Fourier Methods in Imaging

Fourier Methods in Imaging introduces the mathematical tools for modeling linear imaging systems to predict the action of the system or for solving for the input. The chapters are grouped into five sections, the first introduces the imaging “tasks” (direct, inverse, and system analysis), the basic concepts of linear algebra for vectors and functions, including complex–valued vectors, and inner products of vectors and functions. The second section defines "special" functions, mathematical operations, and transformations that are useful for describing imaging systems. Among these are the Fourier transforms of 1–D and 2–D function, and the Hankel and Radon transforms. This section also considers approximations of the Fourier transform. The third and fourth sections examine the discrete Fourier transform and the description of imaging systems as linear "filters", including the inverse, matched, Wiener and Wiener–Helstrom filters. The final section examines applications of linear system models to optical imaging systems, including holography.
• 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 two–dimensional linear imaging systems, and will also serve as a reference for engineers and scientists.


Spis treści:
Series Editor’s Preface
Preface
1 Introduction
1.1 Signals, Operators, and Imaging Systems . . . . . . . . . . . . . . . . . . . .
1.2 TheThreeImagingTasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 ExamplesofOptical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 ImagingTasks inMedical Imaging . . . . . . . . . . . . . . . . . . . . . . . .
2 Operators and Functions
2.1 Classes of ImagingOperators . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Continuous and Discrete Functions . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Vectors with Real–Valued Components
3.1 Scalar Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 VectorSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Complex Numbers and Functions
4.1 ArithmeticofComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Graphical Representation of Complex Numbers . . . . . . . . . . . . . . . .
4.3 ComplexFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Generalized Spatial Frequency – Negative Frequencies . . . . . . . . . . . .
4.5 Argand Diagrams of Complex–Valued Functions . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Complex–Valued Matrices and Systems
5.1 Vectors with Complex–Valued Components . . . . . . . . . . . . . . . . . . .
5.2 Matrix Analogues of Shift–Invariant Systems . . . . . . . . . . . . . . . . . .
5.3 MatrixFormulationof ImagingTasks . . . . . . . . . . . . . . . . . . . . . . .
5.4 Continuous Analogues of Vector Operations . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 1–D Special Functions
6.1 Definitionsof 1–DSpecialFunctions . . . . . . . . . . . . . . . . . . . . . . .
6.2 1–DDiracDeltaFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 1–DComplex–ValuedSpecialFunctions . . . . . . . . . . . . . . . . . . . . .
6.4 1–DStochasticFunctions–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 2–D Special Functions
7.1 2–D Separable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Definitionsof 2–DSpecialFunctions . . . . . . . . . . . . . . . . . . . . . . .
7.3 2–DDiracDeltaFunctionanditsRelatives. . . . . . . . . . . . . . . . . . . .
7.4 2–DFunctionswithCircularSymmetry . . . . . . . . . . . . . . . . . . . . . .
7.5 Complex–Valued2–DFunctions . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 SpecialFunctionsofThree(orMore)Variables . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Linear Operators
8.1 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Shift–InvariantOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Linear Shift–Invariant (LSI) Operators . . . . . . . . . . . . . . . . . . . . . .
8.4 CalculatingConvolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5 Properties of Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6 Auto correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7 Crosscorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.8 2–DLSIOperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.9 Crosscorrelations of 2–D Functions . . . . . . . . . . . . . . . . . . . . . . . .
8.10 Autocorrelationsof 2–DFunctions . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 Fourier Transforms of 1–D Functions
9.1 Transforms of Continuous–Domain Functions . . . . . . . . . . . . . . . . . .
9.2 Linear Combinations of Reference Functions . . . . . . . . . . . . . . . . . .
9.3 Complex–ValuedReferenceFunctions . . . . . . . . . . . . . . . . . . . . . .
9.4 TransformsofComplex–ValuedFunctions . . . . . . . . . . . . . . . . . . . .
9.5 FourierAnalysisofDiracDeltaFunctions . . . . . . . . . . . . . . . . . . . .
9.6 InverseFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7 FourierTransformsof 1–DSpecialFunctions . . . . . . . . . . . . . . . . . .
9.8 Theorems of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . .
9.9 Appendix: Spectrum of Gaussian via Path Integral . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Multidimensional Fourier Transforms
10.1 2–DFourierTransforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Spectra of Separable 2–D Functions . . . . . . . . . . . . . . . . . . . . . . .
10.3 Theorems of 2–D Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Spectra of Circular Functions
11.1 TheHankelTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 InverseHankelTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Theorems of Hankel Transforms . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 HankelTransformsofSpecialFunctions . . . . . . . . . . . . . . . . . . . . .
11.5 Appendix: Derivations of Equations (11.12) and (11.14) . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 The Radon Transform
12.1 Line–IntegralProjectionsontoRadialAxes . . . . . . . . . . . . . . . . . . .
12.2 Radon Transforms of Special Functions . . . . . . . . . . . . . . . . . . . . .
12.3 Theorems of the Radon Transform . . . . . . . . . . . . . . . . . . . . . . . .
12.4 Inverse Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 Central–SliceTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6 ThreeTransformsofFourFunctions . . . . . . . . . . . . . . . . . . . . . . .
12.7 Fourier and Radon Transforms of Images . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 Approximations to Fourier Transforms 13.1 Moment Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.1 FirstMoment–Centroid. . . . . . . . . . . . . . . . . . . . . . . . .
13.2 1–D Spectra via Method of Stationary Phase . . . . . . . . . . . . . . . . . .
13.3 Central–Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4 WidthMetricsandUncertaintyRelations . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 Discrete Systems, Sampling, and Quantization
14.1 Ideal Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Ideal Sampling of Special Functions . . . . . . . . . . . . . . . . . . . . . . .
14.3 InterpolationofSampledFunctions . . . . . . . . . . . . . . . . . . . . . . .
14.4 Whittaker–Shannon Sampling Theorem . . . . . . . . . . . . . . . . . . . . .
14.5 Aliasingand Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.6 “Prefiltering” toPreventAliasing . . . . . . . . . . . . . . . . . . . . . . . . .
14.7 RealisticSampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.8 RealisticInterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.9 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.10DiscreteConvolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 Discrete Fourier Transforms 511
15.1 Inverse of the Infinite–Support DFT . . . . . . . . . . . . . . . . . . . . . . . .
15.2 DFToverFiniteInterval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3 FourierSeriesDerivedfromFourierTransform . . . . . . . . . . . . . . . . .
15.4 EfficientEvaluationof theFiniteDFT. . . . . . . . . . . . . . . . . . . . . . .  
15.5 PracticalConsiderationsforDFTandFFT . . . . . . . . . . . . . . . . . . . .
15.6 FFTs of 2–D Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.7 DiscreteCosineTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16 Magnitude Filtering
16.1 ClassesofFilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2 Eigenfunctions of Convolution . . . . . . . . . . . . . . . . . . . . . . . . . .
16.3 PowerTransmissionofFilters . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.4 LowpassFilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.5 Highpass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.6 Bandpass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.7 Fourier Transform as a Bandpass Filter . . . . . . . . . . . . . . . . . . . . .
16.8 Bandboost and Bandstop Filters . . . . . . . . . . . . . . . . . . . . . . . . .
16.9 WaveletTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17 Allpass (Phase) Filters
17.1 Power–Series Expansion for Allpass Filters . . . . . . . . . . . . . . . . . . .
17.2 Constant–PhaseAllpassFilter . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3 Linear–PhaseAllpassFilter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.4 Quadratic–Phase Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.5 AllpassFilterswithHigher–OrderPhase . . . . . . . . . . . . . . . . . . . . .
17.6 Allpass Random–Phase Filter . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.7 Relative Importance of Magnitude and Phase . . . . . . . . . . . . . . . . . .
17.8 ImagingofPhaseObjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.9 ChirpFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 Magnitude–Phase Filters
18.1 TransferFunctionsofThreeOperations . . . . . . . . . . . . . . . . . . . . .
18.2 FourierTransformofRampFunction . . . . . . . . . . . . . . . . . . . . . . .
18.3 CausalFilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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– “InverseFiltering” . . . . . . . . . . . . . . . . . . . . . . . .
19.3 Optimum Estimators for Signals in Noise . . . . . . . . . . . . . . . . . . . .
19.4 Detection of Known Signals – Matched Filter . . . . . . . . . . . . . . . . . .
19.5 Analogiesof InverseandMatchedFilters . . . . . . . . . . . . . . . . . . . .
19.6 ApproximationstoReciprocalFilters . . . . . . . . . . . . . . . . . . . . . . .
19.7 InverseFilteringofShift–VariantBlur . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20 Filtering in Discrete Systems
20.1 Translation, Leakage, and Interpolation . . . . . . . . . . . . . . . . . . . . .
20.2 AveragingOperators– LowpassFilters . . . . . . . . . . . . . . . . . . . . .
20.3 Differencing Operators – Highpass Filters . . . . . . . . . . . . . . . . . . . .
20.4 Discrete Sharpening Operators . . . . . . . . . . . . . . . . . . . . . . . . . .
20.5 2–DGradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.6 PatternMatching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.7 ApproximateDiscreteReciprocalFilters . . . . . . . . . . . . . . . . . . . . .
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 Fraunho