Image Estimation By Example

Image Estimation By Example - IMAGE ESTIMATION BY EXAMPLE:...

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Unformatted text preview: IMAGE ESTIMATION BY EXAMPLE: Geophysical Soundings Image Construction Multidimensional autoregression Jon F. Claerbout Cecil and Ida Green Professor of Geophysics Stanford University with Sergey Fomel Stanford University c February 2, 2011 Contents 1 Basic operators and adjoints 1 1.0.1 Programming linear operators . . . . . . . . . . . . . . . . . . . . . . 3 1.1 FAMILIAR OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Adjoint derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Transient convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 Internal convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.4 Zero padding is the transpose of truncation . . . . . . . . . . . . . . 11 1.1.5 Adjoints of products are reverse-ordered products of adjoints . . . . 12 1.1.6 Nearest-neighbor coordinates . . . . . . . . . . . . . . . . . . . . . . 12 1.1.7 Data-push binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.8 Linear interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.9 Spray and sum : scatter and gather . . . . . . . . . . . . . . . . . . 15 1.1.10 Causal and leaky integration . . . . . . . . . . . . . . . . . . . . . . 16 1.1.11 Backsolving, polynomial division and deconvolution . . . . . . . . . 19 1.1.12 The basic low-cut lter . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.1.13 Smoothing with box and triangle . . . . . . . . . . . . . . . . . . . . 22 1.1.14 Nearest-neighbor normal moveout (NMO) . . . . . . . . . . . . . . . 24 1.1.15 Coding chains and arrays . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2 ADJOINT DEFINED: DOT-PRODUCT TEST . . . . . . . . . . . . . . . . 28 1.2.1 Denition of a vector space . . . . . . . . . . . . . . . . . . . . . . . 28 1.2.2 Dot-product test for validity of an adjoint . . . . . . . . . . . . . . . 29 1.2.3 The word adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.2.4 Matrix versus operator . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.2.5 Inverse operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 CONTENTS 1.2.6 Automatic adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 Model tting by least squares 35 2.1 HOW TO DIVIDE NOISY SIGNALS . . . . . . . . . . . . . . . . . . . . . 35 2.1.1 Dividing by zero smoothly . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.2 Damped solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.1.3 Smoothing the denominator spectrum . . . . . . . . . . . . . . . . . 36 2.1.4 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.5 Formal path to the low-cut lter . . . . . . . . . . . . . . . . . . . . 40 2.1.6 The plane-wave destructor . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2 MULTIVARIATE LEAST SQUARES . . . . . . . . . . . . . . . . . . . . . 45 2.2.1 Inside an abstract vector . . . . . . . . . . . . . . . . . . . . . . . . . ....
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This note was uploaded on 02/10/2011 for the course GEOL 7320 taught by Professor Stewart during the Spring '11 term at University of Houston - Downtown.

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Image Estimation By Example - IMAGE ESTIMATION BY EXAMPLE:...

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