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 TheRadonTransform ............................ 12 1.3 Mathematical Morphology . 12 1.4 EdgeDetection....................................... 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 PatternRecognition .................................. 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 NoiseModeling .................................. 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 GaussianBayesModel ............................ 69 3.5.4 Maximum Likelihood with Poisson Noise . 69 3.5.5 PoissonBayesModel.............................. 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 WaveletCLEAN ................................. 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 ThePrinciple.................................... 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 VisualQuality ................................... 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 TheLiftingScheme............................... 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 UseofaModel................................... 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 ThePrinciple.................................... 188 7.5.2 TheParameters.................................. 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 ErrorAnalysis .................................. 204 8.2.4 Correlation Length Determination . 205 8.2.5 Creation of Random Catalogs . 205 8.2.6 Examples ....................................... 206 8.3 FractalAnalysis................................... 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 WaveletAnalysis .................................... 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 FastClusterAnalysis ................................ 231 9.4 Nearest Neighbor Finding on Graphs. 233 9.5 Cluster-Based User Interfaces . 234 9.6 ImagesfromData ................................... 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 ........................