Fourier Analysis of Stochastic Sampling Strategies for Assessing Bias and Variance in Integration

Each pixel in a photorealistic, computer generated picture is calculated by approximately integrating all the light arriving at the pixel, from the virtual scene. A common strategy to calculate these high-dimensional integrals is to average the estimates at stochastically sampled locations. The strategy with which the sampled locations are chosen is of utmost importance in deciding the quality of the approximation, and hence rendered image. We derive connections between the spectral properties of stochastic sampling patterns and the first and second order statistics of estimates of integration using the samples. Our equations provide insight into the assessment of stochastic sampling strategies for integration. We show that the amplitude of the expected Fourier spectrum of sampling patterns is a useful indicator of the bias when used in numerical integration. We deduce that estimator variance is directly dependent on the variance of the sampling spectrum over multiple realizations of the sampling pattern. We then analyse Gaussian jittered sampling, a simple variant of jittered sampling, that allows a smooth trade-off of bias for variance in uniform (regular grid) sampling. We verify our predictions using spectral measurement, quantitative integration experiments and qualitative comparisons of rendered images.</jats:p

k-d Darts: Sampling by k-Dimensional Flat Searches

We formalize the notion of sampling a function using k-d darts. A k-d dart is a set of independent, mutually orthogonal, k-dimensional subspaces called k-d flats. Each dart has d choose k flats, aligned with the coordinate axes for efficiency. We show that k-d darts are useful for exploring a function's properties, such as estimating its integral, or finding an exemplar above a threshold. We describe a recipe for converting an algorithm from point sampling to k-d dart sampling, assuming the function can be evaluated along a k-d flat. We demonstrate that k-d darts are more efficient than point-wise samples in high dimensions, depending on the characteristics of the sampling domain: e.g. the subregion of interest has small volume and evaluating the function along a flat is not too expensive. We present three concrete applications using line darts (1-d darts): relaxed maximal Poisson-disk sampling, high-quality rasterization of depth-of-field blur, and estimation of the probability of failure from a response surface for uncertainty quantification. In these applications, line darts achieve the same fidelity output as point darts in less time. We also demonstrate the accuracy of higher dimensional darts for a volume estimation problem. For Poisson-disk sampling, we use significantly less memory, enabling the generation of larger point clouds in higher dimensions.Comment: 19 pages 16 figure

Gap Processing for Adaptive Maximal Poisson-Disk Sampling

In this paper, we study the generation of maximal Poisson-disk sets with varying radii. First, we present a geometric analysis of gaps in such disk sets. This analysis is the basis for maximal and adaptive sampling in Euclidean space and on manifolds. Second, we propose efficient algorithms and data structures to detect gaps and update gaps when disks are inserted, deleted, moved, or have their radius changed. We build on the concepts of the regular triangulation and the power diagram. Third, we will show how our analysis can make a contribution to the state-of-the-art in surface remeshing.Comment: 16 pages. ACM Transactions on Graphics, 201

The Evolution of the Number Density of Large Disk Galaxies in COSMOS

We study a sample of approximately 16,500 galaxies with I_(ACS,AB) ≤ 22.5 in the central 38% of the COSMOS field, which are extracted from a catalog constructed from the Cycle 12 ACS F814W COSMOS data set. Structural information on the galaxies is derived by fitting single Sérsic models to their two-dimensional surface brightness distributions. In this paper we focus on the disk galaxy population (as classified by the Zurich Estimator of Structural Types), and investigate the evolution of the number density of disk galaxies larger than approximately 5 kpc between redshift z ~ 1 and the present epoch. Specifically, we use the measurements of the half-light radii derived from the Sérsic fits to construct, as a function of redshift, the size function Φ(r_(1/2), z) of both the total disk galaxy population and of disk galaxies split in four bins of bulge-to-disk ratio. In each redshift bin, the size function specifies the number of galaxies per unit comoving volume and per unit half-light radius r_(1/2). Furthermore, we use a selected sample of roughly 1800 SDSS galaxies to calibrate our results with respect to the local universe. We find the following: (1) The number density of disk galaxies with intermediate sizes (r_(1/2) ~ 5-7 kpc) remains nearly constant from z ~ 1 to today. Unless the growth and destruction of such systems exactly balanced in the last eight billion years, they must have neither grown nor been destroyed over this period. (2) The number density of the largest disks (r_(1/2) > 7 kpc) decreases by a factor of about 2 out to z ~ 1. (3) There is a constancy—or even slight increase—in the number density of large bulgeless disks out to z ~ 1; the deficit of large disks at early epochs seems to arise from a smaller number of bulged disks. Our results indicate that the bulk of the large disk galaxy population has completed its growth by z ~ 1 and support the theory that secular evolution processes produce—or at least add stellar mass to—the bulge components of disk galaxies