Radon and Abel Transforms

96 Pages·2000·1.7 MB·English

Deans, S.R. “Radon and Abel Transforms.”
The Transforms and Applications Handbook: Second Edition.
Ed. Alexander D. Poularikas
Boca Raton: CRC Press LLC, 2000
8
Radon and Abel
Transforms
Stanley R. Deans 8.1 Introduction
University of South Florida Organization of the Chapter • Remarks about Notations
8.2 Definitions
Two Dimensions • Three Dimensions • Higher Dimensions •
Probes, Structures, and Transforms • Transforms between
Spaces, Central-Slice Theorem
8.3 Basic Properties
Linearity • Similarity • Symmetry • Shifting • Differentiation •
Convolution
8.4 Linear Transformations
8.5 Finding Transforms
8.6 More on Derivatives
Transform of Derivatives • Derivatives of the Transform
8.7 Hermite Polynomials
8.8 Laguerre Polynomials
8.9 Inversion
Two Dimensions • Three Dimensions
8.10 Abel Transforms
Singular Integral Equations, Abel Type • Some Abel Transform
Pairs • Fractional Integrals • Some Useful Examples
8.11 Related Transforms and Symmetry, Abel and Hankel
Abel Transform • Hankel Transform • Spherical Symmetry,
Three Dimensions
8.12 Methods of Inversion
Backprojection • Backprojection of the Filtered Projections •
Filter of the Backprojections • Direct Fourier Method • Iterative
and Algebraic Reconstruction Techniques
8.13 Series
Circular Harmonic Decomposition • Orthogonal Functions on
the Unit Disk
8.14 Parseval Relation
8.15 Generalizations and Wavelets
8.16 Discrete Periodic Radon Transform
The Discrete Version of the Image • A Discrete Transform • The
Inverse Transform • Good News and Bad News
Appendix A: Functions and Formulas
Appendix B: Short List of Abel and Radon Transforms
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8.1 Introduction
The Austrian mathematician Johann Radon (l887-1956) wrote a classic paper in 1917, “Über die Bestim-
mung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten” (on the determination
of functions from their integrals along certain manifolds) [Radon, 1917]. This work forms the foundation
for what we now call the Radon transform. English translations are available in the monograph by Deans
[1983, 1993] and the translation by Parks [1986]. The problem of determining a function f (x, y) from
knowledge of its line integrals (the two-dimensional case), or a function f (x, y, z) from integrals over
planes the (three-dimensional case) arises in widely diverse fields. These include medical imaging,
astronomy, crystallography, electron microscopy, geophysics, optics, and material science. In these appli-
cations the central aim is to obtain certain informaton about the internal structure of an object either
by passing some probe (such as x-rays) through the object or by using information from the source itself
when it is self-emitting, such as an organ in the body that contains a radioactive isotope, or perhaps the
interior of the Earth when motions occur. Comprehensive reviews of these and other applications are
contained in Brooks and Di Chiro [1976], Scudder [1978], Barrett [1984], Chapman [1987], and Deans
[1983, 1993].
The general problem of unfolding internal structure of an object by observations of projections is
known as the problem of reconstruction from projections. Many situations arise when it is possible to
determine (reconstruct) various structural properties of an object or substance by methods that utilize
projected information and leave the object in an essentially undamaged state. The Radon transform and
it inversion forms the mathematical framework common to a large class of these problems. This problem
of reconstructing a function from knowledge of its projections emerges naturally in fields so diverse that
those working in one area seldom communicate with their counterparts in the other areas. This was
especially true prior to the advent of computerized tomography in the 1970s. As a consequence, there is
an interesting history of the independent development of applications of the Radon transform by indi-
viduals who were not aware of the original work by Radon in 1917, or of contemporary work in other
fields. Those interested in pursuing these historical matters can consult Cormack [1973, 1982, 1984],
Barrett, Hawkins, and Joy [1983], and Deans [1985, 1993].
Also, the Radon transform has varying degrees of relevance in three Nobel prizes: (Medicine 1979,
Allan M. Cormack and Godfrey N. Hounsfield) [DiChiro and Brooks, 1979, 1980], [Cormack, 1980],
and [Hounsfield, 1980]; (Chemistry 1982, Aaron Klug) [Caspar and DeRosier, 1982]; (Chemistry 1991,
Richard R. Ernst) [Amato, 1991].
As short a time as a decade ago, the Radon transform was known by very few engineers and scientsits.
Only those working directly on reconstruction from projections in one of the major areas of application
had knowledge of this transform.Today, the Radon transform is widely known by working scientists in
medicine, engineering, physical science and mathematics. It has made its way into the image processing
texts [Kak, 1984, 1985], [Kak and Slaney, 1988], [Jain, 1989,], [Jähne, 1993], and is widely appreciated
in many diverse areas; among the best known include: medical imaging [Herman, 1980], [Macovski,
1983], [Natterer, 1986], [Swindell and Webb, 1988], [Parker, 1990], [Russ, 1991], [Cho, Jones, and Singh,
1993]; optics and holographic interferometry [Vest, 1979]; geophysics [Claerbout, 1985], [Chapman,
1987], [Ruff, 1987], [Bregman, Bailey, and Chapman, 1989]; radio astronomy [Bracewell,1979]; and pure
mathematics [Grinberg and Quinto, 1990], [Gindikin and Michor, 1994].
The purpose of this chapter is to review (and illustrate with examples) important properties of Radon
and Abel transforms and indicate some of the applications, along with important sources for applications.
Because the Abel transform is a special case of the Radon transform, most of the discussion is for the
more general transform. This is especially important to keep in mind for applications where the Abel
transform can be used. Section 8.10 is devoted to Abel integral equations and Abel transforms. The
formal connection between Abel and Radon transforms is made in Section 8.11; the reader primarily
interest in Abel transforms may want to look at those two sections first.
The overall goal is to provide the reader with basic material that can be used as a foundation for
understanding current research that makes use of the transforms. A conscientious attempt is made to
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present essential mathematical material in a way that is easily understood by anyone having a basic
knowledge of Fourier transforms. In keeping with this goal, the emphasis will be on the two-dimensional
and three-dimensional cases. The extension to higher dimensions will be mentioned at various times,
especially when the extension is rather obvious. For the most part, derivations are kept as simple and
intuitive as possible. Reference is made to more rigorous discussions and abstract applications. The same
policy is followed for highly technical problems related to sampling and numerical implementation of
inversion algorithms. These are ongoing research problems that lie a level above the basic treatment
presented here. Section 8.1.1 contains a brief summary of how the chapter is organized. An attempt is
made to cross reference the various sections, so the reader interested in a given topic can go directly to
that topic without having to read everything that precedes. Finally, it is to be noted that liberal use is
made of material contained in books by the author on the same subject [Deans, 1983,1993].
8.1.1 Organization of the Chapter
Section 8.2 is devoted mainly to fundamental definitions, concepts, and spaces. The definitions are given
several ways and for various dimensions to make it easier for the reader to make connection with usage
in the current literature. The section on probes, structure, and transforms outlines the connection of the
Radon transform to physical applications. A very important theorem known as the central-slice theorem
serves to relate three spaces of special importance: feature space, Radon space and Fourier space. A proof
is provided for the two-dimensional case and an example is given to illustrate how a function transforms
among the three spaces.
Some of the most basic properties of the Radon transform are presented in Section 8.3 and compared
with the corresponding properties for the Fourier transform. These properties are used many times
throughout the sections that follow.
A brief, but important, discussion of the Radon transform of a linear transformation is in Section 8.4.
This provides the foundation for powerful methods to calculate transforms of various functions. In
Section 8.5 this idea is combined with the basic properties to illustrate, by several examples, just how
the Radon transform works when applied to certain special functions. These examples are selected to
bring out subtle points that emerge when actually computing a transform.
More advanced topics on derivatives and the transform are in Section 8.6. This work serves as
background for transforms involving Hermite polynomials in Section 8.7 and Laguerre polynomials in
Section 8.8.
The important problem of inversion is initiated in Section 8.9. Details are given for two and three
dimensions, and the foundation is provided for some of the currently utilized inversion methods outlined
in sections that follow.
Abel transforms and Abel-type integral equations are discussed in Section 8.10. Four different types
of Abel transforms are defined along with the corresponding inverses. Interrelationships among the
transforms are illustrated along with several useful examples. A rule is given to establish a method for
finding Abel transforms from extensive tables of Riemann-Liouville and Weyl (fractional) integrals. The
way the Radon and Fourier transforms relate to the Abel and Hankel transforms is developed in
Section 8.11. An important observation is that the Abel transform is a special case of the Radon transform.
Examples are given to demonstrate the connection for specific cases.
The earlier work on inversion is supplemented in Section 8.12 by some methods that form the basis
for modern algorithms for numerical inversion of discrete data using backprojection and convolution
methods. Diagrams that clearly illustrate the various options are included in this section.
Series methods for inversion are discussed in Section 8.13, with emphasis on two and three dimensions.
Special attention is given to functions defined on the unit disk in feature space. Several examples are
provided to illustrate both techniques and the connection with earlier sections.
The Parseval relation for the Radon transform is given in Section 8.14 for the general n-dimensional
case. A useful example in two dimensions serves to highlight the difference between the Fourier and
Radon cases.
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Extensions and emerging concepts are mentioned briefly in Section 8.15. An especially exciting area
involves the use of the wavelet transform to facilitate inversion of the Radon transform.
Finally, Appendix A contains a compilation of formulas and special functions used throughout the
chapter, and a list of selected Radon and Abel transforms appears in Appendix B .
8.1.2 Remarks about Notation
The Radon transform is defined on real Euclidean space for two and higher dimensions. Many results
are just as easy to obtain for the n-dimensional transform as for the two-dimensional transform. However,
most illustrations (and applications) of the transform are easier in two or three dimensions. Consequently,
several equivalent notations are appropriate for vectors. Various notations are given here and the policy
throughout the entire discussion is to change freely from one notation to the other with absolutely no
apology.
Both component and matrix notations will be used. In component notation, all of the following
expressions are used,
( ) ( ) ( )
x=r= x, y x= x ,x y= y , y .
1 2 1 2
In matrix notation these would be:
     
x x y
x=r=  x= 1 y= 1 .
y x  y 
2 2
Similar notations are used for three dimensions by appending z or x or y . For the n-dimensional
3 3
case we use:
( ) ( )
x= x ,…,x y= y ,…, y ,
1 n 1 n
or the equivalent matrix form. When there is no confusion about which variables are being integrated,
the abbreviated notation
∫ ( ) ∫∞ ∫∞ ( )
f x dx≡ L f x ,…,x dx …dx
1 n 1 n
∞ −∞
will be used for integration over all space.
8.2 Definitions
In a discussion of the Radon transform it is convenient to identify three spaces. These spaces are designated
by feature space, Radon space, and Fourier space.
Feature space is just Euclidean space in two, three, or n dimensions, designated by 2D,3D, or nD. This
is where the spatial distribution f of some physical property is defined. Radon space and Fourier space
designate the spaces for the corresponding transforms of this distribution. Functions in feature space
that represent the distribution are designated by f (x, y), f (x, y, z), and f (x ,...,x ), depending on the
1 n
dimension of the transform. For the purposes of this presentation, these functions f are selected from
some nice class of functions, such as the class of infinitely differentiable (C∞) functions with compact
support or rapidly decreasing C∞ functions [Schwartz, 1966]. This assumption serves well for the current
discussion; however, it can be relaxed in more general treatments (Gel’fand, Graev, and Vilenkin, 1966],
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FIGURE 8.1 Cordinates in feature space used to define the Radon transform. The equation of the line is given by
p = x cos φ + y sin φ.
[Lax and Phillips, 1970, 1979], [Helgason, 1980], [Grinberg and Quinto, 1990], [Mikusin´skiand Zayed,
1993], [Gindikin and Michor, 1994].
The transformation from one space to another can be represented symbolically as a mapping operation.
Let (cid:2) be the operator that transforms f to Radon space. If the corresponding function in Radon space
˘
is designated by f, the mapping operation is expressed by:
f˘ = (cid:2)f. (8.2.1)
In a similar way, the transformation to Fourier space is written:
f˜ = (cid:3)f. (8.2.2)
These operations will be made more precise in the next sections where explicit definitions are given for
various dimensions.
8.2.1 Two Dimensions
The Radon transform of the function f (x, y) is defined as the line integral of f for all lines l defined by
the parameters φ and p, illustrated in Figure 8.1. There are several ways this can be expressed. In terms
of integrals along l,
( ) ∫∞ ( )
f˘ p,φ = f r dl, (8.2.3)
−∞
where r = (x, y) is a general position vector. Another way to write this is to define the unit vector ξξξξ =
(cos φ, sin φ) and the perpendicular vector ξξξξ′ = (–sin φ, cos φ), then the position vector is given by r =
p ξ + t ξ′ and (note that r2 = p2 + t2)
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( ) ∫∞ ( )
f˘ p, ξξ = f pξξ+tξξ′ dt. (8.2.4)
−∞
An equivalent definition making use of the delta function (see Chap[ter 1) is most convenient for the
current discussion,
( ) ∫∞∫∞ ( ) ( )
f˘ p,φ = f x, y δ p−xcosφ−ysinφdxdy. (8.2.5)
−∞ −∞
Note that due to the property of the delta function and the fact that the normal form for the equation
of the line l is given by p = x cos φ + y sin φ, the integral over the plane reduces to a line integral in
agreement with the previous definitions. A slightly different form proves especially useful for generali-
zation to higher dimensions. In terms of the vectors r and ξξξξ,
( ) ∫∞∫∞ ( ) ( )
f˘ p,ξ = f r δ p−ξξ⋅r dxdy , (8.2.6)
−∞ −∞
where ξξξξ · r = ξx + ξy = x cos φ + y sin φ.
1 2
˘
It is important to understand that f is not defined on a circular polar coordinate system. The appro-
priate space is on the surface of a half-cylinder. Consider an infinite cylinder of radius unity. Let the
parameter p measure length along the cylinder from –∞ to +∞, and let the angle φ measure the angle of
rotation with respect to an arbitrary reference position. A point on an arbitrary cross section of the
cylinder is represented by (p, φ) as illustrated in Figure 8.2.
˘
Observe that from the definition of the transform, if f is known for –∞ < p < ∞, then only values of
φ in the range 0 ≤ φ < π are needed. To verify this, recall that the delta function is even δ(x) = δ(–x),
and the change φ → φ + π corresponds to ξξξξ→ –ξξξξ. Hence, the coordinates (–p, φ) and (p, φ + π) denote
˘
the same point in Radon space. Likewise, the function f is completely defined for 0 ≤ p < ∞ and 0 ≤ φ < 2π.
˘
More will be said about properties of f in Section 8.3.
FIGURE 8.2 Coordinates in Radon space on the surface of a cylinder.
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Now, suppose we unroll the half-cylinder in Figure 8.2. The resulting surface is a plane with points
represented by (p, φ) on a rectangular grid. It is convenient to let p vary along the vertical axis and φ
along the horizontal axis, restricted to the range 0 to π. This construction is especially useful for
˘
illustrations because the values of f can be represented as a surface in the third dimension perpendicular
to this plane. Also, note that for most practical applications the object of interest in feature space does
˘
not extend to infinity. Suppose f (r) = 0 for (cid:2)r(cid:2) > R, where R is finite. It follows that f = 0 for (cid:2)p(cid:2) > R, and
p varies on a finite interval.
To help interpret (8.2.6) let f (x, y) represent the density (in 2D) for some finite mass distributed
throughout the plane. (Here we are considering a special case of the more general result in nD discussed
by Gel’fand, Graev, and Vilenkin [1966]. If (cid:4)(p, ξξξξ) denotes the total mass in the region ξξξξ · r < p, then
( ) ( ) ( ) ( )
∫∫ ∫∫
(cid:4) p,ξξ= f x, y dxdy= f x, y (cid:5) p−ξξ⋅r dxdy,
ξξ⋅r<p
∂(cid:5)(p)
where (cid:5)(·) denotes the unit step function. Now from the relation = δ(p) for generalized functions,
∂p
the above equation becomes
( )
∂(cid:4) p,ξξ ∫∞ ∫∞ ( ) ( ) { ( )}
= f r δ p−ξξ⋅r dxdy=(cid:2) f x, y . (8.2.7)
∂p −∞ −∞
This result shows that if f (x, y) denotes a density with which a finite mass is distributed throughout
space, its Radon transform is
( )
( ) ∂(cid:4) p,ξξ
f˘ p,ξξ= .
∂p
where (cid:4)(p, ξξξξ) is the mass in the half-space ξξξξ · r < p, and the derivative with respect to p is assumed to
exist. It is important to observe that to have complete knowledge of the Radon transform one must know
the mass distribution for all values of the variables p and ξξξξ. If the transform is found for only selected
values of these variables, we may call the result a sample of the Radon transform. The next example
illustrates this idea.
Example 1
Find a sample of the Radon transform for the case shown in Figure 8.3, for the case where the mass in
proportional to the area. For simplicity, let the proportionality constant be unity. The equation of the
line specified in the figure is x = p and the angle is φ = 0. The required sample is found from
f˘= ∂A ,
∂p
where A is the area in the neighborhood of the line x = p. This example is simple enough to yield, by
simple calculus for finding areas, an explicit expression for A as a function of p,
( ) ∫p
A p =2 1−x2 dx.
0
It follows that f˘ = 2 1−p2.
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FIGURE 8.3 A semicircle of unit radius. The equation of the line is x = p.
In this example, it is worth noting that although a sample of the Radon transform is found, the result
has relevance to the entire Radon transform for circular symmetry. More will be said about this in several
˘
of the sections that follow. Also, observe that f depends on how A changes with p where the derivative
is taken, and not on how much area lies to the left or right of the line x = p.
From (8.2.3) the Radon transform can also be defined by
( ) ( )
∫
f˘ p,φ = f x, y ds, (8.2.8)
ξξ⋅r=p
where the integration is taken along the line ξξξξ · r = p and ds is an infinitesimal element on the line.
Observe specifically that each line can be uniquely specified by the two coordinates φ and p.
In terms of rotated coordinates of Figure 8.4, Equations (8.2.5) and (8.2.8) can be expressed in the
form (with x = p cos φ – t sin φ, y = p sin φ + t cos φ)
( ) ∫∞ ( )
f˘ p,φ = f pcosφ−t sinφ, psinφ+t cosφ dt . (8.2.9)
−∞
This reflects a rotation of the coordinate axes by φ such that the p axis is perpendicular to the original
line ξξξξ · r = p. The above equation can also be interpreted as follows: if f (p, t) is the representation of
φ
˘
f (x, y) with respect to the rotated coordinate system, then f (p) is the integral of f (p, t) with respect
φ φ
to t for fixed φ. That is
( ) ∫∞ ( )
f˘ p = f p,t dt , (8.2.10)
φ φ
−∞
where f (p, t) = f (p cos φ – t sin φ, p sin φ + t cos φ). The interpretation given here covers those cases
φ
where the Radon transform is treated as a function of a single variable p with the angle φ = Φ viewed
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FIGURE 8.4 Rotated coordinates so the line of integration (dashed) is perpendicular to the p axis.
as a parameter. In this case the functions of p for various values of Φ are called the projections of f (x, y)
at angle Φ.
8.2.2 Three Dimensions
The definition given by (8.2.6) is easy to extend to three dimensions. Let the line l be replaced by a plane,
and let the vector ξξξξ be a unit vector from the origin such that the vector pξξξξ is perpendicular to the plane.
That is, the perpendicular distance from the origin to the plane is p and the vector ξξξξ defines the direction.
Now, the equation of the plane is given by p = ξξξξ · r, where the position vector is extended to three
dimensions, r = (x, y, z). The Radon transform of this function is given by
( ) ∫∞∫∞∫∞ ( ) ( )
f˘ p,ξξ= f r δ p−ξξ⋅r dxdydz . (8.2.11)
−∞ −∞ −∞
Here, it is understood that the integral is over all planes defined by the equation p = ξξξξ · r.
8.2.3 Higher Dimensions
The extension to higher dimensions is accomplished by defining the position vector r = (x ,...,x ),
1 n
extending the unit vector ξξξξ to n dimensions, and integrating over all hyperplanes with equation given
by p = ξξξξ · r,
( ) ∫∞ ∫∞ ( ) ( )
f˘ p,ξξ= L f r δ p−ξξ⋅r dx …dx . (8.2.12)
1 n
−∞ −∞
Although we do not emphasize use of the transform in higher dimensions in this discussion, it should
be noted that the nD version is just a natural extension of the 3D transform. And, as might be expected,
most of the major properties and theorems are just logical extensions of the corresponding results for
two and three dimensions [Ludwig, 1966], [Helgason,1980].
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