End-to-end Sampling Patterns

Sample patterns have many uses in Computer Graphics, ranging from procedural object placement over Monte Carlo image synthesis to non-photorealistic depiction. Their properties such as discrepancy, spectra, anisotropy, or progressiveness have been analyzed extensively. However, designing methods to produce sampling patterns with certain properties can require substantial hand-crafting effort, both in coding, mathematical derivation and compute time. In particular, there is no systematic way to derive the best sampling algorithm for a specific end-task. Tackling this issue, we suggest another level of abstraction: a toolkit to end-to-end optimize over all sampling methods to find the one producing user-prescribed properties such as discrepancy or a spectrum that best fit the end-task. A user simply implements the forward losses and the sampling method is found automatically -- without coding or mathematical derivation -- by making use of back-propagation abilities of modern deep learning frameworks. While this optimization takes long, at deployment time the sampling method is quick to execute as iterated unstructured non-linear filtering using radial basis functions (RBFs) to represent high-dimensional kernels. Several important previous methods are special cases of this approach, which we compare to previous work and demonstrate its usefulness in several typical Computer Graphics applications. Finally, we propose sampling patterns with properties not shown before, such as high-dimensional blue noise with projective properties

Projections of determinantal point processes

Let $\mathbf x=\{x^{(1)},\dots,x^{(n)}\}$ be a space filling-design of $n$ points defined in $[0{,}1]^d$. In computer experiments, an important property seeked for $\mathbf x$ is a nice coverage of $[0{,}1]^d$. This property could be desirable as well as for any projection of $\mathbf x$ onto $[0{,}1]^\iota$ for $\iota<d$ . Thus we expect that $\mathbf x_I=\{x_I^{(1)},\dots,x_I^{(n)}\}$, which represents the design $\mathbf x$ with coordinates associated to any index set $I\subseteq\{1,\dots,d\}$, remains regular in $[0{,}1]^\iota$ where $\iota$ is the cardinality of $I$. This paper examines the conservation of nice coverage by projection using spatial point processes, and more specifically using the class of determinantal point processes. We provide necessary conditions on the kernel defining these processes, ensuring that the projected point process $\mathbf{X}_I$ is repulsive, in the sense that its pair correlation function is uniformly bounded by 1, for all $I\subseteq\{1,\dots,d\}$. We present a few examples, compare them using a new normalized version of Ripley's function. Finally, we illustrate the interest of this research for Monte-Carlo integration

Evolution of galaxies due to self-excitation

These lectures will cover methods for studying the evolution of galaxies since their formation. Because the properties of a galaxy depend on its history, an understanding of galaxy evolution requires that we understand the dynamical interplay between all components. The first part will emphasize n-body simulation methods which minimize sampling noise. These techniques are based on harmonic expansions and scale linearly with the number of bodies, similar to Fourier transform solutions used in cosmological simulations. Although fast, until recently they were only efficiently used for small number of geometries and background profiles. These same techniques may be used to study the modes and response of a galaxy to an arbitrary perturbation. In particular, I will describe the modal spectra of stellar systems and role of damped modes which are generic to stellar systems in interactions and appear to play a significant role in determining the common structures that we see. The general development leads indirectly to guidelines for the number of particles necessary to adequately represent the gravitational field such that the modal spectrum is resolvable. I will then apply these same excitation to understanding the importance of noise to galaxy evolution.Comment: 24 pages, 7 figures, using Sussp.sty (included). Lectures presented at the NATO Advanced Study Institute, "The Restless Universe: Applications of Gravitational N-Body Dynamics to Planetary, Stellar and Galactic Systems," Blair Atholl, July 200

Objective Classification of Galaxy Spectra using the Information Bottleneck Method

A new method for classification of galaxy spectra is presented, based on a recently introduced information theoretical principle, the `Information Bottleneck'. For any desired number of classes, galaxies are classified such that the information content about the spectra is maximally preserved. The result is classes of galaxies with similar spectra, where the similarity is determined via a measure of information. We apply our method to approximately 6000 galaxy spectra from the ongoing 2dF redshift survey, and a mock-2dF catalogue produced by a Cold Dark Matter-based semi-analytic model of galaxy formation. We find a good match between the mean spectra of the classes found in the data and in the models. For the mock catalogue, we find that the classes produced by our algorithm form an intuitively sensible sequence in terms of physical properties such as colour, star formation activity, morphology, and internal velocity dispersion. We also show the correlation of the classes with the projections resulting from a Principal Component Analysis.Comment: submitted to MNRAS, 17 pages, Latex, with 14 figures embedde