Image processing and data analysis The multiscale …

1 Image processing and data analysisThe multiscale approachJean-Luc StarckCentre d Etudes de SaclayFionn MurtaghUniversity of UlsterAlbert BijaouiObservatoire de la C ote d AzurContentsPrefacevii1 The wavelet multiscale methods .. Some perspectives on the wavelet transform .. The wavelet transform and the Fourier transform .. Applications of the wavelet transform .. The continuous wavelet transform .. Definition .. Properties .. The inverse transform .. Examples of wavelet functions .. Morlet s wavelet .. Mexican hat .. Haar wavelet .. The discrete wavelet transform .. Multiresolution analysis .. Mallat s horizontal and vertical analyses .. Non-dyadic resolution factor .. The `a trous algorithm .. Pyramidal algorithm .. Scaling functions with a frequency cut-off .. Discussion of the wavelet transform.

2 Multiresolution and median transform .. Multiresolution median transform .. Pyramidal median transform .. Iterative pyramidal median transform .. Non-iterative pyramidal transform with exact recon-struction .. Conclusion on multiscale median transforms .. Multiresolution and mathematical morphology .. Multiresolution .. Pyramidal morphological transform .. Conclusions on non-wavelet multiresolution approaches 502 Multiresolution support and Noise modeling .. Definition of significant coefficients .. Gaussian noise .. Poisson noise .. Gaussian and Poisson noise .. Poisson noise with few photons or counts .. Other types of noise .. Multiresolution support .. Definition .. Multiresolution support from the wavelet transform . Algorithm .. Gaussian noise estimation from the multiresolutionsupport.

3 Concluding remarks on the multiresolution supportand noise .. Filtering .. Convolution using the continuous wavelet transform . Wiener-like filtering in wavelet space .. Hierarchical Wiener filtering .. Adaptive filtering .. Examples .. Multiresolution Image comparison .. 823 Introduction to deconvolution .. Regularization using multiresolution support .. Noise suppression based on the wavelet transform .. Noise suppression based on the multiresolution support Regularization of Van Cittert s algorithm .. Regularization of the one-step gradient method .. Regularization of the Richardson-Lucy algorithm .. Convergence .. Examples from astronomy .. Conclusion on regularization using the multiresolutionsupport .. multiscale entropy and Image restoration .. Image restoration using the maximum entropy method Formalism of maximum entropy multiresolution.

4 Deconvolution using multiscale entropy .. Experiments .. Another application of multiscale entropy: filtering.. Conclusion on multiscale entropy and restoration .. Image restoration for aperture synthesis .. Introduction to deconvolution in aperture synthesis.. CLEAN and wavelets .. Experiment .. Observations of two evolved stars .. Conclusion on interferometric data deconvolution .. 1264 1D signals and Euclidean data analysis of 1D signals: spectral analysis .. Spectral analysis .. Noise determination and detection criteria .. Simulations .. Problems related to detection using the wavelet trans-form .. Band extraction .. Continuum estimation .. Optical depth .. The multiresolution spectral analysis algorithm .. Wavelets and multivariate data analysis .. Wavelet regression in multivariate data analysis .

5 Degradation of distance through wavelet Degradation of first eigenvalue through wavelet The Kohonen map in wavelet space .. Example of SOFM in direct and in wavelet spaces .. K-means and principal components analysis in waveletspace .. Multiresolution regression and forecasting .. Meteorological prediction using the `a trous method .. Sunspot prediction using `a trous and neural Dynamic recurrent neural network architecture .. Combining neural network outputs .. 1595 Geometric Image distortions .. Geometrical distortions .. Radiometrical distortions .. Geometrical Image registration .. Deformation model .. Image correction .. Ground control points .. Registration using the wavelet transform .. Registration of images from the same satellite detector Registration of images from different sensor detectors Registration of images with a pyramidal algorithm.

6 Application .. SPOT data .. MSS data .. SPOT versus MSS data .. SPOT with different imaging directions .. Astronomical Image registration .. Field of view distortion estimation in ISOCAM images Error analysis .. Conclusion on registration .. 1976 Disparity analysis in remote Definitions .. Disparity .. Matching .. Extraction of ground control points .. Disparity mapping .. Kriging .. Variogram .. Kriging as an interpolator .. Disparity mapping with the wavelet transform .. Image registration .. Application to real images .. Conclusion on disparity analysis .. 2267 Image Pyramidal median transform and compression .. Compression method .. Quantized multiresolution coefficients .. Quadtree and Huffman encoding .. Noise compression .. Image decompression .. Examples and assessments.

7 Image transmission over networks .. Conclusion on Image compression .. 2428 Object detection and point The problem and the data .. Median transform and Minkowski operators .. Conclusion on astronomical object detection .. Clustering in point patterns .. Example 1: excellent recovery of Gaussian clusters .. Example 2: diffuse rectangular cluster .. Example 3: diffuse rectangle and faint Gaussian Conclusion: cluster analysis in constant time .. 2539 multiscale vision Artificial vision and astronomical images .. Object definition in wavelet transform space .. Choice of a wavelet transform algorithm .. Bases of object definition .. Significant wavelet coefficients .. Scale-by-scale field labeling .. Interscale connection graph .. An object as a tree .. Object identification .. Partial reconstruction .. The basic problem.

8 Choice of the scale number .. Reconstruction algorithm .. Numerical experiments .. Applications to a real Image .. multiscale method .. INVENTORY method .. Comments .. Vision models and Image classes .. 273A Variance Mean and variance expansions .. Determination of centered moments .. Study of the expansions for the variance .. Variance-stabilizing transformation ..277B Software information279C Acronyms283 Index305viCONTENTSP refaceThere is a very large literature on the theoretical underpinnings of thewavelet transform. However, theory must be complemented with a sig-nificant amount of practical work. Selection of method, implementation,validation of results, comparison with alternatives these are all centrallyimportant for the applied scientist or engineer. Turning theory into prac-tice is the theme of this book. Various applications have benefited from thewavelet and other multiscale transforms.

9 In this book, we describe manysuch applications, and in this way illustrate the theory andpractice of suchtransforms. We describe an embedded systems approach to wavelets andmultiscale transforms in this book, in that we introduce andappraise ap-propriate multiscale methods for use in a wide range of application provides an illustrative background for many of the exam-ples used in this book. Chapters 5 and 6 cover problems in remote 3, dealing with noise in images, includes discussion on problems ofwide relevance. At the time of writing, the authors are applying these meth-ods to other fields: medical Image analysis (radiology, for mammography;echocardiology), plasma physics response signals, and 1 provides an extensive review of the theory and practice of thewavelet transform. This chapter then considers other multiscale transforms,offering possible advantages in regard to robustness.

10 The reader wishingearly action may wish to read parts of Chapter 1 at first, anddip into itagain later, for discussion of particular Chapter 2, an important property of images noise is of the lessons learned in regard to noise is 3 describes deconvolution, or Image sharpening and/or includes drawing various links with entropy-based smoothness 4 covers (i) spectral analysis and (ii) general themes in multivari-ate data analysis . It is shown how the wavelet transform can be integratedseamlessly into various multivariate data analysis methods. Chapter 5 coversimage registration, in remote sensing and in astronomy. Chapter 6 deals withstereo Image processing in remote sensing. Chapter 7 describes highly effec-tive Image compression procedures based on multiscale transforms. Chapter8 deals with object detection in images and also with point pattern cluster-ing.