Fourier Analysis and Imaging

ISBN-13: 9780306481871 / Angielski / Twarda / 2004 / 689 str.

Ronald Newbold Bracewell

Opis

As Lord Kelvin said, "Fourier's theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics." This has remained durable knowledge for a century, and has extended its applicability to topics as diverse as medical imaging (CT scanning), the presentation of images on screens and their digital transmission, remote sensing, geophysical exploration, and many branches of engineering. Fourier Analysis and Imaging is based on years of teaching a course on the Fourier Transform at the senior or early graduate level, as well as on Prof. Bracewell's 1995 text Two-Dimensional Imaging. It is an excellent textbook and will also be a welcome addition to the reference library of those many professionals whose daily activities involve Fourier analysis in its many guises.

Recenzje

"What is striking about the book is how much text there is and how rich that text is in supplementing the formal structure that is presented. Indeed, this is a hallmark of Bracewell's writing and why, in my opinion, he is held in such high esteem. ... I highly recommend it to anyone who has even a passing interest in the hows and whys of image creation, manipulation, or display."
(Brian H. Kolner, UC Davis, CA) 

Spis treści

1 Introduction.- Summary of the Chapters.- Notation.- Teaching a Course from This Book.- The Problems.- Aspects of Imaging.- Computer Code.- Literature References.- Recommendation.- 2 The Image Plane.- Modes of Representation.- Some Properties of a Function of Two Variables.- Projection of Solid Objects.- Image Distortion.- Operations in the Image Plane.- Binary Images.- Operations on Digital Images.- Reflectance Distribution.- Data Compression.- Summary.- Appendix: A Contour Plot Programt.- Literature Cited.- Further Reading.- Problems.- 3 Two-Dimensional Impulse Functions.- The Two-Dimensional Point Impulse.- Rules for Interpreting Delta Notation.- Generalized Functions.- The Shah Functions iii and 2III.- Line Impulses.- Regular Impulse Patterns.- Interpretation of Rectangle Function of f(x).- Interpretation of Rectangle Function of f(x,y).- General Rule for Line Deltas.- The Ring Impulse.- Impulse Function of f(x,y).- Sifting Property.- Derivatives of Impulses.- Summary.- Literature Cited.- Problems.- 4 The Two-Dimensional Fourier Transform.- One Dimension.- The Fourier Component in Two Dimensions.- Three or More Dimensions.- Vector Form of Transform.- The Corrugation Viewpoint.- Examples of Transform Pairs.- Theorems for Two-Dimensional Fourier Transforms.- The Two-Dimensional Hartley Transform.- Theorems for the Hartley Transform.- Discrete Transforms.- Summary.- Literature Cited.- Further Reading.- Problems.- 5 Two-Dimensional Convolution.- Convolution Defined.- Cross-Correlation Defined.- Feature Detection by Matched Filtering.- Autocorrelation Defined.- Understanding Autocorrelation.- Cross-Correlation Islands and Dilation.- Lazy Pyramid and Chinese Hat Function.- Central Value and Volume of Autocorrelation.- The Convolution Sum.- Computing the Convolution.- Digital Smoothing.- Matrix Product Notation.- Summary.- Literature Cited.- Problems.- 6 The Two-Dimensional Convolution Theorem.- Convolution Theorem.- An Instrumental Caution.- Point Response and Transfer Function.- Autocorrelation Theorem.- Cross-Correlation Theorem.- Factorization and Separation.- Convolution with the Hartley Transform.- Summary.- Problems.- 7 Sampling and Interpolation in Two Dimensions.- What is a Sample?.- Sampling at a Point.- Sampling on a Point Pattern, and the Associated Transfer Function.- Sampling Along a Line.- Curvilinear Sampling.- The Shah Function.- Fourier Transform of the Shah Function.- Other Patterns of Sampling.- Factoring.- The Two-Dimensional Sampling Theorem.- Undersampling.- Aliasing.- Circular Cutoff.- Double-Rectangle Pass Band.- Discrete Aspect of Sampling.- Interpolating Between Samples.- Interlaced Sampling.- Appendix: The Two-Dimensional Fourier Transform of the Shah Function.- Literature Cited.- Problems.- 8 Digital Operations.- Smoothing.- Nonconvolutional Smoothing.- Trend Reduction.- Sharpening.- What is a Digital Filter?.- Guard Zone.- Transform Aspect of Smoothing Operator.- Finite Impulse Response (FIR).- Special Filters.- Densifying.- The Arbitrary Operator.- Derivatives.- The Laplacian Operator.- Projection as a Digital Operation.- Moire Patterns.- Functions of an Image.- Digital Representation of Objects.- Filling a Polygon.- Edge Detection and Segmentation.- Discrete Binary Objects.- Operations on Discrete Binary Objects.- Union and Intersection.- Pixel Morphology.- Dilation.- Coding a Binary Matrix.- Granulometry.- Conclusion.- Literature Cited.- Problems.- 9 Rotational Symmetry.- What Is a Bessel Function?.- The Hankel Transform.- The jinc Function.- The Struve Function.- The Abel Transform.- Spin Averaging.- Angular Variation and Chebyshev Polynomials.- Summary.- Table of the jinc Function.- Problems.- 10 Imaging by Convolution.- Mapping by Antenna Beam.- Scanning the Spherical Sky.- Photography.- Microdensitometry.- Video Recording.- Eclipsometry.- The Scanning Acoustic Microscope.- Focusing Underwater Sound.- Literature Cited.- Problems.- 11 Diffraction Theory of Sensors and Radiators.- The Concept of Aperture Distribution.- Source Pair and Wave Pair.- Two-Dimensional Apertures.- Rectangular Aperture.- Example of Circular Aperture.- Duality.- The Thin Lens.- What Happens at a Focus?.- Shadow of a Straight Edge.- Fresnel Diffraction in General.- Literature Cited.- Problems.- 12 Aperture Synthesis and Interferometry.- Image Extraction from a Field.- Incoherent Radiation Source.- Field of Incoherent Source.- Correlation in the Field of an Incoherent Source.- Visibility.- Measurement of Coherence.- Notation.- Interferometers.- Radio Interferometers.- Rationale Behind Two-Element Interferometer.- Aperture Synthesis (Indirect Imaging).- Literature Cited.- Problems.- 13 Restoration.- Restoration by Successive Substitutions.- Running Means.- Eddington’s Formula.- Finite Differences.- Finite Difference Formula.- Chord Construction.- The Principal Solution.- Finite Differencing in Two Dimensions.- Restoration in the Presence of Errors.- The Additive Noise Signal.- Determination of the Real Restoring Function.- Determination of the Complex Restoring Function.- Some Practical Remarks.- Artificial Sharpening.- Antidiffusion.- Nonlinear Methods.- Restoring Binary Images.- CLEAN.- Maximum Entropy.- Literature Cited.- Problems.- 14 The Projection-Slice Theorem.- Circular Symmetry Reviewed.- The Abel-Fourier-Hankel Cycle.- The Projection-Slice Theorem.- Literature Cited.- Problems.- 15 Computed Tomography.- Workingfrom Projections.- An X-Ray Scanner.- Fourier Approach to Computed Tomography.- Back-Projection Methods.- The Radon Transform.- The Impulse Response of the Radon Transformation.- Some Radon Transforms.- The Eigenfunctions.- Theorems for the Radon Transform.- The Radon Boundary.- Applications.- Literature Cited.- Problems.- 16 Synthetic-Aperture Radar.- Doppler Radar.- Some History of Radiofrequency Doppler.- Range-Doppler Radar.- Radargrarnmetry.- Literature Cited.- Problems.- 17 Two-Dimensional Noise Images.- Some Types of Random Image.- Gaussian Noise.- The Spatial Spectrum of a Random Scatter.- Autocorrelation of a Random Scatter.- Pseudorandom Scatter.- Random Orientation.- Nonuniform Random Scatter.- Spatially Correlated Noise.- The Familiar Maze.- The Drunkard’s Walk.- Fractal Polygons.- Conclusion.- Literature Cited.- Problems.- Appendix A Solutions to Problems.