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Improved total variation-based CT image reconstruction applied to clinical data

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 Phys. Med. Biol. 56 1545 (http://iopscience.iop.org/0031-9155/56/6/003) View the table of contents for this issue, or go to the journal homepage for more

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IOP PUBLISHING

PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 56 (2011) 1545–1561

doi:10.1088/0031-9155/56/6/003

Improved total variation-based CT image reconstruction applied to clinical data Ludwig Ritschl1 , Frank Bergner1 , Christof Fleischmann2 and Marc Kachelrieß1 1 Institute of Medical Physics (IMP), University of Erlangen-N¨ urnberg, Henkestr. 91, 91052 Erlangen, Germany 2 Ziehm Imaging GmbH, Donaustraße 31, 90451 N¨ urnberg, Germany

E-mail: [email protected]

Received 24 August 2010, in final form 12 January 2011 Published 16 February 2011 Online at stacks.iop.org/PMB/56/1545 Abstract In computed tomography there are different situations where reconstruction has to be performed with limited raw data. In the past few years it has been shown that algorithms which are based on compressed sensing theory are able to handle incomplete datasets quite well. As a cost function these algorithms use the 1 -norm of the image after it has been transformed by a sparsifying transformation. This yields to an inequality-constrained convex optimization problem. Due to the large size of the optimization problem some heuristic optimization algorithms have been proposed in the past few years. The most popular way is optimizing the raw data and sparsity cost functions separately in an alternating manner. In this paper we will follow this strategy and present a new method to adapt these optimization steps. Compared to existing methods which perform similarly, the proposed method needs no a priori knowledge about the raw data consistency. It is ensured that the algorithm converges to the lowest possible value of the raw data cost function, while holding the sparsity constraint at a low value. This is achieved by transferring the step-size determination of both optimization procedures into the raw data domain, where they are adapted to each other. To evaluate the algorithm, we process measured clinical datasets. To cover a wide field of possible applications, we focus on the problems of angular undersampling, data lost due to metal implants, limited view angle tomography and interior tomography. In all cases the presented method reaches convergence within less than 25 iteration steps, while using a constant set of algorithm control parameters. The image artifacts caused by incomplete raw data are mostly removed without introducing new effects like staircasing. All scenarios are compared to an existing implementation of the ASD-POCS algorithm, which realizes the step-size adaption in a different

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© 2011 Institute of Physics and Engineering in Medicine

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way. Additional prior information as proposed by the PICCS algorithm can be incorporated easily into the optimization process. (Some figures in this article are in colour only in the electronic version)

1. Introduction Iterative image reconstruction algorithms constitute a wide field of research in computed tomography. Due to the emergence of the theory of compressed sensing (Donoho 2006) in the past few years, many publications have addressed this topic (Persson et al 2001, Sidky et al 2006, 2009, Song et al 2007, Sidky and Pan 2008, Chen et al 2008, Yu and Wang 2010, Bian et al 2010). The algorithms presented seem to treat different problems of missing or undersampled data quite well. First some of the basic facts of compressed sensing in context with computed tomography and prior work will be summarized. Unconstrained iterative image reconstruction minimizes the 2 -norm ||Rf (r) − p||22 .

(1)

Here R is the x-ray transform, f (r) is the reconstructed image and p are the measured raw data. In the case of incomplete raw data, the inverse problem (1) is underdetermined and has an infinite number of possible solutions. Then it makes sense to incorporate a priori knowledge into the iteration process, which means that the cost function (1) has to be extended. One way to create such a cost function is based on the idea of compressed sensing. The quintessence of compressed sensing is that a signal, in this case the image f (r), can be completely reconstructed with a high probability with less samples than required by the Nyquist criterion, if a sparsifying transformation of the signal f (r) is known. That means most entries of the vector f (r) are zero. This fact can be approximated by the 1 -norm ||f (r)||1 of the vector f (r). Incorporating this into the process of image reconstruction yields the following constrained convex optimization problem: min ||f (r)||1 subject to ||Rf (r) − p||22 < .

(2)

characterizes the raw data consistency. The best reachable value of will be denoted with opt . The value of opt > 0 is not known before performing the iteration process. It depends on effects like noise, scatter, beam hardening and misalignment in the raw data which lead to a higher inconsistency and to larger values of opt . Choosing in equation (2) close to opt is recommended to guarantee a meaningful image content. Otherwise the influence of the 1 cost function might become too strong. This fact has also been extensively discussed in Bian et al (2010). The two constraints on the image f (r) are defined in two different spaces, the image domain and the raw data domain. The linear x-ray transform R interacts between these spaces. Applying R is realized by performing a forward projection. RT corresponds to a backprojection. These steps are computationally very demanding and can lead to long reconstruction times. So our main goal is to reduce the required number of forward- and backprojection steps to a minimum, to reach clinically acceptable reconstruction times. In the past different methods were presented to find a solution to equation (2). Sidky et al proposed different implementations of a hybrid algorithm, the so-called ASD-POCS framework (Sidky et al 2006, 2009, Sidky and Pan 2008). This algorithm treats the raw data fidelity and the sparseness constraint separately in an alternating manner. The raw data fidelity is minimized by an SART (Andersen and Kak 1984), the cost function with a gradient descent.


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There the two minimization procedures are adapted to each other by comparing the size of the change in the image due to the SART and the gradient descent sets. Chen et al use an extended cost function which considers a priori information in the form of a prior image (PICCS) (Chen et al 2008). They also use alternating gradient descent and SART, but without any adaptive control. Song et al (2007) and Yu and Wang (2010) use the unconstrained formulation (3) min ||f (r)||1 + μ||Rf (r) − p||22 of equation (2). That enables us to solve the resulting cost functional with a standard method like a conjugate gradient solver (Boyd and Vandenberghe 2004). For every value of in equation (2) there exists a value of μ in equation (3) which yields the same result. Main problem when using the unconstrained formulation is the choice of μ, which requires knowledge about the raw data consistency. Due to this we will focus on solving the constrained optimization problem. In this study we use the image gradient as sparsifying transformation. The resulting cost function ||∇f (r)||1 is also known as total variation (TV). The method presented here is an improved algorithm to solve the problem of constrained TV minimization. It is based on the ideas of the ASD-POCS framework, which means that we also use an alternating direction method to minimize (2). The main innovation of the proposed method is improved automatic adaption of the step size of both minimization steps. In real world applications the value opt of the reachable raw data consistency is unknown before performing a reconstruction. So it must be our aim to create an algorithm which does not need a fixed value for as input. Additionally the current value of during the optimization must not be ignored. Only a small value of which should be close to the unknown value opt can guarantee useful image content. In the literature there was only one adaptive method reported, which does not require explicit a priori knowledge of opt (Sidky et al 2009). As explicitly mentioned by the authors, one direction of future research could be an optimized parameter choice of the automatic adaption. With the method proposed we want to present one possible way to do so. To have a clear differentiation between the original ASD-POCS algorithm and the improved version, we will denote the proposed method with iTV (improved total variation constrained reconstruction). 2. Method To minimize the cost functional (2) iTV treats the two terms of equation (2) separately. The raw data fidelity term is minimized using SART. Therefor the projection data were divided into subsets. For precise definition of the implementation used, the following indices and parameters are defined: i := index which represents a volume voxel r j := index which represents an element of one projection k := index which represents a projection inside a subset Rkj,i := systemmatrix of the kth projection, which maps f on p k

NProj := total number of projections NSub := number of subsets ν := index which represents a subset s(ν) := number of projections in the νth subset β := relaxation parameter

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fiν+1 = fiν + β

k∈s(ν)

1 T k j (R )i,j

(RT )ki,j

i

Rkj,i fiν − pjk

i

Rkj,i

, ν ∈ [0, NSub [.

(4)

The two vector divisions in the right fraction of equation (4) are calculated component by component. The operator Rk treats only the projection pk. Note that all elements fi of the defined volume have to be processed inside each subiteration ν. In this study we chose NSub = NProj which leads to a subset size s(ν) = 1. Updating all subsets ν leads to the SART , so the index n represents the iteration number of a whole complete SART update fn+1 SART update step. In the following n will denote a so-called outer iteration. Due to Censor and Elfving (2002) and Jiang and Wang (2003), SART minimizes a weighted norm LW 3 instead of the Euclidian norm in equation (2). But according to the results in Sidky and Pan (2008) and Chen et al (2008) we could observe a straight decrease of the unweighted raw data norm by the SART, so we will use this expression in our algorithm. SART to minimize After each SART update the gradient descent method is applied on fn+1 the TV. An iteration of the gradient descent is given by TV TV TV fn+1,m+1 (r) = fn+1,m (r) − α · ∇ ∇fn+1,m (r)1 . (5) TV SART As initialization image fn+1,0 (r) we use fn+1 (r). The index m describes a gradient descent iteration, which is the inner iteration. The parameter α is calculated individually in each update step by a backtracking line search to ensure a minimization of the TV (Boyd and Vandenberghe 2004). The major problem when combining these two procedures is to ensure the optimal balance between the SART and the gradient descent step. The main goal of iTV is to converge to a small value of so that the raw data constraint is satisfied, whereas the cost function, in this case the TV, should be kept at a low value. To reach this the following scheme for an automatic adaption is proposed: let

n = ||Rfn (r) − p||22 SART be the value of the fidelity term at the outer iteration n. Then n+1 is the optimized value after the SART step. After applying an inner iteration step of the gradient descent the value TV is given by n+1,m+1 TV 2 TV n+1,m+1 = Rfn+1,m+1 (r) − p2 .

To ensure a constant decrease of the raw data constraint in each outer iteration the gradient descent must be stopped if the condition TV SART n+1,m < (1 − w)n+1 + wn

with

w ∈ [0, 1[

(6)

does not hold any more. Here w is a user-defined constant parameter defining an upper bound of the raw data fidelity that must not be exceeded by the gradient descent. The restriction SART w ∈ [0, 1[ leads to a continuous decrease of implying the assumption that n+1 < n . The way the gradient descent is adopted here has one big disadvantage. Every calculation TV requires a forward projection, which is computationally demanding. So it is of n+1,m+1 desirable to find another way which has the same effect, but which bypasses the stepwise TV . This can be reached by using the convex properties of the 1 -norm. calculation of n+1,m+1 3

LW =

k,j

2 1 Rkj,i fi − pjk . k i Rj,i


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Figure 1. This image illustrates how the gradient descent step size is adapted by iTV. A blue line represents regions of constant total variation whereas the minimum value is in the center. The arrows depict the path generated by the gradient descent, in this case for M = 2. The dotted line SART TV implies all linear combinations of fn+1 and fn+1,M . All values of 0 < λ 1 lead to reduced TV.

A convex function f (x) with x, a ∈ D, where D is a convex set is defined by f ((1 − λ)x + λa) < (1 − λ)f (x) + λf (a) ∀ λ ∈ ]0, 1[ . If f (x) < f (a), then f ((1 − λ)x + λa) < f (a) ∀ λ ∈ ]0, 1[

(7)

follows directly. TV at each step m to prove if (6) is still valid, iTV performs a Instead of computing n+1,m+1 TV is fixed number M of gradient descent iterations. Here we choose M = 30. The value n+1,M only calculated once at the end of the iteration loop. Now the variable λ ∈ ]0; 1] is introduced which enables us to adapt the TV step size in the image domain, so that no further forward projection is required: SART TV (r) + λfn+1,M (r) fn+1 (r, λ) = (1 − λ)fn+1

is the final update image of one iTV iteration. Due to (7) the TV of the image fn+1 (r, λ) will be reduced for all values of λ (figure 1): ||∇f TV (r)||1 ||∇fn+1 (r, λ) < ||∇f SART (r) . n+1,M

1

n+1

1

Using the linearity of the x-ray transform, λ is determined analytically by solving the quadratic equation SART + wn . ||Rfn+1 (r, λ) − p||22 = (1 − w)n+1

In the case of a solution with λ > 1, λ has to be reset to 1. This corresponds to TV SART < (1 − w)n+1 + wn , n+1,M

(8)

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TV Figure 2. Here the development of during one iteration of iTV is shown. In this example n+1,M SART is above the user-defined upper bound (1 − w)n+1 + wn , which is marked by the red line. Now λ is reduced as proposed by equation (8).

Figure 3. A flow chart showing how the single steps of iTV are connected.

which means that the gradient descent leads after M iterations to a value of below the upper bound. Figure 2 shows the single steps of the iTV algorithm with the according values of . Additionally figure 3 shows a flow chart of the algorithm. As stopping criterium for the algorithm a fix number N of iterations or a value lim can be used with stop if

SART < lim . n − n+1

There are three parameters which control the algorithm: the stopping criterium as mentioned above, the SART relaxation parameter β and the introduced value w. 3. Results It was our aim to proof the algorithm using different datasets of clinical interest. One case is the loss of data caused by the high attenuation of metal implants. The second scenario covers limited view angle tomography. Another important case we demonstrate is angular undersampling, which has been proposed as a method to reduce patient dose while preserving


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image quality, if a TV-based algorithm is used for image reconstruction (Sidky et al 2006). The next example shows interior tomography reconstructions of fully truncated projection data. This problem can be solved by applying compressed sensing to the reconstruction as shown in Yu and Wang (2009). For all reconstructions the parameters were chosen in the following way: β = 0.8 w = 0.8 lim = 0.01 2SART − 1 . Here is defined relative to the SART update step size at the start of the iTV algorithm, which should be larger than the following step sizes. That allows applying the algorithm where the magnitude of is unknown. The parameters β and w have an influence on the convergence speed toward low values of and on the size of the TV value reached inside the -radius. Choosing small values for β, that means slower convergence of , and a large value for w, which ensures a sufficient expansion space for the gradient descent, leads to results, which are a good approximation of equation (2). For an optimal tradeoff between the quality of the approximation of equation (2) and convergence to a low value application-specific parameter tuning can be done, but in general we see no necessity to do so. The ability of iTV to reach good results with a fixed set of parameters in different situations and with different datasets will be shown in this section. To demonstrate the effectiveness of iTV and to have a comparison between existing algorithms, all datasets have also been reconstructed with the ASD-POCS algorithm. We used the implementation proposed in Sidky et al (2009). This version of the ASD-POCS algorithm tries to adapt the step sizes by comparing their size in the image domain. If the sparsity step has a larger Eucledian norm than the raw data step, it is reduced. The constraint of convergence to a small ≈ opt is only controlled by the choice of the SART relaxation parameter. For a more detailed description we refer the reader to the original paper. The parameters of the ASD-POCS algorithm were tuned as similarly as possible to the iTV parameters to allow an objective comparison. We used the same number M = 30 of gradient descent iteration steps. The only parameter which differs is the relaxation parameter β of the SART update. We set it to β = 1 in the ASD-POCS case, because it is the only way to enforce stronger raw data consistency. As the stopping criterium we used the number of iterations N after which iTV reached the stopping criterium. Additionally we show ASD-POCS images for N = 50, which is quite a large iteration number compared to the values of iTV. After the last SART update step of ASD-POCS the gradient descent was skipped, as proposed by the authors. The effect of this step will be discussed later. To evaluate the image quality and the convergence of the algorithms the difference images fdiff (r) relative to a full sampled reconstruction ffull(r) are shown. An exception is the metal implant scan. Here the ground truth image is not available. The value |fdiff (r)| = |(fN (r) − ffull (r))| E= r

r

was used to quantify the result. Additionally, the value of N and the value of the TV of the final images fN (r) are mentioned. Because both algorithms depend on their parameters and on the application also, it is difficult to evaluate them. In this paper we put the main focus on the convergence toward low values of . The second criterium is the value of the TV inside the reached radius of , which means how well is the solution of (2) approximated. This fact should also be reflected by the image error E, assuming that the image gradient is an adequate sparsifying transform of the underlying data. To get an impression of the progress of both algorithms the trajectories of and of the TV are plotted for one example.

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L Ritschl et al Table 1. Image error E, raw data fidelity N and total variation of the metal datasets.

E N TV

FBP

LI-MAR

– 12.34 153.22

– 13.61 143.93

ASD-POCS (N = 7)

ASD-POCS (N = 50)

–

–

2.73 141.38

2.69 141.62

iTV

2.62 136.52

Table 2. Image error E, raw data fidelity N and total variation of the sparse sampled datatsets.

E N TV

FBP

ASD-POCS (N = 21)

ASD-POCS (N = 50)

iTV

4.82 21.61 168.76

0.34 1.34 142.14

0.34 1.33 142.23

0.33 1.31 142.76

3.1. Metal implants The dataset used here shows a hip with two metallic implants from a clinical CT scanner (Somatom Sensation 16, Siemens Healthcare, Forchheim, Germany). The problem while scanning objects with metal implants is the high attenuation property of metal. That means most of the x-rays which were absorbed by metal do not consist of useful low contrast information. The projection values in the metal trace were considered as corrupted and were not used for image reconstruction. The resulting images and their evaluation, including a standard linear interpolation-based metal artifact reduction method (LI-MAR) (Kalender et al 1987), are shown in figure 4 and table 1. In the case of N = 50 the ASD-POCS implementation did not lead to much improvement compared to N = 7. The stopping criterium was reached after N = 7 iterations. The FBP reconstruction suffers under strong metal artifacts. The standard metal artifact correction method is able to remove them, but introduces new streak artifacts. The iTV reconstruction is nearly free of metal and streak artifacts. Also, here low contrast details are clearly visible. The ASD-POCS reconstruction shows good low contrast characteristics but still suffers from streak artifacts, which leads to a higher TV value.

3.2. Angular undersampling In our case angular undersampling was achieved by skipping projections in the angular direction. The dataset is an abdomen scan performed with a clinical CT scanner (Somatom Sensation 16). The undersampled data consist of 145 equally spaced projections, which complies with an undersampling factor of 4. The convergence criterium was reached after 21 iterations. Additionally a filtered backprojection of the full sampled data (580 projections) is shown (figure 5). The FBP reconstruction (145 projections) shows strong streak artifacts due to the insufficient sampling rate. The iTV algorithm is able to remove most of these artifacts without visible introduction of unwanted smoothing effects. The ASD-POCS algorithm reaches the same image quality as iTV for N = 21 and N = 50. The values reached for all algorithms are shown in table 2.


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Figure 4. The upper image shows an FBP (filtered backprojection) reconstruction of the complete dataset. In the middle image one can see a reconstruction of the data corrected by a standard linear interpolation-based scheme. Below there are the iTV and ASD-POCS reconstructions. All images are windowed C = 0 HU/W = 400 HU.

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Figure 5. The upper image shows an FBP (filtered backprojection) reconstruction of the undersampled dataset. In the middle images one can see reconstructions of the undersampled data using iTV and ASD-POCS. At the bottom there is an FBP reconstruction of the fully sampled dataset. In the right column the difference images relative to the fully sampled FBP reconstruction are shown. All images are windowed C = 0 HU/W = 400 HU.


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Table 3. Image error E, raw data fidelity N and total variation of the limited angle datatsets.

E N TV

FBP

ASD-POCS (N = 24)

ASD-POCS (N = 50)

iTV

8.32 17.12 174.89

4.24 13.19 149.23

4.22 13.17 149.34

2.54 3.40 166.60

Table 4. Image error E, raw data fidelity N and total variation of the interior tomography datatsets.

E N TV

FBP

Extrapolation

ASD-POCS (N = 22)

ASD-POCS (N = 50)

iTV

8.92 17.41 234.51

1.74 12.96 126.74

2.12 4.97 108.84

2.16 5.01 108.42

0.84 1.91 124.82

3.3. Limited view angle The limited angle problem can arise in different situations. One important case is a temporal resolution improvement of dynamic CT scans. Here it can be helpful to integrate additional prior knowledge into the cost functional (Chen et al 2008). To simulate a limited view angle scan, only 160◦ of the available angular range was used for reconstruction. The dataset was also acquired with a Somatom Sensation 16 CT scanner. The FBP reconstruction shows strong artifacts. The iTV reconstructed image has good soft tissue contrast and much less artifacts. ASD-POCS was able to remove most of the artifacts but the image looks quite smoothed and patchy. This is also reflected in the low value of the TV, but the relatively high value of . The number of iterations was N = 24. All images and their evaluation are shown in figure 6 and table 3. Also in this example ASD-POCS did not show any improvement for N = 50 iterations. 3.4. Interior tomography Interior tomography means that the image has to be reconstructed from fully truncated projection data. To truncate the used dataset (Somatom 16), 25% of the projection data were cut off at each side of the detector. The resulting field of measurement is marked green in the images (figure 7). In addition to the compressed sensing-based methods and the FBP reconstruction, an extrapolation-based detruncation method (Ohnesorge et al 2000) was used to compare the results. As expected the FBP reconstruction shows very strong cupping due to the truncated projections. The both TV-based methods remove these artifacts efficiently, whereby iTV reaches better low contrast characteristics than the ASD-POCS reconstruction. iTV reached the stopping criterium after N = 22 iteration steps. The extrapolation method also yields good results but not of the quality of iTV. This example demonstrates quite well the ability of compressed sensing to produce quantitatively correct region of interest tomography reconstructions. The values of E were only evaluated in the field of measurement 4. 4. Computational cost The computationally most expensive steps of both implementations are the forward and backprojection. These can be efficiently accelerated using methods which have been proposed

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Figure 6. The upper image shows an FBP (filtered backprojection) reconstruction of the incomplete dataset. Below one can see the reconstructions using ASD-POCS and iTV. At the bottom there is an FBP reconstruction using the full dataset. In the right column the difference images relative to the fully sampled FBP reconstruction are shown. All images are windowed C = 0 HU/W = 500 HU.


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Figure 7. The upper image shows an FBP (filtered backprojection) reconstruction of the truncated dataset. Below one can see the reconstructions using the extrapolation method, ASD-POCS and iTV. At the bottom there is an FBP reconstruction using the full dataset. In the right column the difference images relative to the fully sampled FBP reconstruction are shown. The field of measurement is marked green. All images are windowed C = 0 HU/W = 1000 HU.

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in the past (Kachelrieß et al 2007, Knaup and Kachelrieß 2007). The gradient descent method, which is used for minimizing the TV, is also very appropriate for parallel computing as shown in Sidky and Pan (2007). The two forward-projection steps, which are required for calculating SART TV and n+1,M have additionally been sped up by down sampling of the volume and detector n+1 resolution by a factor of 2. This still leads to sufficient numerical accuracy in the calculation of these values. This additional part causes higher computational cost of an inner iTV iteration. To get an impression of the whole computational cost of one iteration of iTV and one iteration of the proposed ASD-POCS implementation, we compared the complexity of a forward or backprojection to the complexity of M iteration gradient descent. We assumed that the number of memory accesses in the innermost loop are the limiting factor. Of course this fact depends on the hardware used, but it is a good assumption to get an approximative estimate of the performances. To update one voxel r during a backprojection four detector values are read out, if a linear interpolation-based scheme like in our method is used. Additionally the updated voxel must be read out and rewritten to memory. This leads to six memory accesses in the innermost loop. The complexity of the used Joseph forward projector (Joseph 1982) is the same. So performing a forward or backprojection of N projections to a volume of N3 voxels has the complexity of OBP/FP = 6 · N 4 .

Calculating one iteration of gradient descent to reduce the TV of the image f (r) requires reading the voxel itself and six neighboring voxels for each voxel r (Sidky et al 2009). The final value is written back to the memory. This leads to eight memory accesses in the innermost loop. The complexity of the gradient descent is OGD = 8 · N 3 · M · NLS ,

where NLS represents the number of linesearch steps in each iteration. This number differs in every step and depends on things like the image content and the start value of α in equation (5). For our implementations we observe that an average value of NLS between 1 and 2 represents the behavior of the algorithms quite good. The total complexity of one iteration iTV and ASD-POCS is the sum of three forward or backprojections, M iterations gradient descent, and two downsampled forward projections in the case of iTV. The three forward or backprojections are caused by the SART. The expression i Rkj,i in equation (4) can be stored in the memory and is only computed once. This leads to the following approximations of the computational cost: OASD−POCS = 3 · 6 · N 4 + 8 · N 3 · M · NLS OiTV = 3 · 6 · N 4 + 8 · N 3 · M · NLS + 2 · 6 · N 4 · (0.5)3

= 3.25 · 6 · N 4 + 8 · N 3 · M · NLS .

5. Discussion None of the images reconstructed with iTV shows the staircase effect, which is caused by overregularization by the TV. As expected and already shown in previous papers, compressed sensing-based reconstruction leads to superior image quality compared to FBP in the evaluated situations. The adaption technique of iTV leads in all presented cases with a fixed parameter set to equal or superior results than the ASD-POCS implementation used. To understand the reason for the improved image quality the progress of three important values of both algorithms


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Figure 8. Raw data error at iteration number n in the case of limited angle tomography.

Figure 9. Total variation of the image fn (r) at iteration number n in the case of limited angle tomography.

is shown. The underlying dataset is the limited angle scan shown in section 3.3. Figure 8 shows the raw data error n at each iteration step n. While iTV reaches a strong decrease of , the ASD-POCS algorithm seems to converge toward oscilllating behavior between two fixpoints at much higher values of than iTV. This behavior has also been observed for higher iteration numbers of ASD-POCS (N = 50), so the image content does not visibly change compared to the number N of iTV. This fact can also be seen in figure 9, which shows the TV of the image fn at iteration step n. With a decreasing radius of , the reachable minima of the TV get more bounded. This yields to the constant increasing TV-norm of iTV. In the last

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Figure 10. Reduction factor λ of the gradient descent step at iteration number n in the case of limited angle tomography.

iteration the ASD-POCS algorithm skips the gradient descent step. That leads to a big step toward lower but ignores the sparsity constraint. Also one can see that the value reached for is nevertheless above the value reached by iTV. The third plot in figure 10 shows the value TV . This variable can of the variable λ which is the linear combination weight of fnSART and fn,M be seen as the source of the differences between the two implementations. As clearly shown in this section the proposed pure image-based calculation of λ in the ASD-POCS algorithm fails to enforce stronger raw data consistency in every iteration step. Both methods try to approximate the constrained minimization problem (2) in a heuristical way. To find an exact solution, higher order interior point methods should be used. In those cases the computational effort is orders of magnitude higher. The field of finding exact solutions of constrained 1 -norm minimization is still a very active field of research (Becker et al 2009). iTV is not restricted to a pure TV-based constraint. An integration of any further prior information as proposed in Chen et al (2008) can be done in a straightforward way. Additionally we observe that for meaningful evaluation of algorithms in the field of compressed sensing it seems to be essential to use real measured datasets of patients, where a certain complexity of the image content is given. 6. Conclusion We presented a new method for optimized parameter adaption within the ASD-POCS framework for sparsity constrained image reconstruction. Thereby the step-size adaption of the sparsity constraint, which is defined in the image domain, is transferred into the raw data domain using the convex properties of the 1 -norm and the linearity of the x-ray transform. This enables us to minimize the raw data and the sparsity constraint separately while assuring the right weight between both minimization steps. The proposed method iTV is able to handle different problems arising in CT image reconstruction. Here the examples of angular undersampling, metal implants, limited view angle tomography and interior tomography were


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shown. In all the situations shown, iTV leads to significant artifact reduction without visible oversmoothing of low contrast areas in the image. All results were compared to a standard FBP reconstruction and a previously proposed implementation of the ASD-POCS algorithm. Acknowledgments This study was supported by Ziehm Imaging GmbH, N¨urnberg, Germany. The high-speed image reconstruction software RayConStruct-IR was provided by RayConStruct GmbH, N¨urnberg, Germany. References Andersen A H and Kak A C 1984 Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm Ultrason. Imaging 6 81–94 Becker S, Bobin J and Candes E J 2009 NESTA: a fast and accurate first-order method for sparse recovery SIAM J. Imaging Sci. 4 1–39 Bian J, Siewerdsen J H, Han X, Sidky E Y, Prince J L, Pelizzari C A and Pan X 2010 Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT Phys. Med. Biol. 55 6575–99 Boyd S and Vandenberghe L 2004 Convex Optimization (Cambridge: Cambridge University Press) Censor Y and Elfving T 2002 Block iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem SIAM J. Matrix Anal. Appl. 24 40–58 Chen G-H, Tang J and Leng S 2008 Prior image constrained compressed sensing (PICCS): a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets Med. Phys. 35 660–3 Donoho D L 2006 Compressed sensing IEEE Trans. Inf. Theory 52 1289–306 Jiang M and Wang G 2003 Convergence of the simultaneous algebraic reconstruction technique (SART) IEEE Trans. Imaging Process. 12 957–61 Joseph P M 1982 An improved algorithm for reprojecting rays through pixel images IEEE Trans. Med. Imaging MI-2 192–6 Kachelrieß M, Knaup M and Bockenbach O 2007 Hyperfast parallel-beam and cone-beam backprojection using the cell general purpose hardware Med. Phys. 34 1474–86 Kalender W A, Hebel R and Ebersberger J 1987 Reduction of CT artifacts caused by metallic implants Radiology 164 576–7 Knaup M and Kachelrieß M 2007 Acceleration techniques for 2D parallel and 3D perspective forward and backprojections Proc. 1st Workshop on High Performance Image Reconstruction and the 9th Int. Meeting on Fully 3D Image Reconstruction pp 45–8 Ohnesorge B, Flohr T, Schwarz K, Heiken J P and Bae K T 2000 Efficient correction for CT image artifacts caused by objects extending outside of the scan field of view Med. Phys. 27 39–46 Persson M, Bone D and Elmquist H 2001 Total variation norm for three-dimensional iterative reconstruction in limited view angle tomography Phys. Med. Biol. 28 853–66 Sidky E Y and Pan X 2007 Few-view, cone-beam CT image reconstruction by GPU-accelerated total variation minimization Proc. 1st Workshop on High Performance Image Reconstruction and the 9th Int. Meeting on Fully 3D Image Reconstruction pp 60–3 Sidky E Y and Pan X 2008 Image reconstruction in circular cone-beam computed tomography by total variation minimization Phys. Med. Biol. 53 4777–807 Sidky E Y, Kao C and Pan X 2006 Accurate image reconstruction from few-views and limited-angle data in divergentbeam CT J. X-Ray Sci. Technol. 14 119–39 Sidky E Y, Pan X, Reiser I S, Nishikawa R M, Moore R H and Kopans D B 2009 Enhanced imaging of microcalcifications in digital breast tomosynthesis through improved image-reconstruction algorithms Med. Phys. 36 4920–32 Song J, Liu Q H, Johnson G A and Badea C T 2007 Sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro-CT Med. Phys. 34 4476–83 Yu H and Wang G 2009 Compressed sensing based interior tomography Phys. Med. Biol. 54 2791–805 Yu H and Wang G 2010 A soft-threshold filtering approach for reconstruction from a limited number of projections Phys. Med. Biol. 55 3905–16