R. N. Bracewell

The fast Hartley transform

A fast algorithm has been worked out for performing the Discrete Hartley Transform (DHT) of a data sequence of N elements in a time proportional to Nlog2N. The Fast Hartley Transform (FHT) is as fast as or faster than the Fast Fourier Transform (FFT) and serves for all the uses such as spectral analysis, digital processing, and convolution to which the FFT is at present applied. A new timing diagram (stripe diagram) is presented to illustrate the overall dependence of running time on the subroutines composing one implementation; this mode of presentation supplements the simple counting of multiplies and adds. One may view the Fast Hartley procedure as a sequence of matrix operations on the data and thus as constituting a new factorization of the DFT matrix operator; this factorization is presented. The FHT computes convolutions and power spectra distinctly faster than the FFT.

Discrete Hartley transform

The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable. The inverse DHT is identical with the direct transform, and so it is not necessary to keep track of the +i and −i versions as with the DFT. Also, the DHT has real rather than complex values and thus does not require provision for complex arithmetic or separately managed storage for real and imaginary parts. Nevertheless, the DFT is directly obtainable from the DHT by a simple additive operation. In most image-processing applications the convolution of two data sequences f1 and f2 is given by DHT of [(DHT of f1) × (DHT of f2)], which is a rather simpler algorithm than the DFT permits, especially if images are. to be manipulated in two dimensions. It permits faster computing. Since the speed of the fast Fourier transform depends on the number of multiplications, and since one complex multiplication equals four real multiplications, a fast Hartley transform also promises to speed up Fourier-transform calculations. The name discrete Hartley transform is proposed because the DHT bears the same relation to an integral transform described by Hartley [ HartleyR. V. L., Proc. IRE30, 144 ( 1942)] as the DFT bears to the Fourier transform.

Fast two-dimensional Hartley transform

The fast Hartley transform algorithm introduced in 1984 offers an alternative to the fast Fourier transform, with the advantages of not requiring complex arithmetic or a sign change of i to distinguish inverse transformation from direct. A two-dimensional extension is described that speeds up Fourier transformation of real digital images.

Fourier analysis and imaging

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.- Eddingtons 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 Drunkards Walk.- Fractal Polygons.- Conclusion.- Literature Cited.- Problems.- Appendix A Solutions to Problems.

Searching for nonsolar planets

Abstract Existing instruments are unable to detect planets about stars other than the Sun but such detection would be important for the theory of origin of our solar system and in the search for extraterrestrial intelligence. Infrared offers an advantage of about 10 5 over visible light as regards the ratio of power received from star and planet. Infrared interferometry from Earth orbit would allow discrimination against the stellar infrared by the placement of an interference null on the star and a spinning infrared interferometer would modulate the planetary emission to permit extraction by synchronous detection from the background level. The limit to sensitivity will be set by thermal emission from the zodiacal light particles near the Earths orbit unless the interferometer is launched out of the ecliptic or out to the orbit of Jupiter, in which case instrumental limitations will dominate. Technological developments in several fields will be required as also with astrometry, spectroscopic radial velocity measurement, and direct photography from orbit, three approaches with which infrared interferometry should be carefully compared.

Observation of sea-ice dynamics using synthetic aperture radar images: automated analysis

Two techniques for automated sea-ice tracking, image pyramid area correlation (hierarchical correlation) and feature tracking, are described. Each technique is applied to a pair of Seasat SAR sea-ice images. The results compare well with each other and with manually tracked estimates of the ice velocity. The advantages and disadvantages of these automated methods are pointed out. Using these ice velocity field estimates it is possible to construct one sea-ice image from the other member of the pair. Comparing the reconstructed image with the observed image, errors in the estimated velocity field can be recognized and a useful probable error display created automatically to accompany ice velocity estimates. It is suggested that this error display may be useful in segmenting the sea ice observed into regions that move as rigid plates of significant ice velocity shear and distortion. >

Radio Interferometry of Discrete Sources

Salient features of the theory and practice of radio interferometry are presented with special attention to assumptions and to the specifically two-dimensional aspects of the subject. The measurable quantity on an interferometer record is defined as complex visibility by generalization from an analogous quantity in optical interferometry. Subject to conditions on antenna size and symmetry, the observed complex visibility is equal to the normalized two-dimensional Fourier transform of the source distribution, with respect to certain variables S. and S, which are defined. This transform is in turn identically equal to the complex degree of coherence Γ between the field phasors at the points occupied by the interferometer elements. The correlation between the instantaneous fields, and that between the instantaneous intensities are less general parameters which are, however, deducible from Γ. A theorem is proved according to which only certain discrete stations on a rectangular lattice need be occupied for full determination of a discrete source distribution. Procedures in interferometry are discussed in the light of this result and an optimum procedure is deduced. Current practice is considered over-conservative, e.g, independent data in the case of the sun are obtainable only at station spacings of about 100 wavelengths on the ground, a fact which has not hitherto been taken into account.

Image reconstruction over a finite field of view

The reconstruction of images from profiles has recently received wide attention as the result of the development of x-ray scanners that produce tomograms, or density maps of thin slices through an organ, without the presence of confusing contributions from background or foreground objects situated out of the slice of interest. This very important application to medical imagery now invites further development along many lines. Reduction of scanning time is necessary, both to increase the amount of work that one instrument can do and to permit work on moving organs such as the lungs and heart. Reduction of x-ray dosage is also desirable.

A three-dimensional DFT algorithm using the fast Hartley transform

A three-dimensional (3-D) Discrete Fourier Transform (DFT) algorithm for real data using the one-dimensional Fast Hartley Transform (FHT) is introduced. It requires the same number of one-dimensional transforms as a direct FFT approach but is simpler and retains the speed advantage that is characteristic of the Hartley approach. The method utilizes a decomposition of the cas function kernel of the Hartley transform to obtain a temporary transform, which is then corrected by some additions to yield the 3-D DFT. A Fortran subroutine is available on request.

Aspects of the Hartley transform

Known in the signal processing literature mainly as a computational technique, the Hartley transform has proven to possess attributes in the physical world of experiment, especially in optics and microwaves. The Fourier transform, which has constituted the standard approach to spectral analysis, is understood both in the domain of numerical analysis and in the world of natural phenomena, gaining its central significance from the ubiquity of sinusoidal natural behavior; it also gains appeal from the convenience offered by complex variable analysis, which is a basic part of technical education. Computers, however, prefer real numbers, an attribute that first opened a niche for the Hartley transform. As applications have multiplied, and others open up, it will be helpful to understand the Hartley idea from more than one point of view. Several topics, including multidimensional transforms, the complex generalization, and microwave phase measurement by amplitude measurement only, are discussed with the intention of throwing light on the numerical and physical relationships between the Hartley and Fourier transforms. >