💻 Giuliano Ciccarese

Let me use this great card Fattoretto Agency | SEO e AI per eCommerce prepared for me to remind all the SEO nerds close to Milan 🇮🇹 that next week I'll be a speaker at their Summit dedicated to the future of the industry. 💃 #seo

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Unit Disk Uniform Sampling https://ift.tt/3QezM2o Discover the optimal transformations to apply on the standard [0,1] uniform random generator for uniformly sampling a 2D disk Photo by Vitolda Klein on Unsplash 0. Introduction In my previous article on Unit Disk and 2D Bounded KDE, I discussed the importance of being able to sample arbitrary distributions. This is especially relevant for applications like Monte Carlo integration, which is used to solve complex integrals, such as light scattering in Physically Based Rendering (PBRT). Sampling in 2D introduces new challenges compared to 1D. This article focuses on uniformly sampling the 2D unit disk and visualizing how transformations applied to a standard [0,1] uniform random generator create different distributions. We’ll also explore how these transformations, though yielding the same distribution, affect Monte Carlo integration by introducing distortion, leading to increased variance. 1. How to sample a disk uniformly? Introduction Random number generators often offer many predefined sampling distributions. However, for highly specific distributions, you’ll likely need to create your own. This involves combining and transforming basic distributions to achieve the desired outcome. For instance, to uniformly sample the interval between a and b, you can apply an affine transform on the standard uniform sampling from [0,1]. In this article, we’ll explore how to uniformly sample points within the 2D unit disk by building on the basic [0,1] uniform sampling. For readability, I’ve intentionally used the adjective “unit” in two different contexts in this article. The “unit square” refers to the [0,1]² domain, reflecting the range of the basic random generator. Conversely, the “unit disk” is described within [-1,1]² for convenience in polar coordinates. In practice, we can easily map between these using an affine transformation. We’ll denote u and v as samples drawn from the standard [0,1] or [-1,1] uniform distribution. Rejection Sampling Using the [0,1] uniform sampling twice allows us to uniformly sample the unit square [0,1]². A very simple approach known as “Rejection Sampling” consists in sampling the unit square and rejecting any sample falling outside the disk. Rejection sampling of the inner disk inside the unit square. Valid (green) and invalid (red) samples - Figure by the author This results in points that follow a uniform 2D distribution within the disk contained in the unit square, as illustrated in the figure below. The density maps in this article are generated by sampling many points from the specified distribution and then applying a Kernel Density Estimator. Methods for addressing boundary bias in the density are detailed in the previous article “Unit Disk and 2D Bounded KDE”. Left: Density of the 2D Disk Rejection Sampling estimated by Kernel Density Estimation on 10000 samples. Right: Corresponding 1D density profile along the...

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