU solver implementations with performance competitive with the best published hand-tuned, application-specific GPU solvers, and 1--2 orders of magnitude beyond a general-purpose auto-generated solver.
Implicitizing rational surfaces is a fundamental computational task in Algorithmic Algebraic Geometry. Although the resultant of three moving planes corresponding to a ¼-basis for a rational surface is guaranteed to contain the implicit equation of the surface as a factor, ¼-bases for rational surfaces are difficult to compute. Moreover, ¼-bases often have high degrees, so these resultants generally contain many extraneous factors. Here we develop fast algorithms to implicitize rational tensor product surfaces by computing the resultant of three moving planes corresponding to three syzygies with low degrees. These syzygies are easy to compute, and the resultants of the corresponding moving planes generally contain fewer extraneous factors than the resultants of the moving planes corresponding to ¼-bases. We predict and compute all the possible extraneous factors that may appear in these resultants. Examples are provided to clarify and illuminate the theory.