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.

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.

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

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.

4.1 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.

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.

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].

4.4 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].

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.

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.

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.