20th Iranian Conference on Electrical Engineering, (ICEE2012), May 15-17,2012, Tehran, Iran

MRI Reconstruction Through Compressed Sensing Using Principle Component Analysis (PCA)

Jafar Zamani, Abbas Nasiraei Moghaddam

Dept. of Biomedical Engineering Amirkabir university of Technology

Tehran, Iran zamani.jafar@aut.ac.ir,

nasiraei@aut.ac.ir

Abstract- Compressed Sensing (CS) is a theory with potential to

reconstruct sparse images from a small number of random

samples in the frequency domain, and with the aim of increasing

achievable acceleration factors along with improved SNR and

fidelity. Therefore we minimize an objective function defined as

weighted sum of non-zero elements, error and total variation

(TV). The accuracy and speed of the reconstruction depends on

how we choose and update the aforementioned weights and how

to solve the minimization problem. In this study, we proposed the

Principle Component Analysis (PCA) for weighting the sparsity

in the CS formulation. Considering the dimension reduction

property of PC A, it is suitable for weighting the sparse transform

in the CS algorithm. In the proposed implementation the weight

of sparsity is updated at each iteration using the norm Ll of

PCA significant components, which quantifies the sparsity at that

stage. Results were compared with the zero-filling (ZF) and low

resolution (LR) techniques. Compared to CS without using the

sparse weighted, the proposed method took 15% higher SNR and

reached 10% higher correlation with the original image.

Keywords- Compressed Sensing; K-space; Principle Component Analysis; Sparse ; random under-sampling

I. INTRODUCTION

Magnetic Resonance Imaging (MRI) is a relatively slow imaging modality in which we acquire the spatial frequency component of the structure in the K-space and then reconstruct the image by Fourier transform [1-3]. Acceleration of the acquisition is necessary for many clinical applications of MRI. The compressed sensing method has been recently proposed as a remedy to this problem. Compressed sensing (CS) is an efficient tool that accelerates the data acquisition in MRI through the significant reduction of required measurements for image reconstruction. The performance of this technique highly depends on the sparsifying transformation that provides a basis in which the image has a sparse presentation [4, 5]. In this approach three conditions must be considered 1) Sparsity: A sparse representation of the image must be possible in some known transform domain 2) Incoherent under-sampling: K­space under-sampling should be randomized so it generates noise-like aliasing interference in that transform domain. 3) Non-linear Reconstruction: A non-linear algorithm is required for the reconstruction of the image with emphasis on minimization of norm L 1 image in known sparse transform and

Hamidreza Saligheh Rad

Dept. of Medical Physics and Biomedical Engineering Tehran University of Medical Sciences (TUMS)

And Research Center for Science and Technology in Medicine

(RCSTIM), Imam Khomeini Hospital Complex, Keshavarz Blvd

Tehran, Iran hamid.saligheh@gmail.com

error [6]. For example the wavelet transform is useful for brain images because the coefficient of wavelet is very sparse in comparison to image in pixel domain. For MR angiography (MRA) the identity transform is suitable because of MRA intrinsic sparsity in pixel domain which makes it a remarkable application for CS algorithm [7, 8]. For K-space under­sampling in this scheme, a random sampling scheme with a probability density function (PDF) such as Gaussian, partial Fourier or Bernoulli, is required to satisfy the incoherency condition [9, 10]. Incoherency is technically defmed as relatively small height of side lobes compared to the main lobe of the point-spread function (PSF) and strongly depends on the sampling scheme [9]. Since the vicinity of the center of the K­space determines the contrast of the MRI image, thus the sampling pattern contains highly dense sampling near the center of K-space. The image reconstruction in CS comprises solving a constrained optimization problem in which the sparsity is maximized while the error between K-space under­sampled data and the Fourier transform of the reconstructed image is kept limited [4, 5]. In CS we minimize an objective function defined as weighted sum of non-zero elements (to emphasize sparsity), error (to underline fidelity) and total variation (TV) (to denoise the image). There are several ways for non-linear reconstruction of the image, e.g. greedy algorithms such as orthogonal matching pursuit, iterative thresholding and conjugate gradient. In this work, we use a non-linear conjugate gradient (NLCG) optimization method for image reconstruction [11, 12].

In recent years an adaptive basis such as singular value decomposition (SVD) and PCA was utilized for sparse transform [13]. This sparse basis is general for all kind of images. The outcomes of these methods were not superior to others in terms of MES, SNR or similarity to original image. For this reason we proposed an alternative method for CS based on PCA analysis in which the weight of norm L I of the image was adaptively updated at each iteration with respect to reduction of dimensions.

The outline of this paper is as follows: Section II briefly introduces compressed sensing theory and the method for data acquisition, K-space under-sampling. Section III describes proposed method, Section IV shows the Results and Section V is dedicated to conclusion and future works.

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II. MATERIALS AND METHODS

The theory of compressed sensing is introduced by Candes (2006) [14] and Donoho (2006) [15]. In traditional digital signal processing to reconstruct a signal from its samples, the signal needs to be sampled with a rate at least twice as much as its bandwidth (Nyquist rate). CS is a sampling technique to reconstruct signals with a rate below the Nyquist rate. CS is a theory with potential to reconstruct sparse images from a small number of random samples in the frequency domain, and with the aim of increasing achievable acceleration factors along with improved SNR and artefacts level of images obtained from incomplete K-space data. For this purpose the signal is sparse by using the appropriate known sparse transform. In CS theory, an unknown signal m that has a sparse representation in a basis transform domain (If!), can be reconstructed from incomplete measurement y [16]. Which is related to the signal through a linear transformation y = n m, if If! and n (sensing matrix does not depend on m) are incoherent [10, 16]. To recover signal (m), we should solve constrained optimization problem as following:

Minimize II \!f(m) II 0 s.t. <1> m = y. (1)

Since using LO results in a NP-hard problem, L I

optimization is employed to reconstruct the signal in the CS theory [16].

Minimize II \!f(m) III s.t. <1> m = y. (2)

The acquired data (y) in MRI is a subset of K-space. That is, the sensing matrix (9) consists of Fourier transformation of image as well as the under-sampling scheme. Gaussian or Bernoulli PDF satisfies the CS condition on requiring a sensing matrix with a small restricted isometry constant. K-space is randomly under-sampled only in phase encoding direction. The readout direction is fully sampled, because it does not increase the imaging time [5].To reconstruct an image in CS algorithm a constrained optimization problem is solved that includes minimization of norm L 1 of reconstructed image in sparse domain to maximize the sparsity while it penalizes the error between K-space under-sampled data and reconstructed image. The constrained optimization problem is converted to an unconstrained problem through this formulation:

where a and � are the regularization parameters; a is to tradeoff between the sparsity and fidelity, and � is to weight the total variation (TV), the added term for denoising. For reconstruction of the MR images, equation (3) is written as follow:

arg minm (II \!f( m ) 111+�TV(m)+ A II Fum-y Ill). (4)

In which Fu, y, and A under-sampled Fourier operator, K­space under-sampled data, and the fidelity weighting, respectively. Also II x 111=Lil Xii represents the norm Ll and

II x 112=(Lil Xii 2)112 represents the norm L2. The TV is the difference of image pixels. For angiography we may choose the sparse transformation as the identity transform [17]. The TV is the difference of image pixels.

A. Data Acquisition

Using thin maximum intensity projections (MIP) of clinical high-resolution 3D CE MRA images, obtained on a 3T scanner, two sets of realistic 2D images were reconstructed. These images were then considered as the ground truth for this study.

B. K-Space Under-Sampling

Due to importance of K-space center, and incoherent under­sampling in CS, the sampling scheme should satisfy the incoherency and restricted isometry property (RIP) requirements. Therefore, the vicinity of the K-space centre was chosen completely and other places were sampled using a random Gaussian probability density function to keep only 33% of the data (3-fold less). Fig. 1 illustrates the under­sampling scheme for the k-space used in this paper.

Figure 1. k-space under-sampling with random Gaussian probability function.

III. PROPOSED METHOD

We employed an adaptive approach, based on PCA, to adjust the weight of sparsity (norm Ll of the image) in each iteration of reconstruction algorithm to emphasize sparsity with corresponding sparsifying. PCA is a mathematical technique that uses an orthogonal transformation to convert a set of observations of correlated variables into a set of values of linearly uncorrelated variables called principal components. Using the PCA adaptive approach as the weight of sparsity, the constrained optimization is converted into the minimization of the following expression:

arg minm( II u Xs Xv 11, 11 \!f(m ) 11,+�TV(m)+ A II Fm-y 1122). (5)

Matrices u, s and v, are the PCA elements ([u, s, v) =PCA (m)).The norm Ll of PC A in the first iteration is the dominant weight with a great value of (3000) but in the last iteration, when the image sparsity is achieved this number becomes very small (0.89). This means the great emphasize on sparsity in early iterations is gradually shifted towards fidelity in late iterations. Using the NLCG for image reconstruction, the final image was reconstructed iteratively.

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IV. RESULTS

We used SparseMRI free source code [5] in MATLAB version 7.10 on computer with Core 2 Duo CPU 2.2GHz for both aforementioned MRI data. The results of the proposed algorithm were compared to those of CS without sparse weighting with adaptive basis, Zero Filling (ZF) with CS data, Low Resolution (LR) with only center of K-space data and original image with full K-space data. To quantity the comparison, we obtain the Signal to Noise Ratio (SNR) and Peak SNR (PSNR) as well as correlation that measure the similarity between reconstruction results and the original image. The PSNR is equal to:

PSNR=IOxlog1ol /MSE. (6)

Figs. 2 and 3 show comparison of reconstructed images obtained from the proposed method with the original one (with full K-space), ZF and LR reconstruction methods. In Fig.3 results for traditional CS without using the sparse weighted with PCA or other adaptive algorithm were illustrated. Qualitatively it shows that the employed CS-PCA method has less noise and higher resolution than the CS, ZF and LR methods. TABLE I shows the calculated numerical results including SNR, PSNR, MSE, computational time and correlation between original image and reconstructed image for each method.

TABLE I. METHODS.

COMPARISON RESULTS FOR USED ALGORITHM AND ORIGINAL IMAGE. THE RESULTS SHOW THAT PROPOSED METHOD OUT PERFORMS OTHER

------- SNR PSNR MSE Time(s) Correlation Fig. 2 Original 25.45 45.027 -- -- --

ZF 17.64 35. 174 0.067 -- 42% LR 19.86 40.45 2.4xlO-4 -- 64%

CS-PCA 22.04 4 1. 186 1.3x 10-7 46.2 85% Fig. 3 Original 23. 12 40.460

-- -- --

ZF 16.48 26.35 0.0 183 -- 18% LR 19.01 30. 133 0.0032 -- 56%

CS-PCA 2 1.05 38.436 3.0 1 x I 0.5 25.25 68%

CS 19.68 32.527 0.002 1 24.7 59%

Figure 2. ,original image from full K-space data, B) CS image without the sparsity weighing with 33% randomly under-sampled data, C) ZF-CS data, and D) LR, the CS-PCA Image IS Similar to ongmallmage. Also proposed method better illustrate the details of image that Shown in the red circles.

Figure 3.. left to right A) Original image from full K-space data, B) LR C) ZF-CS data D) CS image with 33% randomly under-sampled data, and E) CS-PCA Image With 33% randomly under-sampled data. The CS-PCA shows the details of the image such section marked with a red circle and with higher SNR compared to other methods.

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V. CONCLUSION AND FUTURE WORKS

In this study we proposed a method for reconstruction of MR images using CS based on PCA method. In this method the norm L 1 of the PCA components of the reconstructed image was applied at each iteration of NLCG algorithm as the weight of sparsity. The norm Ll PCA in first iteration is the dominant factor but it decreases fast in the following iterations. That is the weight of sparsity changed adaptively with the image sparsity in each iteration. The results show the efficiency and accuracy of the proposed method compared to other conventional methods. Compared to CS without using the sparse weighted, the proposed method took 15% higher SNR and reached 10% higher correlation with the original image. Proposed method is able to illustrate the details of Contrast­Enhanced MR Angiography (CE-MRA) images in a variety of flow dynamic conditions. In particular the vascular stenosis and high temporal resolution required for depiction of vessel flow dynamics are achievable through this acceleration method. Diagnostic level in particular the vascular stenosis and high temporal resolution required for depiction of vessel. For further acceleration of MRI acquisition the best approach is the combination of CS and parallel imaging [18] through a chaotic K-space under-sampling scheme that satisfies these two methods.

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