The conventional Nyquist sampling depends on the highest amount of rate of alteration of a signal. In this sampling scheme, samples of the signal are captured at double the highest frequency in the signal. However, this method restricts the efficient compression of a signal. Since this scheme places an enormous burden on an encoder side which acquires a vast number of samples of the signal and keeps only a few significant samples that are required to characterize the signal. Furthermore, these methods include complicated multiplications, an exhaustive coefficient search, and sorting procedure along with the arithmetic encoding of the significant coefficients with their locations. Consequently, it results in a vast storage requirement and power consumption.
On the contrary, compressed sensing is emerging as the most recent sampling scheme, which allows compression and signal reconstruction from the minimum number of measurements. In this scheme, the signal acquisition and compression are performed simultaneously at the encoder end. The signal is recovered back with a higher probability of success by using different optimization algorithms. Thus, CS results in a significant reduction in storage requirement and further reduces power consumption.
Compressed sensing has been implemented in diverse fields, including medical imaging, radar imaging, cameras, coding theory, geophysics, and astronomy.
CS-based biomedical imaging has been shown enormous interest and growth in recent times. Recently, researchers Wang, Bresler, and Vasilis [1] had reported an in-depth survey and success on the application of CS in MRI, CT, PET, SPECT, optical imaging, and ultrasound imaging. The researchers Lustig et al. [2] had been successfully used CS to MRI. Thus, CS-based MRI could speed up the data acquisition process by reducing the scan time, and this allows us to examine a higher number of patients.
The dictionary learning-based reconstruction of MR images is one of the recent developments and shows great potential in medical applications. The researchers Ravishankar and Bresler [3] reconstructed MR images based on the dictionary learning approach. Furthermore, they [4] had successfully proposed learning of doubly sparse transform for the images.
Further, the application of CS in radar imaging has been an additional growing field of interest. Yang et al. [5] had successfully designed the segmented recovery scheme for CS-based SAR (Synthetic Aperture Radar) imaging. Bu et al. [6] had developed a CS algorithm for SAR imaging. They had reconstructed the data of good quality with limited observations and thus results in the reduction of storage requirement. Deng et al. [7] had successfully proposed CS-based image coding. They had been achieved a robust performance against the lossy channel compared to conventional coding methods.
Li and Qi [8] proposed a nonlocal Douglas-Rachford (NLDR) algorithm, based on Douglas-Rachford splitting to solve low-rank optimization problems constrained by the CS measurements. Shen et al. [9] sparse Bayesian dictionary learning based compressed sensing-based inpainting of aqua moderate resolution imaging.
Furthermore, some researchers had been implemented and successfully tested the real-time hardware for CS-based applications. For example, the single-pixel camera based on compressed sensing had developed by Duarte et al. [10]. Further, Nagesh and Li [11] had developed color imaging architecture based on the combination of single-pixel CS camera and Bayer color filter. Similarly, single-pixel CS was applied for remote sensing by researcher Ma Jianwei [12], which results in the reduction of storage requirement and the computational cost of imaging.
The author Liquan Zhao et al. [13] has implemented compressed sensing for monitoring the images of the transmission line. These images are compressed and reconstructed using a compressed sensing technique which, reduces the overall operational cost of a system.
In a nutshell, the Nyquist sampling put an enormous burden on the encoder side due to a massive number of samples of an acquired signal, particularly for audio/speech, ECG, image, and video signals. This fact inspired the study of compressed sensing as a potential solution for sampling, compression, and reconstruction of a signal. Intuitively, sparsity represents a large amount of energy concentration in a few numbers of coefficients. Several real-world signals such as speech and image are sparse or compressible in some transform domain. For example, images are compressible in basis algorithms such as JPEG and JPEG2000. The compressed sensing uses this sparsity property to compress and recover the signal effectively.
Traditionally, the random sensing matrices widely employed for signal compression in CS. The random Gaussian sensing matrices are entirely unstructured. Therefore, these matrices resulted in an enhanced computational complexity and increased memory storage requirement. Hence, the practical implementation of random Gaussian sensing matrices is costly. Further, sensing technologies need structured measurement matrices to accomplish different applications. Thus, the Toeplitz measurement matrix is one of the structured class matrix and widely used in a different field of applications, such as MRI [14], Synthetic Aperture Radar (SAR) [15], and channel estimations [16]. The Toeplitz matrices possess some exceptional features, such as these matrices generated with a smaller number of entries. Moreover, different techniques are available to speed up the matrix multiplication, which further may result in fast signal reconstruction. So far, the work carried out under the statement that the Toeplitz matrices generated using randomly drawn entries. Recently, Dirksen et al. [17] proposed partial Gaussian circulant matrices for 1-bit compressed sensing. Furthermore, Jie et al. [18] proposed compressed sensing matrices using vector spaces for signal processing.
In the literature, so far, different methods are proposed for the optimization of random Gaussian measurement matrices [19, 20]. However, these methods randomly draw entries to generate random Toeplitz matrices and further optimize them with proper optimization methods. Nevertheless, during an optimization process, these methods lose control over the structure of measurement matrices. Also, researchers Abolghasemi, Jarchi, and Sanei [20] proposed the gradient-decent-based method to optimize mutual coherence. In this method, the modified cost function is followed by the gradient-descent minimization method to optimize the sensing matrices iteratively. This method shows the robustness in handling complex values. The researcher Duarte-Carvajalino and Sapiro [21] proposed the non-iterative way to calculate t-averaged mutual coherence. This method intended to make a Gram matrix closer to the identity matrix. However, this method ignores the negative eigenvalues and thus presents the problem of complex values which, causes the algorithm to fail.
It had proved that if sensing matrices satisfy restricted isometry property (RIP) [22], then there has been a high probability of superior quality signal reconstruction. On the contrary, the RIP is impractical to evaluate. Therefore, another way to satisfy the RIP and guarantee the exact reconstruction of a signal is to compute the mutual coherence (μ) between the sensing matrix (Φ) and the sparsifying matrix (Ψ). The mutual coherence (μ) of a dictionaryDM × N = ΦM × N × ΨN × N is defined as the biggest absolute and normalized inner product among different columns of D [19] and given by the equation (1).
$$ Minimize\mu (D)=\sqrt{N.}{\max}_{1\le i,j\le N,i\ne j}\frac{\left|{d}_i^T{d}_j\right|}{\left\Vert {d}_i\right\Vert .\left\Vert {d}_j\right\Vert } $$
(1)
where N is the length of the input signal. From linear algebra, \( 1\le \mu \left(\Phi, \Psi \right)\le \sqrt{N} \).
Thus, the minimization of mutual coherence may be one of the effective ways to boost the recovery performance of compressed sensing matrices [23].
This paper proposes the optimization of Toeplitz sensing matrices based on evolutionary algorithms such as Genetic Algorithm (GA), Simulated Annealing (SA), and Particle Swarm Optimization (PSO) algorithm for image compression. The minimization of mutual coherence may be one of the effective ways to boost the recovery performance of compressed sensing matrices. Thus, this paper proposed the minimization of the mutual coherence (μ) between the sensing matrix (Φ) and the sparsifying matrix (Ψ) using evolutionary algorithms. The proposed optimization approach provides the best random Toeplitz vector, which consequently minimizes the mutual coherence (μ) between the sensing matrix (Φ) and the sparsifying matrix (Ψ). Further, the Toeplitz measurement matrix generated using the best random Toeplitz vector and finally applied for image compression. Furthermore, this proposed approach retains the structure of the Toeplitz matrix and improves the image recovery performance of compressed sensing.
Since the novel approach of the proposed optimization method is based on an evolutionary algorithm, and hence, it is entirely different from the state-of-the-art optimization methods. Until now, all the state-of-the-art optimization methods use non-evolutionary approaches for optimization of sensing matrices and thus tend to lose the structure of sensing matrices. Therefore, it is not practicable to compare proposed evolutionary approaches directly with non-evolutionary approaches of state-of-the-art methods. Thus, rather than, we have compared the performance of the proposed optimized Toeplitz sensing matrices based on evolutionary algorithms with non-optimized Toeplitz sensing matrices.
The main contributions of the proposed work are as follows:
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1.
We proposed a novel approach for the optimization of Toeplitz sensing matrices based on Evolutionary algorithms.
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2.
We proposed the first approach for the optimization of Toeplitz sensing matrices based on the Genetic Algorithm.
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3.
We proposed the second approach for the optimization of Toeplitz sensing matrices based on the Simulated Annealing (SA) Algorithm.
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4.
We proposed the third approach for the optimization of Toeplitz sensing matrices based on the Particle Swarm Optimization (PSO) Algorithm.
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5.
We investigated the signal reconstruction performance using Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP) algorithm for GA, SA, and PSO-based optimization approaches.
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6.
Finally, GA, SA, PSO-based optimization approaches exhibit a significant reduction in the mutual coherence (μ) and thus improved the recovery performance of 2D images compared to non-optimized Toeplitz sensing matrices.
The organization of the paper is as follows: Section 1 presents the formulation of an optimization problem. Section 2 elaborates on the proposed optimization method based on evolutionary algorithms. Section 4 presents results and discussion. Finally, Section 5 presents the conclusions.
