Accelerating CT Iterative Reconstruction Using ADMM and Nesterov s Methods

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1 Accelerating CT Iterative Reconstruction Using ADMM and Nesterov s Methods Jingyuan Chen Meng Wu Yuan Yao June 4, Introduction 1.1 CT Reconstruction X-ray Computed Tomography (CT) is a conventional medical imaging modality. Figure 1 shows the basic hardware configuration of CT: a circular gantry composed of x-ray source and multi-row detectors rotates around the patient and takes projection images per small incremental angel over 360 degrees. The underlying physics is the attenuation of x-ray intensity along the path due to the human tissue, and each detector pixel outputs line integrals of the attenuation coefficients. Therefore, to reconstruct a tomographic image from the raw projection data (a.k.a sinogram), the most straightforward yet less premium method is the inverse radon transform [THH + 05]. Cross-section raw data y Y. Yao Lateral view Reconstruction A 1 CT Image Y. Yao x Figure 1: L: CT scanner geometry, R: CT data collection and reconstruction 1

2 1.2 Penalized Weighted Least Squares (PWLS) Inverse radon transform is essentially a matrix inverse problem but direct least-squares solution typically yields noisy reconstructions. Alternatively, model-based statistical reconstruction methods has been proved to be a better approach in terms of image quality [TSBH07]. It assumes that the detection noise follows a Poisson distribution and the maximum likelihood solution to the model can be approximated as a penalized weighted least squares (PWLS) problem: minimize f(x) = 1 2 I w i (Ax y) 2 i + h(x), (1) i=1 where x R J is the image to reconstruct, y R I is the collected raw projection data, A R I J is the system matrix of CT, i.e. the forward projector that maps the image x to the sinogram y, w R I is the weighting vector computed from the variance of y (I and J are on the order of ). The penalty function of piecewise difference h(x) is h(x) = β J j=1 k S(j) Ψ(x j x k ), (2) where S(j) denotes the neighboring pixels of j, β is a tuning parameter, and Ψ(x) is the Huber penalty function defined as: { t δ/2, if t δ Ψ(t) = (3) t 2 /2δ, if t < δ. In the cost function (1), the weighted least-squares component enforces data consistency, which often produces a sharp but noisy image, while the penalty function applies smoothing constraints to the PWLS algorithm. These two functions are balanced by the tunning parameter β. Due to the enormous projection matrix and the dynamic range of the statistical weighting, there is no efficient way to solve the inner least-squares problem. The primary goal of this project is to implement advanced optimization algorithms learned from EE 364B and other literatures to accelerate the convergence speed of the PWLS solvers. 2 Methods 2.1 ADMM Alternating direction method of multiplier (ADMM) is a simple but powerful tool for solving large-scale optimization problem, and can be aptly applied to the proposed PWLS CT reconstruction [RF12, BPC + 10]. 2

3 With parameter substitution, the problem in (1) becomes: { A W 1/2 A y W 1/2 y, and can be modified by variable splitting as [NF14]: minimize subject to 1 2 z y h(x) Ax = z We further included Augmented Largrangian (AL) to solve the constrained minimization problem: L A (x, u, d; ρ) 1 2 z y h(x) + ρ 2 Ax z d 2 2, (6) where d and ρ > 0 are the corresponding AL penalty parameters. ADMM alternatively minimizes the objective functions with respect to x and z followed by a gradient descent of d: x 1) argmin{h(x) + ρ Ax 2 z(k) d (k) 2 2} x z 1) argmin{ 1 z 2 y ρ 2 Ax1) z d (k) 2 2} (7) z d 1) = d (k) Ax 1) + z 1). Due to the quadratic loss function, the z-update in (7) has a simple closed-form solution: z 1) = (4) (5) ρ ρ + 1 (Ax1) d (k) ) + 1 y, (8) ρ + 1 The x-update could be re-written as the proximal mapping of h: x 1) prox (ρ 1 t)h(x (k) ta (Ax (k) z (k) d (k) )). (9) Let s 1) denotes the search direction of the proximal gradient x-update in (9) and combining (8) with d-update of (7) yields s 1) ρa (Ax (k) z (k) d (k) ), (10) z 1) + ρd 1) = y. (11) Substituting (8) to (11) into (7) and rearranging it leads to the linearized ADMM iterations [NF14]: s 1) = ρa (Ax (k) y) + (1 ρ)g (k) x 1) prox (ρ t)h(x (k) (ρ 1 t)s 1) ) (12) 1 g 1) = ρ ρ+1 A (Ax 1) y) + 1 ρ+1 g(k), where g A (z y) is the backprojection of the split residual. The net computational complexity of (12) per iteration includes one multiplication by A, one multiplication by A, and one proximal mapping of h that can be solved iteratively without using A or A by Fast Iterative Shrinkage/Thresholding Algorithm (FISTA) [BT09]. 3

4 2.2 ADMM with Ordered Subset Let l(x) 1 z 2 Ax 2 2 denote the quadratic data-fitting term, we assume that l can be decomposed into M smaller quadratic functions l 1,..., l M using the Ordered Subset (OS) [EF99] of the projection data: l(x) M l 1 (x) M l M (x), (13) so that the subset gradients approximate the full gradient of l(x). The OS approach basically takes advantage of the data redundancy embedded in the CT system (the acquired data at adjacent angles has considerable overlaps, see Figure 1), and downsamples the projection operation A and the collected y to approximate the full data-fitting. Replacing l A (Ax (k) y) with M l m and incrementally performing (12) M times in a complete iteration, we reached the OS-accelerable ADMM updates (ADMM+OS): s M ) = ρm l m (x m M ) ) + (1 ρ)g m M ) x M ) ) M ) prox (ρ t)h(x m 1 M ) (ρ 1 t)s g M ) = ρ M l M ρ+1 (x ) ) + 1 ρ+1 g m M ). In the implementation, M sub-iterations require one complete projection (A) and backprojection (A ). 2.3 Nesterov s First-order Method Nesterov s fast first-order algorithm uses previous iterates to provide momentum towards the optimum. Combined with OS and separable quadratic surrogate (SQS) method, it has a very fast convergence rate [EF02]. Let C(x) 1 2 u y h(x) denote the overall cost function of the PWLS problem, the proposed approach minimizes equation at each updating step: (14) x (n+1) argmin{c(x (n) ) + C(x (n) ) (x x (n) ) + 1 z 2 x x(n) 2 D}, (15) where D is a precomputed diagonal matrix [EF02]. Initialized with x (0) = v (0) = z (0), t 0 = 1, the OS implementation scheme is as follows: t km+ = (1 + (1 + 4t 2 km+m ))/2 x M ) = [z m M) D 1 M Ψ m (z k+ m M )] + km+m x M ) = [z (0) D 1 t n M Ψ (n)m (z n M )]+ z M ) 1 = (1 t km+ )x n=0 M ) 1 + t km+ v M ). (16) 4

5 3 Simulations The CT data was simulated based on a numerical thorax phantom as collected from the GE LightSpeed TM multi-slice CT system. The size of the projections data was , and the reconstructed image was pixels. The performance of the proposed methods was compared to other state-of-art optimization solvers such as gradient descent (GD+OS), preconditioned gradient descent (PGD+OS), both of which were accelerated by OS [KPTF13]. The high-level algorithms were written in MATLAB TM and the projection/backprojection was implemented in C. The numbers of the ordered subsets were 10 for the ADMM+OS and Nesterov+OS algorithms, and 20 for the GD+OS and PGD+OS algorithms. 4 Results Figure 2 shows the reconstructed image using filtered back projection (FBP), a commonly used analytical reconstruction approach, and various iterative implementations of PWLS solvers. The reconstructed images using the PLWS method has less streak artifact than the FBP reconstructed image. The second row shows the differences between the images using proposed algorithms after 30 iterations and converged PWLS reconstruction. The ADMM+OS and Nesterov+OS provide more closed PWLS solution than the ADMM algorithm in (7). (a) FBP (b) ADMM (c) ADMM+OS (d) Nesterov+OS (e) Converged (f) Diff ADMM (g) Diff ADMM+OS (h) Diff Nesterov+OS Figure 2: (a)-(d): Reconstructed images using different algorithms, (e) converged result using Nesterov+OS after 500 iterations, (f)-(h): difference images of (b)-(d) and (e). 5

6 GD+OS PGD+OS Nesterov+OS ADMM ADMM+OS GD+OS PGD+OS Nesterov+OS ADMM ADMM+OS f(x (k) ) 10 8 x k x * / J k (a) k (b) Figure 3: Convergence speed of various implemented approaches. (a) Object function v.s. No. of iterations; (b) RMS differences with the converged PWLS Figure 3 shows the convergence speed of different PWLS solvers. For all the solvers, each iteration involves one complete projection and back-projection. The ADMM+OS is the fastest algorithm in the first 10 iterations, and the Nesterov+OS becomes the fastest afterward. 5 Discussion and Conclusion In this pedagogical project, we ve implemented and compared various approaches to accelerate the convergence speed of iterative CT reconstruction. In the real data simulation, all the adopted methods achieved acceptable results, and the convergence speed was proved to be further reduced by OS. Among the tested algorithms, Nesterov+OS was the fastest one while ADMM+OS had similar performance but was more fluctuating, which might be attributable to the alternating scheme of ADMM, and may likely be optimized by proper choices of step size and the tuning parameters in AL. Acknowledgement The authors gratefully acknowledge professor Steve Boyd and the TAs for valuable discussions on the project, as well as all the peer reviewers for their feedbacks that have substantially improved the quality of our work. References [BPC + 10] Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein. Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Found. Trends Mach. Learn., 3(1):1 122,

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