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Handbook of Astronomical Data Analysis - …

Jean-Luc Starck and Fionn Murtagh Handbook of Astronomical data Analysis Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest Table of Contents Contents .. i Preface .. vii 1. Introduction to Applications and Methods .. 1. Introduction .. 1. Transformation and data Representation .. 4. Fourier Analysis .. 5. Time-Frequency Representation .. 6. Time-Scale Representation: The Wavelet Transform .. 8. The Radon Transform .. 12. Mathematical Morphology .. 12. Edge Detection .. 15. First Order Derivative Edge Detection .. 16. Second Order Derivative Edge Detection .. 19. Segmentation .. 20. Pattern Recognition .. 21. Chapter Summary .. 25. 2. Filtering.

Preface When we consider the ever increasing amount of astronomical data available to us, we can well say that the needs of modern astronomy are growing by

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1 Jean-Luc Starck and Fionn Murtagh Handbook of Astronomical data Analysis Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest Table of Contents Contents .. i Preface .. vii 1. Introduction to Applications and Methods .. 1. Introduction .. 1. Transformation and data Representation .. 4. Fourier Analysis .. 5. Time-Frequency Representation .. 6. Time-Scale Representation: The Wavelet Transform .. 8. The Radon Transform .. 12. Mathematical Morphology .. 12. Edge Detection .. 15. First Order Derivative Edge Detection .. 16. Second Order Derivative Edge Detection .. 19. Segmentation .. 20. Pattern Recognition .. 21. Chapter Summary .. 25. 2. Filtering.

2 27. Introduction .. 27. Multiscale Transforms .. 29. The A Trous Isotropic Wavelet Transform .. 29. Multiscale Transforms Compared to Other data Trans- forms .. 30. Choice of Multiscale Transform .. 33. The Multiresolution Support .. 34. Signi cant Wavelet Coe cients .. 36. De nition .. 36. Noise Modeling .. 37. Automatic Estimation of Gaussian Noise .. 37. Filtering and Wavelet Coe cient Thresholding .. 46. Thresholding .. 46. Iterative Filtering .. 47. ii Table of Contents Experiments .. 48. Iterative Filtering with a Smoothness Constraint .. 51. Haar Wavelet Transform and Poisson Noise .. 52. Haar Wavelet Transform .. 52. Poisson Noise and Haar Wavelet Coe cients .. 53. Experiments.

3 56. Chapter Summary .. 59. 3. Deconvolution .. 61. Introduction .. 61. The Deconvolution Problem .. 62. Linear Regularized Methods .. 65. Least Squares Solution .. 65. Tikhonov Regularization .. 65. Generalization .. 66. CLEAN .. 67. Bayesian Methodology .. 68. De nition .. 68. Maximum Likelihood with Gaussian Noise .. 68. Gaussian Bayes Model .. 69. Maximum Likelihood with Poisson Noise .. 69. Poisson Bayes Model .. 70. Maximum Entropy Method .. 70. Other Regularization Models .. 71. Iterative Regularized Methods .. 72. Constraints .. 72. Jansson-Van Cittert Method .. 73. Other iterative methods .. 73. Wavelet-Based Deconvolution .. 74. Introduction .. 74. Wavelet-Vaguelette Decomposition.

4 75. Regularization from the Multiresolution Support .. 77. Wavelet CLEAN .. 81. Multiscale Entropy .. 86. Deconvolution and Resolution .. 88. Super-Resolution .. 89. De nition .. 89. Gerchberg-Saxon Papoulis Method .. 89. Deconvolution with Interpolation .. 90. Undersampled Point Spread Function .. 91. Multiscale Support Constraint .. 92. Conclusions and Chapter Summary .. 92. Table of Contents iii 4. Detection .. 95. Introduction .. 95. From Images to Catalogs .. 96. Multiscale Vision Model .. 100. Introduction .. 100. Multiscale Vision Model De nition .. 101. From Wavelet Coe cients to Object Identi cation .. 101. Partial Reconstruction .. 104. Examples .. 105. Application to ISOCAM data Calibration.

5 109. Detection and Deconvolution .. 113. Conclusion .. 115. Chapter Summary .. 116. 5. Image Compression .. 117. Introduction .. 117. Lossy Image Compression Methods .. 119. The Principle .. 119. Compression with Pyramidal Median Transform .. 120. PMT and Image Compression .. 122. Compression Packages .. 125. Remarks on these Methods .. 126. Comparison .. 128. Quality Assessment .. 128. Visual Quality .. 129. First Aladin Project Study .. 132. Second Aladin Project Study .. 134. Computation Time .. 139. Conclusion .. 140. Lossless Image Compression .. 141. Introduction .. 141. The Lifting Scheme .. 141. Comparison .. 145. Large Images: Compression and Visualization .. 146. Large Image Visualization Environment: LIVE.

6 146. Decompression by Scale and by Region .. 147. The SAO-DS9 LIVE Implementation .. 149. Chapter Summary .. 150. 6. Multichannel data .. 153. Introduction .. 153. The Wavelet-Karhunen-Lo`eve Transform .. 153. De nition .. 153. Correlation Matrix and Noise Modeling .. 154. Scale and Karhunen-Lo`eve Transform .. 156. iv Table of Contents The WT-KLT Transform .. 156. The WT-KLT Reconstruction Algorithm .. 157. Noise Modeling in the WT-KLT Space .. 157. Multichannel data Filtering .. 158. Introduction .. 158. Reconstruction from a Subset of Eigenvectors .. 158. WT-KLT Coe cient Thresholding .. 160. Example: Astronomical Source Detection .. 160. The Haar-Multichannel Transform .. 160.

7 Independent Component Analysis .. 161. Chapter Summary .. 162. 7. An Entropic Tour of Astronomical data Analysis .. 165. Introduction .. 165. The Concept of Entropy .. 168. Multiscale Entropy .. 174. De nition .. 174. Signal and Noise Information .. 176. Multiscale Entropy Filtering .. 179. Filtering .. 179. The Regularization Parameter .. 179. Use of a Model .. 181. The Multiscale Entropy Filtering Algorithm .. 182. Optimization .. 183. Examples .. 184. Deconvolution .. 188. The Principle .. 188. The Parameters .. 189. Examples .. 189. Multichannel data Filtering .. 190. Background Fluctuation Analysis .. 192. Relevant Information in an Image .. 195. Multiscale Entropy and Optimal Compressibility.

8 195. Conclusions and Chapter Summary .. 196. 8. Astronomical Catalog Analysis .. 201. Introduction .. 201. Two-Point Correlation Function .. 202. Introduction .. 202. Determining the 2-Point Correlation Function .. 203. Error Analysis .. 204. Correlation Length Determination .. 205. Creation of Random Catalogs .. 205. Examples .. 206. Fractal Analysis .. 211. Table of Contents v Introduction .. 211. The Hausdor and Minkowski Measures .. 212. The Hausdor and Minkowski Dimensions .. 212. Multifractality .. 213. Generalized Fractal Dimension .. 214. Wavelet and Multifractality .. 215. Spanning Trees and Graph Clustering .. 220. Voronoi Tessellation and Percolation .. 221. Model-Based Clustering.

9 222. Modeling of Signal and Noise .. 222. Application to Thresholding .. 224. Wavelet Analysis .. 224. Nearest Neighbor Clutter Removal .. 225. Chapter Summary .. 226. 9. Multiple Resolution in data Storage and Retrieval .. 229. Introduction .. 229. Wavelets in Database Management .. 229. Fast Cluster Analysis .. 231. Nearest Neighbor Finding on Graphs .. 233. Cluster-Based User Interfaces .. 234. Images from data .. 235. Matrix Sequencing .. 235. Filtering Hypertext .. 239. Clustering Document-Term data .. 240. Chapter Summary .. 245. 10. Towards the Virtual Observatory .. 247. data and Information .. 247. The Information Handling Challenges Facing Us .. 249. References .. 250.

10 Appendix A: A Trous Wavelet Transform .. 269. Appendix B: Picard Iteration .. 275. Appendix C: Wavelet Transform using the Fourier Transform 277. Appendix D: Derivative Needed for the Minimization .. 281. Appendix E: Generalization of the Derivative Needed for the Minimization .. 285. Appendix F: Software and Related Developments .. 287. vi Table of Contents Index .. 289. Preface When we consider the ever increasing amount of Astronomical data available to us, we can well say that the needs of modern astronomy are growing by the day. Ever better observing facilities are in operation. The fusion of infor- mation leading to the coordination of observations is of central importance.


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