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A low-complexity 2-point step size gradient projection method with selective function

evaluations for smoothed total variation based CBCT reconstructions

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2014 Phys. Med. Biol. 59 6565

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6565

Physics in Medicine & Biology

A low-complexity 2-point step size gradient projection method with selective function evaluations for smoothed total variation based CBCT reconstructions

Bongyong Song1, Justin C Park2, and William Y Song3

1 Department of Radiation Medicine and Applied Sciences, University of California San Diego, La Jolla, CA 92093, USA2 Department of Radiation Oncology, University of Florida, Gainesville, FL 32610, USA3 Department of Medical Physics, Sunnybrook Health Sciences Centre, Department of Radiation Oncology, University of Toronto, Toronto, ON M4N 3M5, Canada

E-mail: william.song@sunnybrook.ca

Received 28 March 2014, revised 11 September 2014Accepted for publication 16 September 2014Published 16 October 2014

AbstractThe Barzilai–Borwein (BB) 2-point step size gradient method is receiving attention for accelerating Total Variation (TV) based CBCT reconstructions. In order to become truly viable for clinical applications, however, its convergence property needs to be properly addressed. We propose a novel fast converging gradient projection BB method that requires ‘at most one function evaluation’ in each iterative step. This Selective Function Evaluation method, referred to as GPBB-SFE in this paper, exhibits the desired convergence property when it is combined with a ‘smoothed TV’ or any other differentiable prior. This way, the proposed GPBB-SFE algorithm offers fast and guaranteed convergence to the desired 3DCBCT image with minimal computational complexity. We first applied this algorithm to a Shepp–Logan numerical phantom. We then applied to a CatPhan 600 physical phantom (The Phantom Laboratory, Salem, NY) and a clinically-treated head-and-neck patient, both acquired from the TrueBeam™ system (Varian Medical Systems, Palo Alto, CA). Furthermore, we accelerated the reconstruction by implementing the algorithm on NVIDIA GTX 480 GPU card. We first compared GPBB-SFE with three recently proposed BB-based CBCT reconstruction methods available in the literature using Shepp–Logan numerical phantom with 40 projections. It is found that GPBB-SFE shows either faster convergence speed/time or superior convergence property compared to existing BB-based algorithms. With the CatPhan 600 physical phantom, the GPBB-SFE algorithm requires only 3 function evaluations in 30 iterations

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and reconstructs the standard, 364-projection FDK reconstruction quality image using only 60 projections. We then applied the algorithm to a clinically-treated head-and-neck patient. It was observed that the GPBB-SFE algorithm requires only 18 function evaluations in 30 iterations. Compared with the FDK algorithm with 364 projections, the GPBB-SFE algorithm produces visibly equivalent quality CBCT image for the head-and-neck patient with only 180 projections, in 131.7 s, further supporting its clinical applicability.

Keywords: GPBB-SFE, compressive sensing, iterative reconstruction, CBCT, IGRT

(Some figures may appear in colour only in the online journal)

1. Introduction

One of the major applications of cone-beam computed tomography (CBCT) imaging modality is its use in Image Guided Radiation Therapy (IGRT). The CBCT on-board linear accelera-tors (Jaffray et al 2002, Oldham et al 2005, Amies et al 2006, Thilmann et al 2006, Yoo et al 2006) are widely used just before/during RT treatment to allow precise target localization and treatment guidance. The imaging, registration, and correction procedure enables better/more accurate control of radiation dose delivery within the body.

On the other hand, CBCT uses ionizing x-rays to image and there is a legitimate con-cern of hazardous radiation exposure to patients (Song et al 2008). Due to this, the excessive use should be prohibited and the benefits-versus-harm ratio should be carefully weighed and debated for each treatment, especially for pediatric patients. This concern has now become an issue of central importance in North America, resulting in collaborative efforts by academia, clinical practice, industry, and regulatory and funding agencies for developing lower dose imaging technologies (McCoollough et al 2011).

Due to the analytic nature, the currently-standard FDK reconstruction algorithm (Feldkamp et al 1984) allows a convenient batch computation of projection data, filtering, and backpro-jection, which is very advantageous for near real-time reconstruction of the large 3DCBCT image. As FDK is a generic solution that works for any types of CBCT images, it has rather stringent requirements on the number of projections and the radiation dose in each projection, which significantly limit the room for dose reduction. It requires sufficient number of projec-tions to avoid aliasing artifact and requires sufficient amount of dose (in mAs) to obtain the desired spatial resolution while keeping the image noise at an acceptable level.

Iterative reconstruction is one of the most promising enablers for low-dose CBCT recon-struction (McCoollough et al 2011). This approach typically casts the restriction problem into a convex optimization problem where an iterative correction loop ensures the solution converges to the desired 3D image. Typically, the corrective loop compares synthesized pro-jection data (from the current 3D image estimates) with real measurement data and determines the correction to the current 3D image to minimize the discrepancy. Furthermore, various forms of prior knowledge, such as image and noise statistics, can be effectively incorporated into the optimization problem to promote the characteristics of the desired image. As a result, a high quality image can be obtained from a limited number of projections and/or reduced radiation dose per projection (Thibault et al 2007, Chen et al 2008, Leng et al 2008, Sidky and Pan 2008, Tang et al 2009, Wang et al 2009, Bian et al 2010, Choi et al 2010, Szczykutowicz and Chen 2010, Yu and Wang 2010, Jørgensen et al 2011, Ritschl et al 2011, Niu and Zhu 2012, Park et al 2012, Kuntz et al 2013).

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Formulating an optimization problem based on the compressive sensing (Candes et al 2006, Donoho 2006, Donoho et al 2006) approach is one of the most successful iterative reconstruc-tion approaches in recent years. The compressive sensing theory states that sparse images (in some known transform domain) can be accurately reconstructed from fewer measurements/projections since they can be represented with fewer numbers (in the transform domain) than the amount of pixels in the original image, resulting in reduced imaging dose without com-promising image quality. Total variation (TV) minimization formulation which was originally introduced as an efficient tool for image de-noising (Rudin et al 1992), has been particularly useful in exploiting the prior knowledge of minimal variation in x-ray attenuation character-istics across human body (Chen et al 2008, Leng et al 2008, Sidky and Pan 2008, Tang et al 2009, Bian et al 2010, Choi et al 2010, Szczykutowicz and Chen 2010, Yu and Wang 2010, Jørgensen et al 2011, Ritschl et al 2011, Niu and Zhu 2012, Park et al 2012, Kuntz et al 2013).

In spite of the promising compressive sensing theory, the immediate application to clini-cal practice is not straight-forward. Due to the iterative nature of solving the TV-based com-pressed sensing formulation, image reconstruction requires multiple iterations of forward and backward projections of large datasets, which are well known to be computationally expen-sive. In order for this technology to become more practical and attractive, the iterations must converge to the desired image in clinically feasible time frame (e.g. <1 min). This challenge can be primarily addressed by leveraging recent advancements in parallel computing hard-ware with efficient parallel-programming implementation of large data forward and backward projections. There have been many efforts (Xu and Mueller 2005, Xu and Mueller 2007, Jia et al 2010, 2011, Park et al 2011, Park et al 2012) to take advantage of the massive parallel com-putation power of graphics processing units (GPU). This approach reduced the computational time from several hours to few minutes (Jia et al 2010, 2011, Park et al 2012). In addition, development of the computationally efficient iterative reconstruction algorithms that quickly converge to the desired solution is another important domain of innovations to address the computational challenge. An ideal algorithm would be characterized by i) simple computa-tions (especially in the parallel computation environment) in each iterative step and ii) guar-anteed convergence to the desired 3D image.

In this paper, we propose an iterative reconstruction algorithm that exhibits these two prop-erties. To do this, we revisit the Barzilai–Borwein (Barzilai and Borwein 1988, Birgin et al 2000, Figueiredo et al 2007) framework of which computational effectiveness in CT/CBCT reconstructions has been witnessed in several very recent studies (Jørgensen et al 2011, Niu and Zhu 2012, Park et al 2012). We will show that existing approaches require more than necessary computations (in terms of function evaluations) in each iterative step and/or do not guarantee convergence to a desired image. We then propose an algorithm that requires at most one function evaluation in each iterative step while offering guaranteed convergence property to a desired image. As a result, we get a clinically reasonable patient image in 30 iterations for a total reconstruction time of 132 s using a single GPU card (NVIDIA GTX 480, Santa Clara, CA). Comparison of our novel approach with the FDK and other published BB-based com-pressed sensing techniques are presented in detail with numerical and physical phantoms, and a head-and-neck clinical patient data. Notations: Matrices are denoted by boldface uppercase letters and vectors are denoted by boldface lowercase letters.

2. Material and methods

The Barzilai–Borwein (BB) 2-point step size gradient method is getting a recent attention for accelerating Total Variation (TV) based CBCT reconstructions (Jørgensen et al 2011, Niu and

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Zhu 2012, Park et al 2012). Due to its fast convergence and low memory/storage requirements, it is ideally suited for solving large scale problems such as 3D- and 4D-CBCT reconstructions.

The main problem is to solve the following constrained convex optimization problem of the form:

λ= − + ≥x Ax b x xf TV    s.tmin ( ) ( ) .  0x

22

(1)

where x = unknown CBCT volume image of size N, A = Radon transform operator, b = mea-sured projections data obtained from the x-ray imager, λ = regularization constant, and TV() = Total Variation (TV) operator. Among many variants, the specific TV term we used in this study is defined as:

∑= + − + + −

+ + −

[ ] [ ]

[ ]

xTV x i j k x i j k x i j k x i j k

x i j k x i j k

( ) ( 1, , ) ( , , ) ( , 1, ) ( , , )

( , , 1) ( , , )i j k, ,

2 2

2(2)

where i, j, and k corresponds to Left–Right (LR), Anterior–Posterior (AP), and Cranial–Caudal (CC) coordinates in 3D volume space, respectively. Here, elements of vector x are indexed by 3D coordinates for notational simplicity. Alternatively, the TV term can be expressed as:

∑=x D xTV ( )i j k

i j k

, ,

, , 2 (3)

by properly defining the difference matrices ∈ ℜ ×Di j kN

, ,3 for all x(i, j, k) according to the

three difference terms in the square root in equation (2) (Jensen et al 2011). The first term in equation (1) is the fidelity term, which enforces fidelity of x with the measured projection data b. The second term (the regularization term) promotes sparsity inherent in the x-ray attenua-tion characteristics of the human body.

The BB algorithm is essentially a gradient based algorithm of the form:

α= −++x x g[ ]n n BB n n1 , (4)

where λ= ∇ = − + ∇g x A Ax b xf TV( ) ( ) ( )n nT

n n and ∇ xTV ( )n denote the gradient of xf ( ) and a sub-gradient of xTV ( ) at =x x ,n respectively, and [•]+ = max[•, 0]. The main recipe for enabling fast convergence to an optimal solution lies in cleverly choosing the step size α nBB, based on x and g in the past 2 steps as:

α =−

− −−

− −

x x

x x g g[ ] [ ]n

n n

n nT

n nBB,

1 22

1 1(5)

An alternative form of the BB step size also exists in the form of:

α =− −

−′ − −

−

x x g g

g g

[ ] [ ].n

n nT

n n

n nBB,

1 1

1 22 (6)

The original 2-point step size algorithm was formulated for ‘unconstrained quadratic’ problems (Barzilai and Borwein 1988) and the theory has expanded to further improve the convergence speed of the BB framework. For example, the Adaptive Barzilai Borwein (ABB) algorithm that adaptively chooses one of the two possible BB step sizes above, often offers superior convergence performance compared to the original BB algorithm (Zhou et al 2006). For more general functions, the BB framework has evolved to be applicable to general dif-ferentiable functions on closed convex sets (Birgin et al 2000). One of the most interesting properties of the BB algorithm is its non-monotonic convergence. Unlike many gradient pro-jection algorithms where convergence is guaranteed through strict monotonic decrease of the

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objective function, i.e. <+x xf f( )   ( )n n1 for all n, the BB algorithm requires only a loose sense of monotonicity to ensure convergence to an optimal solution. Specifically, for a given integer parameter ≥K 1, define:

…− − +{ }x x xf f f fdef max ( ),   ( ),   , ( ) .n n n n Kmax, 1 1 (7)

Each step in the BB algorithm only requires sufficient decrease of +xf ( )n 1 compared to f ,nmax, not to xf ( )n (Birgin et al 2000). The specific application of this approach to solve equation (1) for CT was analyzed by Jørgensen et al (2011) and their algorithm for finding an appropriate BB step size starting from equation (5) to ensure convergence is summarized in figure 1(a). It can be seen that the backtracking line search loop for finding the appropri-ate step size αn at interation n does not enforce strict monotonic decrease of the objective

Figure 1. Illustrations of (a) the step-size determination algorithm in conventional GPBB-NMLS method, and (b) the proposed step-size determination algorithm in our GPBB-SFE algorithm.

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function xf ( ). This non-monotone line search (NMLS) method is referred to as GPBB-NMLS in this paper. In general, this backtracking line search involves ‘multiple’ function evaluations, each requiring a computationally expensive forward projection operation. The overall GPBB-NMLS algorithm is summarized in figure 2(a).

On the other hand, the GPBB algorithm without backtracking line search (Park et al 2012) focused on empirical performance of a BB algorithm that is very similar to the original work by Barzilai and Borwein (Barzilai and Borwein 1988). In this case, the overall computation was greatly reduced due to absence of the line search effort as shown in figure 2(b). Although theoretical convergence was not demonstrated, the authors reported extremely fast 3DCBCT reconstruction experimental results in GPU implementations which are attractive for on-line IGRT.

The Adaptive Barzilai Borwein (ABB) adaptively chooses either equation  (5) or (6) to compute the BB step size according to the following rule:

αα α α κα

=<′ ′⎧

⎨⎩

,     if  /

,    otherwise,             n

n n n

n

BB, BB, BB,

BB,(8)

where κ ∈ (0, 1) is an optimization parameter of the algorithm. It has been shown that ABB converges without requiring the line search for unconstrained quadratic problems (Zhou et al

Figure 2. Illustration of computational processes required at each iteration for six different algorithms: (a) GPBB-NMLS, (b) GPBB, (c) GPABB (d) GPBB-SFE, (e) ASD-POCS, and (f) GP.

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2006). In order to ensure convergence even for more general functions such as problem (1), Zhou et al (2006) suggested to combine ABB with some form of non-monotone line search methods. In this paper, we only incorporated the non-negativity constraint in (1) by adding a gradient projection step to ABB, which is referred to as GPABB in this paper and illustrated in figure 2(c). By doing this, GPABB would still maintain its fast convergence property (i.e. no line search) although it is not guaranteed to converge for solving our CBCT problem (1).

In this paper, we propose a novel BB-based algorithm that theoretically converges to an optimal solution. One salient feature of the proposed algorithm is that it requires at most one function evaluation in an iterative step, which greatly reduces the overall computa-tional complexity for iterative 3DCBCT reconstructions. To do this, we first address the non-differentiability issue of TV defined by equation  (2). In order to take advantage of the established convergence framework for differentiable functions (Birgin et al 2000), we modify equation (1) to:

λ= − + ≥εx Ax b x xf TV    s tmin ( ) ( ) . .  0x

22

(9)

where εTV x( ) is a smoothed approximation of TV (Vogel and Oman 1996).

∑

ε ε

= + − + + −

+ + − + −ε xTV x i j k x i j k x i j k x i j k

x i j k x i j k

( ) [ ( 1, , ) ( , , )] [ ( , 1, ) ( , , )]

[ ( , , 1) ( , , )] .i j k, ,

2 2

2 2(10)

Other types of smoothed version of TV, such as the Huber function, can also be considered (Huber 1964, Jensen et al 2011).

The main computational reduction in our proposed method is enabled by leveraging the fact that there exists a small scalar αth such that, the objective function in equation (9) decreases sufficiently whenever α α≤nBB, th and xn is not a stationary point. (Bertsekas 1999, Proposition 2.2.3(a)). This implies that backtracking line search is not required whenever α α≤ .nBB, th In these steps, the proposed algorithm requires ‘zero’ function evaluation. When α α> ,nBB, th the objective function may or may not satisfy the sufficient decreases criteria (w.r.t. f nmax, ) that is required for ensuring convergence (Birgin et al 2000). If the evaluated objective function does satisfy the sufficient decrease criteria, α nBB, can still be used as an appropriate step size. In this case, the function may or may not decrease w.r.t. fn since it is compared with f nmax, rather than f .n On the other hand, if the objective function does not satisfy the sufficient decrease cri-teria, we simply use the step size αth without conducting the computationally expensive back-tracking line search. As a result, the number of function evaluation is ‘no greater than one’. The proposed step size determination algorithm is summarized in figure 1(b) and the overall algorithm is depicted in figure 2(c). It can be seen that, unlike GPBB-NMLS where an inner loop is required for backtracking line search, the proposed algorithm does not have an inner loop. Due to the selective function evaluation (SFE) nature of the proposed algorithm (i.e. one function evaluation only when α α>nBB, th), we call the proposed algorithm GPBB-SFE. It can be easily seen that the proposed algorithm satisfies all the conditions for the prototype BB algorithm SPG1 developed by Birgin et al (2000) where its convergence is established. As a result, GPBB-SFE is an instantiation of SPG1 with a special property—at most one function evaluation per iteration.

It is found that the BB step sizes computed by equation (5) vary significantly from itera-tion to iteration. As an example, we illustrated the BB step sizes computed for reconstruct-ing a 2D Shepp–Logan numerical phantom in figure 3. In this case, more than two orders of magnitude variation was observed, which gives ample opportunities for GPBB-SFE to

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move on to the next iteration without conducting a function evaluation. Detailed performance analysis of the GPBB-SFE for this numerical phantom is described in section 3 along with real patient results.

In the proposed GPBB-SFE, properly choosing the step size threshold αth is very impor-tant. In fact, it can be shown that we can analytically find a proper threshold. It is well known that, for a Lipschitz continuous function with Lipschitz constant L, αth can be chosen to be less than or equal to 1/L (Bertsekas 1999, Proposition 1.2.3(a)). The fidelity term in equation (9) is quadratic where the Lipschitz constant is given by A .2

2 In the regularized TV term, the curvature is highest around its zero where the second-order Taylor series approximation can be shown to be ε Dx1/ 2

2 whose Lipschitz constant is given by D 22 where ∈ ℜ ×D N N3 is a

matrix by stacking all Di j k, , matrices together. These results suggest that, in theory, the desired threshold can be analytically set to λ ε+( )A D1/ / .2

222 Unfortunately, the dimensions of A

and D are so large that computing the matrix 2-norms are computationally prohibitive. This forces us to find αth empirically. Fortunately, this value is only system dependent (i.e. A and D, and independent of unknown geometry of x) which makes this process simpler. More discus-sions on this aspect are found in section 4.

In this paper, we consider two additional algorithms in the literature to be compared with GPBB-SFE. First, the adaptive-steepest-descent-projections-onto-convex-set (ASD-POCS) method proposed by Sidky and Pan (Sidky and Pan 2008) was implemented where the for-ward–backward splitting technique is employed. ASD-POCS individually minimizes the fidelity and the smoothed TV terms in equation (9) in each iteration as summarized in fig-ure 2(e). Second, we have implemented a fixed step-size Gradient Projection (GP) method that attempts to simultaneously minimize both terms in equation (9) in a single step as summarized in figure 2(f).

Figure 3. The BB step sizes computed using equation (5) while reconstructing the 2D Shepp–Logan numerical phantom. A total of 40 projections in fan-beam geometry were used for the reconstructions.

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In order to quantitatively compare the algorithms described in figure  2, we define Normalized Squared Error (NSE) that measures the mean-squared percent error from the ground truth pixel values:

∑

∑=

−

×( )

( )

x x

xNormalized Squared Error(%) 100

i j k

i j k i j k

i j ki j k

, ,

, , , ,Ground Truth 2

, ,, ,Ground Truth 2 (11)

where xi j k, , corresponds to the voxel values in the reconstructed volume x and xi j k, ,Ground Truth

refers to the ground truth values of the Shepp–Logan phantom used.In our implementation of this CBCT GPBB-SFE algorithm, the entire code is structured

and implemented in C with CUDA programming environment (NVIDIA, Santa Clara, CA) to utilize the massive parallel computational capability of GPU hardware. We used a single GTX 480 card (~$400US) that consists of 448 processing cores with 1.215 GHz clock speed and 1,536 MB memory. In terms of CPU, we used Intel Core™ i7 with 2.68 GHz clock speed, 12.0 GB DDR3 RAM, on a 64-bit Window 7 OS. In order to fully utilize the parallel com-puting capability, we adopted detector-driven/ray-driven projection and voxel-driven back-projection implementation approaches (Pratx and Xing 2011).

3. Results

To evaluate the performance of our GPBB-SFE algorithm, we first considered a 256  ×  256 2D Shepp–Logan numerical phantom. A total of 40 projections in fan-beam geometry were used for the reconstructions. We have compared it with five other algorithms introduced in section 2—ASD-POCS, GP, GPABB, GPBB-NMLS and GPBB. A sufficiently small fixed step size α( )fixed should be chosen to ensure monotonic convergence of the GP algorithm, and this value was used as αth for GPBB-SFE.

Some important parameters for each algorithm in figure 2 are summarized below:

• λ = 3.0, ε = 0.001 for all algorithms • ASD-POCS (αconst = 0.0001, δ = 0.001) • GP (αfixed = 0.0001) • GPABB (κ = 0.3) • GPBB-NMLS (σ = 0.02, β = 0.7, K = 3) • GPBB-SFE (αth = 0.0001, σ = 0.02, K = 30)

Note that the fixed step size αfixed that was chosen for GP was used as αth for GPBB-SFE. Also note that K is set to 30 for GPBB-SFE which is significantly higher than K = 3 for GPBB-NMLS. Since ̂fmax in GPBB-SFE is recomputed whenever α α> ,nBB, th K become important only when α α≤nBB, th for K consecutive iterations. A large value of K in GPBB-SFE allows the algorithm to run without any function evaluations for large number of iterations as long as α nBB, continues being less than or equal to α .th Impact of using larger values of K for GPBB-NMLS is also investigated as discussed later. GPABB’s adaptation threshold κ was set to 0.3 (instead of κ = 0.5 that Zhou et al 2006 used in their numerical examples) because it showed best convergence performance among values of κ ranging from 0.1 to 0.9 in steps of 0.1.

Figure 4 compares the NSE performance of different algorithms as the iteration progresses. It can be seen that all four BB approaches (GPBB-NMLS, GPBB, GPABB and GPBB-SFE) show noticeably superior convergence performance compared to ASD-POCS and GP. In order to reach NSE below 0.01, GPBB-NMLS, GPBB, GPABB and GPBB-SFE require 150, 128,

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149 and 117 iterations, respectively. In contrast, ASD-POCS and GP require more than 200 iterations. This demonstrates the utility of the BB approaches that cleverly utilize past two step estimates to determine the new step size. We found that the advantage of using a larger K value for GPBB-NMLS was mixed. Specifically, when we used K = 30 the required number of iterations for reaching NSE below 0.01 was 137 which is smaller than 150 (required for GPBB-NMLS with K = 3). On the other hand, GPBB-NMLS with K = 3 performed better than with K = 30 as the number of iterations further increases (i.e. crossing of the NSE lines). From these observations, K = 3 recommended by Jørgensen et al (2011) seems to be a reasonably good choice for GPBB-NMLS.

Although the three BB approaches show more or less comparable convergence perfor-mance in the sense of NSE with respect to the number of iterations, their computational com-plexity per iteration is quite different. Over the 200 iterations, GPBB-NMLS, GPBB-SFE, and GPBB conduct 342, 117 and 0 function evaluations, respectively. This means that GPBB-SFE requires only 1/3 of function evaluations that GPBB-NMLS does, which greatly reduces the computational burden. Of course, the GPBB requires least computations as it never needs to evaluate the objective function. However, it should be noted that the GPBB’s empirical con-vergence results without guaranteed (theoretic) convergence property may not be sufficient for clinical adoption.

The reconstructed Shepp–Logan images after 50 iterations are compared in figure  5. It is observed that all BB methods (GPBB-NMLS, GPBB, and GPBB-SFE) fairly accurately reconstruct the Shepp–Logan phantom in 50 iterations whereas ASD-POCS and GP appar-ently require more iterations to reconstruct the image.

Figure 4. Comparison of the NSE performance of five different algorithms as a function of iterations in reconstructing the 2D Shepp–Logan numerical phantom. A total of 40 projections in fan-beam geometry were used for the reconstructions.

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We remark that our implementations of the ASD-POCS and GPBB-NMLS may not have been optimized as the ones originally proposed and implemented. Although we attempted to make the fairest comparison by best implementing the original ideas of the ASD-POCS and GPBB-NMLS using the published information, it is rather difficult to reproduce the same optimizations and implementations as well as the same experimental setup. As a result, our evaluations of the past algorithms may not represent their best possible convergence perfor-mance and image quality.

We then used the CatPhan 600 physical phantom (The Phantom Laboratory, Salem, NY), and clinically-treated head-and-neck patient, both acquired from the TrueBeam™ system (Varian Medical Systems, Palo Alto, CA) to evaluate GPBB-SFE performance for 3DCBCT reconstructions. For TrueBeam™ scans, in total, 364 projections were acquired in a 200° rota-tion, in a full-fan mode. The imager has 1024  ×  768 pixels with 0.388  ×  0.388 mm resolution. This was down-sampled to 512  ×  384 pixels with 0.776  ×  0.776 mm for reconstructing a 512  ×  512  ×  70 3D image. In order to confirm the dose reduction capability of the compres-sive sensing approach that utilizes the smoothed TV in equation (10), we used only 60 evenly space projections that are subsampled from the entire 364 projections. For both experiments, the following parameters are applied:

• λ = 0.0025, ε = 0.001 • GPBB-NMLS (σ = 0.02, β = 0.7, K = 3) • GPBB-SFE (αth = 2.0, σ = 0.02, K = 30)

Upper part of table 1 compares the number of function evaluations in 30 iterations and the average computation time per iteration in three BB based algorithms used for reconstructing

Figure 5. The reconstructed images of the 256  ×  256 2D Shepp–Logan phantom after 50 iterations using the respective five algorithms. A total of 40 projections in fan-beam geometry were used for the reconstructions.

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the CatPhan 600 physical phantom. (In terms of image quality, all three algorithms produced very similar images and thus we provide the comparison of computation times only.) For a given iterative algorithm, the stopping criterion (i.e. 30 iterations in this case) was chosen such that the algorithm stops at the point where −+x x x/n n n1 2 2 becomes less than a preconfigured threshold. We used 0.01 for the threshold as we observed a smaller threshold (thus requir-ing more iterations) does not offer noticeable image quality improvement. This happened in about 30 iterations in the experiments in table 1, and we presented 30 iteration results for all cases. In any gradient based algorithms, a single iteration involves at least one forward and one backward projection calculations, which are required for computing the gradient

λ= − + ∇ εg A Ax b xTV( ) ( ).nT

n n Each function evaluation λ− + ε( )A b xx TV ( )n n22 adds one

more forward projection, which is computationally expensive. In 30 iterations, the proposed GPBB-SFE algorithm requires only 21 function evaluations compared to 44 function evalua-tions required for GPBB-NMLS. This is more than 50% reductions in the number of function evaluations. The resultant reconstruction time per iteration in GPBB-SFE is 2.23 s which is noticeably smaller than the 3.34 s required for GPBB-NMLS. The overall GPBB-SFE results are quite close to those of GPBB which could be considered as the fastest BB based algorithm (along with GPABB) requiring no function evaluation. Furthermore, the proposed GPBB-SFE has theoretic convergence property which simpler GPBB algorithm lacks. These results sug-gest that GPBB-SFE offers two important advantages for clinical adoption.

The reconstructed 3D CatPhan 600 physical phantom volume (after 30 iterations) is com-pared with the standard FDK reconstructed images using all 364 projections and the FDK reconstructed images with the same subsampled 60 projections in figure  6. It is clearly observed that the GPBB-SFE images in figure 6(c) are noticeably better than 60-projection-based FDK reconstructed images in figure 6(b) with minimal aliasing artifacts. Furthermore, their spatial and contrast resolution is quite comparable to the standard FDK reconstructed images in figure 6(a) with improved noise performance. It is also observed that the iterative reconstruction using GPBB-SFE has successively removed the Feldkamp artifacts (dark and bright shadows) in figures 6(a) and (b), which is inherent in the FDK reconstruction method. The overall reconstruction time using the GPBB-SFE was 100.4 s in our implementation.

The next experiment was conducted for a clinically-treated head-and-neck patient acquired from the TrueBeam™ system (Varian Medical Systems, Palo Alto, CA). 180 evenly spaced projections were used for all algorithms. Lower part of table  1 compares GPBB-SFE,

Table 1. Computational complexity comparison of three BB algorithms for CBCT reconstruction for (i) the CatPhan 600 physical phantom (The Phantom Laboratory, Salem, NY) using 60 projections acquired from the TrueBeam™ system (Varian Medical Systems, Palo Alto, CA), and (ii) a head-and-neck patient using 180 projections acquired from the TrueBeam™ system (Varian Medical Systems, Palo Alto, CA).

GPBB-NMLS GPBB-SFE GPBB

CatPhan 600 Physical Phantom

# of function evaluations in 30 iterations

44 21 0

Time/Iteration (sec) 3.34 2.23 1.97Head-and-Neck Patient # of function evaluations in

30 iterations32 13 0

Time/Iteration (sec) 6.35 4.39 3.53Note: Guaranteed

convergenceGuaranteed convergence

No theoretic convergence

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GPBB-NMLS and GPBB used for this patient imaging in terms of the number of function evaluations in 30 iterations and the average computation time per iteration. The proposed GPBB-SFE algorithm requires only 13 function evaluations compared to 32 function evalu-ations required for GPBB-NMLS. This is approximately a 60% reduction in the number of function evaluations. The reconstruction time per iteration in GPBB-SFE is 4.39 s which is sufficiently smaller than 6.35 s required for GPBB-NMLS. As in the CatPhan 600 physical phantom results, the overall GPBB-SFE results are quite close to those of GPBB. Higher reconstruction times in this case compared to the CatPhan 600 phantom results (upper part of table 1) are due to the three-fold increase in number of projections.

The reconstructed 3D head-and-neck patient volume (after 30 iterations) is compared with the 180-projection FDK reconstructed image and the standard 364-projection FDK recon-structed image in figure  7. In comparison to the 180-projection FDK reconstructed image in figure 7(a), the GPBB-SFE image in figure 7(b) shows aliasing artifact reduction, noise reduction, and improved contrast so that the overall image quality is visually similar to the 364-projection FDK reconstructed image in figure 7(c). The entire reconstruction time in our GPU implementation was 131.7 s. These reconfirm the promising results in both the image quality and the reconstruction time for popular IGRT applications, e.g. patient setup.

We also presented GPBB-SFE image with 364 projections (after 30 iterations) in fig-ure 7(d) to see if the proposed algorithm can be used to primarily improve the image quality w.r.t. the standard 364-projection FDK reconstructed image (while less emphasizing dose reduction benefit). It displays a visually better quality image than that of the FDK by enabling less noise, streaking artifacts reduction around bones, etc. Although beyond the scope of this study, whether GPBB-SFE could be used (in the current form or with further improvements) for other important applications such as tumor contouring, treatment re-planning, etc, is an interesting question to further investigate.

Figure 6. Spatial and contrast resolution slices of the reconstructed CatPhan 600 phantom using (a) FDK with 364 projections, (b) FDK with 60 projections, and (c) GPBB-SFE with 60 projections after 30 iterations. The window and level were kept the same for all images.

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4. Discussion

The proposed GPBB-SFE algorithm brings forth i) guaranteed convergence to the desired CBCT image which would be very important for clinical applications, and ii) fast convergence to a desired solution with lower complexity per iteration, which greatly improve the practi-cal value of the algorithm. To do these, the proposed GPBB-SFE introduces an additional parameter αth which is not required in the conventional BB methods. As pointed out earlier, calculation of the desired value λ ε+( )A D1/ /2

222 may not be computationally feasible (the

dimension of the system matrix A for reconstructing the CatPhan 600 physical phantom is 18 350 080  ×  11 796 480). If a too small threshold is chosen, the algorithm would degenerate to GPBB-NMLS. If a too large threshold is chosen, the algorithm would degenerate to GPBB whose convergence is not guaranteed. As a result, a proper value of αth needs to be empirically chosen to be set as large value as possible and, at the same time, to ensure monotonic decrease of the objective function. This generally involves learning through experience, i.e. learn the appropriate value of αth while applying different values and observing the results. (This is sim-ilar to the empirical step size determination in the fixed step-size gradient descent algorithm.)

Figure 7. Comparison of reconstructed images for the head-and-neck example patient using (a) FDK with 180 projections, (b) GPBB-SFE with 180 projections after 30 iterations, (c) FDK with 364 projections, and (d) GPBB-SFE with 364 projections after 30 iterations. The window and level were arbitrarily chosen for best display and kept the same for all images.

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One thing to note is that the desired value λ ε+( )A D1/ /22

22 is essentially a function of A

and D, which are solely determined by the number of projections, the detector dimension, and the 3D volume dimension. These quantities are independent of the unknown image x, mAs setting, etc. Therefore, for a given CBCT setup (# of projections, detector dimension, 3D vol-ume dimension), one should be able to estimate a proper value of αth as a result of a sequence of empirical studies. Developing a systematic way of accurately estimating these variables is an interesting area for further research.

A more challenging task is to optimize the regularization parameter λ which has depend-ency not only on the system parameters but also on other factors. It needs to be carefully chosen considering system parameters such as A and D, scan settings such as mAs, and the anatomical structure of the unknown 3D image x. While attempting to find an optimal value for the regularization parameter λ, we clearly observed the noise-contrast tradeoff. That is, for the given set of system parameters, the higher the λ, the less noisier yet blurrier the images, whereas the smaller the λ, the sharper yet noisier the images. In order to choose the best value(s) in our experiments, we applied a range of regularization parameters and identified the one(s) that results in the best image(s) in terms of reducing the noise while preserving the contrast. Of course, there have been considerable amount of studies and discussions in the literature to systematically choose λ (Zhu and Xing 2009, Wang et al 2011, Niu and Zhu 2012, Park et al 2012), but much more innovations are required to properly select λ in an automated and robust manner while considering aforementioned factors. This will greatly improve clinical applicability of any TV-regularization-based CBCT reconstruction algorithms.

There are several important applications where GPBB-SFE or similar algorithmic enhancements are particularly valuable to the research and the clinical community. For a given CT/CBCT setup, two main ways to reduce the imaging dose of CT/CBCT scans are to reduce the number of projections and/or to lower the mAs setting per projection. Each approach has associated pros and cons. The reduced mAs method (while maintaining the number of projections at the standard setup, e.g. 364 projections) has some reported advantages over the reduced projection method as it eliminates the streaking artifacts that can arise due to a small number of projections (Tang et al 2009). Furthermore, this method requires only the mAs setting changes and, as a result, can be enabled without upgrading the legacy CBCT scanning devices where the desired sub-sampled scanning is not sup-ported. On the other hand, one of the main advantages of the reduced projection method is to relax the computational requirements for iterative CBCT reconstructions. Since the column dimension of the system matrix/projection operator A is proportional to the num-ber of projections, a reduced projection method minimizes the computational burden of each forward and backward projection operation. In this regard, continued enhancements of lower complexity iterative CBCT reconstruction algorithms such as GPBB-SFE are even more important if the reduced mAs method is desired for any reason. For the same reasons, combining the GPBB-SFE with the ABB algorithm (Zhou et al 2006) to further enhance the convergence speed or optimizing ABB algorithm for the CBCT reconstruction problem could be a very interesting follow-up research.

It is important to analyze the performance of GPBB-SFE to other related image recon-struction problems. Since the performance of a given iterative algorithm may vary from one problem to another (Fletcher 2001), the proposed GPBB-SFE may have different implications to the related problems such as various statistical image reconstruction problems (Thibault et al 2007, Tang et al 2009), 4DCBCT problems (Leng et al 2008, Kuntz et al 2013), etc. Among others, application to 4DCBCT problems would be very interesting and important.

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Due to the additional dimension in the 4DCBCT, multiple 3DCBCT images representing different phases are required to be reconstructed. This implies that the overall size of the reconstruction volume/number of unknowns is increased by a factor of P for reconstructing P phase 4DCBCT images (e.g. P = 10). For these even larger scale problems, there are stronger needs for algorithmic innovations such as GPBB-SFE along with continued improvements in the computation H/W performance (Pratx and Xing 2011). Furthermore, as many recent innovations in 4DCBCT incorporate advanced problem formulations to incorporate stronger prior information22,23 (Leng et al 2008, Kuntz et al 2013) and motion estimate information (Park et al 2013), evaluation and enhancement of the proposed GPBB-SFE for these problems would be very timely and important.

Finally, we remark that GPU hardware performance is rapidly being advanced. Although our current implementation is based on NVIDIA GTX 480 which has 448 processing cores, more recent GPU hardware offers even more massive parallel processing capabilities. For instance, NVIDIA GTX 690 has 3072 processing cores which can theoretically reduce the reconstruction speed by factor of ~1/6 with proper implementation optimizations. This implies that, when combined with more advanced GPU hardware and implementation, our GPBB-SFE algorithm would be able to complete a CBCT reconstruction in less than 30 s which should be very attractive for use in on-line IGRT applications.

5. Conclusion

In this paper, we proposed a novel, fast, smoothed TV based CBCT reconstruction algorithm using Barzilai–Borwein step size. The proposed algorithm exhibits theoretic convergence property while requiring at most one function evaluation in each iterative step. The major finding was that a clinically viable head-and-neck image can be obtained within ~132 s while simultaneously cutting the dose by one-half. Further reconstruction speed acceleration is anticipated with enhanced GPU hardware and implementation. This makes our GPBB-SFE algorithm potentially practical for use in on-line IGRT.

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