KTH Royal Institute of Technology

Bachelor thesis

Reconstruction inComputerized Tomography

using Entropy Regularization

Author:Martin SjöbergTheoharis Charitidis

Supervisor:Axel Ringh

Abstract

Inverse problems are a class of problems that often appear in science. They canbe thought of as calculating the cause that corresponds to measured data, andit is the opposite (i.e. inverse) of a direct problem, where data is calculated froma known source. In Computerized Tomography (CT), which is an importantnon-invasive imaging method in medicine, inverse problems arise when wewant to reconstruct the inner structure of the physical body from indirectmeasurements. In this thesis we investigate different methods for CT imagereconstruction. We compare the reconstruction quality of using Entropy asa regularizer to Tikhonov - and Total Variation (TV) regularization and theanalytic Filtered Backprojection approach, and also investigate the sensitivityof entropy regularization with respect to priors.

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Contents

1 Introduction 3

2 Background 42.1 Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 The Radon Transform . . . . . . . . . . . . . . . . . . 42.2 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Filtered backprojection . . . . . . . . . . . . . . . . . . . . . . 72.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Tikhonov regularization . . . . . . . . . . . . . . . . . 82.4.2 Total Variation regularization . . . . . . . . . . . . . . 82.4.3 Entropy regularization . . . . . . . . . . . . . . . . . . 9

3 Method 93.1 ODL library . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Results 124.1 Comparing reconstructions with different methods . . . . . . . 124.2 Entropy Priors . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Discussion 16

A Python code utilizing ODL 18

2

1 Introduction

Inverse problems arise in many areas of science and mathematics when onewants to infer properties of something which is not directly observed. This hasapplications in many areas including signal processing, astronomy and medicalimaging. Simply put it is the problem of finding a cause from an effect, whichis the opposite of a direct problem where the result is calculated from the cause.

Computerized tomography (CT) is a method used in a number of industriesand is also an active field of research. It is a non-invasive method of looking atthe inner structure of objects, which is useful in industry for internal inspectionof components, and one common application area is in medical imaging. Thetype of CT in focus here is X-ray Tomography, which is one of several types [8,p. 41-61].

The reconstruction problem of CT is an ill-posed inverse problem, andthere are different ways to try and solve it. One important algorithm is theFiltered Backprojection algorithm (FBP) [8, p. 81], which is an approach basedon approximating an inverse to the Radon transform analytically. However,the drawbacks of FBP are for example its sensitivity to noise and it’s tendencyto produce artifacts (e.g. if the data is incomplete) [10]. Alternative methodshave been suggested, methods that are normally computationally more expen-sive but can be more robust regarding noise or incomplete data. One way toformulate such methods is to formulate the reconstruction problem as an op-timization problem which can be solved by iterative methods. In this settingthe ill-posedness can be mitigated with regularizing cost functions, where dif-ferent cost functions can be used to promote different structures and features.The main focus of this thesis is to evaluate the use of entropy as a regular-izer, first by comparing the maximum entropy regularization (MER) to theTikhonov- and Total Variation (TV) regularizations, secondly by comparingMER to relative entropy by introducing different priors.

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2 Background

2.1 Tomography

Tomography aims at creating images of sections in an object. The term ismostly present in industrial and scientific applications but also in the medicalfield. The basic technique works by shooting rays through the analyzed objectfrom several angles which are then captured by a detector on the opposite side(see figure 1) . This method is based on what is called the Radon Transform [7].

Figure 1: Illustration of how tomographic measurements are made, (from [7]p.28)

2.1.1 The Radon Transform

The Radon Transform is the foundation of the reconstruction process in to-mography. The concept is that by measuring line integrals from several anglesthrough a body, which is defined by a function f , one can calculate the originalfunction f . In the case of tomography the function f corresponds to the imagewe want to reconstruct and the line integrals correspond to the rays that travelthrough the analyzed object.

If we shoot one ray with known intensity, I0, and measure the intensity onthe opposite side of the body, I, the ratio between them can be approximatedby the following line integral

logI0I≈∫ ba

f(x)dx, (1)

where f(x) is the attenuation along the path of the ray, a and b are theendpoints of the path [7, p. 24]. The forward operator in CT, called theRadon transform (see figure 2) is based on this, and defined as

R(f)(~n⊥, x) =

∫ ∞−∞

f(x+ t~n⊥)dt, (2)

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where x is the point where the ray crosses the horizontal axis, and α is theangle ~n makes with the horizontal axis. The integration is along the ray, whichis perpendicular to ~n, i.e., parallel to ~n⊥.

Figure 2: Illustration of how the Radon Transform conceptually works fromen.wikipedia.org.

2.2 Inverse Problems

In the following subsection we present a general theory for inverse problems.For a more thorough introduction see, e.g , [7].

Unlike direct problems, which are finding the results from a cause, inverseproblems are formulated as to find the cause from a result. For instance, if animage is inverted then the inverse problem would be to invert it back to theoriginal. This implies that one can move seamlessly between the inverted imageand the original without any loss of information or error. This would workin practice because the example problem is well-posed. The direct problemis formulated as Ax = m and the inverse problem is formulated as to find x,that is solving Ax = m for x. A is called the forward map and in the contextof tomography it is the Radon transform, sometimes referred to as the raytransform.

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Figure 3: Illustration of color inversion of an image(from pexels.com).

However, the methods used in this paper are formulated for ill-posed in-verse problems, which are inverse problems that are more difficult to solve. Tounderstand this better we use the notion of a well-posed problem, introducedby Jasques Hadamard (1865-1963). The inverse problem Ax = b is said to bewell-posed if

A1: Existence. There is at least one solution.

A2: Uniqueness. Only one solution exists.

A3: Stability. Solution depends continuously on data.

An ill posed problem is one which fails to fulfill at least one of the aboveconditions.

Example 1: Consider the equation:

Ax = b ⇐⇒[0 10 0

] [x1x2

]=

[b1b2

], (3)

and b2 6= 0. Since 0 · x1 + 0 · x2 = 0 6= b2, a solution does not exist. In thiscase it fails to satisfy A1. This occurs in the case of tomography as well, wherenoise is prevalent in the measurement.

Example 2: We consider the equation:

Ax = b ⇐⇒[1 0 10 1 0

]x1x2x3

= [b1b2

], (4)

where b1, b2 are constants. It is easily seen that any vector of the form

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x1x2x3

=b1 − sb2

s

, (5)for all s ∈ IR is a solution. Thus this system does not have a unique solution,hence violating A2.

We consider the general mathematical formulation of an inverse problem,which is to solve the following equation for f:

m = Af + �, (6)

where A : X 7→ Y is the ray transform, mapping the image f ∈ X intothe data space Y (represented in figure 4b as a sinogram) thus producing themeasured data m. It should be noted that the direct problem must be wellposed - A must be well defined, single valued and continuous. The vector � ∈ Ycorresponds to the error caused by noise in the measurement and m ∈ Y isthe measured data.

If f is the image then arguably the solution could be approximated as

A−1m ≈ f, (7)

however, reconstructing the CT image using (7) will not work because the raytransform, A, is not invertible.

2.3 Filtered backprojection

Initially in X-ray tomography the filtered backprojection (FBP) algorithm wasused to create images from a finite data set. Even today FBP holds a strongpresence in the industry of tomography, and that is arguably for its accuracy,easy implementation and computation speed.

The downside of FBP is that it needs evenly spaced angles in its measure-ments to work properly. If one of the data sets are missing or get corruptedthe end result will form line artifacts that affect the overall image quality.

FBP is an analytical approach unlike other methods described later. Sim-ply put this algorithm can be thought of as trying to solve the reconstructionproblem by approximating the inverse of the forward map connected to thespecific space, i.e., it is a pseudo-inverse of the ray transform. [10]

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2.4 Optimization

As described earlier, the naive reconstruction (7) will not suffice to solve theinverse problem. Instead reconstruction methods can be formulated using opti-mization. A broad class of methods are captured by the following optimizationproblem:

minf||Af −m||22 + αF (f), (8)

where F (f) is the regularizing functional and α > 0 is the regularizationparameter, meaning that it tunes the balance between the regularizing termand the data-matching term. Note that the problem at hand is a convex oneif F is convex. What separates the different methods in this context is thechoice of the regularization functional F (f), which can depend on our priorinformation about the solution. Having information about the solution is oftenreferred to as the prior [7].

2.4.1 Tikhonov regularization

Tikhonov regularization is a classical method for solving ill-posed inverse prob-lems. This is because of the method’s ability to provide smoothing and imple-ment known properties of the solution.

The classical Tikhonov regularization is formulated as

minf≥0||Af −m||22 + α||f ||22, (9)

where α > 0 acts as the regularization parameter, which as stated earlier tunesthe balance between the regularizing term and the data-matching term.

2.4.2 Total Variation regularization

Total Variation (TV) is another well known method for solving ill-posed in-verse problems. Unlike Tikhonov and FBP, which produce fairly smooth re-constructions, TV has the ability to minimize noise and be edge preserving.This comes with the cost of losing fine details but gives more definition alongthe sharp edges of a reconstruction [7].

The mathematical formulation for Total variation is

minf≥0||Af −m||22 + α||∇f ||1, (10)

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where ∇f is the spatial gradient of f. The regularizing functional in TV sharessome similarity to the one in Tikhonov regularization, but here it is the 1-norm of the spatial gradient of f instead. The regularization parameter α > 0acts as the ”tuning” parameter here as well - it can be modified as to decidehow much we ”trust” the original data. This is a non-smooth problem dueto the 1-norm, wherefore we need to use a solver able to handle non-smoothoptimization.

2.4.3 Entropy regularization

The mathematical notion of Entropy, closely related and sometimes used inter-changeably with Kullback Leibler divergence (KL) [6], was introduced by CEShannon [2,9] in the field of information theory. The use of the so called max-imum entropy regularization (MER) has been investigated in solving inverseproblems [2, 5], including image reconstruction. [10]

According to Tseitlin et al. [10] there have been a small number of con-flicting reports on the use of entropy for magnetic resonance imaging (MRI).In that setting they conclude that it can be better than FBP in suppressingbaseline noise and it also can produce reconstructions without the character-istic artifacts of FBP using few projection angles and/or projections withouta uniform angle distribution. [10]

Here, MER is formulated as

minf≥0||Af −m||22 + αH(f), (11)

where H is the functional:

H(f) =

∫f(x)− g(x) + g(x) log( g(x)

f(x))dx if f > 0

+∞ otherwise, (12)

with prior g ≡ 1. Inserting a prior g(x) > 0 corresponds to the relativeentropy of g with respect to f . We consider the use of MER and relativeentropy separately, using different priors g(x) in the latter method.

3 Method

The goal of this thesis is twofold. First we compare the quality of reconstruc-tion of Filtered Backprojection- (FBP), Tikhonov-, Total Variation- (TV) andEntropy regularization. Secondly, we test the Entropy regularization’s sensi-tivity with respect to priors.

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3.1 ODL library

The Operator Discretization Library (ODL1) is an open-source Python libraryfor prototyping reconstruction methods for inverse problems, which will beused for all computation and reconstruction [1].

3.2 Approach

For the specifics of how ODL is used, the complete code can be seen in Ap-pendix A. A more in depth view of the package with example code, etc., canbe found in the documentation.2 ODL works on a high level and the built insolver used, the Douglas-Rachford primal-dual splitting algorithm [3], can beutilized using minimalistic code. We also utilize the ASTRA toolbox [11] forcomputing the ray transform.

The phantom we use is the Shepp-Logan phantom, which can be seen inFigure 4a and the corresponding data generated by the ray transform can beseen in Figure 4b. This data is generated using a uniform angle partition of 180angles over π radians, along with a signal-to-noise level of approximately 5 %white Gaussian noise. This noisy data can be seen in Figure 4c. We also use adetector partition of 256 ’bins’. After the reconstruction methods are defined,we try different regularization parameters and select the best reconstructionsby visual assessment. The side-by-side comparisons can be seen in the resultssection.

The second part of this paper is focused on testing entropy regularization’ssensitivity to priors. This is done by using different priors, where some arecreated by modifying the original Shepp-Logan phantom (e.g. by removingor adding some features) and one reconstruction is done using the Forbildphantom as a prior, which can be considered a ’bad’ prior. This is done toinvestigate if the algorithm performs well even though the prior is bad.

1https://github.com/odlgroup/odl2https://odlgroup.github.io/odl/

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https://github.com/odlgroup/odlhttps://odlgroup.github.io/odl/

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Figure 4: The Shepp-Logan Phantom (a), the corresponding data (b) and datawith additive white Gaussian noise (c).

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4 Results

As stated above, white Gaussian noise is added to the data to yield approx-imately a signal-to-noise ratio of 5 %. This is used for all reconstructions.Below, the best reconstructions are presented for FBP, Tikhonov, TV andEntropy respectively.

4.1 Comparing reconstructions with different methods

In figure 5 below is a side-by-side comparison of FBP, Tikhonov, TV andMaximum Entropy respectively. As can be seen in figure 5a, FBP does notsuppress much of the noise and the noise is also present somewhat uniformlyover the entire image independently of underlying density of the phantom.The three smaller dots in the lower region is not very distinctive and if we didnot know they were there to begin with we would not be able to tell for sure.This can be seen in strong contrast to Tikhonov (figure 5b) where that methoddoes a lot better at suppressing the noise in the regions with low density, e.g.on the outside of the body and the inside of the two bigger ellipses inside thephantom. However, Tikhonov does not handle the noise as good in regionswith higher density, i.e., the lighter gray areas. As with FBP, this makes itdifficult to discern the smaller dots from the surrounding noise, therefore wecan not suggest it is better than FBP in that perspective.

Looking at figure 5c, Total Variation does a tremendous job at suppressingnoise all over and smoothing out the image, which is essentially what wasexpected. The strong edges between areas are well preserved when they differenough in contrast, but when the difference between density is smaller it doesnot perform as well. This can be seen by the three smaller dots being difficultto discern from one another.

Lastly we look at the Maximum Entropy in figure 5d. There is a decentamount of noise suppression, especially on the outside of the phantom, but thenoise present inside the body is almost on the same level as with Tikhonov.The main drawbacks that can be observed is a bad edge definition which looksalmost like a ’glow’, especially visible on the outside rim of the body wherethe stark contrast between the white and the black meet. This also makesthe edges of features inside the body worse defined than for all the otherreconstructions.

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Figure 5: Reconstructions using FPB (a), Tikhonov (b), TV (c) and MER (d).

4.2 Entropy Priors

Earlier we used the so called maximum entropy for reconstruction (see figure5d). This means setting the prior in (12) to a constant 1, i.e., g(x) ≡ 1. Simplyput it means a prior that is white on the entire domain, which can be seen asa ’dumb’ prior in the sense that it does not use any prior information aboutthe body in study. Now we investigate the use of a few different priors to seehow much it can affect the reconstruction. We use both priors that are closeto the Shepp-Logan phantom (figure 4a) but with a few features changed andsome that do not resemble the phantom as much. Below (figures 6-9) are allpriors tested side-by-side to their resulting reconstruction.

Looking at the first one in figure 6, the Forbild phantom is used as a priorand the reconstruction is seen in 6b. This prior is obviously differing fromthe Shepp-Logan phantom and the effect on the reconstruction is clearly seen.

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There are artifacts introduced from the prior that does not result in a goodreconstruction.

In figure 7 a prior is used were the three small dots in the lower half ofthe phantom has been removed. The major thing to notice is the difference insharpness, contrast and edge-preservation that occurs compared to maximumentropy (compare figure 5d and 7b). On the other hand, the three small dotsare still hard to see.

A prior with another feature removed can be seen in figure 8a. This issimilar in result to the one used above, the difference is that now we removedone of the medium sized dots in the middle. The same pros can be seen here,e.g. a major improvement in edge-preservation and contrast. Even in this casethe dot is visible to some degree, but it is hard to tell if one would notice it ifit was not known beforehand that it is supposed to be there.

Lastly we use the same phantom and prior as in figure 7, but we inter-change how they are used, i.e., use figure 7a as phantom and the Shepp-Loganphantom as prior. This means that the measurement data is different, be-cause we use figure 7a as the data generating phantom. Thus, the data doesnot contain the three small dots, i.e., they do not exist in the actual body;they are only introduced in the prior. This can be thought of as using priorinformation that the body used to contain the three dots but do not anymore.In this case the reconstruction shows the dots clearly, and thus introduces falsestructures in the reconstruction.

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Figure 6: Reconstruction (b) using the Forbild phantom (a) as prior.

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Figure 7: Reconstruction (b) using a modified SL phantom (a) as prior.

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Figure 8: Reconstruction (b) using a modified SL phantom (a) as prior.

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Figure 9: Reconstruction (b) using figure 7a as the data generating phantomand the standard Shepp-Logan phantom (a) as prior.

5 Discussion

From the results it can be concluded that each method managed to highlightdifferent features from the original Shepp-Logan phantom. However, the moreestablished FBP failed to produce satisfactory results with the applied noiseand goes to show that it probably does not suffice for the given context, evenif it is the most established method for image reconstructions [10]. MoreoverMER did not deliver satisfactory results when no prior was used, instead weobserved (as seen in figure 5c) that TV gave a reconstruction that delivers themost noise free, and edge defined result.

This can be explained by that the regularizing functional used in TV istrying to minimize the spatial gradient, which in simple terms means adjacentpixels should have similar values. It is useful if we know we are looking at asmall number of jumps between different materials since that makes the gradi-ent inherently small on most parts of the image, for example if we are lookingat bone versus soft tissue; then it’s easy to see the outlines and structure ofthe bone. On the other hand it suppresses the smaller details so it might notbe ideal when trying to see structure in materials close in density. As can beseen in Figure 5c it loses the definition between the three dots at the bottomwhich just have a slight higher density than the surrounding ’tissue’.

Tikhonov did a decent job at suppressing noise on the parts with zeroattenuation (the black portions in the phantom, Figure 4a), but the noiseremaining on the inside isn’t considerably better than FBP. The edges are ofcourse not as sharp as with TV since Tikhonov punishes sharp edges more

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than TV.As mentioned, MER did not deliver satisfactory results regarding edge

definition, but managed to suppress the noise on the zero attenuation parts(as with Tikhonov). However as it stands, MER without a prior is not suitablein this context due to the blurriness and unclear definition.

There have been some reports that conflict with each other regarding theutility of MER in magnetic resonance imaging (MRI). Constable and Henkel-man [4] came to the conclusion that MER ’does not work’ in the context ofMRI reconstruction. On the other hand Tseitlin et.al. [10] mention that ithas been demonstrated that when MER applied to a 2D image, baseline noisewas suppressed and the images could be reconstructed from smaller numberof projections without any unwanted effects.

In the case of using priors in connection with MER the results tell us thatthere is much potential to produce a good quality reconstruction. It is clearthat new information appears in the reconstruction when the prior is wellmatched to the measurement data. However, using priors for MER does notproduce a well defined reconstruction when the prior is not well matched tothe original data, producing artifacts (as seen in figure 6b). This is also seen infigure 9b, where false positive structures are introduced in the reconstruction.We thus conclude that if one is not certain about the accuracy of the prior, itis better to use MER or other methods.

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A Python code utilizing ODL

This is the code we used for reconstructing using the different methods.

import numpy as np

import odl # the Operator Discretization Library

# Choice between Filtered Backprojection (FBP), Tikhonov, Total

Variation(TV) and Entropy

regularization = ’Entropy’

# The regularization parameter

a=1

# The reconstuction space

space = odl.uniform_discr(min_pt=[-20, -20], max_pt=[20, 20],

shape=[512, 512])

# Number of projection angles and their distribution

angle_partition = odl.uniform_partition(0, np.pi, 180)

# Detector partition

detector_partition = odl.uniform_partition(-30, 30, 256)

# The detector geometry

geometry = odl.tomo.Parallel2dGeometry(angle_partition,

detector_partition)

# The ray transform

ray_trafo = odl.tomo.RayTransform(space, geometry)

# The phantom

phantom = odl.phantom.shepp_logan(space, modified=True)

# Generate data

data = ray_trafo(phantom)

# Adding white Gaussian noise, approximately 5% signal-to-noise ratio

raw_noise = odl.phantom.white_noise(ray_trafo.range)

data += raw_noise/raw_noise.norm()*data.norm()*0.05

if regularization == ’FBP’:

# There are different filters in the fbp_op function, but we used

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the default one

fbp_recon = odl.tomo.fbp_op(ray_trafo)(data)

fbp_recon.show(’FBP’)

else:

A = ray_trafo

f = odl.solvers.IndicatorBox(space, 0)

# the data-matching term

g1 = odl.solvers.L2NormSquared(A.range).translated(data)

if regularization == ’Tikhonov’:

B = odl.IdentityOperator(space)

# 2-norm squared of f

g2 = a * odl.solvers.L2NormSquared(space)

sigma = [0.2, 0.5]

elif regularization == ’TV’:

B = odl.Gradient(space)

# 1-norm of the spatial gradient of f

g2 = a * odl.solvers.L1Norm(B.range)

sigma = [0.1, 0.02]

elif regularization == ’Entropy’:

B = odl.IdentityOperator(space)

# the Kullback-Leibler functional

g2 = a*odl.solvers.KullbackLeibler(space)

sigma = [0.2, 0.5]

else:

raise RuntimeError(’Unknown regularization’)

lin_ops = [A, B]

g_funcs = [g1, g2]

x = A.domain.zero()

call = (odl.solvers.CallbackPrintIteration()&

odl.solvers.CallbackShow(title=regularization + ’, alpha = ’

+ str(a),display_step=100))

odl.solvers.douglas_rachford_pd(x,f,g_funcs, lin_ops,

tau=0.1, sigma=sigma, niter=1000,

callback=call)

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References

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[2] G. Le Besnerais, J.F. Bercher, and G. Demoment. A new look at entropyfor solving linear inverse problems. IEEE Transactions on InformationTheory, 45(5):1565–1578, 1999.

[3] R.I. Bot and C. Hendrich. A douglas–rachford type primal-dual methodfor solving inclusions with mixtures of composite and parallel-sum typemonotone operators. SIAM Journal on Optimization, 23(4):2541–2565,2013.

[4] R.T. Constable and R.M. Henkelman. Why MEM does not work in mrimage reconstruction. Magnetic resonance in medicine, 14(1):12–25, 1990.

[5] I. Csiszar. Why least squares and maximum entropy? An axiomaticapproach to inference for linear inverse problems. The annals of statistics,pages 2032–2066, 1991.

[6] S. Kullback and R.A. Leibler. On information and sufficiency. The annalsof mathematical statistics, 22(1):79–86, 1951.

[7] J.L. Mueller and S. Siltanen. Linear and nonlinear inverse problems withpractical applications. SIAM, 2012.

[8] F. Natterer and F. Wübbeling. Mathematical methods in image recon-struction. SIAM, 2001.

[9] C.E. Shannon and W. Weaver. The mathematical theory of communica-tion. 1948.

[10] M. Tseitlin, A. Dhami, S.S. Eaton, and G.R. Eaton. Comparison ofmaximum entropy and filtered back-projection methods to reconstructrapid-scan EPR images. Journal of Magnetic Resonance, 184(1):157–168,2007.

[11] W. van Aarle, W.J. Palenstijn, J. De Beenhouwer, T. Altantzis, S. Bals,K.J. Batenburg, and J. Sijbers. The ASTRA toolbox: A platform for ad-vanced algorithm development in electron tomography. Ultramicroscopy,157:35 – 47, 2015.

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IntroductionBackgroundTomographyThe Radon Transform

Inverse ProblemsFiltered backprojectionOptimizationTikhonov regularizationTotal Variation regularizationEntropy regularization

MethodODL libraryApproach

ResultsComparing reconstructions with different methodsEntropy Priors

DiscussionPython code utilizing ODL