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2003 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Fourier Analysis and Imaging

verfasst von: Ronald Bracewell

Verlag: Springer US

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
1. Introduction
Abstract
Telecommunication by radio shrank the world to a global village, and the satellite and computer have made imagery the language of that village. The creation of images was once mainly in the hands of artists and scribes. Their art works—the famous stone-age cave paintings, the engravings on stone, bone and tooth, and the paintings on bark and skin that have been lost—go back to the distant past. Their inscriptions are also images, though of a different kind. They are also very old. as witnessed by hieroglyphic writing on the walls of tombs and on papyrus in Egypt. In modem times artists and scribes were joined by photographers and printers as creators of images. A variety of artisans, playing a secondary role as artists, created a third kind of image: ornament manifested on pottery, in woven fabrics and carpets, in tile patterns, and in painted friezes. Builders and cartographers created a fourth kind of image, once relatively rare but nevertheless going back to antiquity: construction plans and maps.
Ronald Bracewell
2. The Image Plane
Abstract
Picture plane is a traditional name in descriptive geometry for the location of a two-dimensional image. It corresponds with the plane of the grid used by Albrecht Dürer (1471–1528) as illustrated in Fig. 2-1 and with the film plane in a camera. However, not all images arise from projection: image plane is a more general term for the domain of a plane image. Images may be presented in many ways, such as by photography, television, or printing, and there are other modes of presentation that may be equivalent to images. Examples of equivalents include an array of numbers, a voltage waveform (from which a television picture could be generated), or a contour diagram of the kind introduced in 1728 by M. S. Cruquius and most familiar (Fig. 2-2) in the representation of surface relief on maps. One chooses according to end use: a passport photo rendered in grey levels is more appropriate than the equivalent contour diagram or matrix of numerical values, if it is a matter of identifying the bearer. All of these forms, in one way or another, are equivalent to a single-valued scalar function of two variables. and therefore the theory of such functions must be basic to an understanding of images. even though a full understanding of images may go farther.
Ronald Bracewell
3. Two-Dimensional Impulse Functions
Abstract
Just as in one dimension we need entities that are concentrated at a point in time, so in two dimensions we find it useful to introduce corresponding entities that are concentrated at a point in the (x, y)-plane. But in two dimensions there are two generalizations, the second being concentration on a line. One class of examples is furnished by the familiar mechanical notion of pressure; in mechanics, the oldest branch of physics, the concept that we require is well established in connection with the theoretical idea of a point force or point load applied to a surface. We recall that in one dimension the term impulse itself derives from mechanics, where it signifies the time integral of a force that is applied for a time that is more or less short. It is therefore useful, in introducing the theory of impulse functions on the plane, to appeal to the subject of mechanics, where the essential concepts are already familiar. First we deal with the two-dimensional point impulse or dot and then with a variety of impulse arrays and with impulsive entities that are not restricted to a point but are distributed along straight or curved lines. The δ-notation of P. A. M. Dirac (1902–1987) generalizes to allow convenient representation of the two-dimensional concepts.
Ronald Bracewell
4. The Two-Dimensional Fourier Transform
Abstract
In this chapter the two-dimensional Fourier transform is defined mathematically, and then some intuitive feeling for the two-dimensional Fourier component is developed. Just as we understand that a waveform can be broken down into time-varying sinusoids, so also we can acquire a corresponding physical picture of the decomposition of a single-valued surface in space. A set of useful theorems analogous to the more familiar one-dimensional theorems is presented for reference. Since many of these theorems are of enduring value, it is worthwhile choosing and memorizing those that you think may be useful to have at your fingertips in the future.
Ronald Bracewell
5. Two-Dimensional Convolution
Abstract
Convolution derives its importance for time-dependent signal processing in a fundamental way. Input and output signals need not be related through convolution, but they are if (and only if) the processor is linear and time invariant. Now these two conditions are just those that are met very commonly indeed in analogue systems. Therefore, convolution is exactly what we want for the neat expression of the cause-effect relationship in a large fraction of all signal transmission systems, both natural and artificial.
Ronald Bracewell
6. The Two-Dimensional Convolution Theorem
Abstract
Just as in one dimension, the convolution theorem in two dimensions plays a pervasive role wherever linearity and shift invariance are simultaneously present. In one dimension shift invariance most commonly means time invariance. A time-invariant system has the property that the response to an input impulse is independent of epoch. In other words, if two different input impulses are considered, one shifted in time by any amount with respect to the other, then the responses will be the same, allowing for the time shift. In two dimensions, where the variables represent space, the corresponding attribute of a system is space invariance. Suppose that a television camera is pointed at a blackboard. Then a space-invariant system has the property of imaging the blackboard onto the screen of the television display so that two different white dots of chalk, no matter where they are in the plane of the board, produce appropriately shifted identical images (Fig. 6-1).
Ronald Bracewell
7. Sampling and Interpolation in Two Dimensions
Abstract
Sampling and interpolation in two dimensions is much richer than in one dimension. Not only are there polar coordinates and other coordinate systems in addition to cartesian, but sampling can be done along lines as well as at points. The distinction between point and line sampling will be discussed first. Since sampling at regular intervals plays a major role, we pass on to regular point sampling as expressed in one dimension by the shah function III(x), an entity that was introduced earlier in connection with delta functions. In two dimensions we find the direct generalization 2III(x, y), which occupies a square lattice of points, together with a range of other manifestations of the shah function. Since the Fourier transform of a sampling pattern is a transfer function, this treatment brings us to the brink of digital filtering. Before developing this subject in the next chapter, however, we discuss the sampling theorem and interpolation.
Ronald Bracewell
8. Digital Operations
Abstract
Many of the topics already taken up have underlying implications of numerical evaluation, even where the treatment is presented in the form of continuous analysis. In some cases the computational evaluation is straightforward, as with many integrals, but in other cases ideas enter in that are associated with discrete mathematics. Many of these ideas come to the fore the present chapter, which begins with a variety of frequently needed elementary operations, such as smoothing and sharpening digital images, and continues on to introduce morphological operations, such as dilation and erosion, that are prerequisite to handling binary objects and to following the literature of feature recognition in images. Where it helps to clarify numerical procedures, short segments of computer pseudocode have been supplied, and the opportunity has been taken to point out how algebraic notation such as A ∪ B and A ⊕ B translate directly into the simple logical expressions acceptable to computers.
Ronald Bracewell
9. Rotational Symmetry
Abstract
A television image is rectangular and uses two independent variables in its representation of a three-dimensional scene. However, the television camera has an optical system with component parts, such as the aperture containing the objective lens, which, while distinctly multidimensional, possess circular symmetry. Only one independent variable, namely radius, may be needed to specify the instrumental properties across such an aperture, because the properties may be independent of the second variable, in this case angle. Circular symmetry also occurs in objects that are under study (some astronomical objects, for example) or in objects that influence imaging (rain drops), and is a property of many artifacts. Because of the prevalence of circular symmetry, particularly in instruments, special attention to circularly symmetrical transforms is warranted. This chapter also deals briefly with objects having rotational symmetry, which are of less frequent occurrence, but related.
Ronald Bracewell
10. Imaging by Convolution
Abstract
Many images are constructed by active convolution arising from scanning motion. In others, even though motion is absent, the dependence of the image on the object can be expressed as a convolution integral, at least approximately. Examples of the former arise when the ground is scanned by the antenna of a microwave radiometer carried in an airplane or satellite for remote sensing purposes, when the sky is scanned by a radio telescope as in radio astronomy, or when a photographic transparency is scanned by passing light through a small orifice to a photodetector as in microdensitometry. When an object is photographed, pp the image is produced simultaneously rather than sequentially, but the intervention of the camera lens introduces degradation which also involves two-dimensional convolution. Image construction for the most part is linear, but even with linearity the convolution relation may be perturbed where there is partial failure of space invariance, as with lenses. Not uncommonly (as with overexposure or underexposure) there may also be some degree of nonlinearity. Both linearity and space invariance are prerequisites to the following discussion, which treats cases where there is a simple convolution relation between object and image. First we consider methods such as antenna scanning and photography, where the convolving function is a compact but fuzzy diffraction pattern, and then microdensitometry and other types of imaging, where the convolving function is more or less sharply bounded.
Ronald Bracewell
11. Diffraction Theory of Sensors and Radiators
Abstract
The word diffraction was invented by Francesco Maria Grimaldi (1618–1663) to describe the tendency of light to deviate from propagation in straight lines, for example at a shadow edge, and to break up into dark and light bands. Grimaldi’s work was done at about the same time that Isaac Newton (1642–1727) was studying refraction by a prism.
Ronald Bracewell
12. Aperture Synthesis and Interferometry
Abstract
To record the image of a sunset we point the camera toward the setting sun, and the rays diverging from the various elements of the scene fall on the lens and are focused, element by element, on the film. Some people undoubtedly conceive of a camera as a device that reaches out and captures the distant scene, and years ago it was vigorously contended that vision reached out from the eye much as a hand reaches out in a dark room to explore the surroundings. It is, of course, perfectly all right to describe photography as in the introductory sentence above, but with the caveat that the description involves information about certain entities (the source elements and the propagation medium) that are unascertainable by the camera alone. In many branches of science the output of the observing instruments is the only information—astronomy is one example, and seismic exploration of the earth’s interior is another. Both of these endeavors produce images of inaccessible regions of physical space, and they suggest another, more operational, way of describing image formation.
Ronald Bracewell
13. Restoration
Abstract
Whenever a time-varying quantity has to be measured, there is an inevitable blurring due to the nonzero time interval necessary to make a single measurement. Consequently, the measurement relating to a given instant always lumps together values that occurred during the measurement interval. the time-varying quantity is varying slowly, or if the time resolution is short, the measurement may be very good; but some degree of smoothing is in principle always present. That means that a measured waveform never faithfully represents the original quantity. Therefore, one may ask what correction has to be applied to the measurement. In the development given below it is supposed that convolution is involved, but it is understood that convolution is merely an important case of the more general linear functional. Measurements may also involve a little nonlinearity and a little hysteresis. Even so, as will be seen, there is enough complication in the simple presentation below to suffice for a first study. A further limitation may be mentioned. In some subjects, including astronomy, meteorology, and geophysics, the measurements, or observations, may be all we know about the time-varying quantity; in fact the purpose of the observations may be to find out what is there. In such a case the “time-varying quantity” is just a concept without any reality of its own; only the measurements exist and are available to work with. In experimental subjects, as distinct from the observational, the underlying quantity may be verifiable by alternative methods, but in the final analysis measurement is always conducted to a finite resolution. Procedures for combating the smoothing effect of instrumental intervention are known as restoration.
Ronald Bracewell
14. The Projection-Slice Theorem
Abstract
When a function has circular symmetry, its two-dimensional Fourier transform can be expressed as a Hankel transform in terms of the single radial variable \( q = \sqrt {u^2 + v^2 } \) in the transform plane, as noted earlier.
Ronald Bracewell
15. Computed Tomography
Abstract
When a two-dimensional function f(x, y) is line-integrated in the y-direction or, as we say, is projected onto the x-axis, the resulting one-dimensional function ∫ f(x, y) dy does not contain as much information as the original. But other projections, such as the projection ∫ f(x, y) dy′ onto the x′-axis, where (x′, y′) is a rotated coordinate system, contain different information. From a set of such projections one might hope to be able to reconstruct the original function. The best-known example comes from x-ray computed tomography, where Hounsfield’s brain scanner has had a dramatic effect in radiology and dependent fields such as neurology, but reconstruction of projections was already known in more than one branch of astronomy and has continued to arise in many different contexts. Some indispensable preliminary theory for following the reconstruction techniques is given in the preceding chapter. In this chapter the theory is developed in terms of the x-ray application in order to get the benefit of being able to interpret the equations in physical terms.
Ronald Bracewell
16. Synthetic-Aperture Radar
Abstract
Some of the most striking images of the earth’s surface have been made from aircraft in level flight using radar in a sophisticated modality. The foliage of trees is stripped away and the ridges and gullies of the hard surface are shown in stark relief. For the purpose of topographic survey the technique supersedes airborne stereophotography, which itself had earlier superseded ground survey, in mountainous and forested areas. An example of such an image is shown in Fig. 16-1.
Ronald Bracewell
17. Two-Dimensional Noise Images
Abstract
Noise in an image can be defined as the unwanted part of that image. The noise may be random in some way, as is the pepper-and-salt appearance on a television screen when the station goes off the air, or it may be systematic, as with the ghost seen when an echo of the wanted signal arrives with a time delay after reflection from a hill. When the television image responds independently to sparks in a faulty thermostat in the nearby refrigerator or to a faulty ignition system on a passing motorcycle, the noise exhibits both random and systematic features. In other cases, the wanted signal may be random; thermal microwave or infrared radiation used for mapping the ground is of this nature. As a result, one person’s noise may be another person’s signal and vice versa. Very often it does not matter much what the character of the noise is, only its magnitude is needed, an attitude that is reflected in the term signal-to-noise ratio. As the examples show, the noise in an image need not be independent of, but can be closely connected with, the wanted signal itself. In the latter case if the signal is removed, the noise will change. When the noise is independent, it may be studied on its own in the absence of any wanted signal.
Ronald Bracewell
Backmatter
Metadaten
Titel
Fourier Analysis and Imaging
verfasst von
Ronald Bracewell
Copyright-Jahr
2003
Verlag
Springer US
Electronic ISBN
978-1-4419-8963-5
Print ISBN
978-1-4613-4738-5
DOI
https://doi.org/10.1007/978-1-4419-8963-5