Computed Tomography Lab phyc30021 Laboratory and Computational Physics 3

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1 Computed Tomography Lab phyc30021 Laboratory and Computational Physics 3 By Joshua P. Ellis School of Physics Last compiled 8th August 2017

2 Contents 1. Introduction Aim Projections and Reconstructions Projections Tomographic Reconstruction Back-projection Filtered Back-projection Dealing with Noise A. Fourier Transform and Series 9 A.1. From Series to Transform A.2. A Question of Convention A.3. Extra Dimensions A.4. Derivatives A.5. Convolutions A.6. Discrete Fourier Transform A.6.1. Fast Fourier Transforms B. Interpolating Functions 16 References 18 i

3 1. Introduction (b) A modern X-ray image of an elbow. (a) One of the first medical X-ray image. Taken by Wilhelm Röntgen of his wife s left hand. 1. Introduction Figure 1: X-ray images. In 1895, Wilhelm Röntgen discovered a new form of radiation which, to everyone s great surprise, was able to penetrate through soft tissue but was mostly stopped by bones.1 [1 4] Up until that point, the only way to see inside a person was during surgery or using a cadaver; this new possibility of imaging the inside of a patient without requiring surgery was revolutionary. In what is now one of the first medical images, Röntgen created a radiograph of his wife s left hand (fig. 1a) and since then, medical imaging has improved greatly (fig. 1b) and it is regularly used in hospitals. One disadvantage of X-ray images such as those in fig. 1 is that an opaque object will obstruct everything in front and behind it. In the case of bones this is may not be too much of an issue as there aren t many overlapping bones in the human body and the patient and/or radiograph can be rotated to ensure the desired region isn t obstructed. On the other hand, this may not be possible when imagine soft tissue such as in a radiograph of the abdomen. In order to see what might be in front or behind the opaque object, the X-ray projection has to be taken from a different angle, but the question then becomes: 1He was not the first to discover X-rays, nor was he the first to notice that they were able to penetrate through certain materials; however, he was the first to systematically study them. 1

4 1. Introduction Given a set of X-ray images from different angles, can we reconstruct the full 3d shape of the original object? As you might expect, the answer is yes and is exactly what happens in computed tomography (ct) scans. The process of reconstructing the full 3d object from these different projection angles is called tomographic reconstruction and will be the focus of this lab. Although the term ct scans is now synonymous with X-ray ct scans, it is possible to use different kinds of radiation. Some example well-known examples in the medical field include positron emission tomography (pet) and magnetic resonance imaging (mri) scans. Even outside the medical field, ct scans are used in many other fields from geophysics to atmospheric science to astrophysics Aim In this lab, the mathematical properties of tomographic projection and reconstruction will first be investigated. An algorithm to numerically project images will be implemented, replicating what happens when an X- ray image is taken. Subsequently, the original image will be reconstructed from the projected image using back-projection and filtered back-projection. Once these algorithms are implemented, the effects of other effects that can arise in realistic scenarios can be investigated. This section of the lab is intended to be exploratory and is open-ended. It will be up to the student to decide what they wish to investigate. Some ideas that may be investigated include: The effects of noise during the creation of projection and its effect on the reconstruction; The effects of defects in the detector such as a dead/malfunctioning pixel; Limitations in the angles that can be projected and how that can affect the quality of the reconstructed image; Limitations in the resolution of the detector and how that can affect the quality of the reconstructed image. This list is not exhaustive and other aspects can also be explored. It is not necessary to answer all questions and they are mostly there to help. 2

5 2. Projections and Reconstructions 2. Projections and Reconstructions Before anything can be implemented, the processes involved in creating X-ray projections and subsequently reconstructing the original image from the projections must be understood. In the case of the former process, a simple physical understanding of attenuation is sufficient. As for the latter process, it is much less obvious and the mathematical properties of these properties were first investigated by Johann Karl August Radon (16 December May 1956) [5, 6] Projections X-rays are capable of penetrating through materials and they generally either travel in straight lines, or get absorbed by the material. How strongly they get absorbed depends on the composition of the material which, in general, will vary as a function of position. For an infinitesimal segment dη, the attenuation coefficient µ is given by µ = 1 I di dη, (1) where I is the intensity of the X-ray. In the special case where the material has a constant attenuation factor over some distance d, the overall attenuation factor is I = I 0 e µd. In general though, the attenuation factor depends on many other factors and will vary in space (and possibly even time, though we ll ignore this here); thus, over the full path of the X-ray, the total attenuation is obtained by integrating eq. (1): I = I 0 exp [ ] µ(x, y) dη. (2) When an X-ray image of an object is taken, a projection of the object density is being created along a particular direction. Consider the diagram in fig. 2 with the (x, y) coordinates provide a fixed reference coordinate system. The X-rays are coming from particular direction and, by assumption, they are all parallel. It is convenient to define the (ξ, η) coordinates such that η is parallel to the X-ray, and ξ is perpendicular to the ray. From eq. (2), it becomes clear that we can now define the overall attenuation along an X-ray path, or projected attenuation, as: ( ) I p θ (ξ) = ln = µ(x, y) dη (3) where x and y are implicitly functions of ξ and η (see fig. 2). I 0 3

6 2. Projections and Reconstructions η y ξ θ x p θ (ξ) Figure 2: Diagram illustrating the path travelled by X-rays. The X-rays are travelling along the η direction and being projected on the surface (photographic plate or detector) on the bottom right. The projected attenuation is, ultimately, function of θ and ξ as the integration is done over all η. In a sense, we can think of p θ (ξ) as a polar function, though unlike a true polar function, ξ is allowed to be negative and θ goes over the range [0, π). This integral along perpendicular lines is called Radon transform, named after the Austrian mathematician Johann Radon and the function p θ (ξ) is called the sinogram of µ. In this lab, the Radon transform of µ(x, y) will be denoted by p θ (ξ), but more generally the Radon transform of some function f (x, y) is denoted as R x,y { f (x, y)}(θ, ξ) in analogy to the way Fourier transforms are denoted.1 Question 1: Why is p θ (ξ) called a sinogram? (Hint: plot it) Question 2: Show that the Radon transform is linear and distributive. That is for any functions f and g, and scalars a and b. R(a f + bg) = ar f + brg Question 3: Calculate analytically the Radon transform for the function f (x, y) = { 1 x 2 + y 2 R 2 0 otherwise 1If the variables being transformed are obvious from context, the Radon transform is simply written as R f. 4

7 2. Projections and Reconstructions 2.2. Tomographic Reconstruction In practice, µ(x, y) can t be measured directly without taking apart the object; however, it is possible to pass X-rays through the unknown object and p θ (ξ) for a range of ξ and θ values. The intent is to now reconstruct the original function µ(x, y) based on the measured p θ (ξ) values. This section will make use of Fourier transforms. A summary of some important properties of Fourier transforms is available in appendix A Back-projection In creating the projection, whatever was located at coordinate (x, y) contributes to the projection at p θ (x cos θ+ y sin θ), and if what was located at (x, y) was very opaque, the projected attenuation will have a much large value consistent for all θ. A very simple thing to do is apply the projection backward. That is, if a particular value of p θ (ξ) is very large, then something along corresponding the line η was more opaque and the whole line can be drawn in darker. By going over all values of θ and ξ, opaque regions will have more lines going through it which are darker thereby reconstructing the original µ(x, y). This method of reconstructing the original is called back-projection and mathematically can be expressed as: µ(x, y) := π 0 p θ (x cos θ + y sin θ) dθ. (4) Note that the integral is only over the interval [0, π] since p θ (ξ) is defined for both positive and negative values of ξ. Question 4: Does back-projection give you the original attenuation function µ(x, y)? Filtered Back-projection The back-projection defined in eq. (4) works to some extent, but it doesn t return the original function exactly. Filtered back-projection allows for the original function to be exactly recovered from Radon transform. We begin with the original attenuation µ(x, y) and the identity, µ(x, y) = ˆµ(k x, k y )e ik x dk x dk y (2π) (2π), where ˆµ is the Fourier transform of µ. This first equation simply states that the inverse Fourier transform of the Fourier transform gives the original function. 5

8 2. Projections and Reconstructions The projected attenuation, p θ (ξ), uses coordinate system other than (x, y) used in the Fourier transform above, so we will change coordinates into the (ξ, θ) coordinates. The coordinate transforms in real and Fourier space are the same: k = (k x, k y ) = (k ξ cos θ, k ξ sin θ) and the Jacobian to change from dk x dk y to dk ξ dθ is just k ξ (the absolute value comes from the fact that k ξ can be negative). Performing this change of variables, the integral becomes: µ(x, y) = 1 2π ˆµ(k ξ cos θ, k ξ sin θ)e ik ξ(x cos θ+y sin θ) k ξ dk ξ dθ. (5) (2π) Now let s take a closer look at ˆµ(k ξ cos θ, k ξ sin θ) which is, by definition, the Fourier transform of µ(x, y): ˆµ(k ξ cos θ, k ξ sin θ) = µ(x, y)e ik ξ(x cos θ+y sin θ) dx dy. Notice now that in the exponent, (x cos θ + y sin θ), is exactly ξ, so we change coordinates from (x, y) to (ξ, η): = µ(x, y)e ik ξ ξ dξ dη Where x, y are now implicitly functions of ξ and η. The exponential has no dependence on η, so the integral of η is only integrating µ(x, y) which, quite amazingly, corresponds exactly to the projected attenuation we defined in eq. (3) = p θ (ξ)e ik ξ ξ dξ What remains now is just the Fourier transform of p θ (ξ): = ˆp θ (k ξ ). This quite remarkable identity, that ˆµ(k ξ cos θ, k ξ sin θ) = ˆp θ (k ξ ), is known as the central slice theorem. Armed with this information, the Fourier transform of the projected attenuation can be substituted into eq. (5): µ(x, y) = 1 k ξ ˆp θ (k ξ )e ik ξ(x cos θ+y sin θ) dk ξ 2π (2π) dθ. The integral over k ξ is an inverse Fourier transform of the function k ξ ˆp θ (k ξ ): µ(x, y) = 1 F 1 { k 2π ξ kξ ˆp θ (k ξ ) } (x cos θ + y sin θ) dθ. (6) 6

9 2. Projections and Reconstructions Finally, you ll notice that the remaining integral over θ is just a back-projection. The reason this is called a filtered back-projection is that the frequencies in ˆp θ (k ξ ) aren t all equally passed through. Very large frequencies (corresponding to large k) are emphasized over smaller frequencies. One last observation: in reconstructing µ, the inverse Fourier transform is calculated for a product of two functions, and one of these functions is itself a Fourier transform. This is very similar to the convolution theorem (c.f. appendix A.5) which states that: The convolution of two function f and g is equal to the inverse Fourier transform of the product of the function s respective Fourier transforms: f g = F 1 { (F f )(F g) } This means that instead of back-projection p θ (ξ), the tomographic reconstruction back-projects the convolution of p θ (ξ) with a function f whose Fourier transform is k. Question 5: Go through the derivation for the filtered back-projection and make sure you are comfortable with all the steps. Question 6: Can you find/construct an analytic µ(x, y) and/or p θ (ξ) such that all of the steps above can be done analytically? This will allow for verification of certain steps during the implementation. Question 7: What is the form of the filter f (ξ), given that we know ˆf = k? Question 8: What happens if the filter is changed? The original filter is: ˆp θ (k ξ ) k ξ ˆp θ (k ξ ) but this can be generalized to: ˆp θ (k ξ ) ˆf ( k ξ ) ˆp θ (k ξ ). What happens if you change the filter ˆf? Dealing with Noise A common problem encountered when acquiring data is that it can be noisy. In the case of X-rays in ct scan, some rays might be scattered at different angles, radiation might cause false hits in the detector, the detector may not be perfect, etc. 7

10 2. Projections and Reconstructions If the noise is completely random and has a mean of zero, then a simple way of removing the noise is to average the function over small segments: p avg θ (ξ) = 1 ξ+ L 2 L ξ L 2 p θ (ξ ) dξ. With this in mind, it is possible to recalculate everything from section 2.2.2, but replacing p θ with the new smoothed function. The problem is that is that this requires a lot of numerical integration which can slow things down considerably when in fact, a simple observation can simplify everything. First, observe that the above averaging can be written as, p avg θ (ξ) = p θ (ξ )W L (ξ ξ ) dξ, where W is the a top-hat window function: W L (ξ) = { 1 L L 2 < ξ < L 2 0 else. Rewriting the averaging above in this form, it should become clear that average the function is really just convolving p θ with the window function: p avg θ = p θ W L In the previous section, it was just observed that the filtered back-projection already is convolving p θ with another function in order to recover the original (eq. (6)), and seeing as convolutions as associative and commutative, then it becomes really quite simple to add an extra convolution in the back-projection: µ(x, y) = 1 2π F 1 k ξ { kξ Ŵ ˆp θ (k ξ ) } (x cos θ + y sin θ) dθ. Question 9: Try adding noise to p θ (ξ), and see whether you can remove by applying an appropriate filter/window W. Question 10: Consider using other filters. Some filters can be used to smooth out features; however, other filters can be used to enhance certain features. 8

11 A. Fourier Transform and Series A. Fourier Transform and Series The trigonometric functions sin(nt) and cos(nt), for n N have an important property that: sin(2πnt) sin(2πmt) dt = 1 2 δ nm cos(2πnt) cos(2πmt) dt = 1 2 δ nm sin(2πnt) cos(2πmt) dt = 0 (7a) (7b) (7c) where δ nm is the usual Kronecker delta (that is, δ nm = 1 if n = m, and vanishes for all other values). To make it clear why this is useful, a new operation can be defined by f, g := 1 0 f (t)g(t) dt. (8) Question 11: Verify that eq. (8) satisfies all the necessary conditions for it to be an inner production. These conditions are, for all functions f, g : [0, 1] R and scalar a C: 1. Conjugate symmetry: f, g = g, f ; 2. Linearity in the first argument: a f, g = a f, g ; 3. Positive-definiteness: f, g 0 and f, f = 0 x = 0. The inner product is a generalization of a dot product; however, it works in very much the same way: Two vectors can be described as orthogonal if their dot product vanishes, and similarly two functions can be orthogonal if their inner product (as defined in eq. (8)) vanishes; A set of vectors {b 1,..., b n } can form an orthonormal basis if they satisfy that b i b j = δ i j, and similarly a section of function can also form a basis. It is evident from eqs. (7a) to (7c) that the (infinite) set of functions { } sin(2πnt), cos(2πnt) : n N (9) form an orthogonal basis (and with a tweak in normalization, an orthonormal basis). This generalization of a vector space is called a Hilbert space. What won t be proven here is that this set of trigonometric functions of functions spans all functions f : [0, 1] R where f, f is finite. 9

12 A. Fourier Transform and Series Given a basis {b 1,..., b n }, an arbitrary vector v in the vector space spaced by this basis can be uniquely written as n v = c i b i i=1 where each c i C is some coefficient which encodes how much of v is along b i. Similarly, an arbitrary function f can be decomposed in terms of the basis in eq. (9): f (t) = a a n cos(2πnt) + b n sin(2πnt). (10) n=1 Analogously to the coefficients c i in with the vectors, the coefficients a n and b n encode how much of f is made up from a particular trigonometric function. Question 12: Derive the expression that allows a n and b n to be calculated. Use the orthogonality property of the basis in eq. (9). What about a 0? One last trick that will be observed is that cos(2πnt) = R[e 2πint ], sin(2πnt) = I[e 2πint ]. This allows for both sines and cosines to be combined into one complex function. This can greatly simplify the Fourier series in eq. (10) to be in terms of just one function: f = n= c n exp[2πint]. Note that the sum is now over all integers as this will ensure that f is real (though it also becomes clear that it can be generalized for f mapping to C). There s just one thing that has to be verified: it is necessarily that exp[2πint] form a basis. Intuitively they should, be it is worth verifying that e 2πint, e 2πimt = δ nm. A.1. From Series to Transform The way the Fourier the inner product is defined in eq. (8) only works for functions f : [0, 1] R. A function g 1 : [a, b] R can be remapped onto the interval [0, 1] so that it can be decomposed into a Fourier 10

13 A. Fourier Transform and Series series; equivalently, it is possible to adjust the inner product to work on the interval [a, b]: f, g := 1 b f (t)g(t) dt. b a a If this is the inner product, then the basis functions have to be adjusted slightly to: { exp [ ] 2πint b a } : n Z. The decomposition of some arbitrary function g : [a, b] C remains the very similar: g(t) = n= c n exp [ ] 2πint. b a except that it is become cumbersome to be dividing by b a everywhere. It is possible to define f n = n/(b a) which has the inverse of the units of a, b, so it correspond to a particular frequency. In this way, we can rewrite the sum as g(t) = c f e 2πi f t. In the limit that the interval b a tends to infinity, the sum can be converted into an integral g(t) = f c( f )e 2πi f t d f (11) where c( f ) is no longer a list coefficient, but a smooth function. This smooth function has a special name: it is the Fourier transform of g and is usually denoted by ĝ. Just like with the series, ĝ can be understood as how strong is frequency f in g. Within the series, a particular coefficient could be extracted with: and in the continuous limit, this is still true: c( f ) = g, e 2πi f t = 1 b g(t)e 2πi f t dt b a a ĝ( f ) = g(t)e 2πi f t dt. (12) The operation of taking an (inverse) Fourier transform is often denoted with F and F 1 as: ĝ( f ) = F t {g(t)}( f ) g(t) = F 1 f {ĝ( f )}(t) 11

14 A. Fourier Transform and Series where the subscript indicates the variable that is being Fourier transformed. If the variables in the Fourier transform are obvious from context, then they will often be omitted entirely: ĝ = F g, g = F 1 g. A.2. A Question of Convention The above derivation has defined the Fourier transform and inverse Fourier transform in eqs. (11) and (12): g(t) = ĝ( f )e 2πi f t d f, ĝ( f ) = g(t)e 2πi f t dt. This is fine when working with angular frequency; however, it can be quite convenient to work with angular frequency ω := 2π f instead. In this case, the Fourier transform and its inverse are: g(t) = 1 ĝ(ω)e iωt dω, ĝ(ω) = g(t)e iωt dt. 2π Some people prefer a more symmetric definition of the Fourier transform and its inverse: g(t) = 1 2π which corresponds to a scaling of ĝ by 1/ 2π. ĝ(ω)e iωt dω, ĝ(ω) = 1 2π g(t)e iωt dt. (13) You ll then notice that there s not much of a difference between the transform and its inverse in eq. (13), and some people will then even swap which side has +i and i: g(t) = 1 2π ĝ(ω)e iωt dω, ĝ(ω) = 1 2π g(t)e iωt dt. Depending on the field of study, the convention used in the definition of the Fourier transform can change; however, the choice of convention is not important as long as it remains consistent. A.3. Extra Dimensions The Fourier transform can be straightforwardly generalized to n dimensions: g(x) = ĝ(k)e ik x dn k (2π) n, ĝ(k) = g(x)e ik x d n x. 12

15 A. Fourier Transform and Series Notice that the factor of 2π arises for each dimension. As a result, the n-dimensional version of eq. (13) is: ( ) n ( ) n 1 g(x) = ĝ(k)e ik x d n 1 k, ĝ(k) = g(x)e ik x d n x. 2π 2π When calculating multi-dimensional integrals, it can sometimes be advantageous to perform a change of variables to exploit a symmetry or just simplify the mathematics, and the same is true for Fourier transforms. In some cases, the vector k might be best expressed as k = (k x, k y ), but some other circumstances might favour k = (k r cos θ, k r sin θ). As with all changes of variables, care must be taken to include the appropriate Jacobian. A.4. Derivatives An important and very useful feature of Fourier transforms is the way derivatives are involved: { } dg dg F = dt dt e iωt dt dge iωt = g de iωt dt dt dt = [ ge iωt] + iω ge iωt dt = iωĝ where the first integral vanishes as g 0 when t ±. The fact that g 0 is guaranteed because g, g <. For multi-dimensional Fourier transform, the procedure is similar: { } dg(x) F x = ikĝ dx where A.5. Convolutions A convolution of two functions f, g is: ( ) d d dx := dx, d dy, d. dz ( f g)(x) := f (ξ)g(x ξ) dξ. 13

16 A. Fourier Transform and Series Figure 3: Illustration of issues that arise due the finite number of sampels in a dft. The samples in red are unable to encode some of the high frequency information. Taking the Fourier transform of the convolution: F { f g} = ( f g)(x)e ik x dx = f (ξ)g(x ξ)e ik x dξ dx and performing the change of variable x x + ξ, = f (ξ)g(x)e ik x e ikξ dξ dx = f (ξ)e ikξ dξ g(x)e ik x dx = ˆf ĝ Thus, the Fourier transform of a convolution of two functions is just the product of the Fourier transform of each function. In other words: f g = F 1 { F( f ) F(g) } A.6. Discrete Fourier Transform A discrete Fourier transform (dft) is really just another word of a Fourier series; however, instead of beginning with some function g(t) known for all values of t over some interval, the function is only know for a finite set of t i as depicted in fig. 3. If the samples are all separated by t, then the maximum frequency that can be detected is f max = 1/2 t. This is called the Nyquist frequency. 14

17 A. Fourier Transform and Series Since f max is known, then it doesn t make sense to talk about the Fourier series beyond this frequency: g(t i ) = f max /2 f max /2 ĝ( f n )e 2πi f nt i (14) For similar reasons, there is also a minimum frequency whose wavelength must correspond to the full length of the time sample, and for every other subdivision of the samples, it is possible to extract an associated frequency f n. In other words: f n := N/2 n f max For n = 0, then f 0 is undefined and looks to be infinite which actually corresponds to having a constant. When working with discrete Fourier transforms in computing, the different frequencies are stored in a list. Based on the way frequencies are summed in eq. (14), it seems sensible to sort list the frequencies as [ f N/2,..., f 2, f 1, f 0, f 1, f 2,..., f N/2 ], However, this proves to be quite inconvenient since getting the 0th frequency requires finding the middle of the array, the 1st positive frequency is one element after the middle, etc. As a result, the frequencies are often stored as: [ f0, f 1, f 2,..., f N/2, f N/2,..., f 2, f 1 ], where the 1th frequency is the first element starting from the end, etc. Lastly, you ll notice that the common factor of f max is superfluous and doesn t need to be carried everywhere. Similarly, the algorithm to calculate ĝ( f i ) doesn t care whether t was 1 ms or 3 cm. For this reason, dft algorithms usually is normalized with factor of N. Furthermore, just like different Fourier transform conventions will include normalization factor of 2π in different places, the dft normalization factors of N will change too. As with the Fourier transform, the convention used is unimportant as long as it remains consistent. A.6.1. Fast Fourier Transforms The fast Fourier transform (fft) is, as the name suggests, a way of calculating a dft a lot faster. In the naive approach of calculating a dft, each of the N ĝ( f i ) require accessing all N g(t i ) and as a result it takes O(N 2 ) to calculate the dft. If N = 100, this is not an issue for a computer, but if N = 10 6, calculating a dft can quickly become intractable. A fft utilizes some clever tricks which exploit periodicity of the basis functions in order to reduce some redundant calculations. The resulting complexity of the dft in order O(N log N). This may not seem significant, but for an array of 10 6 elements, the computation time will go from arbitrary units to just This is a speed improvement of five orders of magnitude! In order words, if this fft computation took 1 second to compute, the naive dft would have taken about 1 day to compute. 15

18 B. Interpolating Functions Figure 4: The set of points in (x, y) coordinates does not, in general, correspond to a nice set of points in (ξ, θ) coordinates. B. Interpolating Functions In this lab, we have to change between coordinate systems quite frequently: (x, y) (ξ, η) (ξ, θ). In theory, when everything is over R 2, these changes of coordinates are easy to handle; however in practice, p θ (ξ) is only known for a finite set of θ and ξ. Similarly, µ(x, y) needs to be reconstructed for a set of x and y values. The problem is that a particular point (x i, y i ) will generally fall in between values of (ξ, θ) as illustrated in fig. 4. In order to fix this issue, a way to fill in the gap must be created and there are several methods available. They are explained below in terms of 1d problem, but they can be all generalized into 2d dimensions: Nearest Neighbour Use the value of the closest point that is defined: f (x) = f (x i ) where x i x is minimized Linear Interpolation Connect known points with straight lines: f (x) = f (x i+1) f (x i ) x i+1 x i (x x i ) + f (x i ) 16

19 B. Interpolating Functions Cubic Spline A spline is a (smooth) piecewise polynomial. The polynomial is define by: ( ) ( ) f (t) = 2t 3 + 3t f (x i ) + t 3 2t 3 + t f (x i ) ( + 2t 3 + 3t 2) ( f (x i+1 ) + t 3 t 2) f (x i+1 ) where t is the normalized distance between x i and x i+1 : t(x) := x x i x i+1 x i. This requires knowledge of the derivative of f which can be computed by looking at points around: f (x i ) 1 2 f (x i+1 ) f (x i ) + 1 f (x i ) 1 2 f (x i 1) x i+1 x i 2 x i x i 1 Polynomial Interpolation Given a polynomial of the form p(x) = p n x n + p n 1 x n p 0 and set of n points p(x i ) = y i, then we can form the system of linear equations x0 n x1 n. x n n x n 1 x n x x n... x n 1 x n 1 p n p n 1. p 0 = y n y n 1 which can be easily solved (by a computer) for p i. This then defines the polynomial p(x) that goes through each point p(x i ) = y i.. y 0 17

20 References References [1] W. C. Röntgen, Über eine neue art von strahlen, Vorläufige mittheilung, Sitzungsberichte der Würzburger Physik-medic Gesellschaft (1896). [2] W. C. Röntgen, Über eine neue art von strahlen, Fortsetzung, Sitzungsberichte der Würzburger Physikmedic Gesellschaft (1896). [3] W. C. Röntgen, Weitere beobachtungen über die eigenschaften der x-strahlen, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1897). [4] W. C. Röntgen, On a new kind of rays, Science 3, (1896) /science [5] J. Radon, Über die bestimmung von funktionen durch ihre integralwerte längs gewisser mannigfaltigkeiten, Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse (1917). [6] J. Radon, On the determination of functions from their integral values along certain manifolds, IEEE Transactions on Medical Imaging 5, (1986) /tmi