A Theory for Multiresolution Signal Decomposition: The Wavelet Representation

It is now well admitted in the computer vision literature that a multi-resolution decomposition provides a useful image representation for vision algorithms. In this paper we show that the wavelet theory recently developed by the mathematician Y. Meyer enables us to understand and model the concepts of resolution and scale. In computer vision we generally do not want to analyze the images at each resolution level because the information is redundant. After processing the signal at a resolution r0, it is more efficient to analyze only the additional details which are available at a higher resolution rl. We prove that this difference of information can be computed by decomposing the signal on a wavelet orthonormal basis and that it can be efficiently calculated with a pyramid transform. This can also be interpreted as a division of the signal in a set of orientation selective frequency channels. Such a decomposition is particularly well adapted for computer vision applications such as signal coding, texture discrimination, edge detection, matching algorithms and fractal analysis

Dyadic Wavelets Energy Zero-Crossings

An important problem in signal analysis is to define a general purpose signal representation which is well adapted for developing pattern recognition algorithms. In this paper we will show that such a representation can be defined from the position of the zero-crossings and the local energy values of a dyadic wavelet decomposition. This representation is experimentally complete and admits a simple distance for pattern matching applications. It provides a multiscale decomposition of the signal and at each scale characterizes the locations of abrupt changes in the signal. We have developed a stereo matching algorithm to illustrate the application of this representation to pattern matching

The quijote simulations

The Quijote simulations are a set of 44,100 full N-body simulations spanning more than 7000 cosmological models in the hyperplane. At a single redshift, the simulations contain more than 8.5 trillion particles over a combined volume of 44,100 each simulation follows the evolution of 2563, 5123, or 10243 particles in a box of 1 h -1 Gpc length. Billions of dark matter halos and cosmic voids have been identified in the simulations, whose runs required more than 35 million core hours. The Quijote simulations have been designed for two main purposes: (1) to quantify the information content on cosmological observables and (2) to provide enough data to train machine-learning algorithms. In this paper, we describe the simulations and show a few of their applications. We also release the petabyte of data generated, comprising hundreds of thousands of simulation snapshots at multiple redshifts; halo and void catalogs; and millions of summary statistics, such as power spectra, bispectra, correlation functions, marked power spectra, and estimated probability density functions

Matching pursuits with time-frequency dictionaries

Abstract-We introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions a matching pursuit defines an adaptive time-frequency transform. We derive a signal energy distribution in the time-frequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit isolates the signal structures that are coherent with respect to a given dictionary. An application to pattern extraction from noisy signals is described. We compare a matching pursuit decomposition with a signal expansion over an optimized wavepacket orthonormal basis, selected with the algorithm of Coifman and Wickerhauser. I