The proposed microwave imaging setup with the antenna configuration

The proposed microwave head imaging schematic with sixteen UWB bowtie antennas is shown in Fig. 1. A multilayer phantom of the human head model is simulated in CST software as a region of interest and modeled by applying changes to the internal dielectric properties of the brain, damage or brain tumor inside the brain. All physical and electrical characteristic of the multi-layer head phantom are summarized in Table 1. By arranging the sixteen elements of the bowtie antennas in elliptical geometry around the tissues the simulated microwave imaging system is performed.

Figure 1.

Schematic of the simulated microwave imaging system for brain tumor detection, in CST medium including the multi-layer structure of the designed head phantom and antenna’s positions.

Table 1.

Physical and electrical characteristics of the multi-layer brain phantom

As shown in Fig. 2, a bawtie antenna is used to image the brain based on electromagnetic modality. Many reasons such as compact size, fixed phase point, as well as vertical feeding have made this type of antenna a good option for imaging the brain [Reference Wu, Liu and Zhu19, Reference Jalilvand, Vasanelli, Wu, Kowalewski and Zwick20]. The proposed antenna is a square radiating patch with a dipole placed inside the square loop, which creates a self-complementary structure that increases the impedance bandwidth. The basic antenna structure consists of a radiating patch, a vertical feed line. The radiating patch is printed on a Rogress 5880 substrate with the dimension of 22 × 22 mm², the dielectric constant of 2.2, and the loss tangent of 0.001. Also, the proposed antenna is fed by a 50 Ω microstrip line with a variable length, and by optimizing this feed-line length, suitable impedance matching conditions have been obtained. All dimensions of the antenna are given in Table 2.

Figure 2.

The proposed printed bowtie antenna with matching balun schematic: (a) radiating patch and (b) feed-line.

Table 2.

The proposed slot antenna dimensions

In electromagnetic based brain and breast imaging, due to the high permittivity of the human head environment and the skin effect, a high ratio of the radiation wave to the head does not penetrate in the biological model such as brain and breast [Reference Kaur and Kaur21, Reference Kaur and Kaur22]. Therefore, an appropriate matching medium is needed to be placed between the antenna and the head model to compensate this reflection. In order to decrease the mismatch effects between the antenna a coupling medium is designed based on a parametric sweep of its electrical characteristics. The simulation results show that by selecting $\epsilon_r=20$ and σ = 0.5 S/m as electrical characteristics of the designed coupling medium, the optimum fit between the antenna and the head environment is achieved [Reference Ojaroudi, Bila and Salimi23]. In other words, after placing the antenna in this environment, in the presence of brain model the frequency bandwidth of the radiated wave is enhanced from 1 to 4 GHz. The six samples of the simulated transmission signals of the proposed antennas in different positions inside the designed matching medium are shown in Fig. 3. It can be seen from Fig. 3 that by determining a good choice for permittivity and conductivity of the matching medium all the sixteen antennas are radiated the required time-domain signals.

Figure 3.

Simulated transmission signals of the proposed antennas inside the proposed matching medium.

In order to show the effectiveness of using matching medium structure in radiation characteristics of the proposed antenna, the E-field radiated from the two different antennas for two frequencies are displayed in Fig. 4. It can be seen that the using of appropriate matching medium in the proposed antenna has improved the radiative property, which has increased the penetration of the wave into the brain tissue. Also, due to the increasing in penetration depth, the field distribution is concentrated in the propagation path and the presence of the target is included in the scattered wave.

Figure 4.

Simulated E-field pattern for different antenna: (a) antenna number 1 at 2 GHz, (b) antenna number 1 at 3.5 GHz, (c) antenna number 5 at 2 GHz, and (d) antenna number 5 at 3.5 GHz.

The confocal image reconstruction method

In this section, we took the raw images of the brain, which are formed by the aggregation of signals related to the transmission waves from the phantom of the brain, as input, and then we reached an improved state with transformations to reduce noise and increase the signal-to-clutter ratio. After simulating, the designed imaging setup all of the transmission signals are stored to reconstruct images in MATLAB. In the primary image extraction method, several steps are required to isolate the target response (tumor or damage) from the brain tissue, such as removing the skin effect, cross-linking the antennas, and estimating the permeability of healthy tissue, and so on. The beamforming algorithms for brain tumor detection is an easy and robust technique for tumor detection, which here we used two well-known beamformers which are delay and sum (DAS) and delay-multiply-and-sum (DMAS) for coherent signal integration to approximate the precise location of the tumor [Reference Fatwasauri and Rizkinia1–Reference Naghsh, Ghorbani and Amindavar3]. Then, using the confocal image processing method, the quality of the images is increased to extract the final range and shape of the target, which is a tumor. In addition, for effective permittivity from the point of the average dielectric constant of the multi-layer phantom 38.9 is considered in the proposed study case. In the confocal method, the final image is created by the coherent integration of the acquired signals’ energy at each focal point. Considering the M antenna in the multistatic framework, the energy relationship is defined as follows:

(1)\begin{equation} I(r)=\int_{0}^{T_w}\left[\sum_{n=1}^{M}S_n(t-\tau_n(r))\right]^2dt, \end{equation}

where S_n is the nth backscattered signal. To calculate this energy, the time delay associated with each focal point from all antennas must be calculated. These time delays are calculated as $\tau_n(r)=(2d_n(r))/(vT_s)$, where $d_n(r)=|r-r_n |$ describes the discrete-time distance between the nth transmitting antenna r_n and the focal point r, v describes the average velocity of signal propagation in brain medium, T_w is the window length, and T_s is the sampling interval. Due to reciprocity principle, instead of recording M2 signals, only $M (M -1)/2$ signals are required for the calculation of the energy profile. Through processing raw data, the 2D reconstructed images using the differential imaging scenario by subtracting a healthy brain image from the brain with tumor using DAS and DMAS beamformers are shown in Fig. 5. As illustrated in Fig. 5(b), the reconstructed image result using DMAS beamformer has a good resolution and contrast to determine the expected area of the hemorrhage tumor’s location.

Figure 5.

Reconstructed image from confocal image reconstruction method with: (a) DAS beamforming and (b) DMAS beamforming.

The proposed compressive sensing method for ultra-fast microwave imaging

In this section, we propose a new approach for CS of the reconstructed images. Figure 6 shows the flowchart of the proposed CS as a post-processing stage in a microwave brain imaging algorithm. This automated CAD system could be further used for the reconstruction of images with different selected points. The reconstructed image in the confocal image reconstruction stage is the input for the proposed algorithm.

Figure 6.

Flowchart of the proposed algorithm for tumor detection, based on compressive sensing.

The post-processing CS section is implemented in MATLAB. CS is done based on the sparse signal generations of the acquired signals. First sparse signals are generated by using the random dictionary and Gaussian measurement matrices. For inverse matrix calculation, the Moore-Penrose Pseudoinverse which is based on singular value decomposition is considered. The next step is CS reconstruction. For this purpose, the dictionary and measurement matrices are reused in each pixel [Reference Ojaroudi, Bila and Salimi23]. Figure 7 demonstrates the basic equation of CS y = Ax and the corresponding dimensionality. It is actually a combination of the equation is $f=\psi x$ (sparse representation of signal f) and $y =\Phi f$ (sampling of signal f).

Figure 7.

Matrix operations in compressed sensing.

In order to explain the CS basic principles, assuming that the length of a signal x is N × 1. If the signal is sparse with the factor of K, it can be represented as:

(2)\begin{equation} x=\psi s, \end{equation}

where ψ is an orthogonal and complete dictionary matrix. CS uses the linear operation of measurement matrix (Ψ) and the input signal as follow:

(3)\begin{equation} y=\Phi x. \end{equation}

This theory is based on the assumption that the measurement matrix Ψ is incoherent with the dictionary matrix ψ. By this condition the vector-matrix, s, can be reconstructed from $G=O(K.LogN_t)$. In order to solve this equation the following convex optimization problem is required to be solved [18]:

(4)\begin{equation} min(||s||_{l1}~~~\textrm{subject to}~y=\Phi \psi s). \end{equation}

In this study, L1-Norm based convex optimization is proposed for solving this equation. As shown in Fig. 6, matrix operations in compressed sensing can be written as follow:

(5)\begin{equation} y=\Phi \Psi s=As, \end{equation}

which Φ is a random Gaussian measurement matrix and Ψ is the discrete cosine transform (DCT) matrix.

(6)\begin{equation} A=\Phi \Psi. \end{equation}

As abovementioned, the compressive method is first applied to generate the sparse signals. Then, from these sparse signals, images are obtained from the confocal image reconstruction method. Figure 7 shows six selected samples of the main signals along with the sparse samples of the same signals. As shown in Fig. 8, in the middle frequency range where the changes of the main signals are small, the number of selected samples in the sparse sample also includes a more significant reduction. In other words, an optimal sparsity is applied in such a way that less non-zero elements are used in the range where there are no signal changes. In order to show the potential capability of the CS method, the reconstructed image by using sparse signal is shown in Fig. 9. As can be seen from the result of image reconstruction with the CS method, unrealistic targets have also appeared in the image, which is due to the zeroing of some elements in the thinned signals, which has caused coherence in the formation of the beam. But the intensity of these goals is not enough to be confused with the main goal, in other words, this problem can be easily solved with an adaptive threshold. On the other hand, due to the sparsing of the input signals to the confocal program, the processing time has been improved by 45%, which reveals the major advantage of applying the CS method in detecting brain tumor.

Figure 8.

Sparse signal generation based on compressive: (a) differential scattering parameters and (b) sparsed signals.

Figure 9.

Reconstructed image from confocal image reconstruction method using sparse signals.

In the second stage of compression we used it in image level. For this purpose, we have applied the CS algorithm to the image matrix which contains 5500 pixels. The proposed CS algorithm obtains an average of 1000 random measurements of a 5500-pixel image. In this case, the vector of sparsing coefficients “s” is equal to K = 5, assuming that the image in the DCT domain has a sparse display (not very sparse in practice). Therefore, the original image can be retrieved from its compressed form using basic pursuit method. For the image recovery, the MATLAB toolbox l ₁-MAGIC is used. This toolbox recovers the sparse signals via convex programming, where: x is original signal $(n\times 1)$, y is compressed signal $(m\times 1)$, Φ is measurement matrix $(m\times n)$, Ψ is Basis functions $(n\times n)$, s is sparse coefficient vector (to be determined) $(n\times 1)$, and $A=\Phi \Psi(m\times n)$. By inverting the matrix equation $y=A\times s$ and therefore recover s. S is a k-sparse linear combination of basis functions contained in Ψ. Note also that $y=\Phi x$. The main goal in image reconstruction is completely recovering s from y. Considering that the application method is an online method and that in this dissertation the compact sensor method is applied to both the return signal from the antenna ports and the reconstructed image. This method is based on an inversion of the matrix created with the measurement matrix and the matrix of basic functions, which ultimately leads to convex optimization, which by solving this optimization problem using the soft method $L1$, the ability to thin the signal and image. The extracted simulation results show the potential efficiency of the proposed method in matrix thinning and compact sensors. In order to apply the CS method in image level, after generating sparse signals and creating an image based on these signals, in order to perform the calculations of the relevant matrices, the pseudo-opposite Moore-Penrose operation is used for inversion. In other words, it is a matrix that can act as a partial alternative to the inverse matrix in cases where it is not possible to calculate the inverse matrix directly. According to this method, for each matrix A, there is only one matrix B so that it has the same dimensions as A, so pinv (A) is a costly method for calculating inv (A). If A is not a square matrix, or is square and singular, then inv (A) does not exist. In these cases, pinv (A) has some (but not all) properties of inv (A) [Reference Wu, Zhang, Chen and Yu24].

After inverse calculation of the matrix, the soft program $L1$ is implemented to solve the equation under the condition of minimization. The problem of convex optimization is to find the value of at least one convex function (or at most one concave function) among convex sets. The most important advantage of this type of optimization problem is that each local optimal point is also a global optimal point, and any optimization algorithm that finds a local optimal point actually finds a global optimal point. In mathematics and related disciplines, norm is used in cases where the elements are limited to positive values [Reference An, Wang and Hoorfar25]. Suppose that p-norm is defined as follows:

(7)\begin{equation} ||x||_p=(\sum_{i=1}^{n}|x_i|^p)^{1/p}. \end{equation}

If the value of p is 2, the resulting norm is called the Euclidean norm. To minimize this convex problem, the following optimizer condition is used:

(8)\begin{equation} min x||x|| ~~~~~~~s.t~Ax=b, \end{equation}

(9)\begin{equation} x_p=L1_{eqpd}(x_0,A,A_t,b,P_{dtol},P_{dmax_i},C_{gtol},C_{gmax_i}). \end{equation}

In this relation, x ₀ is the starting point which is a vector N × 1, A is the set to the function that takes an N vector and returns a vector K or a matrix K × N. If A is a function, the algorithm works in “large” mode, solving Newtonian systems through the Conjugate Gradients algorithm. A_t is a function that takes a vector K and returns a vector N. If A is a matrix K × N, A_t is not ignored. b is the vector K × 1 of the observations. $P_{(dtol)}$ is the tolerance for the dual-initial algorithm (the algorithm terminates if the duality gap is less than $P_{(dtol)}$. In this study default is 1e-3 for $P_{(dtol)}$ and $P_{(dmaxi)}$ is the maximum number of initial and duplicate repetitions and the default is 50. C_gtolis the conjugate gradient tolerance and the default value is 1e−8. C_gmaxi is the maximum number of iterations for a conjugate gradient and the default is 200. In the next step, this method was applied to images created by the confocal method.

Figure 10 shows the reconstructed images based on two least squares and the pursuit basis inversion methods in CS for two scenarios of using 500 and 1000 pixels. As shown in Fig. 10(b) using 1000 pixels in pursuit basis inversion method is able to recover the tumor information.

Figure 10.

The reconstructed images are based on two least squares and the pursuit basis inversion methods in compressive sensing for two scenarios of using (a) 500 and (b) 1000 pixels, (i) original image, (ii) compressed image, and (iii) reconstructed image.

Table 3 shows the comparison of processing parameters in two modes without and using CS method. Other post-processing parameters from the comparison of the two images are given in Table 3.

Table 3.

The comparison of processing parameters with and without applying the compressive sensing method