Fast Implementation of Iterative Image Reconstruction

Transcription

1 Fast Implementation of Iterative Image Reconstruction Tobias Schaeffter 1, Michael Schacht Hansen 2, Thomas Sangild Sørensen 3 1 Division of Imaging Sciences and Biomedical Engineering, Kings College London 2 National Heart, Lung, and Blood Institute, NIH, Bethesda, MD, USA 3 Department of Computer Science and Institute of Clinical Medicine, Aarhus University Background Iterative reconstruction is becoming an increasingly important topic in medical imaging. In computed tomography (CT) iterative reconstruction algorithms are becoming standard and they are implemented on current CT-scanners to reduce radiation dose [1]. This is not a new trend; the very first CT machines used an iterative technique, the algebraic reconstruction technique (ART) [2]. In emission tomography in nuclear medicine iterative reconstructions were used in the early days to integrate prior knowledge about the noise statistics into the reconstruction [3] and more advanced iterative algorithms have become a standard in nuclear medicine [4]. Also in MRI there is a trend for advanced reconstruction algorithms to integrate knowledge about coil sensitivities in parallel MRI [5], about motion [6-9] and other prior information into the reconstruction process. All these approaches result in large set of linear equations Ax=b, which need to be solved. A direct inversion of this equation system is impractical, except in the most trivial cases, since it requires a large amount of memory and a prohibitive number of operations. However, this complexity can be reduced by using iterative reconstruction algorithms, where direct inversion of the system matrix is not needed. These algorithms start with an initial guess for the solution and then iteratively progress towards an approximate solution. A variety of such techniques exist for the treatment of large linear systems. For many cases, the so-called conjugate gradient (CG) method [11] particularly suited. It converges to a minimum in N 2 steps (where N is the rank of the matrix) if the matrix A is symmetric positive definite. The rate of convergence can be improved if preconditioning of the matrix A is performed. However, also in well-conditioned systems still around iterations need to be performed. Therefore one of the main limitations of introducing iterative algorithms in image reconstruction has been the long computation time so far hindering the clinical translations of many new reconstruction algorithms. Modern algorithms can make use of acceleration techniques and parallelization to speed up the reconstruction. These techniques make use of the on-going developments in computer processors (CPU) and especially in commodity graphics hardware (GPU). General purpose computation on the GPU is still an emerging research area, but many algorithms have been accelerated significantly compared to their CPU counterparts [12, 13]. One of the most recent GPUs, the Geforce 560 (Nvidia, US), has 352 stream processors for these computations allowing significant acceleration if the computational problem can be solved in parallel. More importantly, both major hardware manufacturers (AMD and Nvidia, US) have released high-level programming environments requiring less knowledge of graphics programming in general. The GPU implementation of MR-reconstruction algorithms has been demonstrated for a number of applications [14-20]. In this presentation the GPU-implementation of an iterative reconstruction algorithm is presented for the example of non-cartesian parallel imaging. However, the basic concepts can be applied to other applications.

2 Parallel Imaging using radial trajectories Non-uniform sampling schemes are more and more used in MR-scanning since they are often more robust against motion and flow. In particular, radial imaging results in more acceptable artifact due to residual motion and changes of the imaging plane during interactive scanning [22]. Furthermore, for radial acquisitions the central part of k-space is fully sampled, which can be used directly for estimating the coil sensitivity and regularization [9]. Non-Cartesian sampling can be combined with parallel imaging techniques such as SENSE to reconstruct images from highly undersampled data [23]. In parallel imaging the data are acquired simultaneously by multiple receiver coils with unique spatial sensitivity profiles S c (r). These profiles encode the unknown image v(r) such that a measured complex value m c (k) sampled by the c-th coil in k-space by: m c (k) = v(r) S c (r) e ikr dr (1) For the discrete case this equation can be written as the linear system of equations: m = Ev (2) using the encoding matrix E: E (γ,κ ),ρ = S γ, (r ρ )e ik κr ρ (3) After noise-decorrelation of the data m one solution of this system is given in ref. 23 by: ( E H E)v = E H m (4) where H is the conjugate transpose of the matrix. In order to solve this equation system (4) by an iterative conjugate gradient optimization, in each step of the iteration multiplications of the encoding matrices E and E H with some temporary vectors have to be performed. These operations are given by equation (3), which are multiplications with the coil sensitivity and Fourier transforms from k-space to image space and vice versa. However, since a non-uniform sampling is used a gridding procedure (including density correction) in combination with a Fast Fourier Transform (FFT) can be used. In Figure 1 it is shown that in each iteration of the conjugate gradient algorithm the residuum image of each coil element undergoes subsequent operations in the image domain and in k-space: re-gridding in k-space from non-uniform to uniform sampling Fast Fourier Transform multiplication with the coil sensitivity (and its conjugate) (inverse) Fast Fourier Transform gridding from uniform to non-uniform sampling

3 Figure 1: Iterative reconstruction using a conjugate gradient algorithm: S coil sensitivity; FFT Fast Fourier Transform; GRID regridding from non-cartesian to cartesian grid or vice versa; Figure adapted from ref. (5) From a computational load perspective most of the operations are simple multiplications and thus the gridding operations and the FFTs are the more demanding. A disadvantage of the proposed algorithm is that the convergence speed depends on how the encoding matrix is conditioned. Recently, a reconstruction algorithm based on Tikhonov regularization has been proposed that reduces the SNR loss due to geometric correlations in the spatial information [24]. The Tikhonov regularization provides a framework for stabilizing the solution of ill-conditioned linear equations by solving an equation system with a least squared estimate: 2 v λ = argmin Ev m 2 + λ 2 v 1 2 { regv 2 } (5) With v reg the inverse of a reference image for regularization and λ the regularization parameter. This approach results in the following equation system [16]: D E H ϒE + λ 2 2 ( v reg )D(D 1 v) = DE H ϒm (6) This equation system is similar to (4) but includes a reference image as a priori information for regularization. Furthermore, the matrix D is a diagonal preconditioning matrix aiming to equalize the diagonal elements of the system matrix on the left side. ϒ is another diagonal matrix that performs density compensation to account the non-uniform sampling in k-space. It should be noted that a similar system needs to be solved for k-t SENSE using a training image data set for regularization. Again in order to solve the equation system by a conjugate gradient optimization, for each iteration different vector matrix multiplications have to be performed. In particular the multiplication of the current search vector with the encoding matrix includes the multiplication the coil sensitivity matrix S followed by a non-equispaced Fourier transform from image space to k-space for each coil, resulting in distinct signals. The conjugate transpose of the encoding matrix E H, consists of a nonequispaced Fourier transform from k-space to image space followed by a

4 multiplication of the complex conjugate of the coil matrix for each coil. Since these computational expensive operations have to be performed for each iteration step, they need be optimized by an efficient implementation. Implementation of Non- equidistant Fourier transform on GPU A description of the implementation and an evaluation of the GPU-based nonequidistant fast Fourier transform (NFFT or gridding) can be found in references [14,16]. Here we summarize the fundamental considerations. The NFFT consists of three steps, namely 1) convolution, 2) FFT, and 3) deapodization. The deapodization step is merely element-wise multiplication of two vectors and thus trivial to implement in parallel. An efficient GPU implementation of the FFT is typically provided by the programming API (e.g. CUDA) and therefore also straightforward to apply. The implementation of the convolution operation on the other hand needs careful consideration for a data-parallel implementation. First, let us consider how a typical sequential CPU implementation would proceed: For both the NFFT and NFFT H implementation we iterate over the acquired samples one by one. For every sample we then iterative over all grid cells within the range of the convolution kernel and compute each convolution contribution. For the NFFT H this contribution is added to the resulting grid cells one by one within the inner loop. For the NFFT on the other hand the contributions of the inner loop are accumulated within the loop and written to the corresponding index in the resulting samples array. For a data-parallel approach these operations must take place in parallel. For the forwards convolution (the NFFT) this is straightforward; we can start a thread for each sample and have each thread perform the corresponding inner loop of the sequential approach. The NFFT H convolution is unfortunately more problematic. If we again parallelized by starting a thread per sample and each sample performs the inner loop of the sequential approach we encounter write conflicts (and undefined behaviour) as multiple threads attempt to update the same Cartesian grid cell in parallel. The solution, in sequential terms, is to interchange the order of the two loops. Correspondingly, in parallel terms, we invoke a thread for each grid cell. Each thread then accumulates the contribution from the samples within the radius of the convolution kernel. This strategy imposes a new challenge however; how to determine (in constant time) which samples that fall within the convolution radius of each cell? We refer the reader to reference [16] for a description of an appropriate pre-processing step. To maximize the performance of the GPU implementation of both the NFFT and NFFT H it is crucial to reduce the obvious bottleneck: the required memory bandwidth. Several strategies can be combined: 1) Use online computation of the convolution kernel instead of predefined lookup tables. Much of the cost of the computation can be hidden by the memory lookup of the corresponding sample or cell. 2) Use on-chip memory to hold accumulation variables and avoid unnecessary writes to main memory. References [2xTMI] describe a strategy in which each thread accumulates the contribution from multiple threads or cells. This reduces memory bandwidth requirements at the expense of parallelism as fewer threads are launched. For small image sizes the reduction in the number of concurrent threads launched is undesired. For parallel MRI one can consider to instead use the sparse on-chip memory to hold the convolution results from the multiple coils.

5 Applications There is wide range of applications that benefit from fast reconstruction. However, reconstructions using the GPU is not always possible due to memory constrains. In the presentation two different applications are demonstrated that benefit from fast reconstruction: 1) Real-time 2D imaging with real-time reconstruction. For this reconstruction computer is connected directly to a streaming output with of a scanner. 2) Offline reconstruction of 3D data that can be separated into a number of independent 2D iterative reconstruction problems. For the real-time cardiac MR-application data was acquired on a Siemens Avanto 1.5 T system using 4 coil-elements. A radial steady state free precession (SSFP) pulse sequence (FOV=300mm, PBW=1500Hz, Flip-angle=60, TR=2.5ms TE=1.2ms) was used to sample projections (each 256 samples) with a golden angular spacing of Figure 2 shows a real-time reconstruction of cardiac function obtained in a clinically referred pediatric patient during free breathing. For each reconstructed frame either 34 or 50 projections were used and a moderate regularization (λ 2 =0.05) was applied. A fixed number of iterations (15 and 20) in the iterative SENSE algorithm ensured that the GPU-reconstruction of all channels was always faster than data acquisition. The temporal fidelity of the reconstruction is demonstrated in a separated plot of one line spatial line over time. Figure 2: Real-time frame of radial SSFP acquisition. The GPU implementation of iterative SENSE reconstruction allows real-time reconstruction, i.e. faster than the data acquisition. a) Sliding window reconstruction using 34 projections and 15 iterations to update an image frame takes 83ms (< 85ms acquisition time). b) Sliding window reconstruction using 50 projections and 20 iterations to update an image frame takes 120ms (125ms acquisition time). For the 3D reconstruction example a radial phase encoding data reconstruction was tested. This encoding scheme combines Cartesian sampling along the readout direction with a radial sampling scheme in the phase encoding plane was suggested [25]. In particular, to reconstruct the 3D volume, initially the fast Fourier Transform (FFT) is applied in the fully sampled readout direction, converting the remaining reconstruction problem into a series (i.e. umber of samples along the readout directions) of two-dimensional radial reconstructions. Cardiac-triggered SSFP imaging (90 flip angle, TR/TE=5.0 ms/2.28 ms) with radial phase encoding was performed during free breathing (6 mm navigator window) to acquire a 3D dataset in end diastole using a resolution of mm 3 (in 4 minutes assuming 100% navigator efficiency) on a 1.5T system (Philips Achieva). The dataset contains 64*128=8192 lines of k-space data using Cartesian sampling in the readout (feet-head) direction (32 receive channels). In the phase encoding plane these were

6 laid out as 64 radial profiles of 256 samples (undersampling 2) each. Data were obtained on Iterative SENSE is required to resolve undersampling along the radial (Rr=2) and angular direction (Ra=4). Usually, this reconstruction is very time consuming; and reconstruction time of 3 minutes/slice using 32 coils was reported for a Matlab implementation resulting in total reconstruction time more than 12 hours [25]. Therefore the performance of GPU-implementation of the iterative SENSE reconstruction was tested. For this each 2D radial iterative SENSE reconstruction was performed on the GPU adapting parts of the implementation using a target matrix size of 256x256 for 256 slices. Furthermore, the coil sensitivity maps were acquired in a separate reference scan and no explicit regularization term was included in the reconstructions. We used 10 iterations of the conjugate gradient solver. The data was reconstructed on an Nvidia Geforce 280 GTX GPU with 1GB of memory. Figure 3 shows three reformatted slices (transversal, coronal, and sagittal view) from the three reconstructed, isotropic 3D volumes. The reconstruction time was reduced to 1.2s per slice. These numbers include data transfers from disc taking roughly 0.5s. In total, the full 3D datasets (256 slices) were each reconstructed in 5 minutes. Figure 3: Multiplanar reformatted slices of a 3D whole heart dataset using an iterative SENSE reconstruction. The GPU-implementation reduced the reconstruction per slice to 1.2 seconds. Conclusion Iterative reconstruction algorithms are becoming increasingly important in advanced MR-acquisition schemes. However, the clinical application is often hindered by long reconstruction times. In particular, the performance growth of individual CPU cores has stagnated and is unlikely to improve notably in the future. The iterative SENSE algorithm presented in this presentation is good candidate for implementation on GPU. This concept can be applied to other iterative algorithms making the application of advanced reconstructions feasible in clinical applications. References [1] Fleischmann D, Boas FE. Computed tomography-old ideas and new technology. Eur Radiol. 2011; 21(3): [2] Macovski A, Herman GT. Principles of reconstruction algorithms. In Newton TH, Potts DG (eds) Radiology of the skull and brain: technical aspects of computed tomography. Mosby, Saint Louis, MO, 1981; [3] Rockmore AJ, Macovski A. A maximum likelihood approach to emission image reconstruction from projections. IEEE Trans Nucl Sci : [4] Vandenberghe S, D'Asseler Y, Van de Walle R, Kauppinen T, Koole M, Bouwens L, Van Laere K,

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