## J ivei2iei2 d

Left image with two lines and Gaussian noise. Right its Radon transform. Fig. 1.5. Left image with two lines and Gaussian noise. Right its Radon transform. which holds if, say, has a vanishing mean J (t)dt 0. We will suppose that is normalized so that J V> ( ) 2 -2d 1. For each a > 0, each b G R and each 0 G 0, 2n , we define the bivariate ridgelet a,b,e R2 R by a, b, e(x) a-1 2 ((x1 cos 0 + x2 sin 0 b) a). (1.25) Given an integrable bivariate function f (x), we define its...

## Djk d21fc

Alternative methods, but based on the same concept, can be found in (Crouse et al., 1998 Moulin and Liu, 1999 Sendur and Selesnik, 2002 Portilla et al., 2003 Kazubek, 2003). Simulation 1 1D Signal Filtering. Fig. 2.12 shows the result after applying the iterative filtering method to a real spectrum. The last plot shows the difference between the original and the filtered spectrum. As we can see, the residual contains only noise. Fig. 2.12. Top real spectrum and filtered spectrum. Bottom both...

Wavelets come into play as a sparsifying transform. Applying a wavelet transform on both sides of (6.18) does not affect the mixing matrix and the model 2 A super-Gaussian distribution is also called a lepto-kurtic distribution, referring to a distribution with a narrow central peak and heavy tails. A typical example is the Laplacian distribution. structure is preserved. Also, moving the data to a wavelet representation does not affect its information content. However, the statistical...

## Introduction

From year to year, the quantity of astronomical data increases at an ever growing rate. In part this is due to very large digitized sky surveys in the optical and near infrared, which in turn is due to the development of digital imaging arrays such as CCDs (charge-coupled devices). The size of digital arrays is also continually increasing, pushed by the demands of astronomical research for ever larger quantities of data in ever shorter time periods. Currently, projects such as the European...

## Voronoi Tessellation and Percolation

A Voronoi tessellation constructs a convex Voronoi cell around each occupied pixel and assigns fluxes to them based on the number of photons in the pixel, the cell area, and the effective exposure time at the pixel location. This of course is an imaging perspective, and mutatis mutandis we may consider continuous spaces, and unit regions. Thus background photons have large cells and low fluxes associated with them, whereas source photons are characterized by small cells and high fluxes. The...

## Limitation of the Two Point Correlation Function Toward Higher Moments

In order to illustrate the limitation of the two-point correlation function, we use two simulated data sets. The first one is a simulation from stochastic geometry. It is based on a Voronoi model. The second one is a mock catalog of the galaxy distribution drawn from a -CDM N-body cosmological model Fig. 8.6. Left the correlation function of the CDM model for linear separation bins. The fit is carried out in 1-50 Mpc separations and the error bars are the standard deviations of 5 random catalog...

## Software Public Domain and Commercial

An excellent first stop is at AstroWeb. Many aspects of astronomy are included, including software. Accessible from cdsweb.u-strasbg.fr astroweb.html, there are nearly 200 links to astronomy software resources. Links to large software packages are included in these lists ESO-MIDAS, AIPS, IRAF, the Starlink Software Collection, the IDL Astronomy User's Library, and information on the FITS storage standard. Statistical software is available at www.astro.psu.edu statcodes. For statistics software...

## Two Point Correlation Function 821 Introduction

The two-point correlation function (r) has been the primary tool for quantifying large-scale cosmic structure (Peebles, 1980). Assuming that the galaxy distribution in the Universe is a realization of a stationary and isotropic random process, the two-point correlation function can be defined from the probability 5P of finding an object within a volume element 5V at distance r from a randomly chosen object or position inside the volume where n is the mean density of objects. The function (r)...

## Ji

Where e is a small number (for example e 10 3). Fig. 2.4 shows a Hale-Bopp Comet image (logarithmic representation) (top left), its histogram equaliza- Fig. 2.2. Wavelet transform of NGC 2997 by the a trous algorithm. Fig. 2.2. Wavelet transform of NGC 2997 by the a trous algorithm. tion (middle row), and its wavelet-log representation (bottom). Jets clearly appear in the last representation of the Hale-Bopp Comet image. 2.2.2 Multiscale Transforms Compared to Other Data Transforms In this...

## Relevant Information in an Image

Since the multiscale entropy extracts the information from the signal only, it was a challenge to see if the astronomical content of an image was related to its multiscale entropy. For this purpose, we used the astronomical content of 200 images of 1024 x 1024 pixels extracted from scans of 8 different photographic plates carried out by the MAMA digitization facility (Paris, France) (Guibert, 1992) and stored at CDS (Strasbourg, France) in the Aladin archive (Bonnarel et al., 1999). We...

## Pc Oxy 0Oxy fDie332

Band-limited the Fourier transform of the object belongs to a given frequency domain. For instance, if Fc is the cut-off frequency of the instrument, we want to impose on the object that it be band-limited These constraints can be incorporated easily in the basic iterative scheme. 3.6.2 Jansson-Van Cittert Method Van Cittert (1931) restoration is relatively easy to write. We start with n 0 and O(0) I and we iterate On+1 n + a(I P * On) (3.34) where a is a convergence parameter generally taken...

## Astronomical Image and Data Analysis

Jean-Luc Starck Service d'Astrophysique CEA Saclay Orme des Merisiers, Bat 709 91191 Gif-sur-Yvette Cedex, France Dept. Computer Science Royal Holloway University of London Egham, Surrey TW20 0EX, UK Cover picture The cover image to this 2nd edition is from the Deep Impact project. It was taken approximately 8 minutes after impact on 4 July 2005 with the CLEAR6 filter and deconvolved using the Richardson-Lucy method. We thank Don Lindler, Ivo Busko, Mike A' Hearn and the Deep Impact team for...

## The Concept of Entropy

We wish to estimate an unknown probability density p(X) of the data. Shannon (1948), in the framework of information theory, defined the entropy of an image X by where X Xi, XN is an image containing integer values, Nb is the number of possible values of a given pixel Xk (256 for an 8-bit image), and the Pk values are derived from the histogram of X Xj k gives the number of pixels with value k, i.e., Xj k. If the image contains floating values, it is possible to build up the histogram L of...