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1. (Department of Electronics & Communication, Malaviya National Institute of Technology, India 2018rec9028@mnit.ac.in)
2. (Department of Electronics & Communication, Malaviya National Institute of Technology, India msalim.ece@mnit.ac.in )

Compressive sensing (CS), Magnetic resonance imaging (MRI), Structural similarity index (SSIM), Feature similarity index measure (FSIM)

## 1. Introduction

Magnetic Resonance Imaging (MRI) is an imaging method that is very popular nowadays. Imaging contrast of soft tissues is very high, and it is the prime reason for its popularity. Our bodies are made up of a large amount of water and hydrogen atoms, and they act like magnetic elements. If the body is placed in an external magnetic field, then these little magnetic entities will align according to the external fields [1]. These little magnetic entities also rotate with an angular frequency known as the Larmor frequency. Further, if radiofrequency is also applied in this Larmor frequency range, then the previously spinning protons will absorb some amount of RF energy and move from lower to higher energy states. This is known as the condition of resonance.

When the external RF source is switched off, these protons are again switched to their initial position, and the energy difference will surge the level of the magnetic resonance signal. We know that the different body tissues release different amounts of energy because of their chemical and physical composition, so the magnetic resonance signal is also different at that level. The main problem associated with MRI is the rate of the acquisition process as it is very slow. Therefore, in the next sub-section, we discuss how the data acquisition process takes place in MRI.

### 1.2 Data Acquisition Strategy for MRI

The data which is acquired in a raw form is stored in the k-space or Fourier domain as a matrix. Therefore, if we want to analyze it in the image, then we have to apply a two-dimensional inverse Fourier transform. Two types of the gradients are used to encode the k-space: one is horizontal, which is based on frequency, and the other is vertical, which is based on the phase. They are named the Frequency Encode Gradient (FEG) and Phase Encode Gradient (PEG).

### 1.3 Types of Weighted Images

If the contrast of a magnetic image is based on the longitudinal relaxation time, then it is termed as a T1 weighted image. Both the repetition time and the echo time are short for this type of weighting strategy. Similarly, T2 weighted images are images that are based on the transverse relaxation time, and the value of repetition time and the echo time are long in this case. The third category is based on the density of hydrogen atoms in the body tissue for a particular volume. In this case, the value of repetition time is large, and the echo time is very short. In the next section, we elaborate on the work that has been done by various researchers in this domain.

## 2. Literature Review

One author proposed an approach that is based on the combination of objective-based and deep learning-based compressive sensing reconstruction approaches [2]. Multi-layer based convolutional sparse coding with iterative thresholding is used to recover the MRI based image. MR images of the knee and brain were used for the experiment. PSNR and SSIM values were mentioned in the results. A method based on enhanced Laplacian Scaled shrinkage and BM3D was proposed by the authors for MR image reconstruction.

Brain, shoulder, chest, and head-based MRI images were considered for analysis [3]. A comparative study based on SSIM, PSNR values, sampling rate, and standard deviation for various algorithms was mentioned. Yiling Liu et al. proposed an improvement for a compressive sensing-based MRI scheme to overcome the structure loss [4]. In this paper, the authors presented the theoretical aspects of the normalized iterative hard thresholding scheme. The contribution of the authors is a quantized framework that can be employed for MRI as well as radio astronomy [5].

A mathematical model was proposed by the authors, which consists of a locally low-rank MRI reconstruction technique. Knee and brain MR images were considered, and the normalized RMSE value was mentioned in the results [6]. Yipeng Liu et al. talked about dynamic MRI. Further, they proposed a hybrid model based on a compressive sensing approach for the highly efficient recovery of MRI images [7]. Experimental justification for the high efficiency and low computational complexity was also mentioned. The prime concern of all the authors was to improve the quality of reconstruction of MRI, and the computational complexity should low. In the next section of the paper, we move towards the mathematical framework for the compressive sensing MRI strategy.

## 3. Mathematical Framework for CS-MRI

Compressive sensing is a technique that can be used to acquire sparse signals by employing a direct method, or the signal may be sparse in some domain [8]. We talk about the key parameters that are responsible for the success of the compressive sensing approach. The first key aspect is that the signal must be sparse, or it may be sparse in some random domains. The second one is the incoherence as we use under-sampling in the compressive sensing, so we have to remove the aliasing problem. This is tackled by using the incoherent representation basis.

### 3.1 Sparsity for Magnetic Resonance Imaging

It is a well-known concept that natural images are not sparse in their domain, but they may be sparse in some other transform domain. We consider a dense n x n spatial-domain image a, and it is compressible to some other domain ${\Psi}$ (DCT or wavelet). If image a is reframed to n x 1 column vectors, then ${\Psi}$ can be considered as a transform matrix.

### 3.2 Mutual Coherence

Our prime concern is the efficient recovery of an image from compressed data, so the measurement matrix ${\Phi}$ must be incoherent in some sparse basis matrix ${\Psi}$ [9]. Let G$_{\mathrm{u}}$ be a measurement matrix and G$_{\mathrm{u}}$= ${\Phi}$ G, where $\Phi \in R^{m\times n}$, in which each row has all zero values except one value. If we want to analyze it mathematically, it is represented as below.

##### (1)
$\mu \left(G_{u},\Psi \right)=\sqrt{n}\max _{1\leq k,j\leq n}\left\| \left\langle \left(G_{u}\right)_{k},\Psi _{j}\right\rangle \right\|$

If Gu and ${\Phi}$ contain elements that are correlated, then incoherence is small. Otherwise, it has a higher value.

##### Table 1. Various input images used for the experiment.
 MR Image T-1 MR Image T-2 MR Image PD MR Image Brain Pelvis Feet Thigh Thorax

### 3.3 Proposed Approach

In this work, we want to find out a strategy that can easily indicate that the compressive sensing approach is an effective framework that can be used to compress MR images and then retrieve them. Therefore, for this purpose, we defined a basic approach that is given in the flow diagram below. As per the literature, when under-sampling is conducted in the k-space by periodic interleaving, it will violate the Nyquist theorem and will produce aliasing in the recovered images. Therefore, the aliasing and incoherence should be less visible in the recovered MR image.

As in the work proposed by Candes et al., fewer samples can also reconstruct images without an aliasing effect. In MR, the signal has spatial redundancy, so in some transform domains, few components have maximum information from the image. Therefore, the compressive sensing approach can easily be applied and efficiently recover MR images with a few samples.

As per the flow diagram, first of all, a raw input image is considered. Then, we convert this image into a corresponding number of N discrete samples. After that, the sparsity of the image is calculated by defining the basis and sensing matrices and generating them. Then, we multiply the image with the basis and sensing matrix to represent it in the basis domain.

The inverse function is applied to recover the original image by using the reconstruction strategy. Lastly, we obtain the magnetic resonance image. To check the quality of the recovered image, we use certain image-quality assurance indexes like SSIM, FSIM, etc. In the next section, we discuss various image quality assessment parameters that will clarify the efficiency of the reconstruction algorithm.

## 4. Result and Analysis

The purpose of this work is to compress various types of MR images with the help of compressive sensing and then to compare the reconstructed images with the original image to calculate the efficiency of the proposed image. For this purpose, a set of images was utilized: T-1, T2, and proton density-based images. Image quality assessment was done by using the PSNR, MSE, SSIM, FSIM, and CVSS. Expressions for all the IQA parameters are mentioned in the upcoming section.

### 4.1 Peak Signal-to-noise Ratio

The peak signal-to-noise ratio is an image quality assessment parameter that is used to analyze the quality of a computed tomography-based image [10]. The calculation of PSNR is conducted using actual MR image samples and the reconstructed MR image samples. A higher value of PSNR denotes better quality of the reconstructed MR image.

##### (2)
$PSNR=\frac{10\times \log _{10}\left(\sum _{m}\left({F_{i}}^{2}\right)\right)}{\sum _{m}\left[F_{i}\left(m\right)-G_{o}\left(m\right)\right]^{2}}$

F$_{\mathrm{i}}$ = MR based image’s actual signal, L$_{\mathrm{o}}$ = MR-based image’s reconstructed signal, m = total number of components.

### 4.2 Compression Ratio (C/R)

The compression ratio is a defined parameter that indicates how many sample components are considered in the reconstruction process out of the total number of components present in that signal or MR image [11]. It is the ratio between the sample utilized in reconstruction and the total number of samples in the MR image.

##### (3)
$\textit{CompressionRatio}=\frac{m}{n}$

m= samples used for reconstruction of MR image, n= actual number of samples in MR image.

### 4.3 Structural Similarity Index Measure (SSIM)

The SSIM index is a mathematical model to find out the similarity between two images. It can be considered as a quality measurement for one image compared to another image that keeps the quality of the image in the main perspective. We can say it is a modified structure of the universal image quality index [12].

##### (4)
$SSIM\left(x,y\right)=\frac{\left(2\mu _{x}\mu _{y}+c_{1}\right)\left(2\sigma _{xy}+c_{2}\right)}{\left(\mu _{x}^{2}+\mu _{y}^{2}+c_{1}\right)\left(\sigma _{x}^{2}+\sigma _{y}^{2}+c_{2}\right)}$

### 4.4 Feature Similarity Index Measure (FSIM)

The basic function of FSIM is to map the similarity between two images. Phase congruency and the gradient magnitude are the two key techniques to find out the value of FSIM [13].

##### (5)
$FSIM=\frac{\sum _{x\in \Omega }S_{L}\left(x\right).PC_{m}\left(x\right)}{\sum _{x\in \Omega }PC_{m}\left(x\right)}$

Here, PC$_{\mathrm{m}}$is the maximum value of phase congruency of both images. ${\Omega}$ represents that the image is in the spatial domain.

### 4.5 Contrast and Visual Salient Similarity (CVSS)

CVSS is also an image quality assessment parameter that is used to find out the quality of a reconstructed image [14]. In this approach, we find out a score value that represents how close the recovered image is to the actual image. The quality score parameter is defined below.

##### (5)
$S=w_{1}\cdot SD\left(LCS\right)+w_{2}\cdot SD\left(GVSS\right)$
##### (6)
Subjected to $w$$_{1} + w$$_{2}$ =1

Here, w$_{1}$ and w$_{2}$ are the weights of the local contrast similarity and visual saliency, respectively.

### 4.6 Mean Square Error (MSE)

MSE is used to calculate the relation of the original and recovered MR images.

##### (7)
$MSE\left(U,V\right)=\frac{1}{n}\sum _{1}^{n}\left(\left(U_{i}-V_{i}\right)^{2}\right)$

Here, U represents the original MR image, V represents the recovered MR image, and n shows the total number of samples. Diagrams that show the value of compression ratio vs. PSNR and SSIM values are shown below. We can easily find out the variation between the three types of weighted images. In the images, plots are given for the compression ratio vs. peak signal-to-noise ratio and structural similarity index measures. Now, we elaborate on various tables consisting of the compression Ratio vs. FSIM, CVSS, and MSE.

To present the efficiency of the proposed algorithm, we compare our work with the work proposed by Tariq Tashan et al. [15]. They presented multilevel MR imaging. As a result of comparison, we found that our proposed approach has significantly enhanced values of image quality assurance matrices.

##### Table 1. Table for C/R vs. FSIM for T1 MR Images.
 C/R FSIM Values for T1 Weighted Image Brain Pelvis Feet Thigh Thorax 500/4096 0.5760 0.6821 0.5872 0.5981 0.6928 1000/4096 0.6391 0.7348 0.6190 0.6888 0.7410 1500/4096 0.6918 0.7894 0.7038 0.7491 0.8029 2000/4096 0.7268 0.8427 0.7331 0.7927 0.8484 2500/4096 0.8023 0.8810 0.8082 0.8506 0.8897 3000/4096 0.8354 0.9237 0.8531 0.8971 0.9235 3500/4096 0.9075 0.9542 0.9090 0.9534 0.9587
##### Table 2. Table for C/R vs. FSIM for T2 MR Images.
 C/R FSIM Values for T2 Weighted Image Brain Pelvis Feet Thigh Thorax 500/4096 0.6221 0.7471 0.6747 0.7031 0.7590 1000/4096 0.6858 0.7913 0.7468 0.7786 0.8111 1500/4096 0.7197 0.8365 0.7728 0.8505 0.8534 2000/4096 0.7673 0.8702 0.8394 0.8904 0.8807 2500/4096 0.8174 0.9099 0.8869 0.9371 0.9198 3000/4096 0.8672 0.9372 0.9260 0.9611 0.9437 3500/4096 0.9052 0.9656 0.9634 0.9775 0.9675
##### Table 3. Table for C/R vs. FSIM for PD MR Images.
 C/R FSIM Values for Proton Density Weighted Image Brain Pelvis Feet Thigh Thorax 500/4096 0.5365 0.7086 0.5669 0.6001 0.6804 1000/4096 0.5876 0.7538 0.6197 0.6945 0.7290 1500/4096 0.6312 0.8070 0.6722 0.7560 0.7830 2000/4096 0.6856 0.8364 0.7262 0.7988 0.8271 2500/4096 0.7387 0.8735 0.7848 0.8477 0.8614 3000/4096 0.7816 0.9186 0.8302 0.8992 0.8905 3500/4096 0.8428 0.9481 0.8942 0.9489 0.9380
##### Table 4. Table for C/R vs. CVSS for T1 MR Images.
 C/R CVSS Values for T1 Weighted Image Brain Pelvis Feet Thigh Thorax 500/4096 0.0908 0.0614 0.1024 0.0994 0.0659 1000/4096 0.0790 0.0466 0.0907 0.0776 0.0575 1500/4096 0.0672 0.0411 0.0746 0.0647 0.0388 2000/4096 0.0442 0.0295 0.0650 0.0508 0.0296 2500/4096 0.0430 0.0203 0.0455 0.0360 0.0203 3000/4096 0.0316 0.0121 0.0351 0.0231 0.0131 3500/4096 0.0168 0.0069 0.0195 0.0091 0.0065
##### Table 5. Table for C/R vs. CVSS for T2 MR Images.
 C/R CVSS Values for T2 Weighted Image Brain Pelvis Feet Thigh Thorax 500/4096 0.0826 0.0434 0.0614 0.0512 0.0429 1000/4096 0.0805 0.0315 0.0403 0.0374 0.0259 1500/4096 0.0609 0.0237 0.0343 0.0250 0.0208 2000/4096 0.0512 0.0178 0.0249 0.0176 0.0152 2500/4096 0.0397 0.0119 0.0158 0.0095 0.0112 3000/4096 0.0299 0.0069 0.0107 0.0054 0.0070 3500/4096 0.0184 0.0039 0.0051 0.0029 0.0035
##### Table 6. Table for C/R vs. CVSS for PD MR Images.
 C/R CVSS Values for Proton Density Weighted Image Brain Pelvis Feet Thigh Thorax 500/4096 0.1333 0.0692 0.1159 0.1122 0.0847 1000/4096 0.1253 0.0497 0.1019 0.0860 0.0664 1500/4096 0.1171 0.0412 0.0898 0.0659 0.0479 2000/4096 0.1003 0.0290 0.0733 0.0491 0.0378 2500/4096 0.0871 0.0251 0.0582 0.0320 0.0275 3000/4096 0.0635 0.0138 0.0430 0.0201 0.0207 3500/4096 0.0436 0.0081 0.0239 0.0084 0.0099
##### Table 7. Table for C/R vs. MSE for T1 MR Images.
 C/R MSE Values for T1 Weighted Image (*10^2) Brain Pelvis Feet Thigh Thorax 500/4096 8.37 7.58 9.96 1.11 5.02 1000/4096 6.08 5.59 6.54 6.97 3.40 1500/4096 4.07 5.59 4.54 4.16 2.34 2000/4096 2.87 2.84 3.07 2.5 1.58 2500/4096 1.90 1.82 2.05 1.55 1.09 3000/4096 1.24 1.18 1.15 0.83 0.74 3500/4096 .63 0.60 0.53 0.37 0.36
##### Table 8. Table for C/R vs. MSE for T2 MR Images.
 C/R MSE Values for T2 Weighted Image (*10^2) Brain Pelvis Feet Thigh Thorax 500/4096 7.39 2.63 2.92 2.91 2.50 1000/4096 5.01 1.83 1.86 1.70 1.50 1500/4096 3.97 1.42 1.44 1.14 1.12 2000/4096 2.80 1.04 0.855 0.715 0.772 2500/4096 2.03 0.717 0.555 0.442 0.509 3000/4096 1.24 0.437 0.349 0.235 0.363 3500/4096 0.660 0.243 0.157 0.107 0.177
##### Table 9. Table for C/R vs. MSE for PD MR Images.
 C/R MSE Values for Proton Density Weighted Image (*10^2) Brain Pelvis Feet Thigh Thorax 500/4096 26.3 9.40 14.2 12.3 8.56 1000/4096 16.0 7.20 10.5 8.16 4.96 1500/4096 10.9 5.40 6.38 4.52 3.83 2000/4096 8.36 3.79 4.26 2.69 2.61 2500/4096 5.33 2.58 2.69 1.56 1.82 3000/4096 3.00 1.55 1.67 0.859 1.12 3500/4096 1.61 0.846 0.744 0.381 0.583
##### Table 10. Comparison with Work Shown by Tariq et al..
 C/R PSNR Values (dB) For Proposed algorithm For algorithm mention in [15] 0.12 38.3 37.1 0.24 40.1 39.2 0.36 43.6 42.1 0.48 45.2 44.2 0.61 46.4 45.5 0.73 47.9 46.7 0.85 48.2 47.3

## 5. Conclusion

Compressive sensing is a technique that can efficiently reduce the number of samples required for a good reconstruction process. This technique was used, and we can see that it was very effective in this work. As MR imaging is a slow and time-consuming process, the patient has to wait for a long time (20-30 minutes) for the imaging process. Spending this much time under a huge magnetic field can also harm the patient’s body and some tissues. Therefore, reducing the number of samples in the k-space for the recovery process can also reduce the required time.

In this work, we analyzed various numbers of image quality assessment matrices, which indicated that even if we reduce the number of samples in the reconstruction process, the quality of the image at a significant value of compression ratio is up to the mark and can easily represent all the information in the MR image. Signal intensities for T1, T2, and proton density-weighted MR images are different, so they will directly affect the reconstruction process too. At the same compression ratio (e.g., 3000/4096), the values of FSIM, SSIM, PSNR, MSE, and CVSS are different for similar body-part images in different weighted images.

As we can see, if the number of samples that are required to reconstruct an image is large enough, then the time required to reconstruct the image is also large. Therefore, if the number of samples is reduced to a certain level, then the time required in the MRI process can also be reduced to a certain level. Table 10 shows a significant comparison of our work with other work [15]. Here, we can check that the PSNR values for our proposed algorithm are sufficiently large compared to the other PSNR values.

### ACKNOWLEDGMENTS

This research was supported by the Visvesvaraya Ph.D. Scheme, MeitY, Govt. of India, with unique awardee number “MEITY-PHD-2946”. Name of Grant Recipient: Vivek Upadhyaya.

### Conflict of Interest

Vivek Upadhyaya received a research grant from the Visvesvaraya Ph.D. Scheme, MeitY, Govt. of India, with unique awardee number “MEITY-PHD-2946”. Dr. Mohammad Salim declares that he has no conflict of interest.

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