A novel weighted compressive sensing using L1-magic recovery technique in medical image compression

Abstract

Recent technological advancement in computing technology, communication systems, and machine learning techniques provides opportunities to biomedical engineers to achieve the requirements of clinical practice. This requires storage and/or transmission of medical images with the conservation of the medical information over the communication channel. Accordingly, medical compression is necessary for efficient channel bandwidth utilization. To solve the trade-off between the compression ratio and the preservation of significant information, compressed sensing (CS) can be used. During image recovery in CS, an optimization algorithm is used, such as greedy pursuit, convex relaxation, and Bayesian framework. In the present work, a convex relaxation optimization called L1-magic is employed, where the objective function can be relaxed to the nearest convex norm, i.e., ℓ1-norm. In addition, the discrete cosine transform is used for recovery by transforming the image from time- to frequency-domain. To improve the medical image recovery, a weighted L1-magic is proposed using a threshold based on the image content, where high weight is given to the significant details in the image. Thus, the significant information in the image (values greater than the threshold) is multiplied by a weight factor according to the image characteristics for a successful recovery process. A comparative study of the proposed weighted L1-magic and orthogonal matching pursuit (OMP), one of the greedy algorithms, was conducted. Different metrics were measured, including the Structural Similarity Index Measure and Peak Signal-to-Noise Ratio (PSNR) to evaluate CS performance using the proposed weighted L1-magic as well as the weighted OMP and the principal component analysis (PCA) as a traditional compression method at different compression ratios (CR). The experimental results on diabetic retinopathy images dataset proved the superiority of the weighted L1-magic method, where for example as 0.4 CR, average PSNR is 19.37, 17.95, and 15.64 using the weighted L1-magic, weighted OMP, and PCA, respectively.

Introduction

The compact depiction of any image compression is to provide a compressed digital image representation without/with loss of the image quality for reliable transmission and storage. Since medical images have massive size, huge bandwidth for transmission with large storage spaces are needed. Lossless and lossy compression techniques of medical images are wholly dissimilar compression forms. In lossless compression, the compressed image can be recovered/re-expand as the original image without any information loss but waste time during the compression process. On the contrary, in the lossy compression, less computational time is required, however, the recovered image is not identical to the original one [1, 2]. Consequently, in the medical domain, image compression techniques that save the bandwidth and speed-up the transmission process while keeping the significant medical information become essential. To solve the tradeoff between the bandwidth requirements and saving the significant information in less computational time, compressed sensing (CS) which is a lossy compression technique proved its efficiency as it keeps a lot of image’s details with less complexity. The CS recovers the images from a few numbers of samples that save time also. It depends on sparsity which determines from prior knowledge of the image structure when a large number of its coefficients are zero or close to zero in a transform domain, such as discrete cosine transform (DCT) or wavelet domain [3, 4].

Several studies employed the CS techniques in medical image compression proving its efficiency [5–8]. However, recent studies confirmed that the visually weighted CS improves the performance of traditional CS. Lee et al. [9] proposed a framework for a visually weighted CS using the sparsity in the wavelet transform data. Furthermore, Yang et al. [10] proposed a weighting scheme based on the frequency components in an image to use the human perception in the sampling matrix of the CS procedure. Gaussian matrixes were employed with 2D-DCT and divided each image into blocks of 8 × 8 and 16 × 16 block size. Xu et al. [11] proposed a perceptual rate-distortion optimized (RDO) CS-based image codec for the block CS. Each block was measured as an optimization problem solved by the Lagrangian relaxation method. Wang et al. [12] designed iterative L1-magic weighted DCT and singular value decomposition (DCT-SVD) transform for CS restoration. An iterative SVD was applied to reserve the original image from insufficient CS calculations, while the traditional CS is capable of providing a good approximation.

The current work proposed a new DCT-based weighted CS to improve the compression performance without dividing the image into blocks or using an iterative SVD. Consequently, the present work proposed a weighted L1-magic reconstruction in the CS to enhance the reconstruction of the compressed image with less complexity by generating the weight from the DCT coefficients. In addition, a threshold value is proposed to determine the used DCT coefficients to be weighted for further use in the recovery process using the L1-magic procedure. Moreover, a comparative study with the orthogonal matching pursuit (OMP)-based CS as well as the PCA as a traditional lossy compression method is included. To evaluate the performance of the proposed DCT-based weighing method, diabetic retinopathy images were used.

The structure of the coming sections is as follows. “Methodology” section addresses the methodology including the proposed method. “Experimental results and discussion” section includes the experiment results with comparative studies. Finally, the conclusion of the current work is presented in “Conclusion” section.

Methodology

Discrete cosine transform

In the present work, the DCT is used before applying the compression using CS to guarantee the sparsity of the image, to use the DCT coefficients in the proposed weighted CS procedure, and also for the recovery process. Typically, there are four forms of DCT, namely from DCT-1 to DCT-4, which vary in the boundary states at the ends of the interval. DCT-2 and DCT-4 are frequently used in image processing. The 2D-DCT can be represented as:

$$x\left(\mathit{pq}\right)=\alpha \left(\text{p}\right)\alpha \left(q\right)\sum _{\text{M}=0}^{\text{M}-1}\sum _{n=0}^{N-1}\text{x}\left(\text{mn}\right)cos\frac{\left(2m+1\right)p\pi }{2N}cos\frac{\left(2n+1\right)q\pi }{2M}$$ 1

where M and N are the two dimensions of the image. In addition, $\text{q}\in \left[0,\text{N}-1\right],\text{p}\in \left[0,\text{M}-1\right]$, $\alpha \left(\text{p}\right)=\sqrt{\frac{1}{\text{N}}}$ for m = 0, and $\alpha \left(\text{p}\right)=\sqrt{\frac{2}{\text{N}}}$ for m ≠ 0, while $\alpha \left(\text{q}\right)=\sqrt{\frac{1}{\text{M}}}$ for n = 0 and $\alpha \left(\text{q}\right)=\sqrt{\frac{2}{\text{M}}}$ for n ≠ 0.

Compressed sensing

In sampling theorem, the signal can be recovered back, when the sampling frequency is greater than twice of the highest frequency component of the signal. The CS can recover the signal/image from a small number of measurements below the Nyquist rate. Based on the CS theory, for exact recovery of the image from a small number of measurements, the image should be sparse in the time-domain or any transform-domain such as DCT or wavelet [1]. For an image x of size $N\times N$, if the image is sparse with K, which has at least K non-zero entries, then it can be expressed as:

$$x=\Psi \theta$$ 2

where Ψ denotes the basis matrix of x, whereas Ө is composed of the corresponding coefficients. In addition, when x has less than $K$ nonzero coefficients on basis Ψ, then Ψ can be regarded as a sparse basis of $x$.

In CS, x is compressed in an orthogonal basis Ψ, then the transform coefficient Ө can be stated as $\theta ={\psi }^{T}x$. If the coefficient is projected onto another observation basis that is unrelated to the sparse basis Ψ, then, a measurement vector y can be obtained with the dimension of M × 1 such that M < N. the compressed image $y_{\mathit{cs}}$ can be expressed as:

$$y_{\mathit{cs}}=\varphi .x=\varphi .\psi .\theta$$ 3

$$y_{\mathit{cs}}=AX$$ 4

where A is the measurement matrix that should satisfy two conditions, namely incoherency between $\varphi$ and $\psi$, and the restricted isometry property(RIP) [3]. It means that there is a unique mapping between any $y_{\mathit{cs}}$ and its corresponding x subject to $y_{\mathit{cs}}=\varphi .x$. Thus, the matrix A must maintain the lengths of K-sparse vectors, which was in x. Typically, the matrix A fulfills the K-order RIP property if the relationship of the constant ${\partial }_k$ in $A$ and the vector with sparseness of K satisfies the following condition:

$$(1-{\partial }_k)||x_i{||}_2^2\le ||A_i{x}_i{||}_2^2\le (1+{\partial }_k)||x_i{||}_2^2$$ 5

Hence, the CS theory states that selecting measurement matrix A as a random such as Gaussian matrix having $M\ge cKlog\left(N/K\right)$, where c is a small positive number which affect the recovery performance [3, 4]. Subsequently, weighted CS can be used to enhance the performance of traditional CS.

L1-magic

The solution of compressed sensing linear equation can be solved by minimizing l₀-norm such that y = Ax. It is possible to relax l₀-norm to l₁-norm, if the signal x is sparse, the problem can be stated as:

$$\underset{x}{\text{min}}{||x||}_{1}s.t.y=Ax$$ 6

where l₁ norm for x is described as:

$${||x||}_1=\sum _i\left|x_i\right|$$ 7

Typically, L1-magic is frequently employed to solve CS recovery problems. It is considered a type of interior point algorithms which used for linear programming, which is expressed as:

$$\underset{x}{\text{min}}〈k_0,x〉s.t.y=Ax{z}_i\left(x\right)\le 0$$ 8

where $\mathbf{y}=R^M$, $\mathbf{x}=R^N,$ A is M x N matrix and i = 1,…,k, $z_i\left(x\right)$ is a linear function given by:

$$z_i\left(x\right)=〈k_i,x〉+q_i$$ 9

where $k_i,d_i\in R^M$. The L1-magic can find the optimal point $\overline{x}$ which is the solution using dual vectors, namely ν* $\in R^m$, and λ* $\in R^M$ that represent Karush–Kuhn–Tucker conditions [13]. L1-magic is an iterative procedure that can solve convex optimization problems which are essential in compressed sensing.

Proposed weighted CS based L1-magic

The weighting concept in CS is employed to enhance the performance of the CS during the recovery procedure using any recovery algorithm, such as the L1-magic. Since the DCT basis has energy compaction property, the energy is more concentrated in some elements compared to the distribution of energy in the remaining parts of the image. Therefore, in the present work, a weight matrix is constructed from the DCT basis matrix of the original image. The proposed weight α is considered a multiplicative factor that multiple by the chosen DCT basis matrix based on the proposed threshold value by looking to the positive and negative values of the DCT matrix from − 2 to 2. Accordingly, a weight factor is selected within the range 0–1, for generating a weighted image based on the DCT basis which is larger than the threshold and provides good PSNR. Assume a weighting matrix q which constructed based on the threshold value, which can be expressed as:

$$\text{Q}=\text{diag}\left(\text{q}\right)$$ 10

The weight matrix $\text{Q}$ is then multiplied by the DCT matrix of the original image to form a weighted image (${\text{x}}_{\text{w}}$) which can be represented as:

$${\text{x}}_{\text{w}}=\text{Q}\Psi \theta$$ 11

where Ψ represents the basis matrix of ${\text{x}}_{\text{w}}$ with N × N dimension, Ө is a column vector with a proper dimension that is composed of the corresponding coefficients, Q is the weighted matrix. Subsequently, the measurement vector y can be obtained with the dimension of M × 1, such that M < N. the compressed signal $y_{\mathit{cs}}$ can be expressed as:

$${\text{y}}_{\text{cs}}=\Phi \text{Q}\Psi \theta$$ 12

$${\text{y}}_{\text{cs}}=\text{AQx}$$ 13

$${\text{y}}_{\text{cs}}=A{\text{x}}_{\text{w}}$$ 14

where $x\in R^N$, $y\in R^M$ and the measurement matrix A is M × N matrix. A is a random matrix called the Gaussian matrix. After compression, the image will be sent to the receiver via a channel. Then, at the receiver, the image is recovered using the L1-magic recovery algorithm. The steps of the proposed system are shown in Fig. 1.

Fig. 1.

Flow diagram of the weighted compressed sensing proposed system

Figure 1 illustrates the flow of the compression process from the transmitter to the receiver. Firstly, the original colored image is converted to a gray scale image. Afterward, a random measurement matrix or sensing matrix is generated to be used in compression. A DCT is then used to transform the image to frequency- domain to support the recovery of the sparse image with frequency recovery algorithms. After extracting the DCT coefficients, the proposed threshold and weight values are determined using trial-and-error to realize a suitable PSNR. This weighing process is applied to DCT image coefficients, where a diagonal weighted matrix is multiplied by the DCT image to create the proposed weighted image. Finally, the weighted image is compressed using CS before transmit it through a channel. At the receiver, the received image is recovered using a recovery algorithm, such as Orthogonal Matching Pursuit (OMP) or L1-magic for convex optimization. The steps of the proposed method can be summarized in the following Algorithm.

Performance metrics

To evaluate the proposed compression process, several metrics are measured and considered, namely the Compression Ratio (CR), PSNR, and SSIM. Typically, the CR is the ratio between the numbers of pixels of the compressed image to a number of pixels of the original image, which is given by:

$$CR=N_2/N_1$$ 15

where $N_1$ is the number of pixels in the original image, and $N_2$ is the number of pixels in the compressed image. For example, $CR=80\%$ means that the number of pixels in the compressed image to the number of pixels in the original image is 80:100. The performance evaluation metrics to be used include the PSNR, which is a measure of the peak error in decibels relative to the peak value of the signal. Generally, the higher the PSNR, the better the quality of the compressed or reconstructed image, which is defined by:

$$psnr=10{log}_{10}\left(\frac{{255}^2}{\mathit{MSE}}\right)$$ 16

where MSE compares the quality of various image compression technique, which is defined by:

$$\text{MSE}=\frac{1}{\text{m n}}\sum _{\text{i}=0}^{\text{m}-1}\sum _{\text{j}=0}^{\text{n}-1}{\left(\text{p}\left(\text{i},\text{j}\right)-\text{q}\left(\text{i},\text{j}\right)\right)}^2$$ 17

where $p\left(i,j\right)$ is the original image, and $q\left(i,j\right)$ is the approximated image or decompressed image and (m, n) is the dimension of the image.

Moreover, the Structural Similarity Index (SSIM) is measured in the present work. It is used to determine the similarity between two images in order to enhance the performance of images. The SSIM evaluates the luminance, contrast and structure of the image, where it can be represented as:

$$SSIM\left(x,y\right)=\frac{\left(2{\tau }_{\text{x}}{\tau }_{\text{y}}+k1\right)\left(2{\alpha }_{\text{xy}}+k2\right)}{\left({{\tau }_{\text{x}}}^2+{{\tau }_{\text{y}}}^2+\text{k}1\right)\left({\alpha }_{\text{x}}^2+{\alpha }_{\text{y}}^2+k2\right)}$$ 18

where ${\alpha }_{\text{x}},{\alpha }_{\text{y}},{\tau }_{\text{x}}$,${\tau }_{\text{y}}$, and ${\alpha }_{\text{xy}}$ are the standard deviations, local means, and cross-covariance for images x and y.

Experimental results and discussion

Weighted L1-magic recovery technique for Diabetic Retinopathy images was proposed for compressed sensing framework. By adding weighting coefficients to image before compression, it can enhance the performance. The proposed weighted CS recovery algorithms depend on the image’ content, where there is important detail and unimportant details. Accordingly, a threshold of − 1.13 value was used as a margin between the significant and the insignificant DCT coefficients of the image. Afterward, the weight factor is used to increase the impact of the significant DCT coefficients and generated the weight image according to the characteristics of the image. In the proposed method, the L1-magic was applied to the proposed weighted image to recover the original image, where the best weight value is 0.98.

Diabetic Retinopathy images, as illustrated in Fig. 2, are used as a case study of the proposed weighted CS recovery algorithm to validate and test the system performance. At the transmitter, the CS was applied on the weighted image before transmitting it through the channel to the receiver. At the receiver, the L-magic is used to reconstruct the original image from the compressed, transmitted image. A MATLAB R2018b on PC that has Intel-Core i5-2410M 2.3 GHz processor with 4 GB of RAM running under the MSWin.7 operating system was used in this study.

Fig. 2.

Sample of diabetic retinopathy images

Compression results using proposed weighted L-magic

In the present work, the Gaussian matrix was used with 2D DCT to be able the image recovery in the frequency domain using the weighted L-magic. The proposed system was tested at different compression ratios (CR), namely 20%, 40%, 60%, and 80%, where a sample of the recovered images at these CR was demonstrated in Fig. 3.

Fig. 3.

Sample images after recovery using proposed weighted L-magic that were compressed at different compression ratios, where a original image, b–e recovered the image of a compressed images at 20%, 40%, 60%, and 80%, respectively

Figure 3 displayed the efficiency of the proposed weighted L-magic in the recovery of the compressed images even at high compression as illustrated at 20%. These results are also reported in Fig. 4 showing the PSNR of the recovered three images in Fig. 3 at different compression ratios, where the X-axis represent the image ID number and the Y-axis represents the PSNR value.

Fig. 4.

PSNR of the recovered sample images in Fig. 2, which compressed at different compression ratios, after recovery using proposed weighted L-magic

Comparative study of using L-magic against OMP and PCA compression algorithms

To evaluate the compression performance using the weighted L-magic, which is a convex relaxation, a comparative study with one of the Greedy algorithms, namely the orthogonal matching pursuit (OMP) using the proposed weight procedure as well as one of the traditional lossy techniques, namely the principle component analysis (PCA) compression method was conducted.

Figure 5 established visually the superiority of the proposed weighted L-magic compared to the weighted OMP and PCA compression methods. The same results were proved by measuring the PSNR and the SSIM performance metrics as reported in Table 1.

Fig. 5.

Comparative results of sample images after recovery from 60% compression ratio, where a original images, b–d corresponding recovered images from the traditional PCA, weighted OMP, and proposed weighted L1-magic

Table 1.

Comparison of 80% compression using the proposed weighted L-magic, weighted OMP, and PCA compression in terms of PSNR and SSIM

Image ID	PCA		Weighted OMP		Weighted L1_magic
	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
4	22.919	0.5319	22.924	0.5975	25.724	0.7435
5	23.779	0.833	23.782	0.5453	28.468	0.7795
6	23.116	0.5298	23.118	0.5225	26.602	0.6849
7	23.936	0.5298	23.939	0.5984	24.623	0.7591
8	25.513	0.549	25.517	0.604	26.324	0.7339
9	23.581	0.5313	23.585	0.5137	24.909	0.7187

Table 1 depicted that the weighted L1-magic is superior to the other techniques, where the weighted method is used to enhance the quality of recovered images and the L1-magic has better results compared to the weighted OMP as well as the traditional PCA. Both the PSNR and SSIM are used to compare the proposed weighted L1-magic to the other two methods. It is established that both the weighted-based recovery techniques enhance are superior to the traditional PCA, while the proposed weighted L1-magic achieved the best results. These results are owing to the generation way of the weighted images by multiplying a weight factor according to the characteristics of the image, so images can be successful recovered. Moreover, Tables 2 and 3 reported the PSNR and SSIM values at different compression ratios.

Table 2.

PSNR of the three methods using DCT transform for a sample of three images at different compression ratios

Image ID	Compression ratio (CR)	20% CR	40% CR	60% CR	80% CR
	Compression method
1	PCA compression	13.07907	15.26196	18.1845	22.6215
	Weighted OMP	14.4802	18.6809	21.8585	23.9865
	Weighted L1-magic	15.5878	19.4961	22.97	27.4298
2	PCA compression	13.54995	15.73284	18.6554	23.09238
	Weighted OMP	14.1572	19.1667	22.2016	25.0608
	Weighted L1_magic	16.1843	20.4438	23.8785	26.1704
3	PCA compression	12.30419	14.4871	17.4097	21.84662
	Weighted OMP	13.5798	17.2769	21.2383	23.6169
	Weighted L1_magic	14.3932	18.8845	22.4779	26.2284

Table 3.

SSIM of the three methods using DCT transform for a sample of three images at different compression ratios

Image ID	Compression ratio (CR)	20% CR	40% CR	60% CR	80% CR
	Compression method
1	PCA compression	0.2848	0.3216	0.4958	0.533
	Weighted OMP	0.3276	0.41	0.4586	0.5864
	Weighted L1-magic	0.3627	0.4815	0.6203	0.6737
2	PCA compression	0.2259	0.3258	0.4976	0.5242
	Weighted OMP	0.3772	0.3457	0.4685	0.5342
	Weighted L1_magic	0.369	0.4722	0.5911	0.7284
3	PCA compression	0.2655	0.3 212	0.4985	0.5107
	Weighted OMP	0.3757	0.3775	0.4457	0.5321
	Weighted L1_magic	0.3954	0.4803	0.5394	0.6352

In addition, the average PSNR and SSIM over the used images in the dataset are reported in Table 4 at different CRs.

Table 4.

Comparison in terms of average PSNR in dB using DCT weighted L-magic, weighted OMP, and PCA

Compression ratio (CR) Compression method	20%	40%	60%	80%
PCA compression	13.461	15.636	18.451	22.850
Weighted OMP	13.902	17.949	21.170	23.952
Proposed Weighted L1-magic	15.478	19.371	22.928	26.264

Finally, Figs. 6 and 7 demonstrated the relation between the CR versus average PSNR and the CR versus the average SSIM, respectively.

Fig. 6.

The relation between the CR versus average PSNR

Fig. 7.

Relation between the CR versus average SSIM

The previous results proved the superiority of proposed DCT-based weighted CS compared to the traditional PCA compression method. Likewise, the weighted CS based L1-magic achieved better PSNR and SSIM compared to using the OMP instead at different compression ratios.

Conclusion

Diabetic Retinopathy is one of the critical diseases that affect the eye. Transmitting and storage of such images is a vital task. In the present work, a weighted L1-magic recovery method was proposed in the CS framework. To enhance the compression performance, a weighting factor is multiplied by the DCT basis of the original image before compression. Consequently, a threshold value was used as a margin between the DCT basis while generating a weight image according to the characteristics of the image. In the proposed method, the L1-magic was applied to the proposed weighted image to recover the original image. The simulation results illustrated that weighted L1-magic achieved the best results in comparison to the weighted OMP and the traditional PCA. Due to the efficiency of the proposed compression method, it is recommended to apply this DCT-based weighted CS procedure in different healthcare applications such as those reported in [14–16]. However, one of the challenges is the determination of the threshold using a trial-and-error procedure; accordingly, it is recommended to automate the selection of the threshold process in the future work. Furthermore, it is suggested to adjust the proposed weighted CS using other transforms, such as the wavelet transform.