Statistical iterative reconstruction using fast optimization transfer algorithm with successively increasing factor in Digital Breast Tomosynthesis

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1 Statistical iterative reconstruction using fast optimization transfer algorithm with successively increasing factor in Digital Breast omosynthesis Shiyu Xu a and Zhenxi Zhang b and Ying Chen a,* a Department of Electrical and Computer Engineering, Southern Illinois University Carbondale, IL USA; b he school of Life Science and echnology, Xi an Jiaotong University, Xi an PR China; * Corresponding author ABSRAC Statistical iterative reconstruction exhibits particularly promising since it provides the flexibility of accurate physical noise modeling and geometric system description in transmission tomography system. However, to solve the objective function is computationally intensive compared to analytical reconstruction methods due to multiple iterations needed for convergence and each iteration involving forward/back-projections by using a complex geometric system model. Optimization transfer (O) is a general algorithm converting a high dimensional optimization to a parallel 1-D update. O-based algorithm provides a monotonic convergence and a parallel computing framework but slower convergence rate especially around the global optimal. Based on an indirect estimation on the spectrum of the O convergence rate matrix, we proposed a successively increasing factorscaled optimization transfer (O) algorithm to seek an optimal step size for a faster rate. Compared to a representative O based method such as separable parabolic surrogate with pre-computed curvature (PC-SPS), our algorithm provides comparable image quality (IQ) with fewer iterations. Each iteration retains a similar computational cost to PC-SPS. he initial experiment with a simulated Digital Breast omosynthesis (DB) system shows that a total 40% computing time is saved by the proposed algorithm. In general, the successively increasing factor-scaled O exhibits a tremendous potential to be a iterative method with a parallel computation, a monotonic and global convergence with fast rate. Keywords: Digital Breast omosynthesis, fast convergence rate, optimization transfer, statistical iterative reconstruction, successively increasing factor. 1. INRODUCION Recent application of statistical iterative reconstruction to digital breast tomosynthesis has shown that statistical iterative reconstruction provides a potential to improve image quality(iq) by suppressing noise and some artifacts while retaining decent resolution. 1 4 According to the signal statistics of mono-energetic X-ray detection and an accurate system model, a possion likelihood function is established, 5,6 which is also known as data-fidelity term. With a prior function incorporated, a relationship among adjacent voxels is modeled to mitigate the reconstruction error induced by the over-fitting to noisy data. 7 he image is then reconstructed by computing an estimate which minimizes the resulting objective function. he main obstacle for clinical application of statistical iterative reconstruction is the fact that to solve the objective function is computationally intensive compared to analytical reconstruction methods due to multiple iterations needed for convergence and each iteration involving forward/back-projections by using a complex geometric system model. Optimization transfer (O) 8 is a typical framework to find out the optimal of the objective function, which converts a high dimensional optimization to a parallel 1-D update by transferring the target optimization to a set of surrogate functions. Compared to the iterative coordinate descent (ICD) algorithm, 9 O based methods, such as expectation maximization (EM) and separable parabolic surrogate allow a parallel computation, but receives a low convergence rate and lead to more iterations. Literature 8 proposed non-separable parabolic surrogates with optimal curvatures, which produces a faster convergence but is required tosolveaninversehessianmatrix. Literatures 10,11 reportedorderedsubsetssps(os-sps),whichgainsaninitial V. 2 (p.1 of 7) / Color: No / Format: Letter / Date: 2/4/2014 9:41:22 AM

2 acceleration but induces a divergence in a limited circle. Even a relaxed OS can not guarantee a global optimal. A exponential power based larger step size was applied to Maximum likelihood expectation maximization (ML- EM) reconstruction in Literature. 12 However without monotonicity, this algorithm has potential problem with stability. In addition, the particular technique is not extended to a penalized statistical method, which is more general in tomographic reconstruction. he idea of over-relaxation was also mentioned in literature 13 for the model based iterative reconstruction (MBIR) with ICD framework, where a relaxation within [1, 2] was to enlarge the step size, however a larger step is not guarantee to produce a faster convergence and might take the risk of divergence. In literature, 14 we studied the contraction mapping of O and resulted in the convergence analysis according to a convergence rate matrix, which led to a successive increasing over-relaxation algorithm with a faster and monotonic convergence. In this paper, we summarize the theorem and propose an algorithm framework and then apply it to our pre-computed penalized likelihood reconstruction (PPL) with an edge-preserved regularizer. 4 Preliminary results will be demonstrated with a simulated Digital Breast omosynthesis system. 2. MEHODS In statistical iterative reconstruction in transmission tomography, the negative log penalized-likelihood function can be written as Ψ(µ) = L(µ)+λR(µ), (1) where the first term on the right-hand side is the data fidelity term according to X-ray signal statistics and system geometry. Function R(µ) is a regularizer which typically penalizes the difference among adjacent voxels. λ plays a role to control the tradoff between pixel precision and noise. hrough minimizing the objective function (1), one can estimate the optimal µ, which is formulized as follows: u = argminψ(µ). (2) µ 0 o solve the problem directly is intractable in real applications. Literature 8 proposed the concept of optimization transfer, where a series of surrogate parabolic functions are conceived which are lower bounded by the objective function, therefore minimizing Eq. (1) is transfered to the minimizations of the surrogate ones. We summarize the O to a algorithm scheme: for each iteration t = 1,...,Niter do Find an surrogate function G(Θ, Φ) satisfying, (1) G(Θ t,θ t ) = Ψ(Θ t ), (2) Ψ(Θ) G(Θ,Θ t ), Θ t Θ, we apply one step of Newton s method on the surrogate, the optimal approximation at (t + 1)-th iteration is written as end for Θ t+1 = Θ t 2 ΘG(Θ t,θ t ) 1 Θ G(Θ t,θ t ), (3) where G(Θ,Θ t ) is one surrogate function lower bounded by Ψ(Θ) at Θ t. he bounded condition leads to a monotonic convergence to the optimal automatically. According to Eq. 3, Θ G(Θ t,θ t ) represents the gradient pointing the direction of the optimal in the solution space. 2 Θ G(Θt,Θ t ) 1, which is the inverse of curvature of the surrogate function, denotes the step size along the gradient. Due to the bounded condition, the curvature of the surrogate can not be smaller than the curvature of the objective function, which is resulting in a conservative step size. However, bounded is not necessary, a non-bounded surrogate is possible to produce a monotonic convergence as well. In literature, 13 an over-relaxation was used to yield larger step. However, a larger step may not lead to a faster convergence and even takes the risk of divergence. o break the limitation of bounded condition and determine a optimal step size with a monotonic convergence. Convergence rate is investigated based on a contraction mapping, which is defined as (Θ,Φ) = Θ 2 ΘG(Θ,Φ) 1 Θ G(Θ,Φ). (4) V. 2 (p.2 of 7) / Color: No / Format: Letter / Date: 2/4/2014 9:41:22 AM

3 Given Θ as the global optimal, he local convergence rate is presented as r = lim t Θ Θ t+1 2 Θ Θ t 2 K(Θ ) 2 = λ max, (5) whereλ max denotes thelargesteigenvalue oftheconvergenceratematrix K(Θ )which was derivedinliterature 14 by fix point theorem with the form of K(Θ ) = 2 ΘG(Θ,Θ ) 1 2 Θ,ΦG(Θ,Θ ). (6) If the largest eigenvalue λ max of the convergence rate matrix is equal to 1 or r = 1, the contraction mapping is said to be sublinearly convergent. o speed up the rate at low convergence, a factor-scaled contraction mapping was proposed R(Θ,Φ) := Θ+ρ((Θ,Φ) Θ) = Θ ρ 2 ΘG(Θ,Φ) 1 Θ G(Θ,Φ), (7) where ρ plays a role to scale the step size. ρ [0,2] was proved to ensure a convergence when Θ is close enough to the optimal Θ. a particular ρ called the optimal relaxation, which yields the fastest convergence rate around Θ, was derived 14 as follows ρ = 2 2 λ max λ min. (8) where λ min is the smallest eigenvalue of the convergence rate matrix K(Θ ). Unfortunately, direct estimation of ρ requires knowledge of the convergence rate matrix K(Θ,Θ ). Without exhausted computation, it is difficult to give a precise estimation. Instead of a direct estimation, we provides an indirect method according to the characteristic of O. Given 0 λ min λ max 1, we admit that the contraction mapping Eq. 4 enjoys a faster convergence at the early stage and tends to endure a slower convergence later, therefore λ max varies from 0 to 1 gradually. One can see that the optimal ρ is changing from 1 to K, where K may be a value larger than 2. In another word, the optimal scaler ρ, which produces the optimal step size to achieve the fastest convergence rate at each iteration, is not to be scaled down gradually, but need to be successively increasing iteration by iteration, which is fundamentally different with our intuition which intends to reduce the scaler to avoid unstability. According to this acknowledge, to mimic the optimal scaler, ρ is preset as 1 and is multiplied by a practical factor a (a > 1) for the following iterations. In addition, the values of the objective function with the ρ scaled estimation and the non-scaled estimation are required to be evaluated and compared at each iteration to ensure a globally faster convergence. herefore, the successively increasing ρ-scaled optimization transfer (ρ-o) based algorithm was summarized as follows: ρ = 1 and a = δ (δ can be adjusted in a specific case) for each iteration t = 1,...,Niter do Θ t+1 = Θ t 2 Θ G(Θt,Θ t ) 1 Θ G(Θ t,θ t ) Θ t+1 NEW = Θt +ρ(θ t+1 Θ t ) Calculate Ψ(Θ t+1 ) and Ψ(Θt+1 NEW ) if Ψ(Θ t+1 NEW ) Ψ(Θt+1 ) then ρ = a and Θ t+1 = Θ t+1 NEW else ρ = 1 and Θ t+1 = Θ t+1 end if end for In the algorithm framework, Θ is computed by any well-known O based algorithm. In our discussion, PPL with an edge preserved regularizer 4 is applied V. 2 (p.3 of 7) / Color: No / Format: Letter / Date: 2/4/2014 9:41:22 AM

4 Figure 1. Geometric configuration of Digital Breast omosynthesis with multiple parallel X-ray beams. he calculation of Θ NEW is updated with a ρ-scaled difference between Θ and the last iterative solution. By comparing the objective functions Ψ(Θ ) and Ψ(Θ NEW ), the strategy of updating current solution and ρ is determined. If Ψ(Θ NEW ) is less than Ψ(Θ ), meaning scaled solution is closer the global optimal, Θ is updated by Θ NEW and ρ is scaled by a multiplier a. Otherwise Θ is used to updates Θ, at the same time ρ is set as 1. herefore, ρ will never worsen the convergence. For each iteration, ρ always produces a faster convergence under the criterion of objective function. For instance, in each iterative step, ρ-o is gaining a speedup by a certain factor σ. After M iterations, an exponential gain of σ M is obtained. hus, a substantial improvement in convergence can be achieved by the ρ-o. o evaluate the computational complexity, given that a representative O based method requires one time forward projection and one time voxel update for each iteration, the proposed method requires one extra voxel update. But this particular update occurs together with the original one, which will not increase computational cost. Ψ(Θ NEW ) and Ψ(Θ ) can be calculated efficiently by reusing the error sinogram and part of the update operation. o demonstrate the performance of the algorithm, we compare our proposed algorithm to separable parabolic surrogate with pre-computed curvature (PC-SPS). 10,11 Both algorithms are started by one-step OS-SPS to gain an initial speed-up. he data was collected with a virtual digital breast tomosynthesis system with the same geometric configuration as a real limited angle X-ray tomography system. Fig. 1 demonstrates the geometric configuration of a Digital Breast omosynthesis referred to the literature. 15 he 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. he 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. he 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. he system provides θ = 48 coverage around. he testing 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. wo 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 radius 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. he 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 system. he projections are generated by a incident value under Poisson distribution and an illumination model V. 2 (p.4 of 7) / Color: No / Format: Letter / Date: 2/4/2014 9:41:22 AM

1.2513 x 1011 1.2513 PC SPS ρ O 1.2513 1.2513 obj value 2 4 6 8 10 12 14 16 18 20 iterations Figure 2. Comparisons of the objective functions of PC-SPS and ρ-o. (a) (b) (c) Figure 3.

(a) shows a focus plane reconstructed by PC-SPS with 20 iterations; (b) shows the difference between the focus plane by ρ-o with 8 iterations and the focus plane by PC-SPS with 20 iterations;(c)

2 presents the comparison between the objective function values from ρ-o and PC-SPS with iteration increasing. One can see that ρ-o shows much faster convergence rate than PC-SPS.

5 x PC SPS ρ O obj value iterations Figure 2. Comparisons of the objective functions of PC-SPS and ρ-o. (a) (b) (c) Figure 3. IQ demonstration of proposed ρ-o. (a) shows a focus plane reconstructed by PC-SPS with 20 iterations; (b) shows the difference between the focus plane by ρ-o with 8 iterations and the focus plane by PC-SPS with 20 iterations;(c) shows the difference between the focus plane by PC-SPS with 8 iterations and the focus plane by PC-SPS with 20 iterations 3. RESULS Fig. 2 presents the comparison between the objective function values from ρ-o and PC-SPS with iteration increasing. One can see that ρ-o shows much faster convergence rate than PC-SPS. For instance, ρ-o with 8 iterations is equivalent to PC-SPS with 20 iterations in terms of objective function value. he new algorithm leads to a 2.5-fold speedup. Fig. 3 demonstrates IQ of ρ-o with less iterations. (a) shows a focus plane reconstructed by PC-SPS with 20 iterations. his result is considered as a convergent reference. (b) shows the difference between the convergent result and the result by the proposed method with 8 iterations. Compared to the difference between the convergent result and the result by PC-SPS with 8 iterations, the proposed method reconstructs more consistent IQ to the convergent one in terms of image structure and noise. With respect to a rough estimation on the computational complexity based on our current implementation and hardware platform, a total 40% computing time was saved by our proposed method with less iterations to achieve the comparable IQ to the convergent one. 4. CONCLUSIONS Statistical iterative reconstruction exhibits particular promising since it provides the flexibility of accurate physical noise modeling and geometric system description in transmission imaging system. However, to solve the V. 2 (p.5 of 7) / Color: No / Format: Letter / Date: 2/4/2014 9:41:22 AM

6 objective function is computationally intensive compared to analytical reconstruction methods due to the more accurate system model and the need for multiple iterations. In this study, we summarize the framework of O based algorithm, which is a general algorithm converting a high dimensional optimization to a parallel 1-D update by transferring the target optimization to a set of surrogate functions. A convergence rate matrix is derived for the rate analysis. Based on the spectrum of the matrix, an optimal scaler was proposed to achieve the fastest convergence rate. Without an expensive direct estimation on the spectrum of the matrix, we provided an indirect method to evaluate an optimal scaler with certain criterion and proposed an successively increasing ρ-o based computing framework. Initial investigation to the proposed algorithm in a simulated DB system demonstrated that fewer iterations with total 40% computing time reduction are needed to achieve a comparable IQ and an equal objective function value compared to PC-SPS. In general, our proposed method exhibits a tremendous potential to be a iterative method with a parallel computation, a monotonic and global convergence with fast rate. Further work will be conducted to present the structure similarity of IQ between ρ-o with less iteration and a convergent one by using root mean square error (RMSE). Instead of computing the objective function value, an alternative method to estimate the optimal ρ precisely is also going to be investigated. ACKNOWLEDGMENS Shiyu Xu wants to thank Dr. Henri Schurz in mathematics department at SIUC due to the meaningful discussions and we also thank Dr. Shaikh Ahmed in ECE department at SIUC for the computational resources. REFERENCES [1] Xu, S., Chen, Y., Lu, J., and Zhou, O., An application of pre-computed backprojection based penalizedlikelihood image reconstruction on stationary digital breast tomosynthesis, Proc. SPIE 8668, 86680v (2013). [2] Xu, S. and Chen, Y., An simulation based image reconstruction strategy with predictable image quality in limited-angle X-ray, Proc. SPIE 8668, 86685p (2013). [3] Xu, S., Schurz, H., and Chen, Y., Parameter optimization of relaxed ordered subsets pre-computed back projection (BP) based penalized-likelihood (OS-PPL) reconstruction in limited-angle X-ray tomography, Computerized Medical Imaging and Graphics 37(4), (2013). [4] Xu, S., Inscoe, C. R., Lu, J., Zhou, O., and Chen, Y., Pre-computed backprojection based penalizedlikelihood (PPL) reconstruction with an edge-preserved regularizer with stationary Digital Breast omosynthesis, Accepted by Proc. SPIE (2014). [5] Lange, K., BAHN, M., and LILE, R., A theoretical study of some maximum likelihood algorithms for emission and transmission tomography, IEEE ransactions on Image Processing MI-6, (1987). [6] Whiting, B. R., Signal statistics in X-ray computed tomography, Proceedings of SPIE 4682, (2002). [7] Stayman, J. W. and Fessler, J. A., Regularization for uniform spatial resolution properties in penalizedlikelihood image reconstruction, IEEE ransactions on Medical Imaging 19, (2000). [8] Erdogan, H. and Fessler, J. A., Monotonic algorithms for transmission tomography, IEEE ransactions on Medical Imaging 18, (1999). [9] Bouman, C. A. and Sauer, K., A unified approach to statistical tomography using coordinate descent optimization, IEEE ransactions on image processing 5032, (2003). [10] Ahn, S. and Fessler, J. A., Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithm, IEEE ransactions on Medical Imaging 22, (2003). [11] Sotthivirat, S. and Fessler, J. A., Relaxed ordered-subset algorithm for penalized-likelihood image restoration, J. Opt. Soc. Am. A 20, (2003). [12] Hwang, D. and Zeng, G. L., Convergence study of an accelerated ml-em algorithm using bigger step size, Phys. Med. Biol. 51, (2006). [13] Yu, Z., hibault, J.-B., Bouman, C. A., Sauer, K. D., and Hsien, J., Fast model-based x-ray C reconstruction using spatially non-homogeneous icd optimization, IEEE ransactions on image processing 20(1), (2011) V. 2 (p.6 of 7) / Color: No / Format: Letter / Date: 2/4/2014 9:41:22 AM

7 [14] Xu, S. and Chen, Y., Local convergence analysis of the optimization transfer and its acceleration in limited view x-ray tomography, IEEE transaction image processing (under submission). [15] Qian, X., Rajaram, R., Calderon-Colon, X., Yang, G., Phan,., Lu, D. S. L. J., and Zhou, O., Design and characterization of a spatially distributed multibeam field emission x-ray source for stationary digital breast tomosynthesis, Medical Physics 36, (2009) V. 2 (p.7 of 7) / Color: No / Format: Letter / Date: 2/4/2014 9:41:22 AM

Carbondale, IL USA; University of North Carolina Chapel Hill, NC USA; USA; ABSTRACT

Carbondale, IL USA; University of North Carolina Chapel Hill, NC USA; USA; ABSTRACT Pre-computed backprojection based penalized-likelihood (PPL) reconstruction with an edge-preserved regularizer for stationary Digital Breast Tomosynthesis Shiyu Xu a, Christy Redmon Inscoe b, Jianping