Informatics in Medicine Unlocked 8 (2017) 21–31
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Informatics in Medicine Unlocked journal homepage: www.elsevier.com/locate/imu
A data-parallelism approach for PSO-ANN based medical image reconstruction on a multi-core system
MARK
⁎
Subramanian Kartheeswaran , Daniel Dharmaraj Christopher Durairaj Research Centre in Computer Science, V.H.N.S.N College (Autonomous), Virudhunagar 626001, Tamil Nadu, India
A R T I C L E I N F O
A BS T RAC T
Keywords: Image reconstruction Filtered back projection Artificial neural networks Particle swarm optimization Multi-core processors
This paper presents the sequential and parallel data decomposition strategies implemented on a Particle Swarm Optimization (PSO) algorithm based Artificial Neural Network (PSO-ANN) weights optimization for image reconstruction. The application system is developed for the reconstruction of two-dimensional spatial standard Computed Tomography (CT) phantom images. It is running on a multi-core computer by varying the number of cores. The feed forward ANN initializes the weight between the ‘ideal’ images that are reconstructed using filtered back projection (FBP) technique and the corresponding projection data of CT phantom. In an earlier work, ANN training time is too long. Hence, we propose that the ANN exemplar datasets are decomposed into subsets. Using these subsets, artificial sub neural nets (subnets) are initialized and each subnet initial weights are optimized using PSO. Consequently, it was observed that the sequential approach of the proposed method consumes more training time. Hence the parallel strategy is attempted to reduce the computational training time. The parallel approach is further explored for image reconstruction from ‘noisy’ and ‘limited-angle’ datasets also.
1. Introduction The complexity of image reconstruction has acquired much concentration in the medical imaging literature. This is due to the continuous search for developments of imaging modalities, varying from X-ray computerized tomography and emission tomography up to acoustic and optical techniques. They all bring different approaches in the human body either morphological or functional. The classic mathematical model of X-ray computerized tomography (CT) assumes that the sensing device evaluates the line integrals of the object attenuation coefficient at some known orientations [1]. As a typical inverse problem, tomography image reconstruction is usually considered as an ill-posed problem [2]. The tomography image reconstruction of an image is formed of deciding an image object f (x, y) from a collection of projections [3] pθ (r ) given by Eq. (1). ∞ ∞
pθ (r ) =
∫∫ −∞ −∞
f (x, y) δ (r −x cos θ−y sin θ ) dxdy. (1)
Where r is taken on the x–y plane such that r =x cos θ−y sin θ . The discrete inverse radon transform is carried out with the Filtered Back Projection (FBP) algorithm. FBP is established on the basis of Fourier slice theorem [4], Fast Fourier summation algorithm for image
⁎
reconstruction [5]. The excellence of reconstruction from entire projections with FBP is widely acceptable, based on the number of angles and sampling points. The projections pθ (r ), accumulated along a group of static fieldgradient orientations in polar grid, are utilized to acquire the sample spin density f (x, y) by Eq. (2) FBP, π
f (x , y )
∫ 0
π
pθ* (r ) dθ =
∞
∫ [∫ 0
Pθ (k ) | k | e−2πikr dk ] dθ ,
−∞
(2)
Here pθ* (r ) is the projection Pθ (k ) filtered by the expression inside the square brackets. The important reconstruction methods such as FBP [6] is convincingly effective and fast, but their quality is low when the projection data are noisy [7,8] and limited [1,9]. An Algebraic Reconstruction Technique (ART) [10–12], Simultaneous ART (SART) [13], Simultaneous Iterative Reconstruction Technique (SIRT) [14] are shows better resolution for medical imaging in real time [9]. However, these techniques are suffered with enormous computational complexity [15], that is corresponding to the square of the image size multiplied by the number of projections [10]. In the framework of soft computing strategies to image reconstruction from projections, artificial neural networks have been utilized as a very popular and important
Correspondance to: Assistant Professor, M.C.A Department, Kalasalingam University, Krishnankoil - 626126, Tamil Nadu, India E-mail addresses:
[email protected] (S. Kartheeswaran),
[email protected] (D.D. Christopher Durairaj).
http://dx.doi.org/10.1016/j.imu.2017.05.001 Received 10 January 2017; Received in revised form 30 January 2017; Accepted 2 May 2017 Available online 17 May 2017 2352-9148/ © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
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projections, coverage angle ranging from 0 to 180º with an incremental value of 10º. The number of projection is proportional to the quality of FBP image. The head image (Fig. 2a) 64 x 64, (Fig. 2c) 128 x 128 reconstructed by FBP with 180 projections are utilized for PSO-ANN. The projections of phantom related to the size of 64 x 64 (Fig. 3a), and 128 x 128 (Fig. 3b). FBP has been realistically useful, but the quality is low when the projection data are noisy and limited [9]. Projection noise crucially limits the capability of a radiologist to categorize between two regions of special spin density and hence to find the talent of the presence of attributes of interest in medical images. Such random noise is simulated either as additive signal reliant noise. Such noise is simulated and added either as additive signal reliant noise, given by pθ (r ) =s + n or multiplicative noise is, specified by pθ (r ) =s ∗ n. Various levels of random noises are added to the projection of the phantom to resemble the noise-added projection data. The noise-added projections (Fig. 3c and d) with FBP based reconstruction data produce distorted images shown in Fig. 4a and b. Many training exemplars are constructed with ‘noise-added’ projection dataset (Fig. 3c, d) and ‘ideal’ target images (Fig. 2a, c). The noise-added projection data and the FBP images are used to construct (Fig. 5) an exemplar dataset. The input training data set comprised 95 sample points of 18 projections. The Shepp-Logan image [64 * 64 pixels] reconstructed by FBP method using these 180 projections are presented to the network as the ‘ideal’ target output. Exemplars are constructed with ‘noise-free’, and ‘noiseadded’ projection dataset. The PSO-ANN has topology with 1710 nodes in the input layer, representing the 18 ‘noise-added’ projections, each with 95 samples scaled uniformly between 0 and 1. The output layer has 4096 nodes, representing FBP image of size 64 ∗ 64 pixels. The FBP images with 180 projections are used as output to exemplar dataset needed for the present work. On the whole of an exemplar creation (Fig. 5), input layer (1710 neurons) represent the 18 projections and output layer (4096 neurons) represent the FBP with 180 projections based reconstructed (64x 64 pixels) spatial image. similarly another type of an exemplar creation for 128 x 128 phantom, input layer (3330 neurons) represents the 18 projections each with 185 samples and output layer (16384 neurons) represents the image of size 128 x 128. Biomedical imaging with long acquisition time and large number of projection data are not chosen for instantaneous studies, because the system under testing has to expend a long time within the imager. Moreover, it can’t be feasible to obtain such a large amount of projection data due to the biological clearance of the imaging agent. In addition, the PSO-ANN training system is constructed with limited-angle projection datasets and the ‘ideal’ target images reconstructed by FBP. The dataset for reconstruction from limited number of projections consisted of nine projections collected at 20◦ apart each. Hence, the structure of the neural network has only 855 nodes in the input layer representing the nine projections each with 95 samples, scaled uniformly between 0 and 1 and 4096 nodes in the output layer forming the two-dimensional spatial image of size 64 x 64 pixels. The reconstruction by FBP images with limited angle projection produces highly distorted and unknown information of image shows in Fig. 4c and d. The next section deals with the sequential and parallel PSO-ANN methodology of the present work.
tool [16]. In an earlier study recurrent neural network algorithm has been utilized for image reconstruction [17], fast tomography reconstruction from limited data has been reported using ANN [14] and the reconstruction technique using BP-ANN has been implemented on a sequential computer [9], which require ANN very long training time. BP-ANN uses the gradient based approach which either trains slowly or get struck with local minimum. Instead of using gradientbased learning techniques, one may apply the commonly used optimization methods such as Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Simulated Annealing (SA), Ant Colony Optimization (ACO) to optimize the network initial weights. The PSO algorithm with artificial neural network (PSO-ANN) has offered fine answers to many problems in biomedical science. PSO is to regulate the weight of a BPANN has a better performance than random search. The particle swarm optimization algorithm is shows to converge global optimum. So in this paper, a hybrid algorithm combining particle swarm optimization (PSO) with back-propagation (BP) algorithm is utilized. PSO has been used with BP-ANN for optimizing the various parameters such as number of hidden nodes, hidden layer sizes, feature subsets, learning rate, momentum, and optimize the network connection weights. This paper presents the application of hybrid model that integrates PSO and BP-ANN (PSO-ANN) for reconstruction of shepp-logan head phantom image by optimizing the connection weights, which require PSO-ANN very long training time. The parallel approach to ANN has been explored in a variety of manner, specifically, training session parallelism, exemplar parallelism, node parallelism, and weight parallelism [18]. Parallel approach of BP-ANN has been exposed to be an efficient resolution to all cases of long training times in sequential ANN training on a cluster computer [19–22]. The training datasets are decomposed into a number of subsets and the subsets are allocated to different computing nodes for parallel processing [23] on a cluster computer [24]. As an extension of the recent work carried out on a sequential computer [9], the present work deals with parallel approach of PSO-ANN decomposition principle. The next section describes standard dataset for the sequential and parallel PSO-ANN. The Section 3 deals with the methodology of the parallel approach of the PSO-ANN. The Section 4 explains the design and implementation details. The results are discussed in Section 5 and concluded in Section 6. 2. Exemplar datasets Initially, the Shepp–Logan head phantom [4,25] has been utilized for testing purposes like CT, MRI, fMRI, PET and SPECT [26]. It is widely used in medical imaging and is the sum of 10 ellipses of varying size and orientation. [27]. The standard shepp-logan head images 64 x 64 (Fig. 1a) and 128 x 128 (Fig. 1b) are bring into play for the present work. FBP is most commonly used algorithm for medical image reconstruction crisis. In fact, the image produced by filtered back projection is identical to the correct image when there are an infinite number of views and an infinite number of points per view. Fig. .2b and d shows reconstruction of phantom head model by FBP with 18
3. Methodology 3.1. Sequential PSO-ANN training for phantom image reconstruction The phantom image reconstruction system is based on a supervised, feed-forward, fully connected artificial neural network which already exists [28–30].
net =
∑ wv Nd v = wT Nd i
Fig. 1. The 2-dimensional standard shepp-logan head phantom images of size 64 x 64 (a), 128 x 128 (b).
(3)
Here Nd1, Nd2, ... Ndn are ANN’s inputs. The w1, w2, ...wn are 22
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Fig. 2. The FBP based reconstruction with 180 projections (a) and (c), with 18 projections (b) and (d).
Fig. 3. The projections of shepp-logan phantom for reconstruction of images in Fig. 1. and the corresponding ‘Noise-added’ projections.
Fig. 4. The results of FBP based reconstruction with ‘noise-added’ projections (a and b) and ‘limited-angle’ projections (c and d).
interconnection weights of the net. We shall indicate Nd=[Nd1, Nd2, ..., Ndn]T input vector, w=[w1, w2, ..., wn]T synaptic connections vector. The output of the ANN can be written in the form of y = f (net-1) = f (wTNd-1). The f is a transfer function of the ANN.
Nd =
pθ (r ) − min( pθ (r )) max( pθ (r )) − min( pθ (r ))
(4)
The projection data pθ (r ) and the corresponding FBP based spatial image f(x, y) are normalized using Eq. (4) and arranged into training exemplars. The datasets are normalized (Nd) uniformly in the ranges between 0 and 1. The size of input projection data pθ (r ) is 1710 * Nd, where ‘Nd’ is the number of normalized sinograms and each sinogram has 18 projections with 95 points. The size of desired output image data are 4096 * Nf, where ‘Nf’ is the number of normalized FBP images and each image has 64 x 64 pixels. The exemplar dataset (D) is constructed from the projection data and FBP image using Eq. (5).
3.1.1. Data normalization Data transformation such as normalization is a data preprocessing method applied earlier in training process. The method for normalization uses the min-max normalization [31]. It carries out a linear transformation on the projection data pθ (r ). The simplified method of min-max normalization using Eq. (4) preserves the relationship among the projection data pθ (r ). 23
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Fig. 5. Exemplar construction for the sequential and parallel PSO-ANN systems.
(5)
D = Nd ⊕ Nf
accurate with respect to the training set t. Then, E (w) = 0; or, homogeneously,
In the present work, BP-ANN with particle swarm optimization algorithm is utilized for training to enable quick convergence. In most of these studies, however, PSO is only used to improve the learning algorithm itself. In this study, PSO is employed not only to improve the learning algorithm, but also to reduce the complexity of weight optimization. The PSO –based BP-ANN structure used in the work. The three-layer network structure often utilizes the log-sigmoid transfer function for BP-ANN weight initialization.
H(Ndt , w*) = Y; t t = 1, 2, 3, …, T
To professionally clarify the global optimization crisis (Eq. (7)), and in this manner train the PSO-ANN to reconstruct the development of phantom image accurately, we utilize PSO, which is a time-tested global optimize so effectively utilized by nature for the evolution of groups. The fitness function (f. f) is taken as
3.1.2. PSO based BP-ANN weight optimization The particle swarm optimization algorithm is to optimize the initial weights and thresholds of the network and makes the optimized BPANN better predict the output. The data processed by the PSO are a set (population) of strings. Population initialization is real-coded and each individual is a real string which consists of connection weights of the input layer and the hidden layer, the threshold of hidden layer, connection weights of hidden layer and output layer, and the threshold of output layer. The individual includes all weights and thresholds of the neural network. Every string has its individual fitness value decided by an objective function (MSE) value of the equivalent point in a search space. Based on fitness values, the PSO chooses the probability of every string in the present generation to be processed by initiating additional deviation to the population, it tolerates the search to continue without losing diversity.
E (w ) =
1 T
(8)
f. f =
1 E (w )
(9)
Where, E (w) is given by Eq. (6). The PSO continues to maximize above fitness function (f. f) resulting in minimization of mean square output error. In this study, the BP-ANN initial weight matrices are optimized by PSO. The population size probability parameters are shown in Table 1. An suitable range of PSO constraints like population size probability, has been originate by multiple model runs for the image reconstruction problem. 3.2. Sequential PSO-ANN with decomposed datasets for phantom image reconstruction There is scope for further improvement in the arrangement of large datasets, such as those used for reconstruction purposes, without
T
∑ ||Rt (w)||22
Table 1 The weight optimization parameters for sequential PSO-ANN.
(6)
t =1
This is our objective function. The subscript 2 in expression (6) indicates the information that we are using L2 or Euclidean norm to form the calculation error. The supervised learning problem for our PSO-ANN model then becomes:
min E (w )
PSO-ANN Parameters Input Hidden Output Transfer functions Error tolerance value Dataset decomposition level Population Size Generation
(7)
w
That is, minimize E(w), find an finest weight vector w that globally minimizes E(w) above the training set. If N includes enough number of hidden units to grow the essential interior representation, w* is *
24
18 x 95 (Acquired Projections) 175 Neurons 64 x 64 (FBP Image) pixels Log-sigmoid e-08 1, 2, 4, 8 30,60,90 80-200
Informatics in Medicine Unlocked 8 (2017) 21–31
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Table 2 Details of exemplar decomposition levels for projections data ( pθ (r )) , FBP images f(x, y), exemplar (D), and 8 cores (C). Projections Data pθ (r ) {1710 * number of sinograms}
FBP Images f(x, y) {4096* number FBP images}
Exemplar {D}
PSO-AN Network
Multi-Core Processor
{(1-214)*n1} {(215-428)*n2} {(429-642)*n3} {(643-856)*n4} {(857-1070)*n5} {(1071-1284)*n6} {(1285-1498)*n7} {(1499-1710)*n8}
{(1-512)*n1} {(513-1024)*n2} {(1025-1536)*n3} {(1537-2048)*n4} {(2049-2560)*n5} {(2561-3072)*n6} {(3073-3584)*n7} {(3585-4096)*n8}
{subset1} {subset2} { subset3} { subset4} { subset5} { subset6} { subset7} { subset8}
{psoannet1} {psoannet2} {psoannet3} {psoannet4} {psoannet5} {psoannet6} {psoannet7} {psoannet8}
{c1} {c2} {c3} {c4} {c5} {c6} {c7} {c8}
be applied using parallel approach on multi-core system.
compromising the achieved accuracy. Dataset decomposition is considered as a realistic solution to solve such data intensive problems [24]. Therefore, in the present work, exemplar datasets could be decomposed into small sizes of subsets. The exemplar datasets (D) are decomposed into d1, d2, and d3 … d8, where 8 is number of subset values (Table 2). Data decomposition principle is based on dividing an exemplar dataset into subsets. Let ‘D’ be a data-parallel problem. If ‘D’ is parallelizable, then D can be decomposed into n sub-problems in Eq. (10). Where n is level of decomposition.
d1 + d2 + …dn = D
n
∑ f (D) = f (d1) + f (d2) + … + f (dn ) i =1
(11)
Where d1, d2, … dn are the decomposition level and D is dataset. A parallel approach is proposed to the existing function of PSO-ANN learning algorithm based phantom image reconstruction system. We focus on implementing the following parameters to be considered to improve the system. The proposed parallel model is in the structure SIMD (Single-Instruction-Multiple-Data) instead of SISD (SingleInstruction-Single-Data) used in the classical versions of PSO-ANN algorithms. The subnet based PSO-ANN training function is called in eight times and, which uses task parallelism to completely harness the processing ability suggested by an eight-core processors. While having a parallel approach of PSO-ANN data decomposition problem complexity of O (n log2 n) [37, 38], which is efficient than O (n log n) ANN algorithms like PSO-ANN. Parallel strategy of PSO-ANN is a parallel weight optimized algorithm because the order of its compare and substitution operations is not data dependent. Here the n is number of decomposition level running on multi-core system. The batch processing is utilized because it assists quicker convergence in large dataset. The ‘matlabpool’ sets up a parallel implementation environment in which ‘parfor’ loops can be executed interactively. The ‘parfor’ loop is useful in conditions that necessitate many loop iterations of an effortless calculation, such as PSO-ANN training task. The iterations of ‘parfor’ loop are executed on labs. A lab is an independent instance of MATLAB that runs in a separate operating system process. The Parallel Computing Toolbox makes ‘parfor’ work well on a multi-core system in a shared memory environment. The subsets of the exemplars are trained in a parallel manner on a multi-core system. After completion of parallel training process, 8 set of trained sub networks are produces.
(10)
Depending upon the size of the subset of data, the structure of the feed-forward sub neural network is determined and constructed. Exemplar datasets are decomposed into eight exemplar subsets. The subnets are constructed with subsets and eight subnets as given in Table 2. Correspondingly the decomposed subnet weights are optimized and a final weight matrix is formed to produce a part of the image. Similarly, when all subnet training is completed, the whole image is obtained during testing phase. In the present work, exemplar datasets are decomposed into 2, 4, and 8 subsets and corresponding subnets are formed. The next subsection deals with a parallel strategy that will be applied on the PSO-ANN system with an intension to improve the speed up performance. 3.3. Parallel data decomposition based PSO-ANN on multi-core systems In an earlier work, it has been described that as the ANN data sizes are too large, the training time is too long [18]. Hence, in the present work, we propose that the ANN datasets are to be decomposed into subsets. Using these subsets, artificial sub neural nets (subnets) are initialized and each subnet initial weights are optimized using PSO. Consequently, it was observed that the sequential approach of the proposed method consumes more training time. Hence the parallel strategy is applied to reduce the computational training time. The parallel decomposition approach is able to simplify reconstruction tasks and is seen improving efficiently. Multi-core systems have a number of processing cores integrated on to a single chip [32]. In general, the processing cores have their individual private L1 cache and share a common L2 cache. In such architecture, the bandwidth between the L2 cache and main memory is shared by all the processing cores. The simplified network architecture utilized in the present work. In traditional programming methods, a large datasets are processed on a single CPU core, while the other CPU cores remain idle. In this scenario, the remaining CPU cores available are idle while the first processor solely bears the load of PSO-ANN based image reconstruction processing the entire dataset. Now, consider the structure, which uses data parallelism to fully harness the processing power offered by an eight-core processor. The technique takes advantage of multi-core processor to exploit data parallelism. The data decomposition principle given in Eq. (15) can
3.3.1. Reconstruction from ‘noise-added’ projections Many training exemplars are constructed with ‘noise-added’ projection datasets and ‘ideal’ target images that are generated by FBP using the ‘noise-free’ acquired projection data. The neural network has a similar structure. The parallel approach of PSO-ANN systems are trained and tested for reconstruction of phantom images also from ‘noise-added’ projections data and its response is discussed in Section 5.2. 3.3.2. Reconstruction from limited-angle projections The ‘limited-angle’ projections are trained in the parallel approach of PSO-ANN. Nine projections are utilized for PSO-ANN. The training exemplar of the system is constructed with limited-angle projection datasets and the ‘ideal’ target images are reconstructed by FBP using 180 acquired projections. The parallel approach of PSO-ANN systems are trained and tested for reconstruction of phantom images also from limited angle projection data and its response is discussed in Sections 5. 2. 3.3.3. Metric for quality assessment of digital images In image quality evaluation the mean square error (MSE) [33] and PSNR [34] are generally utilized to determine the degree of image 25
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quality. Fitness functions used for image reconstruction include: Root Mean Squared error (RMSE), Mean Squared error (MSE) [33]. MSE outperformed for small as well as large size images with different shape complexities. The MSE represents the cumulative squared error between the FPB based spatial images(I1) and the PSO-ANN images(I2), whereas PSNR is an expression for the ratio between the maximum possible value of a signal and the power of distorting noise that affects the quality of its representation. The lower value of MSE, the lower is the error in the reconstructed image.
MSE =
ΣM , N [I1 (M . N ) − I2 (M . N )]2 M *N
Table 3 Pseudo code for the parallel approach of PSO-ANN. 1.Read the MRI Phantom of 18 projections and 95 sample data. 2. Read the FBP (64 x 64) image reconstruct with the 180 projections. 3. Normalize the projections as well as FBP- based image datasets. 4. Constructs the exemplars with ‘noise-free’, ‘noise-added’, and ‘limited-angle’ datasets. 5. Decompose the exemplar datasets into exemplar subsets and corresponding subnets. 6. Initialize the random weights and construct feed-forward subnet with 3 layers. 7. Accumulate the random weights from BP-ANN subnets 8. Choose the multi-core processors for the parallel execution of PSO-ANN. 9. Process the decomposed subnet data into an available number of Matlab workers. 10. Matlab Local Cores/Labs: 10.1. Perform the selection of weight chromosome. 10.2. Objective function of mean square error assigned for actual and targeted weights. 10.3. Each lab/core uses these values to update the weights and thresholds. 10.4. Updates the weight values. 11. Matlab Worker. 11.1. Determine value of E (t) of the last cycle of the minimizing operation. 11.2. Determine whether to continue training or not.
(12)
To estimate the PSNR, the block initially calculates the meansquared error using Eq. (16). In Eq. (12), M and N are the number of rows and columns in the FBP based spatial images, respectively. Then the block computes the PSNR using the Eq. (13).
PSNR = 10log10
R2 MSE
(13)
In Eq. (13), R is the maximum fluctuation in the input image data. For example, if the input image has a double-precision floating point data, then R is 1. If it has an 8-bit unsigned integer data, R is 255, etc.
is no need for parallelization in the PSO-ANN testing phase. The reason is that, only updated weights are mapped in the trained network. Also, there is no need for computational effort. The dataset decomposition value used in the PSO-ANN training phase is used in testing phase also. The reconstruction time of the PSO-ANN based image in testing process is 0.47 s which does not exceed the standard processing time of 2 s. Hence there is no need for parallel processing in testing phase. The results of shepp-logan phantom reconstruction system under sequential and parallel executions are presented and discussed in the next section.
4. Design and implementation 4.1. System overview Initially, the system was developed in Windows operating system on an Intel Core i5 M460 processor 2.53GHz clock speed machine. Due to the high overload of the reconstruction problem, we ported the training process to 16 core processor system. The parallel as well as sequential PSO-ANN is implemented on an AMD OPETRON 16 core processors under Fedora 14 based on the Linux Kernel system. Nevertheless, the code can be run on any common PC with 2GB of RAM under Fedora 12 or higher versions. The difficulty of parallel programming can be solved in C, C++ , Fortran, OpenGL [35], OpenMP, MPI [36], Posix Thread, Cilk++ and GPU CUDA [25,26]. In the present work, parallel programming and Graphical User Interface (GUI) are designed in MATLAB R2010a. The MATLAB Parallel Computing Toolbox (PCT) [39] is utilized for developing parallel programming system. It includes creating tasks and sharing the tasks among the memories, synchronizes the datasets and tasks. PCT quickly solves computationally intensive and data intensive problems. The PSO-ANN parameters like size of the layers, number of hidden neurons, momentum parameter, error tolerance and maximum number of cycles are used interactively. The GUI feature of the system enables the selection of PSO-ANN based reconstructed image and FBPbased reconstructed image. The MSE and PSNR computations are carried out for reconstructions by mouse clicking operations and values are stored in a separate file.The pseudo code for parallel approach of PSO-ANN system is depicted in Table 3.
5. Results and discussion 5.1. Sequential PSO-ANN for phantom image reconstruction In the PSO-ANN system, both the projections as well as two dimensional images are normalized uniformly in the ranges between 0 and 1 and exemplars are constructed for training. The sequential PSO with BP-ANN training phase is executed on a multi-core computer system as described in Section 3.1. The structure of BP-ANN has 1710 nodes in the input layer representing 18 ‘noise-free’ projections each with 95 samples, and 4096 nodes in the output layer representing a two-dimensional image of size 64 x 64 pixels. The FBP based spatial image is produced with 180 projections. Both the ‘noise-free’ projections as well as two dimensional FBP images are normalized uniformly in the ranges between 0 and 1’s and arranged into large exemplars dataset. The number of hidden neurons varies from 125 to 250. The hidden layer size is chosen based on ‘trial and error’ method from the Table 6, to overcome the difficulty of over-training and distortion of the output images. The log-sigmoid transfer function is initialized weights for both input layer into hidden layer and hidden layer into output layer. The sequential PSO-ANN testing phase produces the images as shown in Fig. 6a, b. The maximum generations of PSO has been set to 200. The PSO error tolerance parameter is set to e-08. The population size parameter is set to 30 to 90. These are all the parameters preferred for improving the optimizing performances. The above mentioned PSO-ANN parameters and the corresponding properties play an important role during the initial weight optimizing of BP-ANN. The PSO-ANN based initializing parameters and other input conditions are listed in Table 1. The MSE and PSNR are calculated between the PSO-ANN based reconstructed image and FBP-based reconstructed image. The MSE value is 1.4877 e-06 as well as PSNR value 106.4056. The different level of ‘noise-added’ projections are simulated and also trained in the sequential approach of PSO-ANN. The training time
4.2. PSO-ANN training and testing phases The sequential with exemplar decomposition as well as parallel approach of PSO-ANN system undergoes a training process up to reasonable convergence is achieved. The optimal parameters are obtained on the basis of speed of convergence and quality of the reconstructed image. Subsequent to training process, the random weight values are changed into updated weight values. After the training phase, updated weights are mapped for reconstruction of shepp-logan phantom head images from any corresponding projection dataset. In PSO-ANN testing phase also, normalized projection datasets are decomposed and then subsets of projections data are tested into sequential as well as parallel manner into a multi-core system. There 26
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Fig. 6. The result of reconstructed images 64 x 64 (a) and 128 x 128 (b) obtained by Sequential PSO-ANN system with original projections of phantom, 64 x 64 (c) and 128 x 128 (d) obtained by Sequential PSO-ANN system with ‘noise added’ projections when the exemplars are decomposed into 8 subnets, 64 x 64 (e) and 128 x 128 (f) obtained by parallel PSO-ANN system with ‘noise added’ projections when the exemplars are decomposed into 8 subnets, 64 x 64 (g) and 128 x 128 (h) obtained by parallel PSO-ANN system with ‘limited-angle’ when the exemplars are decomposed into 8 subnets.
obtained by sequential approach of PSO-ANN system with ‘noiseadded’ data. By decomposition of training data, the results will be obtained and presented in next Sections 5. 2.
require more memory space to converge. Also, foremost issue is the training time gradually increases. During the study, it is observed that the training time increases as the number of subnets increases as shown in Fig. 9a. The sequential PSO-ANN with ‘noise-added’ projection testing phase produces the images as shown in Fig. 6c and d when the exemplars are decomposed into 8 subnets. It is also observed that the PSNRs of the images slightly increase as the number of subnets increases. Hence, in a typical case, if the number of subnets is an eight, the training time is nearly 322 seconds, which is very high. Therefore, there is a need for designing parallel training system with decomposed exemplars.
5.2. Sequential PSO-ANN with datasets decomposition for phantom reconstruction The datasets are decomposed into the subsets (2, 3, 4 … 8) of exemplars as shown in Table 2 and the dataset decomposition procedure as described in Section 3.2. The PSO-ANN subnets are initialized and constructed by subsets. The subnets based initial weights are optimized by the genetic algorithm. On the other hand the processing of large size exemplar produces memory intensive problem. Hence, there is a need to design the reconstruction system with decomposed exemplar datasets. The training time obtained by sequential approach of PSO-ANN system with ‘noise-added’ data when the exemplars are decomposed into 8 subnets as shown in Table 4. It is found that the high performance of time consuming values are produced in a ‘sequential with dataset decomposition method’ compared to ‘sequential without dataset decomposition method’. An important feature of the present work is the ability to perform data decomposition approach to reconstruct image better for large size of exemplars. The Table 6 obtains using the exemplars decomposition method corresponds to one subnet, 2 subnets, 4 subnets, and 8 subnets. The PSNR value of the reconstructed image is depending upon the number of subnet values. The rate of slight improvement of PSNR values against the number of subnets is depicting Table 5. The reconstruction results show ideal mapping and excellent reconstruction of data intensive problem. However, the reconstructions based on the large size of the exemplars increase MSE and decrease the PSNR values and
5.3. Parallel approach of PSO-ANN with data decomposition for phantom reconstruction Fortunately, as an artificial neural network have a natural parallel structure; reconstruction of large size exemplars can be exploited with learning about a parallel architectural system. When the parallel approach of PSO-ANN training is performed with decomposed datasets, the updated weight matrices are computed. The output images are produced from the trained PSO-ANN of updated weight values. The exemplar datasets are decomposed as described in the previous section and results are obtained by the parallel approach of PSO-ANN system as shown in Fig. 6e, f, g, and h. The parallel approach of PSO-ANN system is reconstructed images, using 8 subnet level of decomposition on 8 cores for ‘noise-added’ projection as shown in Fig. 6e (64 x 64 phantom) and Fig. 6f (128 x 128 phantom). The noise level 0.0 to 0.5 percent is added to the acquired projections data. The PSO utilizing parameters are 30-population size, 80 generations, MSE, PSNR, training times are depicts in Table 7. A plot of training time values are obtained by sequential and parallel approach of PSO-ANN system with ‘noise-added’ data for the 64 x 64 Shepp Logan Phantom images (Fig. 6e) when the exemplars are decomposed into 1, 2, 4, and 8 subnets shown in Fig. 8. In typical case, if the population size is 90 the sequential training time is nearly 900 Secs, which is very high and display in Table 8. Similarly, PSO-ANN is trained with data decomposition on 8 cores for ‘noise-added’ data and 128 x 128 reconstructed images. The PSO utilizing parameters are 30-population size; 80 generations and values are shown in Table 9 and graphically shown in Fig. 10.
Table 4 Measurement of sequential PSO-ANN trained with different decomposition levels. Decomposition of Exemplar
Training Time (Secs)
MSE e-07
PSNR (dBs)
1 2 4 8
535.21 536.27 584.37 731.79
8.2624 8.2520 8.2400 8.2031
108.9521 108.9652 108.9715 108.9910
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Table 5 Evaluation of PSO-ANN trained with different decomposition levels on multi-core. The PSO utilizing parameters are 30-population size, tolerance value is 1e-8 and 80 generations. Number of core(s)
1
2
Decomposition Level Time (Secs) MSE e-06 PSNR (dBs)
1 156.43 1.4899 106.65
2 182.45 1.4860 106.48
4 225.31 1.4618 106.41
8 322.20 1.4061 106.39
4
1 285.27 1.4927 106.38
2 121.75 1.4867 106.40
4 178.32 1.4708 106.45
8 239.79 1.4218 106.60
1 288.57 1.4927 106.39
8 2 151.07 1.4800 106.42
4 80.83 1.4672 106.46
8 123.20 1.4198 106.60
1 284.58 1.4927 106.39
2 147.41 1.4800 106.42
4 97.57 1.4646 106.47
8 58.73 1.4110 106.63
Table 6 Performance of parallel PSO-ANN system trained with 8 subnets and run on multi-core system with respect to change in hidden layer. Hidden Neurons Size
Time (Secs)
MSE e-06
PSNR (dBs)
125 150 175 200 225 250
59.00 64.07 68.09 71.43 75.54 80.89
1.4110 1.4082 1.4027 1.4179 1.4183 1.4106
106.6144 106.6441 106.6611 106.9510 106.6131 106.6369
Table 7 Performance of PSO-ANN system, having utilizing parameters as 30-population size, and 80 generations. Decomposition of Exemplar
# of Core(s)
Time (Secs)
MSE e-06
PSNR (dBs)
1 2 4 8
Sequential 2 4 8
240.43 146.53 90.06 68.09
1.4877 1.4800 1.4646 1.4027
106.4056 106.4281 106.4701 106.6611
Fig. 8. The training time as achieved by sequential and parallel PSO-ANN system with ‘noise-added’ data (Fig. 11e) when the exemplars are decomposed into 1, 2, 4, and 8 levels. (The PSO-ANN parameters are given in Table 7).
Table 8 Training time and different decomposition level of parallel PSO-ANN approaches implemented with 8 subnet decomposition on 8 cores for ‘noise-added’ data and 64 x 64 reconstructed images. The PSO utilizing parameters are 90-population size and 80 generations.
A plot of training time values are obtained by sequential and parallel approach of PSO-ANN trained with different exemplar decomposition subnet likes 1, 2, 4, 8 on single core (Fig. 7a), 2 cores (Fig. 7b), 4 cores (Fig. 7c), 8 cores (Fig. 7d) for the 64 x 64 Shepp Logan Phantom image. During the study, it is observed that the training time increases as the number of subnets increases as shown in Fig. 7a. Hence, in a typical case, if the number of subnets and cores are an n power 2(n^2) like 2, 4, the training time is feasible seconds shown in
Decomposition of Exemplar
# of Core(s)
Time (Secs)
MSE e-06
PSNR (dBs)
1 2 4 8
Sequential 2 4 8
898.55 611.89 360.75 231.12
1.4422 1.4554 1.4017 1.3121
106.5406 106.5009 106.6483 106.9510
Fig. 7. A plot of training time values versus different exemplar decomposition levels on single core (a), 2 cores (b), 4 cores (c) and 8 cores (d).
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Fig. 7b and c. It is observed that the training time decreases as the number of subnets increases as shown in Fig. 7d. The parallel approach of PSO-ANN by varying the number of cores from 1 to 8 the output images is reconstructed. The MSE, PSNR values are also computed by varying the subnets and the number of cores. The values are presented in Table 5. It is observed that the training time decreases as the number of cores increases. Fig. 12 shows The two-dimensional shepp-logan-phantom 128 x 128 size reconstructed images (Fig. 12a1-a3) obtained by FBP based reconstruction system, (Fig. 12b1-b3) obtained by ART based reconstruction system, (Fig. 12c1-c3) obtained by SART based reconstruction system, (Fig. 12d1-d3) obtained by SIRT based reconstruction system, (Fig. 12e1-e3) obtained by sequential PSO-ANN based reconstruction system and (Fig. 12f1-f3) obtained by parallel PSO-ANN with eight decomposed subnets system, when the projections data are corresponds to ‘noise-free’ data (Fig. 12a1-f1), ‘noise-added’ data (Fig. 12a2f2), and ‘limited-angle’ data (Fig. 12a3-f3). The sequential and parallel approach of PSO-ANN produces relatively equivalent MSE and PSNR values. It is observed that the training time decreases as the number of cores increases. These training values are depicted in Table 8 and graphically shown in Fig. 9. A plot of PSNR values (Fig. 11a), training time values (Fig. 11b) are obtained by FBP, ART, SART, SIRT, sequential PSO-ANN and parallel approach of PSOANN system with ‘noise-added’ data for the 128 x 128 shepp-logan phantom images (Fig. 6f).The reconstruction results show perfect mapping and excellent reconstruction of the presence of circle and ellipse objects in the parallel PSO-ANN system. The FBP reconstructed image shows degradation by wrap-around star artifacts. The time taken for FBP reconstruction of the image (a2) is 1.39 Secs while ART image (B2) takes 27.61 Secs. Thus although ART shows reduced artifact, it suffers due to high reconstruction time. The training times as well as PSNR values are computed different reconstruction algorithm for phantom images shown in Fig. 11a and b, presented in Table 11. The ‘limited-angle’ nine projections are trained in the parallel approach of PSO-ANN. The plot of training time obtained by parallel approach of PSO-ANN system with ‘limited-angle’ data when the exemplars are decomposed into 8 subnets as depicts in Table 10 for the size of 64*64 phantoms as graphically shown in Fig. 13 and Table 12 for 128 * 128 size phantom. In these ‘limited-angle’ projections, the sequential and parallel approach of PSO-ANN produces relatively equivalent MSE and PSNR values. It is observed that the training time decreases as the number of cores increases on ‘limitedangle’ projections as graphically shown in Fig. 14. The limited angle projection of PSO-ANN system also gives better quality of image (Fig. 6g and h) than algorithm of FBP (Fig. 4.c and.d) reconstruction.
Table 9 Performance of both sequential and parallel approach of PSO-ANN trained with data decomposition on 8 cores for ‘noise-added’ data and 128 x 128 reconstructed images. The PSO utilizing parameters are 30-population size and 80 generations. Decomposition of Exemplar
# of Core(s)
Time (Secs)
MSE e-07
PSNR (dBs)
1 2 4 8
Sequential 2 4 8
734.97 479.25 245.48 147.30
8.2582 8.1938 8.1423 8.0532
108.9620 108.9959 109.0233 109.0711
Fig. 9. A plot of training time versus number of cores, when the exemplars are decomposed into 1, 2, 4, and 8 levels. (The PSO-ANN parameters are given in Table 8).
Fig. 10. A graph showing the training time versus number of cores when the PSO-ANN parameters are changed to population size-30 and 80 generations.
Fig. 11. A plot of PSNR values (a) and training time (b) as achieved in various image reconstruction.
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Fig. 12. The two-dimensional shepp-logan-phantom 128 x 128 size reconstructed images (a1-a3) obtained by FBP based reconstruction system, (b1-b3) obtained by ART based reconstruction system, (c1-c3) obtained by SART based reconstruction system, (d1-d3) obtained by SIRT based reconstruction system, (e1-e3) obtained by sequential PSO-ANN based reconstruction system and (f1-f3) obtained by parallel PSO-ANN with eight decomposed subnets system, when the projections data are corresponds to ‘noise-free’ data (a1-f1), ‘noiseadded’ data (a2-f2), and ‘limited-angle’ data (a3-f3). For visual comparison, the images in FBP and Parallel PSO-ANN are also shown here. Table 10 Evaluation measurement of both sequential and parallel strategy of PSO-ANN trained with data decomposition on 8 cores for ‘limited-angle’ data and 64 x 64 reconstructed images. The PSO utilizing parameters are 30-population size and 80 generations. Decomposition of Exemplar
# of Core(s)
Time (Secs)
MSE e-06
PSNR (dBs)
1 2 4 8
Sequential 2 4 8
211.6097 129.8611 82.7303 73.8868
1.7612 1.7612 1.7612 1.7612
105.6727 105.6727 105.6727 105.6727
Table 11 Measurement metrics of training time (seconds) and PSNR (dBs) value for FBP, ART, SART, SIRT, sequential PSO-ANN and parallel approach of PSO-ANN trained with data decomposition on 8 cores for ‘noise-added’ data and 128 x 128 reconstructed images. The PSO utilizing parameters are 30-population size and 80 generations. Reconstruction Algorithms
Training Time (Secs)
MSE e-07
PSNR (dBs)
FBP ART SART SIRT Sequential PSO-ANN Parallel PSO-ANN
1.39 27.61 2.0931 4.7931 734.97 147.30
0.0018 0.0012 0.0011 0.0010 0.00000082582 0.00000080532
75.3533 77.4916 77.6898 77.7710 108.9620 109.0711
Fig. 13. A plot of training time as obtained by sequential and parallel PSO-ANN system with ‘limited-angle’ data for the 64 x 64 Shepp Logan Phantom images (Fig. 11g) when the exemplars are decomposed into 1, 2, 4, and 8 levels. (The PSO-ANN parameters are given in Table 10). Table 12 Assessment of both sequential and parallel strategy of PSO-ANN trained with data decomposition on 8 cores for ‘limited-angle’ data and 128 x 128 reconstructed images. The PSO utilizing parameters are 30-population size and 100 generations.
6. Conclusion The sequential system using dataset decomposition strategy has been developed initially for standard shepp-logan phantom image reconstruction. This strategy led to long training time. In order to reduce the training time, the parallel approach has been devised. It showed better images than FBP in terms of noisy projections data and ‘limited-angle’ projections data and, better performance than sequential PSO-ANN in terms of computational complexity. The parallel system has been tested with ‘noise-free’ and
Decomposition of Exemplar
# of Core(s)
Time (Secs)
MSE e-07
PSNR (dBs)
1 2 4 8
Sequential 2 4 8
442.24 287.06 171.74 106.65
8.0126 8.0024 7.9881 7.9761
109.0911 109.0986 105.1063 105.1129
‘noise-added’ projection data. The system is also tested with ‘limited-angle’ projections data. In all the parallel with data decomposition cases, less training time is reported. The results presented in this paper clearly show that the parallel approach is feasible for reconstruction of acquired data from any standard medical imaging modality.
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Fig. 14. The training time obtained by sequential and parallel PSO-ANN system with ‘limited-angle’ data for the 128 x 128 Shepp Logan Phantom images (Fig. 11h) when the exemplars are decomposed into 1, 2, 4, and 8 subnets. (The PSO-ANN parameters are given in Table 12).
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