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 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Transformation and Data Representation . . . . . . . . . . . . . . . . . . 4 1.2.1 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Time-Frequency Representation . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Time-Scale Representation: The Wavelet Transform . . 8 1.2.4 The Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Mathematical Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Edge Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1 First Order Derivative Edge Detection . . . . . . . . . . . . . . 16 1.4.2 Second Order Derivative Edge Detection . . . . . . . . . . . . 19 1.5 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Pattern Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2. Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Multiscale Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.1 The A Trous Isotropic Wavelet Transform . . . . . . . . . . . 29 2.2.2 Multiscale Transforms Compared to Other Data Trans- forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.3 Choice of Multiscale Transform . . . . . . . . . . . . . . . . . . . . 33 2.2.4 The Multiresolution Support . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 Significant Wavelet Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.2 Noise Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.3 Automatic Estimation of Gaussian Noise . . . . . . . . . . . . 37 2.4 Filtering and Wavelet Coefficient Thresholding . . . . . . . . . . . . . 46 2.4.1 Thresholding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4.2 Iterative Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ii Table of Contents 2.4.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4.4 Iterative Filtering with a Smoothness Constraint . . . . . 51 2.5 Haar Wavelet Transform and Poisson Noise . . . . . . . . . . . . . . . . 52 2.5.1 Haar Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5.2 Poisson Noise and Haar Wavelet Coefficients . . . . . . . . . 53 2.5.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3. Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 The Deconvolution Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3 Linear Regularized Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.1 Least Squares Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.2 Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4 CLEAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.5 Bayesian Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5.2 Maximum Likelihood with Gaussian Noise . . . . . . . . . . . 68 3.5.3 Gaussian Bayes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5.4 Maximum Likelihood with Poisson Noise . . . . . . . . . . . . 69 3.5.5 Poisson Bayes Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.5.6 Maximum Entropy Method . . . . . . . . . . . . . . . . . . . . . . . . 70 3.5.7 Other Regularization Models . . . . . . . . . . . . . . . . . . . . . . . 71 3.6 Iterative Regularized Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.6.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.6.2 Jansson-Van Cittert Method . . . . . . . . . . . . . . . . . . . . . . . 73 3.6.3 Other iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.7 Wavelet-Based Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.7.2 Wavelet-Vaguelette Decomposition . . . . . . . . . . . . . . . . . 75 3.7.3 Regularization from the Multiresolution Support . . . . . 77 3.7.4 Wavelet CLEAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.7.5 Multiscale Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.8 Deconvolution and Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.9 Super-Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.9.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.9.2 Gerchberg-Saxon Papoulis Method . . . . . . . . . . . . . . . . . 89 3.9.3 Deconvolution with Interpolation . . . . . . . . . . . . . . . . . . . 90 3.9.4 Undersampled Point Spread Function . . . . . . . . . . . . . . . 91 3.9.5 Multiscale Support Constraint . . . . . . . . . . . . . . . . . . . . . 92 3.10 Conclusions and Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 92 Table of Contents iii 4. Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 From Images to Catalogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3 Multiscale Vision Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.3.2 Multiscale Vision Model Definition . . . . . . . . . . . . . . . . . 101 4.3.3 From Wavelet Coefficients to Object Identification . . . . 101 4.3.4 Partial Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.3.6 Application to ISOCAM Data Calibration . . . . . . . . . . . 109 4.4 Detection and Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5. Image Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Lossy Image Compression Methods . . . . . . . . . . . . . . . . . . . . . . . 119 5.2.1 The Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.2.2 Compression with Pyramidal Median Transform . . . . . 120 5.2.3 PMT and Image Compression . . . . . . . . . . . . . . . . . . . . . . 122 5.2.4 Compression Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.2.5 Remarks on these Methods . . . . . . . . . . . . . . . . . . . . . . . . 126 5.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.3.1 Quality Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.3.2 Visual Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.3.3 First Aladin Project Study . . . . . . . . . . . . . . . . . . . . . . . . 132 5.3.4 Second Aladin Project Study . . . . . . . . . . . . . . . . . . . . . . 134 5.3.5 Computation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.4 Lossless Image Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.4.2 The Lifting Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.4.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.5 Large Images: Compression and Visualization . . . . . . . . . . . . . . 146 5.5.1 Large Image Visualization Environment: LIVE . . . . . . . 146 5.5.2 Decompression by Scale and by Region . . . . . . . . . . . . . . 147 5.5.3 The SAO-DS9 LIVE Implementation . . . . . . . . . . . . . . . 149 5.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6. Multichannel Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.2 The Wavelet-Karhunen-Lo`eve Transform . . . . . . . . . . . . . . . . . . 153 6.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.2.2 Correlation Matrix and Noise Modeling . . . . . . . . . . . . . 154 6.2.3 Scale and Karhunen-Lo`eve Transform . . . . . . . . . . . . . . . 156 iv Table of Contents 6.2.4 The WT-KLT Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.2.5 The WT-KLT Reconstruction Algorithm . . . . . . . . . . . . 157 6.3 Noise Modeling in the WT-KLT Space . . . . . . . . . . . . . . . . . . . . 157 6.4 Multichannel Data Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.4.2 Reconstruction from a Subset of Eigenvectors . . . . . . . . 158 6.4.3 WT-KLT Coefficient Thresholding . . . . . . . . . . . . . . . . . . 160 6.4.4 Example: Astronomical Source Detection . . . . . . . . . . . . 160 6.5 The Haar-Multichannel Transform . . . . . . . . . . . . . . . . . . . . . . . . 160 6.6 Independent Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . 161 6.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7. An Entropic Tour of Astronomical Data Analysis . . . . . . . . . 165 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.2 The Concept of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.3 Multiscale Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.3.2 Signal and Noise Information . . . . . . . . . . . . . . . . . . . . . . 176 7.4 Multiscale Entropy Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.4.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.4.2 The Regularization Parameter . . . . . . . . . . . . . . . . . . . . . 179 7.4.3 Use of a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.4.4 The Multiscale Entropy Filtering Algorithm . . . . . . . . . 182 7.4.5 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.5 Deconvolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.5.1 The Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.5.2 The Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.6 Multichannel Data Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.7 Background Fluctuation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 192 7.8 Relevant Information in an Image . . . . . . . . . . . . . . . . . . . . . . . . 195 7.9 Multiscale Entropy and Optimal Compressibility . . . . . . . . . . . 195 7.10 Conclusions and Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 196 8. Astronomical Catalog Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.2 Two-Point Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.2.2 Determining the 2-Point Correlation Function . . . . . . . . 203 8.2.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.2.4 Correlation Length Determination . . . . . . . . . . . . . . . . . . 205 8.2.5 Creation of Random Catalogs . . . . . . . . . . . . . . . . . . . . . . 205 8.2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.3 Fractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Table of Contents v 8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.3.2 The Hausdorff and Minkowski Measures . . . . . . . . . . . . . 212 8.3.3 The Hausdorff and Minkowski Dimensions . . . . . . . . . . . 212 8.3.4 Multifractality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.3.5 Generalized Fractal Dimension . . . . . . . . . . . . . . . . . . . . . 214 8.3.6 Wavelet and Multifractality . . . . . . . . . . . . . . . . . . . . . . . . 215 8.4 Spanning Trees and Graph Clustering . . . . . . . . . . . . . . . . . . . . . 220 8.5 Voronoi Tessellation and Percolation . . . . . . . . . . . . . . . . . . . . . . 221 8.6 Model-Based Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.6.1 Modeling of Signal and Noise . . . . . . . . . . . . . . . . . . . . . . 222 8.6.2 Application to Thresholding . . . . . . . . . . . . . . . . . . . . . . . 224 8.7 Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 8.8 Nearest Neighbor Clutter Removal . . . . . . . . . . . . . . . . . . . . . . . . 225 8.9 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 9. Multiple Resolution in Data Storage and Retrieval . . . . . . . 229 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.2 Wavelets in Database Management . . . . . . . . . . . . . . . . . . . . . . . 229 9.3 Fast Cluster Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.4 Nearest Neighbor Finding on Graphs . . . . . . . . . . . . . . . . . . . . . . 233 9.5 Cluster-Based User Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 9.6 Images from Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 9.6.1 Matrix Sequencing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 9.6.2 Filtering Hypertext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 9.6.3 Clustering Document-Term Data . . . . . . . . . . . . . . . . . . . 240 9.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 10. Towards the Virtual Observatory . . . . . . . . . . . . . . . . . . . . . . . . . 247 10.1 Data and Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 10.2 The Information Handling Challenges Facing Us . . . . . . . . . . . . 249 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 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. The methods described in this book can provide effective and efficient ripostes to many of these issues. Much progress has been made in recent years on the methodology front, in line with the rapid pace of evolution of our technological infrastructures. The central themes of this book are information and scale. The approach is astronomy-driven, starting with real problems and issues to be addressed. We then proceed to comprehensive theory, and implementations of demonstrated efficacy. The field is developing rapidly. There is little doubt that further important papers, and books, will follow in the future. Colleagues we would like to acknowledge include: Alexandre Aussem, Al- bert Bijaoui, Fran¸cois Bonnarel, Jonathan G. Campbell, Ghada Jammal, Ren´e Gastaud, Pierre-Fran¸cois Honor´e, Bruno Lopez, Mireille Louys, Clive Page, Eric Pantin, Philippe Querre, Victor Racine, J´erˆome Rodriguez, and Ivan Valtchanov. The cover image is from Jean-Charles Cuillandre. It shows a five minute exposure (five 60-second dithered and stacked images), R filter, taken with CFH12K wide field camera (100 million pixels) at the primary focus of the CFHT in July 2000. The image is from an extremely rich zone of our Galaxy, containing star formation regions, dark nebulae (molecular clouds and dust regions), emission nebulae (H α ), and evolved stars which are scat- tered throughout the field in their two-dimensional projection effect. This zone is in the constellation of Saggitarius. Jean-Luc Starck Fionn Murtagh viii Preface [...]... spaces This is a key aspect of an intelligent streaming system A further area of importance for scientific data interpretation is that of storage and display Long-term storage of astronomical data, we have already noted, is part and parcel of our society’s memory (a formulation due to Michael Kurtz, Center for Astrophysics, Smithsonian Institute) With the rapid obsolescence of storage devices, considerable... effects of atmospheric blurring, or quality of seeing, certainly is of importance There will be a wide-ranging 1.1 Introduction – – – – – – 3 discussion of the state of the art in deconvolution in astronomy in Chapter 3 Compression: Consider three different facts Long-term storage of astronomical data is important A current trend is towards detectors accommodating ever-larger image sizes Research in astronomy... pivotal in Chapter 6 on multichannel data, 8 on catalog analysis, and 9 on data storage and retrieval Hough and Radon transforms, leading to 3D tomography and other applications: Detection of alignments and curves is necessary for many classes of segmentation and feature analysis, and for the building of 3D representations of data Gravitational lensing presents one area of potential application in astronomy... ever-increasing complementarity of professional observational astronomy with education and public outreach Astronomy’s data centers and image and catalog archives play an important role in our society’s collective memory For example, the SIMBAD database of astronomical objects at Strasbourg Observatory contains data on 3 million objects, based on 7.5 million object identifiers Constant updating of SIMBAD... wavelet space of many kind of noise – Poisson noise, combination of Gaussian and Poisson noise components, non-stationary noise, and so on – has been a key motivation for the use of wavelets in scientific, medical, or industrial applications The wavelet transform has also been extensively used in astronomical data analysis during the last ten years A quick search with ADS (NASA Astrophysics Data System,... object position In this schema we start off with raw data (an array of grey-levels) and we end up with information – the identification and position of an object As we progress, the data and processing move from low-level to high-level Haralick and Shapiro (1985) give the following wish-list for segmentation: “What should a good image segmentation be? Regions of an image segmentation should be uniform and... unsuitability of the term if it is composed of raw data It can be useful to classify feature extractors according to whether they are high- or low-level ′ A typical low-level feature extractor is a transformation IRp −→ IRp which, presumably, either enhances the separability of the classes, or, at 22 1 Introduction to Applications and Methods least, reduces the dimensionality of the data (p < p′ )... tractable, or simply to compress the data Many data compression schemes are used as feature extractors, and vice-versa Examples of low-level feature extractors are: – Fourier power spectrum of a signal – appropriate if frequency content is a good discriminator and, additionally, it has the property of shift invariance – Karhunen-Lo`ve transform – transforms the data to a space in which the e features... x-center of gravity of the object, and y = m01 /m00 ˜ gives the y-center of gravity Now we can obtain shift invariant features by referring all coordinates to the center of gravity (˜, y ) These are the central moments: x ˜ m′ = pq x y (x − x)p (y − y )q f (x, y) ˜ ˜ The first few m′ can be interpreted as follows: m′ = m00 = sum of the grey-levels in the object, 00 m′ = m′ = 0, always, i.e center of. .. morphological filters were used for removing “cirrus-like” emission from far-infrared extragalactic IRAS fields (Appleton et al., 1993), and for astronomical image compression (Huang and Bijaoui, 1991) The skeleton of an object in an image is a set of lines that reflect the shape of the object The set of skeletal pixels can be considered to be the medial axis of the object More details can be found in (Soille, . 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 wide-ranging 1.1 Introduction 3 discussion of the state of the art in deconvolution in astronomy in Chapter 3. – Compression: Consider three different facts. Long-term storage of astro- nomical data. Detection of alignments and curves is necessary for many classes of segmentation and feature analysis, and for the building of 3D representations of data. Gravitational lensing presents one area of potential