Shiyu Xu a, Henri Schurz b, Ying Chen a,c, Abstract

Transcription

1 Parameter Optimization of relaxed Ordered Subsets Pre-computed Back Projection (BP) based Penalized-Likelihood (OS-PPL) Reconstruction in Limited-angle X-ray Tomography Shiyu Xu a, Henri Schurz b, Ying Chen a,c, a Department of Electrical and Computer Engineering, Southern Illinois University Carbondale, IL, USA b Department of Mathematics, Southern Illinois University Carbondale, IL, USA c Biomedical Engineering Graduate Program, Southern Illinois University Carbondale, IL USA Abstract This paper presents a two-step strategy to provide a quality-predictable image reconstruction. A Pre-computed Back Projection based Penalized- Likelihood (PPL) method is proposed in the strategy to generate consistent image quality. To solve PPL efficiently, relaxed Ordered Subsets (OS) is applied. A training sets based evaluation is performed to quantify the effect of the undetermined parameters in OS, which lets the results as consistent as possible with the theoretical one. Keywords: limited angle tomography, mono-energenic X-ray, penalized-likelihood, predictable resolution properties, relaxed Ordered Subsets, smoothing parameter Corresponding author address: adachen@siu.edu (Ying Chen ) Preprint submitted to Computerized Medical Imaging and Graphics April 28, 2013

2 1. Introduction Limited angle X-ray tomography is increasingly used in a range of noninvasive anatomical imaging applications. On one hand, few sampling is inevitable due to the constraints of geometric configuration, limitations on image acquisition time, or the necessity to reduce patient radiation dose. On the other hand, the benefits from three dimensional (3-D) reconstruction can be obtained by the limited angle configuration. These benefits include the feasibility in detecting anatomical structure with overlaps and localizing the region of interest. Current clinical applications include: intraoperative imaging for reference with a pre-operative planning CT, angiography, chest tomosynthesis, dental tomosynthesis, cardiac CT, orthopaedic imaging, and, most recently, digital breast tomosynthesis (DBT) [7, 26]. Among current reconstruction techniques, both analytical reconstruction and iterative methods [13] are widely used. One classical analytical reconstruction technique is Filtered Back Projection (FBP)[23] based on Fourier Slice Theorem, which can yield a precise signal reconstruction at a sampling rate satisfying Nyquist-Shannon Theorem, but it may induce reconstruction error from highly incomplete frequency information [5]. Several revised versions of FBP such as adding post-processing filters and interpolating frequency information by a Total-Variation (TV) framework [6] were proposed. One of the iterative methods is Simultaneous Algebraic Reconstruction Technique (SART) developed based on the ART-type (Algebraic Reconstruction Technique type) method in [4, 3]. SART actually applies a Ordered Subsets (OS) method to solve a unweighted least square model, which may lead to over-fitting to the noise data and non-convergence to 2

3 the optimal value. Signal statistics in X-ray Computed Tomography (CT) follows Poisson distribution for mono-energetic CT and compound Poisson for polyenergetic CT [24, 9]. Reconstruction methods such as Maximum Likelihood (ML), Penalized Weighted Least Squares (PWLS) and Penalized Likelihood (PL) with Poisson model were strongly proposed and well studied in [16, 17, 18, 10, 13, 9, 8]. The main benefit from these methods is that the missing data in highly incomplete sampling could be guessed at the maximum probability according to the observation. However, the computational intensity attacks statistical methods. Some accelerated strategies successfully speed up the convergence, such as relaxed Ordered Subsets (OS) [15, 11], Transmission Incremental Optimization Transfer (TRIOT) [2] and recent Alternating Direction Method of Multipliers (ADMM) [20]. In the classical model of X-ray imaging, the Poisson distribution of incident photon number dominates the physical process. Although X-ray detectors are not quanta counters, Poisson distribution still confirms the signal statistics of mono-energetic X-ray detection [9]. The number of photons generated and ultimately detected along a projection follows Poisson distribution which can be described mathematically as P (Y i = y i ) = θy i i e θ i y i!, (1) where Y i is a random variable counting the observed photons on the detector along i-th X-ray; y i is the observation of Y i ; θ i is the expectation value of the random variable Y i. In the classical physical model, θ i can be expressed as θ i = d i e <µ,li>, (2) where d i is the intensity of the incident X-ray beam; µ is a linear attenuation 3

4 coefficient vector to be estimated. Each voxel is assigned an attenuation coefficient and the l i denotes the vector of the intersection length between the i-th X-ray and each voxel. The negative log-likelihood function of all observed photons on the detector can be written as [17] L(µ) = M {d i e <µ,li> + y i < µ, l i >} + c, (3) i by the assumption that {Y i } i [1,M] are i.i.d, where c is a constant and M is the number of X-ray beams. Through minimizing (3), the optimal µ can be estimated. In transmission tomography, Compton scattering which makes X-ray photon deflected from its original path could yield a photon noise randomly adding on the detector. The reconstructed results may over fit the noise data at the convergence. [25] proposed a novel reconstruction method aiming to suppress the Compton Scattering. PL method suggesting to insert a penalty function was also proposed. One can revise (3) by appending a scaled penalty function to the likelihood function. The negative log PL function can be written as Φ(µ) = L(µ) + λr(µ), (4) where the smoothing parameter λ controls the strength of the penalty function. By minimizing (4), the optimal µ is estimated. Generally speaking, to solve the optimization directly is intractable. But, by using the method in [18] or the separable parabolic surrogate in [10, 11], the optimal µ in Φ(µ) can be approached monotonically with optimal solutions of surrogate functions h n (µ, µ n ), each of which is bounded by Φ(µ) at µ n. The main obstacle to apply PL method into the application is the un- 4

5 predictable effect of the smoothing parameter on resolution properties. But thanks to the discussion in [14, 22], the authors proposed a modified quadratic penalty function to eliminate the data-dependent terms in impulse response and noise, such that the effect of the smoothing parameter on resolution properties can be evaluated in advance by studying simulated data. Along with the spirit of the research, we present a simplified version of the modified penalty, which is Pre-computed Back Projection based PL (PPL). A twostep procedure is proposed to perform 3-D reconstructions along with the desired resolution properties. To solve the optimization, relaxed OS separable parabolic surrogate (OS-SPS) algorithm is applied and forms our relaxed OS-PPL. But the undetermined parameters such as relaxation and subsets make the practical resolution properties deviated from the theoretical one. To conquer it, a training set based semi-quantitative evaluation is presented, by which the parameters in OS are tuned to make the results as consistent as possible with the theoretical with less computational cost. 2. Method for characterizing the smoothing parameter λ In (4), the penalty R(µ) can take a general form R(µ) = N j=1 k N j ψ(µ j µ k ), (5) Where N j is the neighbours of the j-th voxel. The function ψ(t) denotes the spatial constraint for adjacent voxels. For a quadratic penalty, ψ can be formulized as follows ψ(µ j µ k ) = 1 2 (µ j µ k ) 2, (6) 5

6 which results in a consistent smoothing on adjacent voxels. Through minimizing (4), the optimal estimation of µ can be shown in the following form u = arg min Φ(µ). (7) µ 0 It s intractable to solve it directly. However, SPS introduced by [10, 11] leads to an iterative solution, which is parallel, monotonic, but suffers slow convergence. The basic idea of SPS is that by constructing a series of separable parabolic functions lower bounded with the objective function, the optimal value can be approached by the solution of the surrogate one at each iteration. By applying SPS on (4) with the quadratic penalty (5), the approximation of (7) at the (n + 1)-th iteration can be written as µ (n+1) j = µ (n) j [ M l ij ( d i e <µ(n),l i > + y i ) i=1 +λ k N j (µ (n) j µ (n) k M )]/[ N (l ij i=1 j=1 d i e <µ(n),l i > ) + 2λ N j ], (8) where N j is the cardinality of the subset N j. The solution sequence of the surrogates converges to the optimal value of the objective function monotonically. Compare to a large curvature of the surrogate function, a small one can yield a faster convergence with bounded condition. By replacing d i e <µ(n),li> in the denominator of (8) as y i, a precomputed curvature in [10] is conceived, which may lead to a faster convergence, yet almost always monotonic decreasing. In practical application, to find a proper smoothing parameter λ in (8) is not trivial. The main reason is that the impulse response and the noise 6 l ij

7 from PL reconstruction are data-sensitive, which means small difference in datasets will yield huge difference on resolution properties, such that the effect of λ is unpredictable. To reduce the data dependence, Fessler et al. proposed a modified penalty function [14] and demonstrated that the impulse response of the reconstructed results is only dominated by λ. The modified penalty is written as R m (µ) = N κ j j=1 k N j ω jk κ k ψ(µ j µ k ), (9) where ω is a weighted coefficient assigned to ψ. κ j is formulized for emission tomography as follows: i=1 κ j = s g2 ij q i j, (10) i=1 g2 ij In X-ray transmission tomography, s i, g ij, q i are translated to s i = 1, i [1, M], g ij = l ij and q i = y i. To reduce the computational complexity, we propose a simplified version as follows: R m (µ) = N j=1 κ 2 j (µ j µ k )2, (11) 2 k N j where κ 2 j = M i=1 l2 ijy i M i=1 l2 ij, (12) since the condition of κ k κ j is obviously held in the neighbours. κ 2 is roughly equivalent to Back Projection (BP) reconstruction on the data y i. The Pre-computed BP based Penalized Likelihood (PPL) method apply κ 2 to absorb the data-related terms in resolution properties, such that the smoothing effect of λ can be evaluated in advance by studying simulated data. With a selective λ, PPL can produce image reconstructions with desired image 7

8 quality. The iterative solution of PPL method is formulized by revising (8) as follows µ (n+1) j = µ (n) j [ M l ij ( d i e <µ(n),l i > + y i ) i=1 +λκ 2 j (µ (n) j k N j µ (n) k where κ 2 can be calculated before the iteration. M )]/[ N (l ij i=1 j=1 d i e <µ(n),l i > ) + 2λκ 2 j N j ], (13) l ij 2.1. Pixel property of PPL Furthermore, to demonstrate the data independence of the impulse response of PPL, we studied the analytical relationship between the impulse response and the smoothing parameter λ for PL method, which is derived in the literature [14] L j (µ) [A T D(y i )A + λr] 1 A T D(y i )Ae j, (14) where L j (µ) denotes an impulse response yielded from an impulse signal at the j-th voxel, which has the form of L j (µ) µ(y i) µ j, (15) µ(y i ) is an estimator of µ on a noiseless measurement y i. A is a coefficient matrix according to the system geometry. If a ray-tracing method is used, A is composed of l ij denoting the length of the intersection between the i-th X-ray and the j-th voxel. D(y i ) is a diagonal matrix with the entry y i. R is the Hessian matrix of R(µ). e j is the j-th unit vector. 8

9 From (14), one can see that the impulse response L j (µ) depends not only on the system geometry and the smoothing parameter λ, but also on the datasets associated with the object and incident X-ray. We substitute R m in (11) to (14). By applying an analogous deduction in the literature [14], L j (µ) has a data independent approximation as follows: L j (µ) [A T IA + λr] 1 A T IAe j, (16) where I is identical matrix. One can see that the data D(y i ) is degraded to I as R m, the Hessian matrix of R m (µ), is transformed to R, the Hessian matrix of R(µ) with a basic quadratic penalty. That means the effect of λ in PPL on the impulse response reconstructed from arbitrary measurements y i is equivalent to the one reconstructed by a penalized-likelihood method with a basic penalty from a uniform background with y i = 1. In other words, the data dependence of pixel property has been eliminated by applying the modified penalty (11) Noise property of PPL Thanks to the studies in literatures [14, 12], the noise property is represented as the covariance on reconstructed voxels Cov(µ) [A T D(θ i )A + λr] 1 A T Cov(y i )A [A T D(θ i )A + λr] 1, (17) where θ i is expressed as (2). From the equation, one can see that the covariance depends on the geometric configuration, smoothing parameter λ and the data. In our model, the Poisson distribution dominates the physical process, 9

10 therefore Cov(y i ) = D(θ i ), since y i, i [1, M] is i.i.d. The variance can be expressed as V ar(µ j ) = (e j ) T Cov(µ)e j. (18) By inducing the modified penalty and applying the similar deduction in the literature [14], we can obtain the formula as follows: V ar(µ j ) (e j ) T [A T D(θ i )A + λr m ] 1 A T D(y i )A [A T D(θ i )A + λr m ] 1 (e j ) V j unit κ 2 j, (19) where V j unit = (ej ) T [A T IA + λr] 1 A T IA[A T IA + λr] 1 (e j ). (20) From (19), one can conclude that the variance of the j-th voxel reconstructed by PPL method on a measurement with a unknown mean and unknown standard deviation can be quantified as V j unit divided by κ2. V j unit denotes the variance of the j-th voxel reconstructed by a penalized-likelihood method with a basic penalty from the measurement y i with a unit mean and unit standard deviation. According to the discussions above, the data dependency of the effect of λ is absorbed by the pre-estimation in the modified penalty, such that desired resolution can be obtained by the proper λ, which is evaluated in advance. This evaluation can be conducted either through simulated data or clinical trial data. In our case, we provide a simulation based two-step procedure for λ selection and image reconstructions. In STEP 1, projections are simulated to generate mappings from λ to resolution properties. This procedure can be executed just at the stage of syste design after the imaging geometry is 10

11 fixed. STEP 1: 1: Set a range of λ and choose a representative focus plane. 2: Tiny balls respectively distributed in nine square regions on the focus plane are used as the reference phantom. These balls are assembled with a small coefficients. Projections are simulated with the reference phantom and a uniform incident value of y i = 1 in the same imaging configuration as the real one. 3: Run (8) with the datasets with each λ and calculate the average Modulation Transfer Function (MTF) and record it in a corresponding table. 4: Simulate projections from Poisson distribution with θ i = 1, i [1, M]. 5: Run (8) with the datasets for each λ and record standard deviation on the focus plane into a corresponding table. In this procedure, λ can be set as [0, 4, 8, 16, 32, 64, 128, 256]. An average MTF on a single plane is applied to represent a frequency response of the whole system, although this response is somewhat space-dependent. Each MTF in the sub-region is calculated across the ball along the scan and the normalized average MTF is estimated by the sum of the sub-regional MTFs dividing the maximum. Furthermore, one can simply evaluate the normalized standard deviation on a square region of the reconstructed focus plane from the projections with poisson noise. For a particular imaging geometry, the values in the tables generated from item 4 and item 7 are fixed and applicable 11

12 Figure 1: Geometric configuration of Digital Breast Tomosynthesis with multiple parallel X-ray beams. for all clinical trails. In STEP 2, before image reconstruction, one can choose a desired λ from the tables to meet the resolution requirement. Then, the reconstruction is preformed along with the determined λ. This step can be summarized as follows STEP 2: 1: Find a λ satisfying the pixel precision and noise reduction in the real application. 2: Run (13) with the imaging projections with chosen λ Simulation results To get a practical illustration of the two-step procedure, we set up a virtual system with the same geometric configuration as a real limited angle X-ray tomography system. Fig. 1 demonstrates the geometric configuration 12

13 of a Digital Breast Tomosynthesis referred to the literature [19]. The detector size is mm by mm with the pixel size of 0.56mm by 0.56mm. O is the origin of the three dimensional coordinate system which is located at the center of the detector. The source to image distance (SID) along Z direction is set as 692.8mm and 25 x-ray beams are positioned in a straight line parallel to the detector plane along the X axis. The middle one of the 25 beams is located on Z axis and the linear spacing between these beams varies to provide a 2 angular spacing around the rotation center T. The system provides θ = 48 coverage around T. The testing phantom has the same structure, but different attenuation coefficient with the reference one in STEP 1. The focus plane is placed at the plane with 40mm away from the detector. The testing dataset is generated by using ray-tracing method with non-uniform incident value of y i 1, i [1, M]. After STEP 1, average Modulation Transfer function (MTF) for λ {0, 4, 8, 16, 32, 64, 128, 256} with the reference phantom are drawn in Fig. 2. Fig. 4 presents the standard deviation versus λ 1/4. According to Fig. 2, table 1 is generated by mapping λ to the value of half magnitude of MTF. The Std. values in table 2 are obtained directly from Fig. 4. Moreover, the drop rate in each table is calculated based on corresponding values. By applying STEP 2, the average MTF of the focus plane reconstructed by PPL with the same range of λ on the testing phantom is presented in Fig. 3. The testing phantom for noise estimation is generated from the poisson distribution with θ i 1. Fig. 5 shows the standard deviation on the focus plane reconstructed by PPL. From Fig. 3, one can see that the MTFs are almost the same as those in 13

14 Modulation Transfer Function(MTF) λ0 λ4 λ8 λ16 λ32 λ64 λ128 λ pp/mm Figure 2: Average MTF of the focus plane reconstructed by the PL method with a basic quadratic penalty from the projections with a uniform incident value of y i = Modulation Transfer Function(MTF) λ0 λ4 λ8 λ16 λ32 λ64 λ128 λ pp/mm Figure 3: Average MTF of the focus plane reconstructed by PPL method from the projections with a non-uniform incident value with y i 1 14

15 Standard deviation versus λ λ0 λ4 λ8 λ16 λ32 λ64 λ128 λ λ 1/4 Figure 4: Standard deviation of the focus plane reconstructed by the PL method with a basic quadratic penalty from the projections with a expected incident value of θ i = Standard deviation versus λ λ0 λ4 λ8 λ16 λ32 λ64 λ128 λ λ 1/4 Figure 5: Standard deviation of the focus plane reconstructed by PPL method from the projections with a expected incident value of θ i 1 15

16 Table 1: Half Width of Half Magnitude of MTF versus λ λ Half Width Pixel Precision Drop (%) Fig. 2, which means the pixel property reconstructed by PPL method is data independent and exactly identical with the one got from PL method with a basic quadratic penalty on the uniform measurement. Fig. 5 shows that the declining trend of the standard deviation of noise is consistent with the one shown in Fig. 4, although the exact standard deviation somewhat depends on the evaluation of κ Method for parameter optimization in Ordered Subsets (OS) framework 3.1. Algorithm of OS method Theoretically speaking, resolution properties are completely dominated by λ in PPL method. But in practical applications, to reach the global optimal is impossible since the iterative method is applied, especially for EM-type algorithm which suffers slow convergence in the neighbours of the 16

17 Table 2: Standard deviation versus λ λ Std. Std. Drop (%) optimum. To accelerate the convergence in limited iterations, OS framework is studied in this section. Ann and Fessler [1] proposed that, OS method could speed up EM-type algorithms in the early iterations by using the sub-gradient to replace the true gradient, However, the OS method usually exhibits limitcycle behavior near the global optimal. Next the authors proposed a proper diminishing relaxation step-size to yield a global convergence. The sufficient conditions on a relaxation for global convergence are the following: a n =, a 2 n <, (21) n where a n is a relaxation parameter at the n-th iteration. Let NS be the number of subsets. Let S 1,..., S NS Each S i has M Si updated by: n denote the subsets. X-ray beams. In each iteration, the relaxation parameter is a n = 1 rn + 1, (22) 17

18 which meets the sufficient conditions of a global convergence. As r, an adjustable parameter, is increasing, a is dereasing which induces a slow initial iteration but a fast convergence to a global optimal. However it is not the true one. One iteration is completed when the algorithm goes through all the projections by going through all the subsets. In each subset, the iterative solution is similar to (13) but gradient and scaled terms are calculated using the subset of data and 1/NS penalty. κ 2 is precomputed before iterations. In our implementation of relaxed OS-PPL, we use linked list RayP ath i to store the sparse matrix A, so that the memory consumption is reduced and accessing l ij in A is also speeding up. We summarize the main steps of our algorithm as follows: for each pixel j = 1,..., N do update κ 2 by (12) end for for each iteration n = 1,..., Niter do update a n by (22) for each subset s = 1,..., NS do for each X-ray i = 1,..., M si do while RayP ath i null do att i + = l ij µ j len i + = l ij end while end for for each X-ray i = 1,..., M si do while RayP ath i null do 18

19 up j + = l ij (d i e att i y i ) down j + = l ij len i y i end while end for µ old = µ for each pixel j = 1,..., N do denom = down j + 2λκ 2 N j /NS nom = up j + λκ 2 k N j (µ old j µ j = [µ old j + a n (nom/denom)] + end for end for end for µ old k )/NS Compared to an OS-SPS method, OS-PPL needs an extra estimation on κ 2. But the computational complexity of the particular estimation is equivalent to the one of BP reconstruction, which can be ignored in the exhausted iterations. Furthermore, one can easily parallelize the algorithm by partitions of a projection image or groups of reconstructed planes Semi-quantitative optimization on training sets The convergence of OS-PPL lead to definite resolution properties according to the discussions in previous sections. However for 3-D reconstruction in a real application, even the relaxed OS can not guarantee the optimal with limited iterations. Moreover, the undetermined parameters in relaxed OS such as relaxation a n and subset may result in a convergence deviated from the true one [1]. Therefore the resolution properties has a certain bias from the theoretical inevitably. Finding an optimal parameter combination 19

20 is essential for sufficient image quality especially in limited iterations. It is intractable to quantify the impact of these parameters on image qualities, since they rely on datasets. However, literature [21] revealed that the convergence of EM-type methods highly depend on the ratio of missing information to complete information. The ratio in our application is associated with the geometric configuration. Unlike the data-sensitivity of the effect of λ, the impact of subsets and relaxation on the algorithm convergence is weakly data-dependent. Therefore these parameters in OS can be evaluated in advance directly based on training sets, which are collected from clinical trails in a real application. The evaluated results are prepared for further usages. To provide a semi-quantitative evaluation, we employ objective function as a function of iteration, noise as a function of contrast and Artifact Spread Function (ASF) to represent the impacts of the parameters. To get a practical demonstration, we design an experiment by using the same system as shown in Fig. 1. The training phantom in the experiment is simulated with a linear attenuation coefficient of 0.005mm 1, a side length of 20cm and a thickness of 2cm. Two focus planes locate at the thickness of 0.5cm and 1.5cm. On each of the planes, two cubes with a linear attenuation coefficients of 0.038mm 1 and 0.08mm 1, a side length of 6cm and a thickness of 0.25cm are located symmetrically. Four tiny ball are arranged vertically between the two cubes on each focus plane with the radii of 2.5mm, 1.5mm, 1.25mm and 0.56mm, and linear attenuation coefficients of 0.02mm 1, 0.025mm 1, 0.05mm 1 and 0.1mm 1. The phantom is placed at 3cm away from the detector surface such that the focus planes appear at the height of 3.5cm and 4.5cm in the 20

21 system. The projections are generated by a incident value under Poisson distribution and an illumination model. λ = 16 is chosen for the experiment. With the Tables 1 and 2, one can predict that, when PPL converges to the optimum, compared to the method with λ = 0, or a ML-EM method, the pixel precision is approximately 2.07 with a drop of 29.65% and the noise is significantly decreased by 64.72%. In this demonstration, we investigate the relaxation a n with r = 1, 4/5, 1/2, 0 and subsets with NS = 1, 5, 25. All relaxed OS-PPLs run 20 iterations with a FBP initialization. For comparisons, a name rule is applied. For example, relaxed OS-PPL with λ = 0, subset of 1 and r = 1/2 is named as OS-PPLλ0-sub1-r12. The curves of objective function as a function of iteration, noise as a function of contrast and Artifact Spread Function (ASF) are plotted to compare the impacts of selective parameter combinations on the convergence, image contrast and the removal of out-of-plane blur. Some representative reconstructions including BP, FBP, SART with a relaxation parameter, and OS-ML with subsets of 25 and r = 1/2 are also presented for a comparison purpose. Fig. 6 presents the convergence of OS-PPL and PPL. Generally speaking, OS-PPLs outperform PPL in terms of convergence in 20 iterations. Among OS methods, they trend to different convergences in spite of the same λ. OS-PPL-λ16-sub5 has a larger value than the other two, while both OS- PPL-λ16-sub25-r45 and OS-PPL-λ16-sub25-r12 approach to a similar one, but the latter yields a faster convergence rate. The observations above justify that (1) the relaxed OS-PPL methods lead to a faster convergence than PPL and (2) the parameters in OS-PPL may result in a divergence from the ture 21

22 x 1011 PPL λ16 OS PPL λ16 sub5 r12 OS PPL λ16 sub25 r45 OS PPL λ16 sub25 r Objective function Iteration Figure 6: Objective function as a function of iteration with OS-PPL-λ x 10 4 OS ML sub25 r12 OS PPL λ16 sub5 r12 OS PPL λ16 sub25 r45 OS PPL λ16 sub25 r12 BP SART Noise Contrast Figure 7: Comparisons of noise versus contrast with iteration increasing between OS-PPL and representative methods one in spite of the same λ. Fig. 7 shows the noise versus the image contrast as iteration increases. The noise is evaluated by standard deviation on a small region near a object. The subtraction between the mean values of the object and the background 22

23 1 0.9 OS ML sub25 r12 OS PPL λ16 sub5 r12 OS PPL λ16 sub25 r45 OS PPL λ16 sub25 r12 BP SART ASF Plane height away from in focus plane (mm) Figure 8: Comparisons of ASF between OS-PPL with 15 iterations and representative methods forms the contrast. Noise level in reconstructed plane by SART and OS- ML increases dramatically with the incremental iterations. In a contrary, noise from OS-PPL tends to decline. Among OS methods, noise at the 20-th iterations approaches to a similar level due to the same λ, but has a slight deviation to each other, which confirms that the parameters in OS-PPL may lead to noise within a perturbation from the theoretical. On the other hand, the image contrast of OS methods settles in a small range around 0.016, where OS-PPL-λ16-sub25-r12 performs the best. ASF is also employed to demonstrate the removal of out-of-plane blur of OS-PPL methods with selective parameters. Figs. 8 and 9 show ASF at the 15-th and 20-th iteration respectively. OS-ML and SART exhibit the best and similar performance. OS-PPLs with all selective combinations have no significant difference with each other. Moreover, OS-PPL with 20 iterations does not improve ASF significantly compared to the one with 15 iterations. 23

24 1 0.9 OS ML sub25 r12 OS PPL λ16 sub5 r12 OS PPL λ16 sub25 r45 OS PPL λ16 sub25 r12 BP SART ASF Plane height away from in focus plane (mm) Figure 9: Comparisons of ASF between OS-PPL with 20 iterations and representative methods Based on the evaluations above, for λ = 16, one can choose subsets of 25 and r = 1/2 as the optimal combination, since they exhibit outstanding convergence and prominent performance of noise versus contrast. Additionally, because 20 iterations do not bring more benefits than 15 iterations in terms of noise, contrast and ASF, one can use OS-PPL with 15 iterations. For practical applications, we perform the two-step procedure together with the semi-quantitative evaluation to obtain a desired image quality. The smoothing parameter λ chosen from the look-up tables generated in STEP 1 dominates the noise level and pixel precision, while the optimal parameter combination in OS evaluated through a training set can make practical results as consistent as possible with the theoretical one in less computational cost. 24

25 4. Experiment In this section, we demonstrate how the two-step procedure works with the semi-quantitative evaluation. The testing phantom used in this section is structurally identical with the training phantom, but the attenuation coefficients of objects have a offset. This represents a practical deviation of a real object from training phantom. Visual comparisons of reconstructed results are reported between relaxed OS-PPL with chosen parameter combination and other representative methods such as FBP, OS-ML and OS-SPS [1]. Furthermore, the curve of contrast versus noise is plotted to check the consistence of the parameter impact on image qualities between testing phantom and training phantom. In STEP 1, λ = 16 is chosen from the look-up tables 1 and 2. Based on the evaluations in Figs. 6, 7 and 8, OS-PPL with NS=25, r = 1/2 and 15 iterations produces a sufficient convergence, prominent image contrast, yet lower computational cost. In STEP 2, OS-PPL with chosen parameter combination is applied. The focus plane reconstructed is shown in Fig. 10. For comparisons, the results from FBP, relaxed OS-SPS with λ = and a quadratic penalty, and relaxed OS-ML are shown in Figs. 11, 12, 13 respectively. First, the image contrast between the on-plane cubes pointed by the arrows of 2 and 3 is much stronger in Figs. 10, 12, 13 than the ones in Fig. 11. Secondly, the edges pointed by the arrow 1 is enhanced clearly but shows obvious artifacts in the FBP result. Among Fig. 10, Fig. 12 and Fig. 13, OS-PPL and OS- SPS shows less sharp edges due to the smoothing effect of λ. But, the noise from OS-PPL and OS-SPS is much lower than the ones reconstructed by OS- 25

26 Figure 10: relaxed OS-PPL-λ16-sub25-r12 reconstruction on the focus plane at the height of 35mm Figure 11: FBP reconstruction on the focus plane at the height of 35mm. 26

27 Figure 12: relaxed OS-SPS reconstruction with λ = weighting a quadratic penalty on the focus plane at the height of 35mm Figure 13: relaxed OS-MLEM reconstruction the focus plane at the height of 35mm. 27

28 3.5 4 x OS ML sub25 r12 OS PPL λ8 sub5 r12 OS PPL λ8 sub25 r45 OS PPL λ8 sub25 r12 OS PPL λ16 sub5 r12 OS PPL λ16 sub25 r45 OS PPL λ16 sub25 r12 BP SART 2.5 Noise Contrast Figure 14: Comparisons of noise versus contrast between OS-PPL and representative methods on the testing phantom. ML. Additionally, although OS-SPS yields comparable resolution properties with OS-PPL, the effect of λ of OS-SPS is unpredicted. λ = is chosen by experiments, which is hard to be decided since its large range and dependence on the incident X-ray energy and objects. Last, the parts pointed by the arrows of 5 and 6 represent out-of-plane blur. Although Fig. 11 seems to show a decent ability to remove the out-of-plane blur, the poor contrast leads to the obstacle to distinguish the objects on focus plane and out of plane. Fig. 14 shows noise versus contrast of OS-PPL with selective parameters on testing phantom. As expected, OS-PPL on both testing phantom and training phantom achieve the same noise at the convergence due to the dataindependent impact of λ, Moreover, the image contrast shown in Fig. 7 and Fig. 14 is quite similar at the convergence. This confirms that the effects of relaxation and subsets on the image quality are weakly data-dependent. 28

29 5. Conclusion In this paper, we proposed training set based pre-evaluations for both λ and parameter combinations in OS. Due to the data sensitivity of the effect of λ, a direct pre-evaluation is not reliable. But by the modified penalty proposed in this paper, the data dependency of the resolution properties is eliminated, such that the two step procedure is applicable. In the first step, pixel and noise level for a range of λ are evaluated and stored in tables; in the second step, the desired image quality can be approached by choosing a proper λ in the look-up tables. Unlike λ, the parameter combination influences the image quality by controlling the convergence rate of OS, which highly depends on the imaging geometry, but weakly relies on the datasets. Therefore, the semi-quantitative evaluation can be conducted directly for an optimal parameter combination which produces a sufficient image quality in limited computational cost. Experimental results illustrate the effectiveness of the two-step procedure and the semi-quantitative evaluation. However, the robustness of the semi-quantitative evaluation needs further validations, since the effect of the parameter combination on image qualtiy still weakly depends on datasets. Further work will be concentrated on performing experiments on real phantom data with a limited angle tomography system. Acknowledgment We thank Dr. Weihua Zhou and Linlin Cong at Southern Illinois University. Part of the simulation and algorithm implementations in this paper is based on their previous work. This work is supported by grant NIH/NCI R01 CA A1. 29

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