CT and Radon

Ada Sedova Math 480 Senior Seminar Thesis Title: The Mathematics of Computed Tomography: Inverse Problems and the Radon Transform Abstract: The use of x-ray “slices” to recover the density of a solid object is known as Computed Tomography, or CT scanning. Tomography is used in medicine, and the first successful combination of mathematics and instrument received a Nobel prize in physiology and medicine in 1979. The mathematical problem involved is called an “inverse problem” in analysis. Techniques for solving this problem using Fourier analysis methods were developed several times in the history of pure mathematics, but were not known in applied sectors in the U.S. until Cormack re-invented a solution in his paper from 1963. A 1917 paper by the mathematician Johann Radon was later found which extensively explored and solved this very problem on a purely theoretical level, and the resulting “Radon transform” has now become the most widely used method for performing the computations in modern tomographic equipment. A history of this inverse problem along with a little known applied Soviet solution from 1958 is presented, including an explanation of the mathematics used in each of the three solutions. Keywords: Fourier analysis, inverse problems, tomography, Radon transform.

1. Introduction: An important advance in medicine in the past half century has been the development of non-invasive imaging techniques of soft tissue for diagnosis. To uncover the internal structure of a solid, such as the human body, without having to disturb the integrity of the body with punctures or incisions, is an improvement in health outcomes for a variety of pathologies, for example, brain diseases (for obvious reasons). Subtleties of this improvement are attributable to reduced rates of infection and other insults of surgery as well as increased early detection of deleterious conditions such as tumors. This “reconstruction problem” occurs not only in medicine but in several other fields of science ranging from molecular biology to astronomy. The recovery of an internal property of an object, such as density, involves some computational analysis which in turn requires specific techniques from applied mathematics. These applied techniques in turn use some deep results from pure mathematics, in the area of Fourier analysis. The mathematical problem is known as an inverse problem because it requires finding an inverse of a function mapping the unknown “density” function to its line integrals along all possible lines in a plane of interest (or hyperplanes in higher dimensions). This problem also intersects the field of integral geometry [6]. Finding the successful sequence of steps such as re-coordinatization of the plane, taking transforms, expressing functions as their Fourier series, and employing convergence factors and convolution, as well as other methods, is necessary for obtaining the solution. Computationally and practically, the problem has just begun when the pure mathematical solution is obtained. It is of course physically impossible to x-ray all possible lines in a plane. Additionally, the x-ray has finite width, the effects of

perturbations can be impossible to correct for, and uniqueness and continuity issues may preclude the solution. This type of phenomena describe what is known as the “technically ill-posed” nature of the problem. Discritization of the projections is necessary, and iterative methods are often employed, at the computational level. Improving accuracy of measurements and noise reduction is a problem for engineering. The Nobel prize in physiology and medicine 1979 was awarded to two mathematical physicists for (independently) solving the mathematical problem and creating a machine to perform the tomography successfully. These physicists could not find any pure mathematical results dealing with this inverse problem. Creating their own methods, Cormack and Hounsfield solved this inverse problem, as did a group of Soviet scientists in “radio-physics” in 1958. The pure mathematical solution had in fact been published, in German in 1917, by Johann Radon. Furthermore, Lorentz is said to have known of a solution at the turn of the 19th century. Gel’fand et. al. were familiar with Radon’s transform in 1966, and described it in their Generalized Functions. It was not until the 1970’s that Cormack learned of Radon’s solution to this very problem [4]. Radon transform theory has since proved to be the effective mathematical method of computing the unknown density function from its integrals along all lines, using the Central Slice Theorem [6]. The disconnection between pure and applied areas of mathematics has created an interesting story of re-discovery and an eventual return to a century-old theoretical technique. This paper will explore the progression of solutions of this mathematical problem from the pre-Dirac-delta-function days of Radon, through the Soviet solution of 1958 and that done by Cormack in 1963. Modern computational methods used in tomography will then be reviewed. 2. Radon, 1917: Physicist Allan McLeod Cormack was asked to be a part-time nuclear physicist for a hospital in S. Africa to fulfill regulations [3]. While exposed to this area of research he began to pose some questions, which resulted in a paper in the Journal of Applied Physics in 1963 entitled “Representation of a function by its line integrals, with some radiological applications.” In this paper, he states the basic problem:

“These considerations suggest that if a solution to the problem can be found at all, it must be sought by considering [the line integral] along all lines intersecting [the plane of interest] and then seeing whether an approximate solution may be found by considering only a finite number of lines… The following problem is thus considered. An unknown, suitably restricted, real function g exists in a finite, two-dimensional domain D and is zero outside D, and the line integrals of g along all straight lines intersecting D are known. Is it possible to determine g?”

This last sentence is followed by the comment, “One would think that this problem would be a standard part of the nineteenth century mathematical repertoire, but the author has found no reference to it in standard works [2].” In fact, this problem was visited by nineteenth century mathematicians. H. Lorentz is said to have known an inversion formula for three dimensions, and other mathematicians studied it after that [4]. The official result for the two and three dimensional cases was published in 1917 by Johann Radon. In 1973, (after realizing its existence), Cormack

cited Radon’s paper as being essential to reconstruction problems, and he did so again in his 1979 Nobel prize address. Radon posed this very problem in the introduction of his paper, “On the determination of functions from their integrals along certain manifolds,” [9]. He describes a point function, f(P), in the plane, integrated “along an arbitrary straight line g,” and designates as F(g) the values of these integrals. He then writes, “The problem that is solved in …this paper is the inversion of this functional transformation. That is, answers to the following questions are given: Is every line-function that satisfies suitable regularity conditions obtainable by this process? If this is the case, is the point function f then uniquely determined by F and how can it be found?” [9]. It has been assumed that Radon was exploring this problem simply out of theoretical mathematical curiosity [4, 8]. However, the x-ray had been discovered by William Roengten in 1895, only 22 years before Radon’s paper was published [6]. Upon consideration of the close ties mathematics and physics maintained at this time, it is reasonable to suspect that Radon may have been considering the practical problem of x-ray reconstruction himself. However, his result can be appreciated as purely mathematical, and his theory was developed along these lines by Gel’fand et al. in 1966 [4, 6]. Radon did not have at his disposal the use of more recent and advanced methods such as incorporation of the delta function, which is used in the modern solution, but he proceeded to solve the problem in the following way: Defining a line, tangent to a circle at the origin, and having the equation

Rotation of the line through all angles, while, keeping the line tangent to a circle, will cover a complete revolution around the circle and the line integrals can give us the circular average of the unknown function. If a circle has center P = (x,y) and radius q, this circular average can be expressed as

Then, by solving an Abel-type integral equation, we can find F and we have the inverse solution. Radon writes, “The value of f is completely determined by F and can be computed as follows:

” [9].

3. The Applied Problem: An Overview: An x-ray beam can be thought of as a collection of photons traveling along a straight line. For such a beam traveling through a homogenous material, the amount of photons was found experimentally to decrease with amount of distance traversed [6, 7]. A classical experiment to test for this phenomenon is the assembly of a series of identical plates (identical in size, thickness and composition). An x-ray beam is emitted by a source, and has a given number of photons initially. Photon detectors passing through each plate are placed on the plates on the side of the plate where the beam exits. The number of photons that “survive” after passing through each plate has been found to decrease at a consistent rate (see Figure 1) [6].

Figure 1. The diminishing of the number of photons in an x-ray beam after passage through identical plates. (From [6]). This loss of photons from the beam is known as attenuation[6]. The attenuation of x-rays is frequently denoted by µ, and is defined as the number of photons lost divided by the distance traveled. Thus if A is the number of photons emitted by the source, B is the number detected upon exiting a homogeneous material, and s is the distance traveled, then

.

Intensity is defined to be the number of photons present per second per unit of cross-sectional area. The intensity, denoted by I, of a beam at a distance s from its point of entry into a homogenous material has been found to closely follow the following

formula: .

This is called the “global version” of the Lambert-Beer law. By differentiating, we can obtain the “local version,”

. The local version works for inhomogeneous materials as long as the intensity function is relatively smooth. From the above equation we can see why µ is often called the attenuation coefficient. From this local formula we can see that the attenuation coefficient is a property of each point in each different material type within an inhomogeneous solid. Thus we can define a point function on a two-dimensional object with varying composition whose attenuation value at that point (x,y) is µ(x,y) [4, 6, 7]. Now, letting L be a line through the object along which the beam travels, and I0 and If be the intensity of the beam emitted and detected, respectively, we can rearrange the previous differential equation, integrate, and perform a change of variables to obtain:

.

The integral of µ along L is the total attenuation of the material along L, and results in what is called a projection. The right side of the equation above can be computed directly from the experimental results from the ratio of intensity emitted to intensity detected. If the source of the x-rays is moved, a series of these projections is obtained, and this is called a profile [4,6,7]. Attenuation is caused by scattering and absorption of the photons by the material. This is closely related to the density of the material. It also depends on the energy of the beam and the types of atoms in the material, among other physical factors. However, we can obtain a good approximation of density from µ, especially when using corrective algorithms for the other factors, (and when we have experience with connecting the values of the attenuation coefficient to certain types of materials, as are most radiologists)[6]. The problem then becomes the following: If we know the value of a sufficient amount of these line integrals, or total “densities” along a x-ray beam path, can we then recover the density of the material at each point in our 2-dimensional object? In other words, knowing enough values for

can we recover µ(x,y)? What is the sufficient number of such lines? In order to begin to answer these questions, we must rigorously define what we mean by “object,” in the mathematical sense [6]. Definition: A two dimensional “object” is a density plot of the function f = f(x,y) that assigns to each point (x,y) a grayscale value representing an attenuation value (density value) for the material at that point. The values can take on any range, with black representing the least attenuation and thus the lowest density, and white representing the greatest values. Choosing units so that the range is from zero to one is often useful. Figure 2 shows an example of an object and the associated grayscale [6].

Figure 2. The 2-dimesional object f representing an abdominal cross section, and the grayscale used to denote the values for f(x,y). (From[6]). The above object is an example of a density plot. In this type of graph, the values of a function of two variables is plotted using the grayscale rather than the three dimensional graph using the triple axis as is often done in the multivariable calculus. The information we need to obtain is more clearly evident in such a graph [6]. To understand how many line integral values are necessary to reconstruct our “density” function, we shall use a thought experiment based on one used by Cormack, although similar reasoning was undoubtedly used by Radon and others. Let our region of interest be a square of side length n, and divide the square into n2 sub-squares. Assume our density function takes on its average values in each sub-square. If we attempt to take the values of the line integrals along parallel lines in each perpendicular direction, one line through each sub-square, we will have 2n equations with n2 unknowns, and because the sum of the values in one direction must equal the sum of the values in the other direction, we see that only 2n-1 of these equations are independent [2]. Now we try to take a larger number of lines, for instance, along the radii of a circle. Letting our object be a circle, and denoting the region between a finite number of radii as n = {0, 1, 2, …}, and let our density function be cos(nπ); we have alternating values of positive and negative one in the regions. If we then take the line integrals perpendicular to each radius through all real numbers on the radius, we have that the function on either side of each radius is symmetric, and thus the value of the line integral is zero for all such lines. However, the density function is not zero anywhere on the object [2]. From these attempts we come to the conclusion that the density function can only be reconstructed from the infinite set of line integrals along all possible lines intersecting the object [2, 9]. This is, of course, a problem in the applied solution, but we can approximate the infinite set with an appropriate sample of lines to within a desired error [6,7]. To find the mathematical solution to the inverse problem, we must consider our inversion to act on the entire infinite set. The set of line integrals of a point-function defining an object, for all lines intersecting the object, has come to be known as the Radon transform. The line integral along one line is called a single Radon projection or x-ray projection [4, 6, 7, 8] . Since the object is a function of two variables, it is useful to consider the Radon transform as a function of two variables. The set up of the x-ray machine provides intuition about how to describe it this way. Taking the perpendicular vector from the origin to our line in question, we define our line uniquely by the (directional) length of

the vector and its angle from the x-axis (or θ = 0 axis) [2,4,5,6,9,11]. Interestingly, Radon also defined his lines in this way, and noted that the set of those lines through the origin which are not definable by this method form a “set of linear measure zero,”[9].

Figure 3. The scanner set-up and the associated definition of a line. (From[6]. The original CT scanners were set-up using a series of parallel beams that rotated together around the object, as in Figure 4. Modern scanners use fan beams, and the newest designs employ a helical x-ray source. The image obtained from a complete scan, or from the line integral values for a complete revolution, is called a sinogram (Figure 5).

Figure 4. The parallel scanning set-up. [8]. Figure 5. A sinogram of a simple object (a square). [6]. We now explore some of the mathematical and technological solutions to this reconstruction problem. 4. Soviet Union, 1958: A group of scientists at the Kiev institute, which included physicists and mathematicians, were considering this problem in 1958 [1]. The experimental design involved a rotating object that was subject to a narrow fan beam of x-rays and whose output was recorded using a film parallel to the axis of rotation [5]. The mathematical problem is posed as such:

Here the scanner arrangement and definitions of the variables are described by a figure reproduced in Figure 6 below.

Figure 6. Fig. 1 from the 1958 Russian tomography article published by a research group from the Kiev Polytechnic Institute, describing the mathematical and physical set-up. [5]. After taking a two dimensional transform of the unknown function, we see that we can consider the inner integral as the line integrals along lines defined by a perpendicular vector where the dot product of this vector and any point on the line is constant. Thus the exponential term is constant with respect to the inner integral and can come out to the outer integral, and we are left with the one dimensional Fourier transform of the line integral transform (known now as the Radon transform). This is essentially what is now called the Central Slice Theorem, or Fourier Slice Theorem, along with other names [6, 7, 11]. Upon the consideration of the smoothness assumptions for our rearranged integral, we see that we can use the inverse Fourier transform formula. The integral for this does not converge absolutely and thus the order of integration can not be interchanged. It is

necessary to use a method of convergence factors. The convergence factor e-δ|r| is used where r is one of the vectors. After removing a singularity by means of the Cauchy principle value for a singularity at zero, we obtain our inversion formula. It is notable that the final result necessarily had to take a form that was passable to an analog computer used at the time. Thus the formula had to be processable separately by a differentiator, an integrator, a multiplier, and an adder, using inherent properties of electrical currents. The final solution is included below:

[5].

4. Cormack, 1963-1964: Cormack approached his problem using the complex Fourier series for the unknown attenuation function, g, as well as the known line integral function, f(ρ,φ). For example,

[2].

After defining the same line as did Radon, by means of the perpendicular vector from the origin, he also uses the symmetry of the arc length along the line on either side of its intersection with the vector to simplify the expression [2]. The appearance of a Chebyshev polynomial of the first kind,

,

allows him to make use of some of this type of polynomial function’s properties.

Here Tn is the Chebyshev polynomial and r is the distance along the line. Cormack then has to solve Abel’s equation,

[2]

His final solution is of the form,

which reflects the advancements in computers since the time of the Russian paper. The integral above does not converge, and thus must be solved using numerical methods. 5. The Modern Version The modern algorithm uses the integral transform methods, the method of convergence factors with the delta function, and the Central Slice Theorem, illustrated in the figure below [11].

Figure 7. Conversion from the 2D F.T. to the 1D F.T. of the Radon transform. [11]. The central slice theorem, similar to the one used by the Russian group, expresses the two dimensional Fourier transform of the unknown function as the one dimensional F.T. of the known Radon transform [5, 6, 7, 11 ]. The inverse solution is found by the inverse F.T., and is assisted by recoordinatizing the plane to lines along which the inner product of the perpendicular vector and the points on the line are constant [8]. A filtering function is used, by means of convolution, to average the noise values of the Riesz potential nature of the backprojection, or inverse object [12]:

The Radon transform

The inverse: backprojection

Backprojection F.T. domain filtering using convolution

Projection Theorem ( F_polar => G)

Change coordinates (Cartesian => polar)

6. The Problem Is Not Solved Now that we have some mathematical solutions, we have only solved half of the problem. The next step is in the computational and applied fields. A problem is said to be

“well-posed (in the sense of Hadamard) if 1. there exists a solution, 2. the solution is unique, 3. the solution depends continuously on the data

If these conditions do not hold, a problem is said to be ill-posed.” (From Michael Renardy and Robert C. Rogers, An Introduction Partial Differential Equations, Springer-Verlag, New York, 1993, p.8).

Due to the frequent lack of unique solutions, and instability of solutions, among other issues, tomographic image reconstruction is ill-posed on the applied, clinical level [4,7,11,12]. We can never know an infinite amount of line integral values, so the problem must be discretized. Many applied algorithms exist to correct details related to physical phenomena and improve approximations [7]. Modern methods of computing, algorithm development, and mechanical technological advancements have reduced scanning times of tens of milliseconds, a significant change from the hours it took for the scanners of Cormack and Hounsfield’s time [6]. References 1. Barrett, H.H., Hawkins, W.G., and Joy, M.L.G., “Historical note on computed

tomography,” found on the web at http://www.radiology.arizona.edu/CGRI/russian/russian_papers.html, University of Arizona, Tucson.

2. Cormack, A.M., “Representation of a function by its line integrals, with some radiological applications,” J. Applied Physics (1963): 34, pp.2722-2727.

3. Cormack, A.M., “Early two-dimensional reconstruction and recent topics stemming from it,” Nobel Lecture, 8 December, 1979. Available at http://nobelprize.org/nobel_prizes/medicine/laureates/1979/cormack-lecture.pdf

4. Deans, S.R., The Radon Transform and Some of its Applications, John Wiley & Sons, New York, 1983.

5. Korenblum, B., Telel'baum, S.I., and Tyutin, A.A., "About one scheme of tomography", Bulletin of the Institutes of Higher Education - Radiophysics, 1

(1958) 151-157. Translated into English from Russian by H. H. Barrett, Department of Radiology and Optical Sciences Center, University of Arizona, Tucson.

6. Markoe, A., Analytic Tomography, Cambridge University Press, NewYork, 2006. 7. Olafsson G., and Quinto, E.T., “An introduction to X-ray tomography and RadonTransforms,” in Olafsson G., and Quinto, E.T., eds., The Radon Transform, Inverse Problems, and Tomography, American Mathematical Society, Providence, 2006. 8. Prestini, E. ,”The Radon transform and computerized tomography,” in Prestini, E., The

Evolution of Applied Harmonic Analysis, Birkhauser, Boston, 2004. 9. Radon, J., “On the determination of functions from their integrals along certain

manifolds,” (translated from the German by R. Lohner, School of Mathematics, Georgia Institute of Technology), in Appendix A of [4] above. Originally published in German in Berichte Sächsische Akademie der Wissenschaften 69 (1917) 262-267.

11. “Tomographic image reconstruction,” published on the web under “meetings” by the American Association of Physicists in Medicine, at

www.aapm.org/meetings/99AM/pdf/2806-57576.pdf 12. Wu, Min. “Lecture 22: Medical Imaging/2nd Course Review,” published on the web

as Course Review for ENEE631 Digital Image Processing, Electrical and Computer Engineering, University of Maryland, College Park. Found at http://www.ece.umd.edu/class/enee631.