Hadamard Transform Optics focuses on Hadamard transform optics and Hadamard encoded optical instruments. The techniques developed to date are described, and a unified mathematical treatment that should facilitate comparisons between different classes of instruments is presented. With this approach, encoded Hadamard transform spectrometers are discussed in very much the same way as encoded imaging devices. The advantages offered by singly and multiply encoded instruments designed for a wide variety of purposes are also considered.
This book is comprised of seven chapters and begins with an introduction to optical multiplexing techniques, as well as the connections with weighing designs, along with the best masks for use in optical instruments and the improvement in signal-to-noise ratio that should be produced by multiplexing. Spectrometers which make use of multiplexing, including the Michelson and Mach-Zehnder interferometers and Golay's multislit spectrometers, are then described. Subsequent chapters deal with the basic theory of Hadamard transform spectrometers and imagers; factors that affect the signal-to-noise ratio; and instrumental considerations and systematic errors in instruments. The final chapter looks at some of the applications of Hadamard transform optics, including image processing and in fields such as astronomy and medicine.
This monograph will be a useful resource for physicists.

Contenu

Preface
Acknowledgments
Chapter 1. An Introduction to Optical Multiplexing Techniques
1.1 Introduction
1.2 Weighing Designs and Optical Multiplexing
1.3 Masks from Hadamard Matrices and S-Matrices
Chapter 2. A Survey of Multiplexing Spectrometers
2.1 Introduction
2.2 Spectrometers which Use Interference Methods
2.3 Spectrometers which Use Masks
2.4 A Survey of Multiplexing Mask Spectrometers
2.4.1 Simple Masking Spectrometer
2.4.2 Golay's Static Multislit Spectrometer
2.4.3 Golay's Dynamic Multislit Spectrometer
2.4.4 Girard's Grill Spectrometer
2.4.5 Mertz's Mock Interferometer
2.4.6 Hadamard Transform Spectrometers
Chapter 3. The Basic Theory of Hadamard Transform Spectrometers and Imagers
3.1 Introduction
3.1.1 Comparison of Different Encoding Schemes
3.2 Analysis of a Singly Encoded Spectrometer
3.2.1 The Basic Equation Relating Measurements and Unknowns
3.2.2 Estimating the Spectrum
3.2.3 The Mean Square Error of the Estimates
3.2.4 Hotelling's Bound on the Mean Square Error
3.2.5 Masks with the Largest Determinant
3.2.6 Masks of 0's and 1's; S-Matrices
3.2.7 Cyclic Masks
3.3 Imagers
3.3.1 Definition of an Imager
3.3.2 Hadamard Transform Imager
3.3.3 Folding the Mask Configuration
3.4 Doubly Encoded Spectrometers and Imaging Spectrometers
3.4.1 Imaging Spectrometer
3.4.2 The S-Matrix Grill Spectrometer
3.4.3 A Second Doubly Encoded Spectrometer
3.4.4 If mn measurements are Made
3.5 More Measurements than Unknowns; Generalized Inverses
3.5.1 Moore-Penrose Generalized Inverse
3.5.2 Singly Multiplexed Spectrometer
3.5.3 Doubly Multiplexed Instrument
3.5.4 Realizing a Hadamard Design with Two Detectors
3.6 Comparison with the Michelson Interferometer
3.6.1 Comparison of Singly Encoded Instruments
3.6.2 Comparison of Doubly Encoded Spectrometers with Wide Aperture Interferometers
3.6.3 Mertz's Mock Interferometer
3.6.4 Rotating Grid Collimator
3.6.5 Comparison of Noise in Different Multiplexing Methods
Chapter 4. Noise, or When to Multiplex and When to Avoid it
4.1 Observed Gain in Signal-to-Noise Ratio
4.2 Light Losses
4.3 Sources of Noise
4.4 Comparison of Noise in Different Instruments
Chapter 5. Instrumental Considerations
5.1 Optics for Singly Encoded Spectrometers
5.2 Pupils and Stops
5.3 Diffraction
5.4 Aberrations
5.4.1 Spherical Aberration
5.4.2 Coma
5.4.3 Distortion
5.4.4 Curvature of Field
5.4.5 Astigmatism
5.4.6 Chromatic Aberration
5.5 Image Defects occurring in Dispersive Spectrometers
5.5.1 Curvature of Slit Image
5.5.2 Anamorphic Magnification
5.6 Optical Limitations in Practice
5.7 Grating Efficiency
5.8 Doubly Encoded and Imaging Spectrometers
5.9 Operation of a Doubly Encoded Spectrometer as an Imaging Spectrometer
5.10 Design of Imaging Spectrometers
5.11 Electronic System Constraints
5.12 Light Gathering Power of Hadamard and Fourier Transform Spectrometers
Chapter 6. Systematic Errors
6.1 Introduction
6.2 Description of Instrument and Spectrum: Simplest Case
6.2.1 Description of Instrument
6.2.2 Description of Spectrum
6.2.3 Operation as a Multiplexing Spectrometer
6.2.4 Example 1: No Diffraction
6.2.5 Example 2: Full Diffraction
6.2.6 Operation as a Monochromator
6.2.7 Recovery of Spectrum
6.2.8 Computing the Spectrum. Inverse Matrices
6.3 Errors Occurring with a Mask which is Moved in Steps
6.3.1 Faulty Mask Alignment
6.3.2 Differences between Slit Width and Step Size
6.3.3 Undercutting in the Mask Pattern
6.3.4 Excessive Gap between Encoding and Blocking Masks
6.3.5 Nonlinearities
6.4 Distortion Introduced by a Continuously Moving Mask
6.4.1 Derivation of Basic Equation
6.4.2 An Ideal Case
6.4.3 If the Mask Velocity or Slit Width is Wrong
6.4.4 Rotating Two-Dimensional Masks
6.4.5 Imaging a Moving Source
6.5 Effect of Drift in Background Level
6.6 Singular Designs
6.6.1 More Measurements than Unknowns
6.6.2 More Unknowns than Measurements
6.6.3 Correction Procedures
Chapter 7. Applications
7.1 Chemical Spectroscopy
7.2 Imaging Spectrometry
7.3 Photoacoustic Spectroscopy and Imaging
7.4 Determining the Sensitivity of Color Film
7.5 Arrays of Detectors
7.6 Hadamard Transforms in Image Processing
7.7 Astronomy
7.8 Medical Applications
7.9 Miscellaneous Applications
Appendix. Hadamard and S-Matrices, Walsh Functions, Pseudo-Random Sequences, and the Fast Hadamard Transform
A.1 Introduction
A.2 Construction of Cyclic S-Matrices
A.2.1 Definition of Hadamard and S-Matrices
A.2.2 The Quadratic Residue Construction of S-Matrices
A.2.3 The Construction of S-Matrices Using Maximal Length Shift-Register Sequences
A.2.4 The Twin Prime Construction for S-Matrices
A.2.5 Other Constructions of Hadamard Matrices
A.3 Hadamard Matrices and Pseudo-Random Sequences
A.4 The Relationship between Hadamard Matrices and Walsh Functions
A.5 Hadamard Matrices and Error-Correcting Codes
A.6 The Fast Hadamard and Fourier Transforms
A.6.1 A Fast Way to Multiply by a Hadamard Matrix
A.6.2 A Fast Way to Multiply by Sn-1
A.6.3 The Fast Fourier Transform
Bibliography
Index