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

A Theory of Content

International audienceSelections are central to image editing, e.g., they are the starting point of common operations such as copy-pasting and local edits. Creating them by hand is particularly tedious and scribble-based techniques have been introduced to assist the process. By interpolating a few strokes specified by users, these methods generate precise selections. However, most of the algorithms assume a 100% accurate input, and even small inaccuracies in the scribbles often degrade the selection quality, which imposes an additional burden on users. In this paper, we propose a selection technique tolerant to input inaccuracies. We use a dense conditional random field (CRF) to robustly infer a selection from possibly inaccurate input. Further, we show that patch-based pixel similarity functions yield more precise selection than simple point-wise metrics. However, efficiently solving a dense CRF is only possible in low-dimensional Euclidean spaces, and the metrics that we use are high-dimensional and often non-Euclidean. We address this challenge by embedding pixels in a low-dimensional Euclidean space with a metric that approximates the desired similarity function. The results show that our approach performs better than previous techniques and that two options are sufficient to cover a variety of images depending on whether the objects are textured.L'opération de sélection est essentielle en traitement d'images, bien que fastidieuse à effectuer a la main. Des techniques à base de marques ont été développées. Cependant celles ci supposent que l'utilisateur fournit des marques exactes a 100%, ce qui impose à l'utilisateur des contraintes supplémentaires. Dans ce papier, nous proposons une technique de sélection qui est robuste par rapport à la qualite des marques fournies par l'utilisateur. Nous utilisons un "conditional random field" dense pour inferrer une selection binaire, de manière robuste à partir de l'entrée utilisateur. Comme ce choix impose une dimensionalité faible des espaces de travail, ainsi qu'une metrique euclidienne, nous utilisons une réduction de dimension empirique basée sur la fonction de similarité entre pixels. Nos résultats montrent qu'a entrée égale, notre méthode surpasse les travaux existants