Fourier Methods in ImagingISBN: 9780470689837
954 pages
June 2010

 Provides a unified mathematical description of imaging systems.
 Develops a consistent mathematical formalism for characterizing imaging systems.
 Helps the reader develop an intuitive grasp of the most common mathematical methods, useful for describing the action of general linear systems on signals of one or more spatial dimensions.
 Offers parallel descriptions of continuous and discrete cases.
 Includes many graphical and pictorial examples to illustrate the concepts.
This book helps students develop an understanding of mathematical tools for describing general one and twodimensional linear imaging systems, and will also serve as a reference for engineers and scientists
Preface.
1 Introduction.
1.1 Signals, Operators, and Imaging Systems.
1.2 The Three Imaging Tasks.
1.3 Examples of Optical Imaging.
1.4 ImagingTasks inMedical Imaging.
2 Operators and Functions.
2.1 Classes of Imaging Operators.
2.2 Continuous and Discrete Functions.
Problems.
3 Vectors with RealValued Components.
3.1 Scalar Products.
3.2 Matrices.
3.3 Vector Spaces.
Problems.
4 Complex Numbers and Functions.
4.1 Arithmetic of Complex Numbers.
4.2 Graphical Representation of Complex Numbers.
4.3 Complex Functions.
4.4 Generalized Spatial Frequency – Negative Frequencies.
4.5 Argand Diagrams of ComplexValued Functions.
Problems.
5 ComplexValued Matrices and Systems.
5.1 Vectors with ComplexValued Components.
5.2 Matrix Analogues of ShiftInvariant Systems.
5.3 Matrix Formulation of ImagingTasks.
5.4 Continuous Analogues of Vector Operations.
Problems.
6 1D Special Functions.
6.1 Definitions of 1D Special Functions.
6.2 1D Dirac Delta Function.
6.3 1D ComplexValued Special Functions.
6.4 1D Stochastic Functions–Noise.
6.5 Appendix A: Area of SINC[x] and SINC2[x].
6.6 Appendix B: Series Solutions for Bessel Functions J0[x] and J1[x].
Problems.
7 2D Special Functions.
7.1 2D Separable Functions.
7.2 Definitions of 2D Special Functions.
7.3 2D Dirac Delta Function and its Relatives.
7.4 2D Functions with Circular Symmetry.
7.5 ComplexValued 2D Functions.
7.6 Special Functions of Three (orMore) Variables.
Problems.
8 Linear Operators.
8.1 Linear Operators.
8.2 ShiftInvariant.Operators.
8.3 Linear ShiftInvariant (LSI) Operators.
8.4 Calculating Convolutions.
8.5 Properties of Convolutions.
8.6 Autocorrelation.
8.7 Crosscorrelation.
8.8 2DLSIOperations.
8.9 Crosscorrelations of 2D Functions.
8.10 Autocorrelations of 2D.Functions.
Problems.
9 Fourier Transforms of 1D Functions.
9.1 Transforms of ContinuousDomain Functions.
9.2 Linear Combinations of Reference Functions.
9.3 ComplexValued Reference Functions.
9.4 Transforms of ComplexValued Functions.
9.5 Fourier Analysis of Dirac Delta Functions.
9.6 Inverse Fourier Transform.
9.7 Fourier Transforms of 1D Special Functions.
9.8 Theorems of the Fourier Transform.
9.9 Appendix: Spectrum of Gaussian via Path Integral.
Problems.
10 Multidimensional Fourier Transforms.
10.1 2D Fourier Transforms.
10.2 Spectra of Separable 2D Functions.
10.3 Theorems of 2D Fourier Transforms.
Problems.
11 Spectra of Circular Functions.
11.1 The Hankel Transform.
11.2 Inverse Hankel Transform.
11.3 Theorems of Hankel Transforms.
11.4 Hankel Transforms of Special Functions.
11.5 Appendix: Derivations of Equations (11.12) and (11.14).
Problems.
12 The Radon Transform.
12.1 LineIntegral Projections onto Radial Axes.
12.2 Radon Transforms of Special Functions.
12.3 Theorems of the Radon Transform.
12.4 Inverse Radon Transform.
12.5 CentralSlice Transform.
12.6 Three Transforms of Four Functions.
12.7 Fourier and Radon Transforms of Images.
Problems.
13 Approximations to Fourier Transforms.
13.1 Moment Theorem.
13.2 1D Spectra via Method of Stationary Phase.
13.3 CentralLimit Theorem.
13.4 Width Metrics and Uncertainty Relations.
Problems.
14 Discrete Systems, Sampling, and Quantization.
14.1 Ideal Sampling.
14.2 Ideal Sampling of Special Functions.
14.3 Interpolation of Sampled Functions.
14.4 Whittaker–Shannon Sampling Theorem.
14.5 Aliasingand Interpolation.
14.6 “Prefiltering” to Prevent Aliasing.
14.7 Realistic Sampling.
14.8 Realistic Interpolation.
14.9 Quantization.
14.10 Discrete Convolution.
Problems.
15 Discrete Fourier Transforms.
15.1 Inverse of the InfiniteSupport DFT.
15.2 DFT over Finite Interval.
15.3 Fourier Series Derived from Fourier Transform.
15.4 Efficient Evaluation of the Finite DFT.
15.5 Practical Considerations for DFT and FFT.
15.6 FFTs of 2D Arrays.
15.7 Discrete Cosine Transform.
Problems.
16 Magnitude Filtering.
16.1 Classes of Filters.
16.2 Eigenfunctions of Convolution.
16.3 Power Transmission of Filters.
16.4 Lowpass Filters.
16.5 Highpass Filters.
16.6 Bandpass Filters.
16.7 Fourier Transform as a Bandpass Filter.
16.8 Bandboost and Bandstop Filters.
16.9 Wavelet Transform.
Problems.
17 Allpass (Phase) Filters.
17.1 PowerSeries Expansion for Allpass Filters.
17.2 ConstantPhase Allpass Filter.
17.3 LinearPhase Allpass Filter.
17.4 QuadraticPhase Filter.
17.5 Allpass Filters with HigherOrder Phase.
17.6 Allpass RandomPhase Filter.
17.7 Relative Importance of Magnitude and Phase.
17.8 Imaging of Phase Objects.
17.9 Chirp Fourier Transform.
Problems.
18 Magnitude–Phase Filters.
18.1 Transfer Functions of Three Operations.
18.2 Fourier Transform of Ramp Function.
18.3 Causal Filters.
18.4 Damped Harmonic Oscillator.
18.5 Mixed Filters with Linear or Random Phase.
18.6 Mixed Filter with Quadratic Phase.
Problems.
19 Applications of Linear Filters.
19.1 Linear Filters for the Imaging Tasks.
19.2 Deconvolution– “Inverse Filtering”.
19.3 Optimum Estimators for Signals in Noise.
19.4 Detection of Known Signals – Matched Filter.
19.5 Analogies of Inverse and Matched Filters.
19.6 Approximations to Reciprocal Filters.
19.7 Inverse Filtering of ShiftVariant Blur.
Problems.
20 Filtering in Discrete Systems.
20.1 Translation, Leakage, and Interpolation.
20.2 Averaging Operators– Lowpass Filters.
20.3 Differencing Operators – Highpass Filters.
20.4 Discrete Sharpening Operators.
20.5 2DGradient.
20.6 Pattern Matching.
20.7 Approximate Discrete Reciprocal Filters.
Problems.
21 Optical Imaging in Monochromatic Light.
21.1 Imaging Systems Based on Ray Optics Model.
21.2 Mathematical Model of Light Propagation.
21.3 Fraunhofer Diffraction.
21.4 Imaging System based on Fraunhofer Diffraction.
21.5 Transmissive Optical Elements.
21.6 Monochromatic Optical Systems.
21.7 ShiftVariant Imaging Systems.
Problems.
22 Incoherent Optical Imaging Systems.
22.1 Coherence.
22.2 Polychromatic Source – Temporal Coherence.
22.3 Imaging in Incoherent Light.
22.4 System Function in Incoherent Light.
Problems.
23 Holography.
23.1 Fraunhofer Holography.
23.2 Holography in Fresnel Diffraction Region.
23.3 ComputerGenerated Holography.
23.4 Matched Filtering with CellType CGH.
23.5 SyntheticAperture Radar (SAR).
Problems.
References.
Index.
Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology
Professor Easton teaches undergraduate and graduate courses in linear systems, optical imaging, and digital image processing at Rochester Institute of Technology. He received a B.S. degree in Astronomy from Haverford College, an M.S. in physics from the University of Maryland, and an M.S. and Ph.D. degree in Optical Sciences from the University of Arizona.
His research interests include the application of digital image processing to text documents and manuscripts. He has contributed to work on the Dead Sea Scrolls and is now part of an imaging team helping scolars to read the original Archimiedes Palimpsest. Professor Easton also conducts research into optical signal processing and computergenerated holography, publishing articles on both.
"This comprehensive textbook represents a practical review of Fourier techniques in imaging methods. It will be very useful for graduate students (in engineering, science, computer science, and applied mathematics) as well as engineers interested in linear imaging systems." (Zentralblatt Math, 2010)
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