The Role of Linear Algebra in Computed Tomography

Transcription

1 The Role of Linear Algebra in Computed Tomography Katarina Gustafsson under the direction of Prof. Ozan Öktem and Emer. Jan Boman Department of Mathematics KTH - Royal Institution of Technology Research Academy for Young Scientists July 10, 2014

2 Abstract In some cases plain X-ray examination from one direction is not good enough and vital information could go undetected, such as a tumor behind a bone. By using Computed Tomography (CT) instead, the possibility of missing this information will get significantly reduced since CT uses X-ray beams from many different directions in order to create a cross-sectional image of the medium, this way objects within the medium cannot cover each other. There are a lot of algorithms for reconstructing a tomography image of a medium. This report will focus on the linear algebraic methods, such as the least squares solution and Kaczmarz s method, for reconstruction of a 2D slice image. Besides describing the theory of how these methods work, we also present a comparison between and within these methods by carrying out simulations in Matlab and an experiment using a CT device.

3 Contents 1 Introduction Background Computed Tomography Introduction to the Mathematics behind CT The Idea Behind the CT Scan Linear Systems and Least Squares Solution An Image as a Vector Linear Equation Systems Least Squares Solution Implementation of the Mathematical Theory Kaczmarz s method Proof of Kaczmarz s Method: The Iteration Scheme A Variation of Kaczmarz s method Investigation Method and Results Simulated Data Comparing Least Squares to Kaczmarz Comparing Kaczmarz to Filtered Back Projection (FBP) Dependency on the Number of Iterations Experimental Data Discussion Comparing the Least Squares and Kaczmarz s Methods Comparing Kaczmarz s Method to FBP Dependency on the Number of Iterations Experimental Data

4 5.5 Future Research Acknowledgements 22

5 Notational Conventions a a vector a length of vector a orthogonal to := defined as R(A) R range of the matrix A belongs to the set of the set of real numbers

6 1 Introduction The aim of this report is to describe the role of linear algebra in Computed Tomography (CT) and how the mathematical models based on linear algebra, the least squares solution and Kazcmarz method, work when applying them to practical use under different conditions. 1.1 Background When carrying out a so-called plain X-ray examination, an object or patient is exposed to X-rays that generate a two-dimensional image. This method is well established and is sufficient for many medical examinations such as imaging a broken bone. On the other hand, such a two-dimensional image is merely a projection of a three-dimensional object, therefore the use of X-ray examination is limited, see figure 1. For instance a tumor located behind a bone could go undetected, leading to failure in obtaining potentially life-saving information. So what if a three-dimensional object could be imaged instead of its two-dimensional projection? Figure 1: A projected image could be misleading [10]. 1.2 Computed Tomography CT is a method that gathers X-ray projection images from different angles and thereafter calculates cross-sectional images of the examined object. Several such cross-sectional images can be stacked to form a three-dimensional image [12]. In this way, one can spot 1

7 tumors behind bones and similar objects which would otherwise be poorly described by plain X-ray examination. A solution to the reconstruction problem, i.e., a way to image the internal structure of some medium, was demanded for a long time within various scientific fields, such as radio astronomy, electron microscopy and nuclear magnetic resonance [1]. Radon solved the reconstruction problem in 1917 without a practical application as a reason and seems to have solved it because of a purely mathematical interest since he does not mention any practical application in his only report regarding reconstruction [11]. Probably because of the lack of application in his publication, his discovery remained unnoticed for a long time. Other researchers rediscovered this independently up until the 1970s when they noticed Radon s publication. This lead to a series of publications of both iterative and analytical methods and eventually to the development of medical tomography [1]. 1.3 Introduction to the Mathematics behind CT To better understand the least squares and Kaczmarz methods, the interaction between X-ray photons and tissue will be modeled. The intensity, I, along a line, l, is defined as the number of photons per second traveling along l. In figure 2, I in is the incident intensity and I out is the transmitted intensity. When passing through a homogeneous medium with the width s, the photon absorption by the medium depends on µ, the attenuation coefficient. That is, every substance has their own property of absorbing a certain amount of photons that pass through it, which µ indicates [3]. Since the intensity is known before the interaction and I out can be measured by a detector, the difference in intensity, i.e., the absorbed amount of intensity, can be calculated. This means that the ability of the medium to absorb energy, µ, can be calculated. This is, however, assuming that the medium is homogeneous which is often not the case [3]. 2

8 I in I out l µ s Figure 2: An X-ray beam following a line, l, passing through a homogeneous medium The Idea Behind the CT Scan By measuring the difference in intensity for each of the X-ray beams passing through the medium from different directions and comparing the measurement data one can determine the amount of absorption, µ, for each location within the medium. A simplified model is presented below. Figure 3: A simplified model of the different absorption values, µ, in a heterogeneous medium. Suppose that figure 3 represents a medium in the shape of a square which is composed of a 3-by-3 grid of smaller squares. Each smaller square can either absorb 1 intensity unit (i.u.) from the X-ray beams, i.e., the white squares, or not absorb any intensity at all, i.e., the black squares. We can think of the white squares as bone and the black squares as air to make the model more tangible. The data we receive from running a CT scan is the total loss of i.u. for each row, column or diagonal that an X-ray beam passes through. That is, if an X-ray beam passes through the first row in figure 3 the data tells us that 2 i.u. have been absorbed in the medium. As we continue to radiate the medium 3

9 with X-ray beams from different angles and positions, we can find out more about the energy absorption for each diagonal, column and row. Eventually, only one possibility exists for where the black and white squares are located [3]. In reality each of the black and white squares are different pixels with a grayscale value based on their respective absorption abilities. By measuring the difference in intensity for many X-ray beams, one may calculate the value for each pixel and then form an image [3]. 2 Linear Systems and Least Squares Solution 2.1 An Image as a Vector Consider pixels in an image to be points on a discrete vector where n is the number of pixels. Every pixel, j, has their own average attenuation coefficient, µ j. Since there might be a variation of the attenuation coefficient within every pixel, the more pixels we have in an image, the more accurate it is. Based on the same principle as in section 1.3.1, we have several X-ray beams each following a line, l i, through a medium, hence crossing pixels which we can see in figure 4. j µ j l i w i,j Figure 4: A line, l i, crossing the pixel, j, with the attenuation coefficient, µ j, where w i,j is the length of the line, l i, through the pixel, j. The data, g i, for each beam is measured when radiating an X-ray beam through the medium and intensity is absorbed. The data for each beam can be expressed by every individual equation of W µ = g, where W is an m n matrix for all w i,j, µ is a vector for all µ j and g is a vector for all g i where m is the number of X-ray beams. 4

10 We can illustrate this the following way: w 1,1 w 1,2 w 1,n µ 1 g 1 w 2,1 w 2,2 w 2,n µ 2 g 2 = w m,1 w m,2 w m,n µ n g m where each individual equation will be w 1,1 µ 1 + w 1,2 µ w 1,n µ n = g 1 w 2,1 µ 1 + w 2,2 µ w 2,n µ n = g 2. (1) w m,1 µ 1 + w m,2 µ w m,n µ n = g m 2.2 Linear Equation Systems A linear equation system is a system of n unknowns and m equations of the type seen above in equation (1). The term system implies that the set of equations is dealt with all together at once and that there are two or more equations and one or more variables. Assuming the same notations as in section 2.1 are used, W is the matrix coefficient of the linear system and g is a non-homogeneous term, i.e., each individual value is different. This means that both w and g are known constants, however µ is the n unknowns. The number of solutions for a linear system will either be exactly one, infinitely many or none. When there is no solution the equations are called inconsistent and if there are one or infinitely many solutions they are called consistent [9]. A system of linear equations is considered overdetermined if there are more equations than unknowns which is usually the case for tomography. That is, the linear system is overdetermined when there are more X-ray beams than pixels. However, if the opposite case is considered, i.e., more pixels than X-ray beams, the linear system would be underdetermined. 5

11 2.3 Least Squares Solution Methods that work theoretically do not necessarily work in reality. Usually some measurement errors occur and have to be taken into consideration before one can move on to traditionally used methods. One way of doing this is to adjust the real results such that it fits the theoretical results. When dealing with scattered points, which theoretically should be located on the same line, a regression can be calculated that finds a line which best describes all of the points at the same time. This can also be applied for vectors onto a matrix, which will be encountered in the least squares solution. One way of thinking is to picture the points or vectors as shadows on the line and the matrix they belong to, respectively. When first dealing with the data it is natural to demand a quadratic matrix, that is the number of data points or X-ray beams, n, is the same amount as the number of pixels, m. This is not reasonable since the measurement data contains errors. The less data one uses, the bigger influence the errors have on the result. The least squares method seeks to introduce an approximate solution for linear equations that lack a solution. When receiving data from tomography, measurement errors have to be taken into consideration. Since the data is contaminated with noise and measurement errors, there is no proper solution. The least squares solution is instead applied in order to find the optimal approximate solution that projects an image which describes all of the data in the best possible way. Consider the system of linear equations W µ = g, which was encountered in section 2.1, where W is an m n matrix. The image we seek, µ, is an n-vector and the tomographic data we measure, g, is an m-vector. This means that m is the number of equations we have, or the number of X-ray beams that we have, and n is the number of unknowns, or the number of pixels in an image. We can only use the least squares solution when m > n, i.e., when the system is overdetermined, which implies that there must be more X-ray beams than pixels. The set of all possible g of the equation W µ = g forms a high-dimensional plane, 6

12 which we denote by R(A), i.e., R(W ) := {g R m : g = W µ for some µ R n }. R(W ) g p r Figure 5: An illustration of the plane R(W ) where p is a projection of g on R(W ) and r is the difference between g and p. If R(W ) can not be defined by g, the most natural approach is to project g onto R(W ). This is seen in figure 5, where the projection is defined as p and the residual, i.e., the difference between g and p, as r. The closer r is to 0, the more accurate the projection is. Note that p is contained in R(W ), so there exists some ˆµ R n such that p = W ˆµ. The least squares solution to the equation W µ = g is precisely ˆµ, which is the solution with the least amount of error. We will now derive the least squares solution algebraically. First, let us introduce the nullspace as the set of all vectors that when multiplied with the matrix equates to zero. N(W ) := {µ R n : W µ = 0} Since the equation does not have a solution, g is not contained in R(W ). By definition g p is orthogonal to the plane R(W ), i.e., g p R(W ) Now, it is well known that R(W ) = N(W T ) where W T is the transpose of W, hence 7

13 g p N(W T ), which is equivalent to W T (g p) = 0. Since p R(W ), there must exist an ˆµ corresponding to p such that p = W ˆµ. This means that W T (g W ˆµ) = 0. Expanding and solving for ˆµ gives us ˆµ = (W T W ) 1 W T g (2) The right hand side of equation (2) is called the normal equation. 3 Implementation of the Mathematical Theory 3.1 Kaczmarz s method Kaczmarz s method is a way of approximating a solution for a linear system and can be applied to Computed Tomography when looking at the pixels as vectors, unlike least squares which uses matrices instead. When dealing with a linear system with at least one solution, the Kaczmarz s method will converge to a solution of the system. The algorithm produces a sequence of vectors for every value of i that satisfies each of the individual hyperplanes for w i µ = g i. These can be illustrated as lines, see figure 6. The different indexes of µ, k, indicates the number of iterations. Iteration signifies the repetition of a process with the aim of approaching a desired goal, in this case the intersection between the hyperplanes. At first, the higher the number of iteration, the closer it will be to the intersection, i.e., the solution. However, at some point the number of iterations will pass the intersection and the result will become worse. Therefore, when applying Kaczmarz s method it is important to take the number of iterations in consideration such that one can get the best possible result [2]. 8

14 µ 2 µ 4 µ 6 µ 7 µ w 1 µ = g 1 w 2 µ = g 2 µ 5 µ 3 µ 1 µ 0 Figure 6: The projection of the vector µ described as points onto the hyperplanes w 1 µ = g 1 and w 2 µ = g 2 until they reach the intersection point µ. After selecting a starting point, µ 0, it is then projected orthogonally onto the hyperplane w 1 µ = g 1 as the new point µ 1 and which would further be allowed to be projected onto the hyperplane w 2 µ = g 2. This can be described the following way: µ 1 := the projection of µ 0 on the hyperplane w 1 µ = g 1. µ 2 := the projection of µ 1 on the hyperplane w 2 µ = g 2. µ 3 := the projection of µ 2 on the hyperplane w 1 µ = g 1.. µ := the intersection of the two hyperplanes w 1 µ = g 1 and w 2 µ = g Proof of Kaczmarz s Method: The Iteration Scheme Consider two arbitrary points on the line w i µ = g i with the coordinates (x 1, y 1 ) and (x 2, y 2 ) respectively. The definition of coordinate multiplication is the following: (x, y) (w 1, w 2 ) = x w 1 + y w 2 x 1 w 1 + y 1 w 2 = g x 2 w 1 + y 2 w 2 = g By subtracting the two equations we get (x 1 x 2 ) w 1 + (y 1 y 2 ) w 2 = 0 which means ((x 1 x 2 ), (y 1 y 2 )) (w 1, w 2 ) = 0. Since (x 1, y 1 ) and (x 2, y 2 ) are located on the line 9

15 w i µ = g i the difference between their coordinates must be a vector that lies on the line w i µ = g i from (x 1, y 1 ) to (x 2, y 2 ). Suppose this vector is called x. When multiplying vectors the following rule applies: x w i = x w i cos(θ), where θ is the angle between x and w i, so if x w i = 0 it means that x w i since cos(90 ) = 0. This means that the vector w i is orthogonal to the hyperplane w i µ = g i. µ 2 µ k+1 w i µ = g i h µ k ω i a w i b µ 1 Figure 7: An illustration of w i µ = g i as a line where µ k+1 is a projection of µ k. In figure 7 we can see that µ k+1 = µ k + h and that h = b a so let us begin by calculating a. a = g i if and only if w i = 1, i.e., w i is a unit vector. Since this is not the case, we must divide by w i which gives us that a = g i w i. Now we shall calculate b. Suppose that b is parallel to w i, both starting from the origin, and is given as a projection of µ k on the line with the unit vector ω i := w i w i, this implies that b = µ k ω i = µ k w i w i, which means that h = b a = µ k w i w i g i w i. Since h has the opposite direction as the unit vector ω i we can write h = h ω i = ( ) ( ) µk w i g i w i w i w i which means h = µk w i g i w i. w i w i w i w i µ k+1 := µ k ( µk w i w i g ) i wi w i w i where i = k mod m + 1. (3) The modulo operation finds the remainder, i, by division of k by m + 1, where i is the 10

16 number of equations, k is the number of iterations and m is the number of X-ray beams A Variation of Kaczmarz s method There exists a variation of Kaczmarz s method which involves an introduction of relaxation parameters. This implies the use of λ i,k for each i and k where it satisfies the interval 0 < λ i,k < 2 and gives us the following variation of equation (3): µ k+1 := µ k λ i,k ( µk w i w i g ) i wi w i w i where i = k mod m + 1. As can be seen in this new formula, λ i,k determines how far µ k should be projected onto w i µ = g i. That is, if λ i,k = 1, the equation will be the same as (3) and µ k will be projected onto w i µ = g i. However, if λ i,k < 1, the projection of µ k will not reach w i µ = g i and if λ i,k > 1, the projection of µ k will pass w i µ = g i. This way, λ i,k, can be changed in order to find an acceptable approximation solution to an indeterminate system of equation and thereby get better results with a more accurate image [2]. 4 Investigation Method and Results In this section the performance of the Kaczmarz s method for tomographic reconstruction is investigated with practical experiments, mostly done by using simulated data tests. For the simulations and the Kaczmarz s method, the publicly available Matlab software AIR Tools, which is described in [4], was used. Finally, a section showing tomographic reconstruction from experimental data is presented. This was executed at KTH-STH where we used a CT device that came with a reconstruction algorithm, which is an in-house implementation of the Filtered Back Projection (FBP) method by Massimo Colareti-Tosti. Since the reconstruction algorithm was not Kaczmarz s method, it could not be tested. 11

17 4.1 Simulated Data All simulated data investigations are performed by using the well-known, but modified with improved contrast, Shepp-Logan phantom. The goal was to recover the phantom from simulated parallel beam tomographic data with different conditions Comparing Least Squares to Kaczmarz In this investigation the least squares solution and Kaczmarz s method on simulated tomographic data in the overdetermined case were compared. Figure 8 shows results for the overdetermined case. The phantom, shown in figure 8(a), is given by pixels. Two sets of simulated parallel beam tomographic data are used, one noise-free shown in figure 8(b) and one with 5% additive Gaussian noise, which is not shown. For both data sets a full angular range [0, 180 ] sampled every 2, i.e., a total of 91 directions, and a simulated 128 pixel line detector is used. Thus data can be seen as a array. To summarize, this tomographic reconstruction problem amounts to solve a linear system with unknowns and equations Comparing Kaczmarz to FBP Here the Kaczmarz s methods is compared against the FBP. The FBP method that is used is the one available in the Matlab Image Processing Toolbox, see [7, 8] for a description of how it is implemented. The results are shown in figure Dependency on the Number of Iterations Within the Kaczmarz method itself reconstruction depend on the number of iterations, see section 3.1. Therefore a choice of the number of iterations must be made when using Kaczmarz method. Here it is shown how the reconstructions depend on the number of iterations in the overdetermined case. The results are shown in figures 10 and 11. The phantom, shown in figure 10(a), is given by pixels. Two sets of simulated parallel beam tomographic data are used, one noise-free and one with 5% additive 12

18 Gaussian noise. For both data sets we have full angular range [0, 180 ] sampled every 1, i.e., a total of 181 directions, and we simulate a 256 pixel line detector, thus data can be seen as a array. To summarize, this tomographic reconstruction problem amounts to solving a linear system with unknowns and equations. 4.2 Experimental Data The CT device, such as the one used in this investigation, is described in detail in [6]. It corresponds to a third generation CT and therefore has an X-ray source that produces a fan beam of X-rays, so a projection of the object can be obtained in one go, without any need of translating the source as in the parallel beam case. The data is recorded by a line detector, represented by a white, dotted line in figure 13(b), with 128 pixels and we cover the full angular range [0, 180 ] with an angular sampling of 1.8, i.e., we have 100 directions. Hence, resulting data has dimensions and the 2D slices have pixels. Before data is recorded, a dark-field correction is preformed to reduce the detector read-out noise. Furthermore, a correction for the non-uniformity of the X-ray source is performed. Finally, recorded tomographic data, which is from a fan-beam source is computationally transformed as parallel-beam data before it is fed to the reconstruction algorithm. This transformation is described in details in [5]. 13

19 (a) 2D phantom. (b) Noise-free sinogram. (c) Least-squares from noise-free data. (d) Kaczmarz from noise-free data. (e) Least-squares from noisy data. (f) Kaczmarz from noisy data. Figure 8: Comparing least-squares solution to Kaczmarz s method in the overdetermined case. In this case the iterations for Kaczmarz was stopped after 10 iterations. 14

20 (a) FBP from noise-free data with X-ray beams from 36 directions. (b) Kazcmarz from noise-free data X-ray beams from 36 directions. (c) FBP from noisy data with X-ray beams from 36 directions. (d) Kaczmarz from noisy data with X-ray beams from 36 directions. (e) FBP from noise-free data with X-ray beams from 9 directions. (f) Kaczmarz from noise-free data with X-ray beams from 9 directions. Figure 9: Comparing FBP to Kaczmarz s method in the overdetermined case. 15

21 (a) 2D phantom. (b) 1st Kaczmarz iteration. (c) 5th Kaczmarz iteration. (d) 10th Kaczmarz iteration. (e) 25th Kaczmarz iteration. (f) 100th Kaczmarz iteration. Figure 10: The Kaczmarz iterates from noise-free data. 16

22 (a) 2D phantom. (b) 1st Kaczmarz iteration. (c) 5th Kaczmarz iteration. (d) 10th Kaczmarz iteration. (e) 25th Kaczmarz iteration. (f) 100th Kaczmarz iteration. Figure 11: The Kaczmarz iterates from noisy data with 5% noise level. 17

23 (a) Sinogram for the walnut. (b) The physical interior of the walnut corresponding to the 2D slice reconstructed in (c). (c) FBP reconstruction based on the data in (a). Figure 12: The top image (a) shows the experimentally measured sinogram. The reconstruction of a 2D slice in (c) corresponds to the photograph of the physical walnut image shown in (b). 18

24 (a) Sinogram for the chicken wing. (b) A photo of the chicken wing with a white, dotted line from where the 2D slice in (c) was reconstructed. (c) FBP reconstruction based on the data in (a). Figure 13: The top image (a) shows the experimentally measured sinogram. The reconstruction of a 2D slice is shown in (c) and a photo of the chicken wing is shown in (b). 19

25 5 Discussion 5.1 Comparing the Least Squares and Kaczmarz s Methods When running this simulation we noticed that Kaczmarz s method is significantly faster than the least squares method. As we can see in the results in figure 8, Kaczmarz s method is also much more reliable since it generates a clear image unlike the least squares method which generates some kind of blur that cannot be distinguished. An explanation to why Kaczmarz s method is faster than the least squares method is obtained by solving the normal equation (2). Since the least squares method uses a W T W matrix unlike Kaczmarz s method, there is a lot more data to handle when using the least squares solution and even though W may be sparse, W T W does not have to be. Furthermore, the W T W matrix might not be invertible or the inverse might be difficult to compute. This can lead to imaging errors, which can be fatal, and a slower calculation, which makes it inefficient to use the least squares method. 5.2 Comparing Kaczmarz s Method to FBP When running simulations without noise in figure 9 both the FBP and the Kaczmarz s method reconstructions work really well. Although, it is possible to distinguish a slightly more enhanced image when using Kaczmarz s method since the contrast seems a bit higher, which at least in this case, makes the reconstructed image more clear when radiating X-ray beams from both 36 and 9 directions. However, when running simulations with noise, the FBP reconstruction is significantly better since there is almost no visible impact on the image unlike Kaczmarz s method where the contrast seems to be very low, hence the gray image in figure 9(d) where one can barely even distinguish the Shepp-Logan phantom. When compared to FBP, Kaczmarz s method is considered ineffective, since FBP s way of dealing with the reconstruction problem is faster. However, we cannot imply that Kaczmarz s method necessarily always reconstructs such an unclear image when it comes 20

26 to noisy data since the number of iterations nor the relaxation parameter were tested in this simulation, which could affect the results. 5.3 Dependency on the Number of Iterations In figure 10 we can see that data without noise can be reconstructed with Kazcmarz s method more accurate as we reach the 100th iteration. However, at some point the reconstruction should get worse as we pass the value closest to the equation intersection, see section 3.1. When dealing with noisy data, as seen in figure 11, the number of iterations barely make any visible difference at all, but once again, other variables could affect the results. 5.4 Experimental Data Unfortunately, Kaczmarz s method could not be used when performing the practical experiments. However, if Kaczmarz s method would have been used in this case, similar results as shown in section could probably be expected. In figure 12 and 13 we can see both the sinogram and the reconstruction in a color map which indicates absorption, or intensity loss. If it is dark blue, the area absorbs almost nothing, i.e., air in this case. If the area is red it absorbs a lot of photons and is probably dense, such as bone or the shell of a walnut. When tomographing a chicken wing, as in figure 13, we have three bones, which we can see as red circles in the reconstruction 13(c) and as three separate, oblong areas in the sinogram 13(a). This phenomenon probably occurs because of the medium being incoherent due to the bones being located at different places within the chicken wing and also because of a large attenuation difference between the bone and the flesh of the chicken. However, when tomographing a walnut, as in figure 12, there are not really any individual areas in the sinogram 12(a) that can be made out to correspond to specific parts 21

27 in the walnut. This is probably because most of the components in the walnut have a high attenuation coefficient, except for the air in-between, which makes the individual parts stand out less. Note that in this investigation we can see that the CT scanner we used produces an somewhat accurate image which we can see when comparing the reconstruction 12(c) to how it really is in 12(b). 5.5 Future Research As mentioned in sections 5.2 and 5.3, the number of iterations and the relaxation parameter were not investigated when comparing Kaczmarz s method to FBP, which could be interesting to do. It might affect the results, leading to Kaczmarz s method being more accurate, especially when the data is contaminated with noise. One could also use a stronger computer with the capacity of reconstructing an image with more pixels than , within a reasonable time. This way one can investigate how the number of pixels affects least squares, Kaczmarz and FBP. Furthermore, it would be interesting to test the simulations in a CT scanner since a simulation could differ from the reality. There is a lot one could investigate and improve with CT scanning. A patient should not be exposed to an unnecessary amount of radiation, however the doctor needs an image accurate enough to see what is requisite, therefore it is important to find the balance between the amount of radiation and the accuracy of images. Another important aspect of Computed Tomography is speeding up the tomography process, since a patient will move, even by just breathing, and thereby generate measurement errors. 6 Acknowledgements I would like to thank my mentors, Professor Ozan Öktem of the Department of Mathematics at the Royal Institute of Technology and Emeritus Jan Boman of the Department of Mathematics at Stockholm University, both of whom developed this project and pro- 22

28 vided great help and guidance throughout it. I would also like to thank my fellow Rays colleague, Martin Nilsson, for the project cooperation, and my counselor Mariam Andersson who helped me through my struggles. Furthermore, I would like to thank Rays for excellence and all of the people involved, including the director of Rays 2014, Philip Frick, who arranged this and made my time at Rays better than ever expected. Lastly, I would like to thank the partners of Rays for excellence, Stockholm Mathematics Center, Erik Johan Ljungberg s Educational Fund and Jacob Wallenberg Foundation, who made this project possible. 23

29 References [1] Deans, Stanley R., Introduction. The Radon Transform and Some of Its Applications ed, New York: Wiley, Print. [2] Feeman, Timothy G., Algebraic Reconstruction Methods. Chapter 9. The Mathematics of Medical Imaging: A Beginner s Guide. New York: Springer, Print. [3] Feeman, Timothy G., X-rays. Chapter 1. The Mathematics of Medical Imaging: A Beginner s Guide. New York: Springer, Print. [4] Hansen, P.-C. and Saxild-Hansen, M., "AIR Tools - A Matlab Package of Algebraic Iterative Reconstruction Methods." Journal of Computational and Applied Mathematics 236 (2012): Print. [5] Kak, A. C. and Slaney, M., Principles of Computerized Tomographic Imaging. Chapter 3. Classics in Applied Mathematics. SIAM, 33 (2001). Print. [6] Kerek, A. and Colareti-Tosti, M. and Valastyán, I., Computed Tomography, lab instructions. Medical Imaging, School of Technology and Health, KTH - Royal Institute of Technology, Print. [7] MathWorks, Matlab Image Processing Toolbox - Reference. MathWorks, Web. 4 July [8] MathWorks, Matlab Image Processing Toolbox - User s Guide. MathWorks, Web. 4 July [9] Petermann, E., Linjär geometri och algebra. Lund: Studentlitteratur, Print. [10] Rabbit projecting a hand. Digital image. ForumKlassika.Ru. N.p., n.d. Web. 26 June < &d= >. [11] Radon, J., On the Determination of Functions from Their Integrals along Certain Manifolds, in: The Radon Transform and Some of Its Applications, Appendix A ed, New York: Wiley, Print. (translation of Radon s 1917 paper by R. Lohner). [12] Romans, Lois E., Foreword. Introduction to Computed Tomography. Baltimore: Williams & Wilkins, Print.