We considered four analyticcal methods for reconstruction method here. The Fourier transformation of is calculated practically by computers using the discrete Fourier transformation with sampled. Thus is obtained only at discrete points located radially on -plane. The discrete inverse Fourier transformation of requires at square lattice points. Since the radially located points and the lattice points are not generally synchronized, the values of at the lattice points have to be estimated from the values at the radially located points by some interpolation.
The error by the interpolation in the frequency domain can yield an artifact, which is a noise not existing in the original image but caused by the processing, spread over the whole image. The artifact causes a severe misjudgment in image-aided medical diagnosis, since such diagnosis should find an object that should not be normally observed, for example a tumor. In second method, back-projection BP , the reconstructed image is blurred by convolution distribution function, , with blurring factor,. Therefore, after back-projection, it is necessary to filter the oversampling in the Fourier space in order to have equal sampling throughout the Fourier space see Figure 9.
This cone filter accentuates values at the edge of the Fourier space and de-accentuates values at the center of Fourier space. However, this method has two problems:. However, from Eq. It means that the DC component of is zero and negative values should appear in. This is a contradiction. The reason is that diverges at and no information on is obtained there. To avoid this advantage, in next method, back-projection and filtering steps is interchanged.
In fourth method, filtered back-projection, first projection data is filtered and then back-projected. This method does not require the inverse Fourier transformation of the spread blurred image, since the Fourier transformation is applied to the projections only. Although this method requires an interpolation between the polar coordinate to the Cartesian coordinate similarly to the Fourier transformation method, no artifact spread over the whole real domain is occurred, since this method carries out the interpolation in the real domain contrarily to the Fourier transformation method.
In general, the most well known of these methods is the filtered back-projection FBP , based on the Central Slice Theorem and easy to be implemented. On the other hand, it does not take into account any of the factors that. Figure 7. Illustration of cone ramp filter. Figure 8. Flow of filtered back-projection FBP method. Figure 9. Therefore, it is necessary to perform radiation interactions correction either before or after the reconstruction. In general, FBP is available in all commercial nuclear medicine imaging systems and the resulting images are adequate for the majority of routine clinical problems.
Defrise, M. In: Townsend, D. Jain A. Prentice Hall, Upper Saddle River. Herman, G. Academic Press, New York. Springer-Verlag, Berlin. Kak, A. In: Ekstrom, M. Bruyant, P. Journal of Nuclear Medicine, 43, Kinahan, P. All rights reserved. McMullen and E. Schulte Abstract Regular Polytopes G. Gierz et al. Continuous Lattices and Domains S.
Finch Mathematical Constants Y. Gasper and M. Rahman Basic Hypergeometric Series, 2ed M. Pedicchio and W. Tholen Categorical Foundations M. Olivieri and M. Vares Large Deviations and Metastability A. Kushner, V. Lychagin, and V. Beineke and R. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Radon transforms.
If you are intrigued with this idea, then, no matter what your background, you will find that at least some portion of this book will provide interesting reading. If this idea is not intriguing, then I would recommend some other publication for your reading pleasure. The unifying idea of tomography is the Radon transform, which is introduced in an informal and graphic way in chapter 1. Reading chapter 1 will give you a good idea of the precise meaning of tomography.
Reading chapter 2 will give you a very good idea of the meaning of tomography and if you read the last few chapters you will have a really good understanding of this idea. However, some of the later chapters will only be accessible to specialists. I tried to write this book with two main ideas in mind. I wanted it to appeal to the broadest possible group of readers and I wanted it to be as comprehensive as possible.
Therefore, chapter 1 has almost no mathematics in it — at least it does not require the reader to have any background beyond a good course in secondary school mathematics. CT computerized tomography scanners are used for medical diagnosis and produce detailed pictures of the human anatomy without opening up the patient. The dedicated reader will learn, in a very graphic way, how a CT scanner works. I hope this chapter will also be interesting to specialists who will see how the Radon transform and integral convolutions correspond to some familiar everyday processes.
Chapter 2 presupposes some knowledge of calculus. I tried to write it so that a second-year undergraduate student in mathematics or the sciences would have enough background to read the chapter. However, most readers of this chapter should have a few more mathematics courses beyond the elementary calculus sequence. This chapter rigorously introduces the Radon transform which is the basis of the rest of the book.
Students in the basic calculus sequence should know these formulas, at least in dimension two. However, and here 1 2 Introduction is where it requires some dedication, I develop the theory in n dimensions, explaining the necessary generalizations as the chapter proceeds.
Chapter 3 requires at least some graduate-level mathematics. This is where we generalize the Radon transform to the k-plane transform. The resulting analysis is much deeper than that of the first two chapters. The minimal requirements are a knowledge of real analysis at least through the general theory of Lebesgue measure and integration and some elementary Fourier analysis and a minimal familiarity with group theory.
Readers having this sort of background should include graduate students in mathematics, even just after the first year, and many scientists and engineers. There are many more ideas necessary to understand this chapter, but I have taken care to at least explain the notation and basic concepts and to provide references for the interested reader whose background does not reach this far.
There is no reason to expect that even a mathematician who is not a full-time analyst would have a good enough working knowledge of Grassmann manifolds, Haar measure, and distribution theory to really understand this chapter. Therefore, I have tried to err on the side of providing more detail, even if some ideas and arguments could be made more succinct when aimed at a specialist in the field.
This will probably annoy most of my colleagues, and for this I hastily offer an apology. However, I think other readers will be thankful for the amount of detail that I have provided. The material in chapter 3 is essentially self-contained beyond the prerequisites that I just mentioned. Any more advanced ideas are described carefully enough that a mathematically literate reader should be able to follow the arguments, although considerable effort may be required.
Except for a set of measure zero, all proofs in this chapter are self-contained. However, in the remaining chapters I do not always provide full proofs. In general, in these chapters I present some basic ideas with full details and then I expand on these ideas. But I do not always give full, or sometimes even any, proofs. When proofs are omitted I always provide appropriate references to the literature.
Sometimes I mention this, but, in general, the lack of a proof indicates that the demonstration may be found in the associated reference. I have provided a brief summary of prerequisites in the introduction to each chapter. Here are the exceptions to the basic policy that I have just outlined. In chapter 1 there is a technical note that requires knowledge of some elementary calculus. In chapter 3, section 3. These results require a much more extensive development of the Riesz potential than I was able to provide.
In fact, I probably would have required another volume just to provide these prerequisites. Therefore, I took the liberty of stating the main ideas and results about Riesz potentials without proof but with references.
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I have tried to make this book a comprehensive treatment of the subject of analytic methods in tomography. Introduction 3 every aspect of this field. You will occasionally find more detail. Also, in these sections, you will occasionally find historical comments. I am not a historian, but I have tried to make these comments as accurate as possible.
Some topics could, maybe even should, be in this volume, but because of space and time restrictions they were not included. One example that comes immediately to mind is the area of impedance tomography. This area is extremely interesting, valuable, and analytic in nature. Similarly I could only give a brief introduction to the relation between tomography and partial differential equations, twistors, several complex variables and D modules.
I hope that my readers will see that any field not included or only briefly treated is of the nature that would probably require a volume all by itself. However, I have tried to make at least passing mention of any area of tomography that is at all related to analysis. Tomography, which may be justifiably defined as the study of the Radon transform, is itself part of the field of integral geometry. Tomography is divided into roughly two fields: geometric tomography and analytic tomography.
Although geometric tomography is mostly concerned with probing the interior structure of geometric objects by using the techniques of geometry, analytic tomography has the same aim but uses techniques that are intimately related to both classical and modern real analysis, and sometimes also to complex analysis. These techniques include Lebesgue integration, the theory of distributions, and Fourier analysis.
Geometric tomography is treated in the excellent book by Gardner , but not at all by me. There are also several texts that deal with the analytic aspects of tomography see section 2.
There is some overlap between this volume and these other texts, but I believe that this volume has a unique emphasis, choice of topics, and point of view. A search of the mathematical literature since that has a relation to tomography or the Radon transform will yield well over 2, publications. This does not take into account publications in other fields such as medicine, physics, and engineering.
I have no idea what the total number of papers on tomography is, but I would not be surprised if it is greater than 10, Although one can trace the origins of tomography further back, the year is generally believed to signal the beginning of tomography. Because of the time and page restraints, I had to narrow this down even more. The result is the set of papers that you see in my bibliography. This is a substantial set of references, but clearly it is not exhaustive. Therefore, I apologize to any colleagues who were not mentioned or who received only a passing mention. This set includes many colleagues who work on the more applied area of the subject.
Their work is interesting and valuable, but unfortunately I could not include every possible reference in this book. The reader should be aware that this is a book on pure mathematics. This is inevitable because mathematical analysis depends on infinite processes, whereas any applied mathematical problem eventually has to deal with a finite process. Not many CT scanners exist that need to handle objects defined by general L p functions. So, if you are planning to build a CT scanner in your garage, this book is not going to tell you how to do it.
Image Reconstruction Techniques
However, I believe that many engineers, physicists, and applied mathematicians will benefit from the theory that is presented here. Tomographers tend to be split into two, probably disjoint, classes. Members of the first class believe that nothing practical can come out of a theorem depending on, say, infinitely many x-ray projections. The other class, of which I am a member, disagrees. However, because I respect the opinions of the first class, I therefore abandon all pretense of presenting practical applications, although in a few places I make some remarks heading in that direction.
This frees me to present the general theory, which happens to be a beautiful mathematical gem. I had an enormous amount of help while writing this book. I thank my wife Ruth Markoe for her support and love and for putting up with me during this project. I also thank my children for the same reasons. I am grateful for the support provided by Rider University in the form of research leaves and financial grants for this project. I am particularly grateful for the help I received from the interlibrary loan department of the Moore Library of Rider University.
This gave me the opportunity to pursue this research in a very pleasant and productive environment. I am especially grateful to Jessica Farris of Cambridge University Press and the anonymous copy editor at TechBooks who had to deal with many fatuous errors on my behalf. My gratitude and a salute go to my students Harry Doctor and Sharon Kobrin who proved that the first two chapters of this book could be read by undergraduates.
Also my thanks go to them for help in proofreading those chapters. Finally, I thank my many colleagues who helped me while I was writing this volume and from whom I learned so much. There are very few mathematical requirements for this chapter, so readers who are not specialists in the field, indeed who are not mathematicians or scientists, should find this material accessible and interesting. Specialists will find a graphic and intuitive presentation of the Radon transform and its approximate inversion. Tomography is concerned with solving problems such as the following.
Suppose that we are given an object but can only see its surface. Could we determine the nature of the object without cutting it open? In an Austrian mathematician named Johann Radon showed that this could be done provided the total density of every line through the object were known. The total density along a line is the sum of the individual densities or amounts of material.
In Wilhelm Roengten discovered x-rays, a property of which is their determining of the total density of an object along their line of travel. For this reason, mathematicians call the total density an x-ray projection. It is immaterial whether the x-ray projection was obtained via x-rays or by some other method; we still call the resulting total density an x-ray projection. We call this process tomography. Tomography can be applied to any object for which we can determine the x-ray projections either by actual x-rays or some other method.
Tomography is used to investigate the interior structure of the following objects: the human body, rocket motors, rocks, the sun microwaves were used here rather than x-rays , snow packs on the Alps, and violins and other bowed instruments. This list could be expanded to hundreds of objects. In this chapter we will see how tomography can be used to obtain detailed information about the human brain from its x-ray projections. Johann Radon tries to figure out what is inside the sphere. Allan M. Cormack and Godfrey N. Hounsfield shared the Nobel Prize in Physiology and Medicine for their contributions to the medical applications of tomography.
The most interesting request for a reprint came from the Swiss Centre for Avalanche Research. The method would work for deposits of snow on mountains if one could get either the detector or the source into the mountain under the snow! We can visualize the discussion up to this point. In figure 1. In the next scene he decides to compute the total density on a single plane through the sphere. He knows that this is not enough information to determine the object, so he successively intersects with more and more planes.
When he has collected the densities on all planes, then he is able to determine the object. How this may be done by using lines through a two-dimensional object is the subject of the remainder of this chapter. You do not need much background in mathematics to read this chapter — some knowledge about triangles and the ability to read a graph is really all that is required. For example,  refers to the article by Cormack that is listed in the references section. This term was first used in diagnostic medicine. Since the discovery of x-rays by Roentgen, diagnosticians have attempted to produce images of human organs without the blurring and overlap of tissue that occurs in traditional x-ray pictures, such as the x-ray of the skull in the accompanying figure.
Courtesy of Ass. We will see that tomography can produce much more detailed pictures from x-ray data. The reference to the fog clearing in the title of this section is from the presentation speech for the Nobel prize for Physiology or Medicine which was awarded, jointly to A. Cormack and G. Hounsfield in The presentation speech containing the preceding quotation was delivered by Professor Torgny Greitz of the Karolinska Medico-Chirurgical Institute and it is interesting to read the excerpts from this speech in Section 1.
Computerized Tomography, also known as CT, refers to the actual process of producing a detailed picture of the interior of an organism by using x-rays. Mathematical tomography refers to the mathematical process by which the picture is obtained. Computerized tomography is accomplished by designing a machine consisting of x-ray sources and x-ray detectors combined with a computer.
The computer uses an algorithm adapted from the field of mathematical tomography to combine the data obtained from the x-rays into a detailed picture of the organs and tissue in a specific slice of a 1. A typical CT scanner. This one is manufactured by the Siemens Corporation. Courtesy of the Siemens corporation. This type of machine is called a CT scanner. This is done without cutting open the body, merely by sending x-rays through the tissue in question. How this is done is explained later in this chapter.
Some forms of tomography were used in diagnostic medicine long before computers were invented see Section 1. A typical method attempted to visualize a section slice of a body by blurring out all the x-rays except those in the focal plane of the desired slice. Early CT scanners also concentrated on a single slice of the body. The desire was to visualize a sliced human body without actually slicing it. In mathematical tomography the slicing refers to the lines or planes that slice through the object of interest.
Figure 1. The circular ring in the CT scanner emits x-rays from a source on one side. These x-rays are detected at the opposite side. Here is a diagram of how this operates. There is the story of the man who brought his sick dog to the veterinarian. Upon examination, the veterinarian pronounced the dog dead.
Is there not a more definitive test that you can do? The cat proceeded to examine the dog. First, the cat only sniffed around the dog who exhibited no reaction. Then the cat hissed at the dog and finally clawed it, all without reaction from the dog. How much do I owe you? That is outrageous! Why is it so much? In this mode x-rays are generated at the source. They form a beam in the shape of a fan and are observed at the detector after passing through the body.
In this way the total density along every line emanating from the source can be computed. By rotating the apparatus, the source and detectors move to new positions. In this way the total density along every line intersecting the body can be determined. These total densities are the input data to an algorithm that reconstructs a picture of the organs and tissue in this slice. Later we will show how these total densities can be used to reconstruct an image of the tissue. Meanwhile, see figure 1. Note the lack of detail of the Figure 1. Left traditional x-ray.
Right Tomographic reconstruction of a brain section. Image on the left courtesy of Ass. CT images. The left image is a cross section of a human brain. The right image is a cross section of a human abdomen. Another set of tomograms is in figure 1. When an x-ray beam is sent through tissue, it experiences more attenuation by heavier tissue than by lighter tissue. For example, the skull is about twice as dense as the gray matter of the brain. Therefore, x-rays are more likely to be absorbed or scattered when passing through the skull than when passing through gray matter.
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Although there are some subtleties with this idea, we can make a working assumption that sending an x-ray beam through an object determines the total density of the object on the line intersected by the x-ray. Before continuing we should mention that older CT machines used a parallel beam geometry. Their mode of operation is illustrated by the following figure. It is 12 1 Computerized Tomography, X-rays, and the Radon Transform simpler to describe the mathematics for parallel beam scanners and from now on we will do so. The method of reconstructing images via parallel beam geometry can be applied to fan beam scanners because, as we already noted, information about any line intersecting the object can be obtained in either geometry.
However, the computational effort in reorganizing the fan beam data into parallel beam data is substantial. There are algorithms that use the fan beam data directly and these will be described in a later chapter. The efficiency of fan beam algorithms together with the efficiency of the fan beam scanning geometry make current CT scanners much faster than older ones.
The latest generation of CT scanners emit x-rays along a helical path and reconstruct three-dimensional pictures. In other situations the unknown quantity might be a function with some given information about its behavior. For example, determine the unknown position of a particle given its acceleration, its initial position, and its initial velocity. In a tomographic problem the given information is a set of x-ray projections of an unknown object. The solution is an exact or approximate representation of the unknown object obtained by mathematical manipulation of the known x-ray data.
In general, a function is a rule that uniquely assigns a value to each element of a given set. The given set is called the domain of the function. In this chapter we assume that the value assigned to an element of the domain is a real number. An example of a function is given by the rule f which assigns x1 to every nonzero real number x. Here the domain is the set of non-zero real numbers. Then, for each x in the domain, f x represents the quantity obtained by applying this rule to x.
However, sometimes we are sloppy and use f x to denote the function f , even though f x actually is a real number representing the value of the function f at x. It is not necessary for the domain of a function to consist of numbers. For example, we can consider the function h, which assigns to every horse its weight in kilograms. In this case the domain is the set of all horses. In the profile of the mountain, the domain was one-dimensional the ground line.
Although the profile of the mountain is two-dimensional, the amount of information needed to determine the profile is only one-dimensional. This is because the function describing the elevation is of the form f x , where x is a single variable that moves along a straight line. In general, a two-dimensional object will require two variables to be completely determined. A general point in the plane is uniquely determined by the coordinates x, y. The density of the tissue at each point is depicted by the amount of gray at that point.
The actual tissue has varying density. The other tissue is less dense and is represented by various shades of gray. To describe this picture, all we need to know is the relative amount of gray to put at each point. This amount can be specified by a real number. Therefore, for all practical purposes, this image can be represented by a function f of two variables: f x, y represents the amount of gray to place at the point x, y to create this picture.
Analytical image reconstruction methods in emission tomography
An object can thus be viewed as a function of two real variables x and y, because we are uniquely assigning a real number a gray value to every point x, y of the plane. Conversely, if we are given a function of two variables, then we can view the associated object by using the value of the function at every point as a gray value. So from the mathematical point of view, there is no difference between a function of two variables and a two-dimensional object.
Tomogram of a human abdominal section. Courtesy of the Siemens Corporation. Also, for this reason we use letters like f and g to represent objects. There are two main ways to exhibit the value of a function at a point. The first way is called a density plot and it attaches a gray level to each point in the domain. An example of a density plot is the tomogram in figure 1. Each gray level represents a specific real number, the lowest values of the function shown in black and the highest values shown in white.
The gray scale presented in the next figure shows how any real number between 0 and 1 can be represented as a gray level. It is not a function, it only serves to establish the correspondence of gray levels to a range of real numbers.
The gray scale plays the same role as the x axis in a graph — it shows how we represent real numbers. One purpose of the gray scale is to establish the range of numbers used in the graph of the object. The range does not have to be from 0 to 1. It could be from any real number a to any larger real number b.
However, the smallest value will always be represented as black and the largest as white. After this example, we will not be fussy about the actual range of values, so we will present objects without the accompanying gray scale. This type of view is called the graph of the object. An example of a graph of an object may be found in figure 1. Graph of a function representing a mountain. Simple objects may be represented by functions that take the value 1, represented by white, at all points that lie on the object, and that take the value zero, represented by black, elsewhere. Therefore, their density graphs will be exactly the shape of the object.
To avoid becoming overly wordy let us agree that when we use a term such as triangular object we really mean the function that is 1 on the triangle and zero elsewhere. Here is the density plot of a square object. This is the three-dimensional analogue of how we represented a mountain earlier. In that case we placed the value f x above the location x on a line. Here we place the height f x, y above each point x, y in the plane.
Note that the graph of the mountain appears three-dimensional as it should. But it actually represents a function whose domain is two-dimensional. Next we show the density plot of the mountain. It is similar to contour plots of mountainous areas and is plotted on a two-dimensional domain. The next figure is a graph of the same cranial section. It appears three-dimensional, but because it represents a transverse section of a cranium, it actually corresponds to a two-dimensional object.
It is clear from these images that for some objects the density plot is a better representation than the graph. This is true for the square object and the brain image. On the other hand other objects may be better represented as graphs, as for the mountain. Most of the objects we need to consider will be represented as density plots.
However, the graphical representation is also useful and will be used if needed. We will investigate this process in more detail later see section 1.
For now we make the working assumption that one can determine the total density, or amount of material, along any line intersecting a specific object. Recall that the total density of an object along a line is called an x-ray projection of the object along that line. It is immaterial whether the knowledge of these x-ray projections was obtained via x-rays or by some other method.
We now examine the situation when the densities are known for all possible lines. The term Radon transform of an object is used to describe the collection of x-ray projections of an object along all possible lines. Given the Radon transform of an unknown object, find the object. This is a typical problem in tomography.
Before solving this problem, let us investigate the Radon transform in more detail. The Radon transform of an object or function7 consists of x-ray projections along all possible lines in the plane. Each line has a specific direction and each direction is uniquely identified by an angle. Conversely, given an angle, we can specify a unique direction. Therefore we can use the terms angle and direction interchangeably. With this in mind, the reader should have no trouble visualizing any angle or direction. We now show that we can think of the Radon transform of an object as a function of two variables.
Although there are many ways of doing this, we choose the method based on the following conceptual diagram of a CT scanner fig. This direction is indicated by the arrow perpendicular to the x-ray beam. We intend to take the x-ray projection along this thick line.
In this way we can obtain 6 7 In mathematics, when a function is defined with a domain consisting of other functions, then the new function is called a transform. Because the Radon transform assigns a set of x-ray projections to an arbitrary object i. More informally, if we know the Radon transform of an object f , then we have a way of knowing the density of f along any line. Later on we will provide a graphic way of describing the Radon transform. It is useful to introduce a variation of this notation when we are dealing only with the x-rays in a single direction.
Supposing that the total density of an 8 9 We also allow s to be negative. Geometry of x-ray projections. Now we take a simple object, a square, and see how to compute its Radon transform. We begin by computing a single x-ray projection. Let f be the function representing the object that is a square of side 2 centered at the origin.
Recall the earlier discussion in which we represented simple objects by functions that take the value 1, represented by white, at all points that lie on the object and that take the value 0, represented by black, elsewhere. The value of the x-ray projection of f can then be easily calculated. This is because the total density at each black point is 0 and at each white point is 1. A single x-ray projection. In CT this computation is automatically derived from the x-ray data.
The x-rays are therefore all vertical and they either intersect the square with the identical length 2 or else they completely miss the square. This is because in this area of the figure, f has the value 0 and hence the 22 1 Computerized Tomography, X-rays, and the Radon Transform Figure 1. The next step in the study of the Radon transform is to devise a way of visualizing the entire Radon transform, instead of a single projection.
The term sinogram is used for the density plot of the Radon transform. Here is a figure showing the sinogram for the square object under consideration. This is illustrated in the next figure. This coincides exactly with the graph of the x-ray projection seen in figure 1. Note that the graph of each individual x-ray projection forms a function of one variable as the theory indicates. You can check the plausibility of the other two x-ray projection graphs in the same manner.
Both plots represent the Radon transform for a section of a human body. One of these gives x-rays for a human abdominal section and the other for a human brain section. But which is which? And 1. Sinogram generated by the Radon transform of the square object, with graphs of three x-ray projections. It is hard to tell directly from the given information. We will determine the answer later. One of these graphs represents the x-rays through a human abdomen, the other represents the x-rays through a human brain.
Which is which? Single x-ray projection. The set of all such x-ray projections forms the Radon transform of the object. The problem in tomography is therefore to use the known x-ray data in the form of the Radon transform to reconstruct the unknown object from which it originates. The sinogram represents the known or given information for this problem. Soon we will give an explanation of how such a reconstruction can be accomplished. The explanation is based on the method used by CT scanners in medical applications.
However, we first introduce an intuitively plausible method, which appears, at first glance, to yield a reasonable reconstruction from x-ray projections in a very simple manner. Let us think of the material in the object f as being made of sand. The next figure illustrates this idea for several directions.
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We now describe the process of backprojection. The process is dual to the operation of taking projections. Instead of starting with an object in the plane and creating a pile of material underneath, we start with a pile of material and create an object in the plane. The object is created by smearing the material back into the plane and for this reason the process is called backprojection. Let g be the function representing the pile of material. A single backprojection. The white stripe consists of the material from the original x-ray projection which has now been backprojected.
Here we started with a blank plane and a pile of material underneath and we created an object, the white stripe, by the process of backprojection, in one direction. This is called backprojection in one direction. The Radon transform operates on a function by integrating it over hyperplanes. The first three chapters are devoted to elementary and graphical introductions to the Radon transform, tomography and CT scanners, a rigorous development of the basic prope rties of the Radon transform, and Grassmann manifolds to the study of k- plane transform.
The remaining chapters are concerned with some more advanced topics. He lives in Lawrenceville, NJ with his wife and the youngest of his three daughters. He graduated with a B. His fields of research include several complex variables, integral geometry and tomography. He is a member of the American Mathematical Society. His hobbies include playing Lacrosse, playing squash and flying. Convert currency. Add to Basket. Condition: New. Language: English. Brand new Book. The remaining chapters are concerned with more advanced topics, such as the attenuated Radon transform and generalized Radon transforms defined by duality of homogeneous spaces and double fibrations.
Questions of invertibility and the range of the Radon transform are dealt with and inversion formulas are developed with particular attention to functions on L2 spaces and some discussion of the case of Lp spaces. Seller Inventory LHB More information about this seller Contact this seller. Book Description Cambridge University Press, Book Description Condition: NEW. For all enquiries, please contact Herb Tandree Philosophy Books directly - customer service is our primary goal.