Volume 53, Issue 12 p. 1452-1471
Research Article
Free Access

Analysis of the Super-Resolution Effect on Microwave Tomography

Nikolai Simonov

Nikolai Simonov

Radio Environment and Monitoring Research Group, ETRI, Daejeon, South Korea

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Soon-Ik Jeon

Soon-Ik Jeon

Radio Environment and Monitoring Research Group, ETRI, Daejeon, South Korea

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Bo-Ra Kim

Bo-Ra Kim

Radio Environment and Monitoring Research Group, ETRI, Daejeon, South Korea

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Seong-Ho Son

Corresponding Author

Seong-Ho Son

Radio Environment and Monitoring Research Group, ETRI, Daejeon, South Korea

Correspondence to: S.-H. Son,

[email protected]

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First published: 29 October 2018
Citations: 4

Abstract

This study investigates the spatial resolution (SR) and super-resolution effect in microwave tomography in terms of single-frequency measured data. The applied method is based on our recently proposed concept of average SR (ASR). We apply truncated singular value decomposition to calculate a regularized forward-modeling matrix and to limit the truncation index by the acceptable level of the imaging noise. A simple relation of the ASR with the truncation index calculates the ASR in the imaging zone. The described method to calculate SR is quite common, and it considers not only the geometrical parameters of the microwave tomography system and object under test but also the noise in the measured data. This method is applicable to the linear and nonlinear considerations of the inverse scattering problem with respect to two- or three-dimensional solutions. In particular, our investigation confirms the conclusion of some other authors that applying nonlinear inverse scattering methods can achieve the super-resolution imaging even when based on far-field measured signals only.

Key Points

  • This paper investigates the super-resolution effect in microwave tomography by utilizing a new method of analysis of spatial resolution
  • The proposed concept of average spatial resolution can examine the dependence of imaging noises on measurement noises
  • This work confirms that the nonlinear consideration has advanced in providing the super-resolution effect, which can occur even with far-field measured signals

1 Introduction

Microwave tomography (MT) is a method of microwave imaging based on multistatic measurement configuration, which often exploits the cylindrical array of transmitting (Tx) and receiving (Rx) antennas (Pastorino, 2010; Rubæk et al., 2007). The main area of potential MT application is the medical imaging of inner permittivity distribution inside bodies. One of the important characteristics of image quality is the spatial resolution (SR), which defines the minimal distance between the distinguishable points of the object or the minimal image size of a small object. For the purpose of SR reduction, MT systems utilize some matching liquid or matching solid materials and the advantage of multi-input/multi-output (MIMO) approach (Rubæk et al., 2007; Simonov et al., 2012a).

Most imaging theories, including optics and the analysis of microwave or synthetic aperture radar (SAR) imaging, utilize the far-field data and linear (or Born) approximation. In the linear consideration, SR cannot be smaller than the half-wavelength Rayleigh (or diffraction) limit in the medium (Born & Wolf, 1997). A more detailed consideration reveals that this limitation is valid if the imaging system can detect only the far field of a body's scattered waves and the linear inverse solver. Overcoming the Rayleigh limit is practically possible if to gather enough information on the near-field components of scattered waves (Cui et al., 2004; Gilmore et al., 2010; Simonov et al., 2012b, 2014). Cui et al. (2004) regarded that the super-resolution phenomenon occurs when the image resolution is less than 0.25 wavelength. In this work, we consider this quarter-wavelength definition of the super-resolution, although some authors use the half-wavelength Rayleigh criterion.

Microwave reconstruction algorithms control the imaging noise and SR with the help of the regularization parameters (Hansen, 1997; Pastorino, 2010). The imaging noise depends on the receiver noise and other uncertainties in the measured data. Regularization smooths images, increases SR, and reduces the dimension of the image space of the image resolution matrix. Some optimal regularization parameter exists, and any attempt to reduce the corresponding SR leads to the catastrophic growth of the image reconstruction noise. Therefore, we can conclude that SR and the super-resolution effect strongly depend on the signal-to-noise ratio of the detected far and near fields. SR also depends on the sensitivity of Rx antennas to the scattered near field, the total number of measured signals, and the values of illumination-sampling spacing (period).

Note that probe-type antennas with an aperture size remarkably smaller than the half-wavelength provide better sensitivity of Rx antennas to the near field, whereas antennas with wider apertures have worse resolution properties because of the space smoothing of the measured electric field. In this study, we limit our investigation to the case of infinitesimal-sized (point-source) antennas.

According to the Whittaker-Kotel'nikov-Shannon sampling theorem (Meijering, 2002; Unser, 2000), the SR is restricted in the sampling period and corresponds to the antenna spacing in the MT system. Therefore, expecting the sampling period to be smaller than the required imaging SR for detecting scattered waves seems natural. However, doing so for electromagnetic scattering is not always necessary. For example, Bucci and Franceschetti (1989), Bucci and Isemia (1997) demonstrated that the spacing of antennas could be larger than the Nyquist period, and the theorem specified above only restricts the period of scattered field sampling near the body surface. In addition, our investigation showed that the MT system overcomes this limitation because of the positive effect of the multi-illumination approach.

Image reconstruction usually provides to the inhomogeneous distribution of SR over an imaging zone (IZ). This fact is obvious in the case of inhomogeneous permittivity distribution. However, the heterogeneous distribution of SR can take place even for homogeneous materials: The resolution can be remarkably smaller near the antennas than in the central area of the antenna array.

In contrast to linear imaging, the nonlinear consideration reveals some unusual properties. Chew and coworkers (Cui et al., 2004) discovered that for the nonlinear inverse techniques, SR could be remarkably smaller than that for the linear approach and that the super-resolution effect is achievable even if the system measures far-field signals only. They used the following explanation for this effect: “even though … only scattered waves corresponding to propagating waves can be measured, the scattered waves contain high resolution information about the scatterer because of the evanescent-propagating waves conversion in a multiply scattered field.”

Okhmatovski et al. (2012) used an original method to noniteratively solve a nonlinear inverse scattered problem. They examined the advantage of using a Veselago lens in the imaging experiment to observe the super-resolution using far-field measured data only.

The concept of SR is not strictly defined because of several reasons. First, the SR definition, which is based on the full width at half maximum (FWHM) of the point spread function (PSF), depends on the profile of this function. In MT, the profiles of the PSF are usually nonregular or nonsymmetric even for the homogeneous background and linear imaging. The situation with the PSF becomes especially complicated in the case of inhomogeneous media, in which the PSF can be especially irregular. The same properties of the PSF take place, particularly in the low-frequency case of electrical capacitance tomography (Lucas et al., 2015).

Second, the methods of estimating SR for SAR, despite their convenience, speed, and visibility, are quite approximated even in the air environment and do not consider the measurement noises properly. Most of them are based on space spectral analysis, consider propagating the scattered waves only, and replace the actual space spectrum profile with the rect function, which is related to the sinc PSF (Cui et al., 2004; Sheen et al., 2001). Other authors (Devaney, 2005; Devaney & Dennison, 2003) considered SR implicitly and calculated the PSF instead, which do not demonstrate regular profiles.

The current study proposes the concept of an average SR (ASR), which helps to solve the above mentioned problems with SR. The important feature of our approach is that it extends the definition of SR to the case of nonlinear imaging in an inhomogeneous body even if it is arranged in the near zone of the array. In addition, the singular value decomposition (SVD) analysis of the sensitivity matrix helps to consider the noises of measurements.

The ASR assumes a homogeneous distribution of SR in the IZ and can be modeled, for example, with the help of sinc-like basis functions (BFs; Simonov et al., 2017). As far as these BFs are centered in the nodes of the grid with spacing equal to SR, their number is equal to the effective rank of the corresponding image resolution matrix (Wiggins, 1972). Despite the property of averaging in volume, the ASR concept is applicable for investigation of more local SR as well, if to consider a smaller investigation domain and then scan this local region over the whole IZ.

This study examines the single-frequency imaging, investigates the dependence of SR on the MT parameters, and demonstrates some super-resolution cases. We consider imaging as an inverse problem in the framework of L2-norm minimization and do not refer to compressive sensing methods, which are based on the L1/L2-norm minimization (Bingchen et al., 2012). However, some studies claimed that a higher resolution could be achieved with the application of compressive sensing-based imaging algorithms (e.g., Yang & Ling, 2017; Zhu, 2011; Zhu & Bamler, 2012).

2 Theoretical Base of SR Investigation

2.1 Determination of the Concept of ASR

As mentioned in section 1, the existing methods to estimate SR in the SAR imaging theory have certain limitations and are not suitable for MT configuration (Figure 1). First, they do not consider explicitly the effect of measurement noise on SR. Second, they utilize a simple estimation of SR based on the spatial spectrum of scattered waves in the far-field zone or consider PSFs.

Details are in the caption following the image
Measurement configuration of the microwave tomography system.

For MT, the PSF-based estimation of SR is limited by the irregular profiles of PSFs because the testing body in MT is arranged in the near zone of the antennas. Consequently, the PSF can have an irregular profile and an uneven distribution of SR in space (Figure 2) despite the homogeneous background and Born (i.e., linear) approximation. Clearly, the PSF profile can be especially irregular in the case of nonlinear imaging in heterogeneous media. Therefore, accurately defining the local SR in MT, especially for nonlinear image reconstructions, is not possible. In this situation, Gilmore et al. (2010) introduced the alternative concept of separation resolution.

Details are in the caption following the image
Example of the heterogeneous distribution of the spatial resolution (SR) over the imaging zone for the case of the microwave tomography imaging algorithm with Tikhonov regularization. The parameters of the ED breast tissue model are presented in Table 1. Tx = transmitting; Rx = receiving.

To overcome these problems, this work introduces the concept of ASR instead of the local SR of the image. The ASR approach is quite general because it is valid for any background with losses and considers the noises of measurements. The developed algorithm calculates the ASR for the linear and nonlinear imaging considerations directly in the entire IZ or in its smaller investigation domains, thus providing an analysis of the local SR.

Bayat and Mojabi (2016) described a similar framework for SR analysis, but they did not derive an ultimate equation for the SR calculation.

Let us apply the measurement configuration for the MT system, as illustrated in Figure 1. The configuration includes a circular array with elements, indicated as bold spots, and an object under test, which are both immersed in the coupling liquid. In this study, we use vertically polarized point Tx and Rx antennas. The array can be planar, scanning in the vertical direction and rotating in the horizontal plane.

The evaluation of ASR is based on the SVD analysis of the Jacobian (or sensitivity matrix), which is used for the iteration image reconstruction by the Gauss-Newton method (e.g., Rubæk et al., 2007, Eq. (7)). The Jacobian J is a Fréchet derivative of vector urn:x-wiley:00486604:media:rds20747:rds20747-math-0001 as a function of the vector of the object contrast distribution C = ε/εb − 1:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0002(1)
Here ε and εbare the complex permittivity of the object and the background, respectively, defined in the nodes of a fine mesh, and urn:x-wiley:00486604:media:rds20747:rds20747-math-0003 and urn:x-wiley:00486604:media:rds20747:rds20747-math-0004 are the vectors of the measured signals of scattered and incident (without the object) waves, respectively. The Jacobian matrix can be defined both for the Born approximation and for the nonlinear consideration (see section 2.5). In the last case, the definition is based on Green's function for the heterogeneous medium in 27 and 29. Jacobian J provides a linear mapping of small variances of contrast δC to the space of the measuring signals:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0005(2)

Note that the equivalent definitions of the sensitivity matrixes exist in many other works (e.g., Cui et al., 2001, Eq. (12); Bayat & Mojabi, 2016, Eq. (13)).

Therefore, the following relation between small receiver noises urn:x-wiley:00486604:media:rds20747:rds20747-math-0006 and small reconstruction noise Cn of the contrast is valid:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0007(3)

The Jacobian is an ill-conditioned matrix, and a direct inverse solution for Cn provides a huge numerical error. To obtain a reasonable approximate inverse of equation 3, different types of regularization are used, providing smooth (space-filtered) solutions (Chew, 1995; Hansen, 1997; Pastorino, 2010). In the present work, we apply a simple truncated SVD (TSVD) regularization (Hansen, 1997; Pastorino, 2010) that provides a fine control of the reconstruction noise.

The SVD of matrix J is its factorization as the product of three matrices:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0008(4)
where U and V are unitary matrices, urn:x-wiley:00486604:media:rds20747:rds20747-math-0009, urn:x-wiley:00486604:media:rds20747:rds20747-math-0010 is a vector of real singular values (Hansen, 1997), and the superscript H is the Hermitian transpose (conjugate transpose) operator. Bayat and Mojabi (2016) also analyzed the TSVD of the sensitivity matrix A, which is equivalent to the Jacobian (equation 3 herein). Using the TSVD method, we replace diagonal matrix S with a truncated diagonal matrix Sr, in which all singular values of urn:x-wiley:00486604:media:rds20747:rds20747-math-0011 with an index of more than r are replaced with zeros. The TSVD corresponds to a filter of singular values with a sharp-cut profile. It can be used to calculate the regularized Jacobian matrix Jr:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0012(5)
Therefore, the corresponding image resolution matrix Rr (Wiggins, 1972) of rank r can be defined as follows:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0013(6)
where the superscript + denotes the Moore-Penrose pseudoinverse and Ir is an identity matrix truncated in the same way as Sr. The image resolution matrix Rr is symmetric, and its columns are the PSFs in column form.

Simple calculations demonstrate that matrix Rr 6 provides imaging with an inhomogeneous distribution of SR in the IZ. Specifically, the SR for points near the antennas is remarkably smaller than that in the center of the IZ. The same property demonstrates the Tikhonov regularization method (Figure 2), which is equivalent to filtering singular values with a smooth profile filter (Pastorino, 2010, Eq. (5.6.5)). The TSVD and the Tikhonov regularization are related to L2-norm minimization. Note that the PSFs of the resolution matrix have an irregular profile in both cases. Nevertheless, the local SR values in Figure 2 are estimated using the FWHM of the PSFs, that is, the Rayleigh criterion (Born & Wolf, 1997).

2.2 Relation Between Measurement and Imaging Noises

Let us replace matrix J in equation 3 with its TSVD approximation Jr 5 and obtain the following simple matrix-vector equality after a simple transformation:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0014(7)
where Cnv = VH · Cn, nu = UH · nm.
Considering the noise variables Cn, Cnv, nm, and nu in the statistical ensemble representation is natural. Then, we define the following variances σm and σc of the measurement and the reconstruction noises, respectively:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0015(8)
urn:x-wiley:00486604:media:rds20747:rds20747-math-0016(9)
where 〈〉 is the ensemble averaging, Nm is the number of all measured signals, and Nf is the number of nodes in the fine mesh of the IZ. Definition 3 proves that the variance σm is equal to the noise-to-signal ratio (NSR) averaged over all the measured signals. When Cnv in equation 9 is replaced with its pseudoinverse solution of 7, we obtain the following formula:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0017(10)
where σi is the singular value that is the diagonal element of Sr 5. Using the white noise model for the noise vector nm 3 of the measured signals, we can conclude that 〈|nu(i)|2 in equation 10 does not depend on index i. Then, quation 8 can help to derive the following equality for any i:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0018(11)
Ultimately, equations 10 and 11 build the main equation for the analysis of ASR:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0019(12)

2.3 Definition of ASR

We now explain our method of ASR evaluation by utilizing equation 12. Choosing an acceptable value of the ratio urn:x-wiley:00486604:media:rds20747:rds20747-math-0020 is necessary. Then, using equation 12, we can find the effective rank r of the resolution matrix Rr 6, which depends on the selected ratio urn:x-wiley:00486604:media:rds20747:rds20747-math-0021. The physical sense of the parameter r is the same as the number of well-resolved (point) targets (Devaney, 2005; Devaney & Dennison, 2003). In other terms, parameter r is a dimension of the image space of the image resolution matrix.

The calculation of ASR is based on the definition of the least square projection matrix (Scharf, 1991Rbr, which has the same rank r as the resolution matrix Rr but provides imaging with a constant SR over the IZ. Naturally, matrix Rbr can play the role of the image resolution matrix (Wiggins, 1972) in the context of microwave imaging. For example, the projection matrix Rbr can be constructed with the help of matrix Br, which contains r columns of smooth sinc-like cardinal cubic spline BFs (Simonov et al., 2017). These functions are centered in the nodes of a coarse rectangular grid with spacing that is equal to the SR and is almost orthogonal. Therefore, Br has rank r as well. The following equation provides the relation between the matrixes Rbr and Br:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0022(13)
where urn:x-wiley:00486604:media:rds20747:rds20747-math-0023 is a dual matrix. Simonov et al. (2017) found that the matrix Rbr provides imaging with a practically constant SR over the inner points of the IZ. However, inside the border layer with thickness surrounding the SR, the boundary of the IZ cuts the PSF, and the SR is not well defined.

Therefore, we can regard SR for Rbr as an average value of SR that is related to matrix Rr.

When the cardinal cubic spline BFs are arranged evenly in the coarse square grid of the IZ, with spacing equal to SR, the following simple equations can be used to calculate the value D of ASR:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0024
urn:x-wiley:00486604:media:rds20747:rds20747-math-0025
where SIZ and VIZ are the transverse section and volume of IZ, respectively. However, these simple equations are rough approximations only, and refining the SR estimation is necessary.
In the 2D case, the ASR estimation can be refined if it is used in the triangular grid with edges D rather than in the square grid because the triangular grid provides a more even distribution of the SR than the rectangular grid. In addition, the nodes that are arranged on the surface of (or near) the IZ must be excluded. To derive the refined equation, we can use a simple equality:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0026(14)
where Ncells is the number of triangles and urn:x-wiley:00486604:media:rds20747:rds20747-math-0027 is the area of the regular triangle. In this case, the number of nodes is equal to r, but Ncells is different:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0028(15)
where Nsurf is the number of nodes on the surface of the IZ (i.e., the outline in the 2D case). In the case of the cylindrical IZ with a radius RIZ and a triangular grid, Nsurf ≅ 4α−1 RIZ/D. Then, D can be represented by the following recursive function:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0029(16)
The 3D consideration can be refined by the triangular prism grid with edges D rather than the cubic grid. This type of grid is natural for the cylindrical configuration of our MT system (Figure 1). Similar to that in the 2D case, we can use a natural equality for the refined equation:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0030(17)
where Ncells is the number of triangular prisms with edges D, and urn:x-wiley:00486604:media:rds20747:rds20747-math-0031. The total number of nodes is equal to r, but Ncells, 3D is different from that in the 2D case. For the phantom and cylindrical IZ with height hIZ (Figure 1) of the cylindrical body, we can obtain the following equation similar to 15:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0032(18)
where Nsec = (hIZ/D + 1) is the number of horizontal sections of the grid and Nnodes = r/Nsec is a number of nodes in each horizontal section of the grid. Ultimately, D can be evaluated from the following recursive function:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0033(19)

2.4 Comparison With SR Based on the PSF

To verify the derived equations 1619, ASR is compared with SR, calculated using the PSF. However, for the near-field zone, the PSF-based estimation of SR is not strictly defined and depends on the function profile as mentioned previously. For example, for the sinc functioning as the PSF, the SR corresponds to the full width at the level 0.6034 rather than at 0.5. In addition, the regular sinc-like PSF profiles only take place for a uniform background and the materials of an object (e.g., see Figure 4). The following methods of SR estimation are certainly more reliable than FWHM for such PSFs:
  1. based on the first zero of the PSF and
  2. based on the integral over area (or volume) of the PSF.

The related equations are exact for the sinc function and for its 2D and 3D extensions. Although the PSF methods of the SR estimation are not correct in the general case, we can apply them to homogeneous media for ASR verification.

2.4.1 Two-Dimensional Imaging

The first-zero SR urn:x-wiley:00486604:media:rds20747:rds20747-math-0034, urn:x-wiley:00486604:media:rds20747:rds20747-math-0035 in both x and y directions and its mean value DFZ can be obtained using the following definitions with 2D PSF:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0036(20)
where (x0, y0) are the coordinates of the maximum PSF. The integral of PSF is used for the following equation, which defines SR DInt:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0037(21)

2.4.2 Three-Dimensional Imaging

In the 3D case, the first-zero approach defines SR urn:x-wiley:00486604:media:rds20747:rds20747-math-0038, urn:x-wiley:00486604:media:rds20747:rds20747-math-0039, urn:x-wiley:00486604:media:rds20747:rds20747-math-0040 in the x, y, and z directions and its mean value DFZ using the following relations:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0041(22)
where (x0, y0, z0) are the coordinates of the maximum PSF. Similar to that in the previous section, SR DInt is defined by the integral of the PSF:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0042(23)

2.5 Calculation of the Jacobian in the Born Approximation

In the linear (Born) approximation, the Jacobian is defined by Green's function of the background material and does not depend on the contrast of the object. The Born approximation is acceptable in the case of a small object contrast.

2.5.1 Two-Dimensional Consideration

The 2D Jacobian matrix J2D in the Born approximation can be calculated by the following equation:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0043(24)
where Δ is the spacing of the fine mesh in the IZ, k is a complex wavenumber in the background, G2D is the matrix of the grid-sampled 2D Green's function (Chew, 1995, Eq. (4.2.1)) for the background, rTx and rRx are the position vectors of the Tx and Rx antennas, respectively, and ri is the position vector related to the pixel with index i.

2.5.2 Three-Dimensional Consideration

The 3D Jacobian matrix J3D in the Born approximation can be calculated by the following equation:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0044(25)
where G is the matrix of the grid-sampled 3D dyadic Green's function (Chew, 1995, Eq. (7.1.19)) in the background and urn:x-wiley:00486604:media:rds20747:rds20747-math-0045 is the vertical unit vector, which relates to the vertical polarization of the antennas.

2.6 Calculation of the Jacobian for Nonlinear Consideration

In conducting the nonlinear analysis, we consider a mutual coupling (or multiscattering) among all pixels or voxels of the imaging object (Chew, 1995; Pastorino, 2010). This step is valid even in the case of a body with a uniform material. This coupling leads to the nonlinear dependence of the Jacobian on the object contrast.

The equations below include multiscattering by the factor of the matrix IGC2D ≡ (I − G2D, mc · C)−1 in 27 or IGC ≡ (I − Gmc · C)−1 in 29.

2.6.1 Two-Dimensional Consideration

The 2D Jacobian matrix urn:x-wiley:00486604:media:rds20747:rds20747-math-0046 for a heterogeneous medium is defined by the following equation:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0047(26)
where urn:x-wiley:00486604:media:rds20747:rds20747-math-0048 is the matrix of the grid-sampled 2D Green's function of the object immersed in the background (in comparison with Pastorino, 2010, Eq. (4.10.15)):
urn:x-wiley:00486604:media:rds20747:rds20747-math-0049(27)
where G2D, mc is the [Nf × Nf] matrix of the Green's function related to the mutual coupling among Nf pixels and C is the [Nf × Nf] diagonal matrix of the object contrast.

2.6.2 Three-Dimensional Consideration

The 3D Jacobian matrix urn:x-wiley:00486604:media:rds20747:rds20747-math-0050 for a heterogeneous medium can be calculated by the following equation:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0051(28)
where Ghg is the matrix of the grid-sampled dyadic Green's functions of the object immersed in the background:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0052(29)
Here we regard Green's dyadic Gmc of the mutual coupling. The diagonal contrast matrix C has [3Nf × 3Nf] sizes considering three components of the electric field vector.

The Jacobian in equations 26 and 28 depends nonlinearly on the object contrast.

3 Characteristics of the SR in the 2D Consideration and Born Approximation

For simplicity, we use the term SR as an equivalent to the term ASR.

3.1 Dependence of SR on the Spacing of Tx Antennas

3.1.1 Computed Results

Figure 3 shows examples of the SR dependence on the spacing of Tx antennas. We imply the arc spacing of the array elements for all simulated results and occasionally omit the word arc for simplicity. We consider the case of a fixed number of Rx antennas distributed evenly over the contour of the 150-mm diameter of the circular array. The background is the coupling liquid PG 90% (90% solution of propylene glycol in water); its dielectric parameters at testing frequencies of 900, 1,500, 2,100, and 2,900 MHz are shown in Table 1. Here ε is a real part of the permittivity, and σ is the conductivity. We utilize the noise ratio urn:x-wiley:00486604:media:rds20747:rds20747-math-0053 in equation 12 to calculate the ASR 16.

Details are in the caption following the image
Computed dependencies of the spatial resolution on the arc spacing of transmitting (Tx) antennas at a fixed spacing of receiving (Rx) antennas at different frequencies for the 2D consideration and Born approximation. The background is propylene glycol (PG) 90%, 15 Rx antennas are used, and the number of Tx antennas varies in the (4 … 2,048) interval. (a) Array diameter of 75 mm. (b) Array diameter of 150 mm.
Table 1. Dielectric Parameters and Diffraction Limits for the PG 90% Background
Frequency (MHz) Bath liquid PG 90%
Ε σ (S/m) Diffraction limit Dlim (mm)
900 21.61 0.72 ~36
1,500 15.23 1.09 ~26
2,100 12.12 1.33 ~21
2,900 9.83 1.56 ~16
  • Note. PG = propylene glycol.

Figure 3 demonstrates that the SRs are below the Rayleigh limits (RLs) for all the considered frequencies, and that the super-resolution occurs if the spacing of the Tx antennas is smaller than some critical values: (1) approximately 3 mm for a 75-mm array diameter and (2) less than 1 mm for a 150-mm array diameter. Figure 3 shows also that for the multistatic setup, the SR can be smaller than the RL even when the spacing of both Tx and Rx antennas exceeds the RL.

Another interesting feature is that SR slowly (logarithmically) decreases when the spacing of Tx antennas tends to 0. Clearly, the MT configuration tends toward setup of the electrical impedance or electrical capacitance tomography. For the last low-frequency system, the SR depends on the spacing of the electrodes but is not limited by the wavelength (Lucas et al., 2015). In any case, the continuing decline of SR is explained by the application of the point-source antennas, but it must be limited with finite apertures of antennas.

3.1.2 Verification of the Computed Results

For verification, we compare the ASR, which is calculated by equation 16, and SR for the PSF in the center of the IZ (Figure 4). The PSFs are calculated on the basis of the image resolution matrix Rr 6.

Details are in the caption following the image
Reconstructed images and profiles of the point object in the center of the imaging zone (point spread function) for the setup of Figure 3 and a frequency of 2,100 MHz. (a) The array diameter is 75 mm, and the spacing of transmitting antennas is 0.1 mm. (b) The array diameter is 150 mm, and the spacing of the transmitting antennas is 0.2 mm.

Figure 5 illustrates the comparison of ASR 16 with the PSF-based SR DFZ 20 and DInt 21. Although the calculated values of SR are close, diversions can be observed in the values DFZ and DInt of the PSF-based SR, thus indicating their principal inaccurate definition.

Details are in the caption following the image
Comparison of the average spatial resolution (ASR) with the point spread function (PSF)-based spatial resolution (SR) for the setup of Figure 3 and frequency of 2,100 MHz: Circles denote the data of the DFZ 20, and crosses denote data of DInt 21. (a) Array diameter is 75 mm. (b) Array diameter is 150 mm. PG = propylene glycol; Tx = transmitting.

Values DFZ and DInt of the PSF-based SR demonstrate interesting property, not smooth and threshold-like dependence on the spacing of Tx antennas. This feature is explained by the fact that the cylindrical symmetry causes some singular vectors of the matrix V in 4 and 5 to have the same spatial pattern but different angles of rotation.

3.2 Equivalency of Information Gathering by Tx or Rx Antennas

3.2.1 Simulated Data

With equations 2429 demonstrating symmetry that relates Tx and Rx antennas, we can expect an equal contribution of the information gathered by Tx or Rx antennas to the SR. The results presented in this section confirm this conclusion, but they also demonstrate a general rule: SR depends mostly on the total number of measurements and weakly depends on the specific numbers of Tx and Rx antennas. We call this property equivalency of information gathering by Tx or Rx antennas (Simonov et al., 2012b). This finding is not directly related to the reciprocity theorem because we compare cases with different positions and numbers of Tx and Rx antennas.

This rule enables, for example, the use of a sampling period over a half-wavelength Nyquist rate and provides the super-resolution at the expense of a small illumination period, as discussed in section 3.1. Note that the same effect takes place in the opposite case, in which the illumination period is relatively large, but the sampling period is small. This finding is also related to the advantage of the MIMO approach in information gathering.

The equivalency concept is especially useful in the general case, in which the multistatic data matrix is sparse, not full. This situation is suitable for the MT system with a plane circular array, which can scan in the horizontal plane, rotating with small-angle periods (Simonov et al., 2017), as well as in the vertical direction. In this case, the Tx antennas have many positions, whereas the Rx antennas only have a small number of corresponding positions equal to the amount of array elements. Bearing in mind similar cases, we can modify the static symmetric multistatic configuration in terms of different numbers and positions of the Tx or Rx antennas. Therefore, applying the following terms is reasonable:
  1. illumination points and illumination period with respect to the Tx antennas and
  2. sampling points and sampling period with respect to the Rx antennas.
We investigate the dependence of the SR on the number of evenly distributed sampling Nsmp and illumination Nill points, keeping the total number Nm = Nsmp · Nill of the measured data constant (Figure 6). We establish that if Nm is fixed, the optimal system configuration with minimal SR corresponds to urn:x-wiley:00486604:media:rds20747:rds20747-math-0054. This finding introduces the following optimal sampling illumination period dopt (or optimal spacing) for the equivalent symmetric multistatic configuration:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0055(30)
Details are in the caption following the image
Dependence of spatial resolution on the spacing of transmitting (Tx) or receiving (Rx) antennas in a circular array. The array diameter is 150 mm, the background is propylene glycol (PG) 90%, the frequency is 2,100 MHz, and the diffraction limit is 21 mm. The spacing of Tx or Rx antennas is normalized by the optimal spacing dopt, as defined in equation 30.

where Ra is the radius of the circular array.

Figure 6 presents some examples of such dependencies in the semilogarithmic scale. We test a circular array with a 150-mm diameter immersed in a PG 90% bath liquid and operating at 2,100 MHz. The calculation of the SR utilizes the parameter urn:x-wiley:00486604:media:rds20747:rds20747-math-0056 in equation 12. Figure 6 shows the SR dependencies for the selected five values of total measurements Nm: 128, 256, 512, 1,024, and 2,048. The dependence of the SR is symmetrical for the illumination and sampling periods, normalized by the optimal spacing dopt 30.

Figure 6 shows that if the number Nm is fixed, the SR almost does not depend on the specific numbers of sampling and illumination points, Nsmp and Nill, in wide diapasons; that is, the SR mostly depends on the total number of measurements. Clearly, this conclusion assumes the condition that the illumination and sampling points spread evenly in the same observation domain.

However, the side parts of the traces in Figure 6 show some deviations of the SR from its minimal values, that is, when numbers Nsmp or Nill are smaller than 4. A more detailed analysis shows that the reason for such deviations is due to the fact that the PSF profiles in this case become irregular because of the lack of circular symmetry in the location of the antennas.

3.2.2 Verification of the Simulated Data

An example of the PSF image for the point object in the center of the measurement setup of Figure 6 is presented in Figure 7. The background used is PG 90%, the frequency is 2,100 MHz, and 32 Tx and 64 Rx antennas are used.

Details are in the caption following the image
Reconstructed image of the point object in the center of the imaging zone (point spread function) for the setup of Figure 6 in the case of 32 transmitting and 64 receiving antennas.

In this case, the computed ASR and PSF-based estimations of the SR, as defined in equation 20, are close: D = 13.5 mm, DFZ = 13.2 mm, and DInt = 13.5 mm. These values are about 0.6 of the RL value (21 mm).

4 Super-Resolution Effects in the 2D Linear and Nonlinear Considerations

4.1 Super-Resolution for a Low NSR

4.1.1 Computed Results

We investigate the dependence of SR on the variance σm of the average NSR (ANSR) of the measured noise using equations 12 and 16. In our case, we choose the contrast variance σc = 0.1 as a reasonable acceptable parameter of the imaging noise.

Figure 8 shows the dependencies of SR on the ANSR in the interval of ( 10−4…1.0 ) for both linear (a) and nonlinear (b) considerations at 3 GHz calculated with equations 24 and 26. We consider the fully multistatic data measurement setup ( NsmpNill) for the array with a diameter of 150 mm immersed in a 1.3 butylene glycol (13BG) background liquid (see Table 1 for its parameters). As mentioned previously, the array elements are the 2D point source and are vertically oriented. We compare three cases for the spacing of the array elements: 23 mm (diffraction limit for the background), 11.5 mm (half of the diffraction limit for the background), and 6 mm (quarter of the diffraction limit for the background) for both linear and nonlinear imaging. The IZ is bounded by the object surface (Figure 1) with a diameter of 100 mm for both linear and nonlinear considerations.

Details are in the caption following the image
Dependences of the spatial resolution (SR) on the variance σm of the average noise-to-signal ration (ANSR) 11. (a) Born approximation. (b) Nonlinear approach. The diffraction limits at 3 GHz are 23 mm for the background and 12 mm for the ED breast phantom. 13BG = 1.3 butylene glycol.

In the case of the Born approximation (Figure 8a) and the ANSR = 1, the SR Dlin is around the background diffraction limit urn:x-wiley:00486604:media:rds20747:rds20747-math-0057, but urn:x-wiley:00486604:media:rds20747:rds20747-math-0058 is a very small ANSR value.

For the ANSR = 1, SR is around the background diffraction limit urn:x-wiley:00486604:media:rds20747:rds20747-math-0059 and for ANSR = 1.0E−4 the super-resolution takes place in both Born and nonlinear considerations. Indeed, in linear approximation, urn:x-wiley:00486604:media:rds20747:rds20747-math-0060, and in the nonlinear consideration (Figure 8b), the SR Dnonlin ≅ 7 mm, which is about 0.15 of the background wavelength and even becomes below the diffraction limit of the material of the object ( urn:x-wiley:00486604:media:rds20747:rds20747-math-0061). The close value of the SR emerges even in the case of the same NSR and an array spacing equal to the background diffraction limit (23 mm), although it is the Nyquist rate that detects the far-field signals only.

The result is consistent with that of Chew and coworkers (Cui et al., 2004): The super-resolution effect can be observed in the nonlinear image reconstruction even if only far-field measured data are used.

4.1.2 Verification of the Computed Results

Figure 9 presents the images of the point object (PSF) placed in the center of the IZ for the linear and nonlinear reconstructions and for the measurement setup, as described in the previous section (see Figure 8) for a particular case: Tx spacing 6 mm, ANSR = 1.0E−4. These images demonstrate that SR in the nonlinear method is remarkably smaller. We compare ASR, which is calculated by 16, with the SR estimations based on the PSF: urn:x-wiley:00486604:media:rds20747:rds20747-math-0062, urn:x-wiley:00486604:media:rds20747:rds20747-math-0063, and DInt equations 20 and 21). The results of the comparison are given in Table 2.

Details are in the caption following the image
Images of the point object (2D point spread function [PSF]) placed in the center of the phantom related to Figure 8: Object position (0, 0), 3 GHz, transmitting spacing 6 mm, noise-to-signal ratio 10E−4. (a) Born approximation. (b) Nonlinear consideration.
Table 2. Computed ASR and PSF-Based SR urn:x-wiley:00486604:media:rds20747:rds20747-math-0064, urn:x-wiley:00486604:media:rds20747:rds20747-math-0065, and DInt as Defined by Equations 20 and 21
Method SR (mm)
urn:x-wiley:00486604:media:rds20747:rds20747-math-0066 urn:x-wiley:00486604:media:rds20747:rds20747-math-0067 DInt D
Born approximation 11.3 11.3 10.5 9.4
Nonlinear reconstruction 10.4 10.4 9.4 8.0
  • Note. Measurement setup of Figure 8: Tx spacing 6 mm, NSR 10E−4, frequency 3 GHz, 13BG background. point spread function; Tx = transmitting; NSR = noise-to-signal ratio; 13BG = 1.3 butylene glycol.

5 Investigation of the Super-Resolution in the 3D Nonlinear Consideration

5.1 Computed Results

In addition to the 2D case, the 3D consideration involves the dependence of ASR on the vertical scan period. Figure 10 gives an example of the analysis at 3 and 6 GHz.

Details are in the caption following the image
Dependence of the spatial resolution (SR) on the vertical scan period of the array with a diameter of 180 mm and 16 elements. The array is immersed into the 1.3 butylene glycol (13BG) liquid, can scan in the vertical direction and in the horizontal plane, and rotates with a period of 4.5°. The body is the ED breast phantom with a cylindrical shape: The diameter is 100 mm, and the height is 40 mm (Figure 2). Comparison computed results for the 3D nonlinear considerations. (a) Frequency of 3 GHz. (b) Frequency of 6 GHz. PSF = point spread function.

Here we test the MT configuration with the antenna array with a diameter 180 mm immersed in the 13BG liquid. This array can scan in the vertical direction and in the horizontal plane, rotating with a period of 4.5°. The illumination period is 7 mm, and the sampling period is 35 mm in the horizontal plane. The object under test is the cylinder of the ED breast tissue phantom with a diameter of 100 mm and height of 40 mm (Figure 1). Table 3 shows the parameters of all the dielectric media.

Table 3. Dielectric Parameters for the Materials 13BG = 1.3 butylene glycol; MF = most fatty; ED = extremely dense breast types (Shea et al., 2010)
Frequency (MHz) 13BG MF ED
Ε σ (S/m) Ε σ (S/m) ε σ (S/m)
3,000 4.34 0.354 6.1 0.175 17.42 0.686
6,000 2.96 0.430 5.6 0.439 15.68 1.709
  • Note. 13BG = 1.3 butylene glycol.

We produce the 3D nonlinear consideration with noise relation urn:x-wiley:00486604:media:rds20747:rds20747-math-0068 in equation 12 to calculate the ASR. The IZ is slightly larger than the object volume. Figure 10 shows the lines of the diffraction limits for the background and the ED medium of the phantom.

Figure 10a indicates that the ASR is close to the super-resolution at 3 GHz with respect to the background materials but is above the RL level in terms of the material of the object. Figure 10b shows that the ASR becomes almost 2 times smaller than the RL at 6 GHz. However, in contrast to the 3-GHz case, it is above the RL level in terms of the material of the object.

5.2 Verification of the Computed Results and Discussion

To verify the obtained ASR data, we compute the image resolution matrix Rr 6 and the PSF for the point in the center of the IZ. Figure 11 shows the horizontal (x, y) and vertical (x, z) sections of the 3D PSF in the case of 3-GHz and 2-mm period of the vertical array scan. The SR is considerably larger in the vertical than in the horizontal direction. This finding can be explained by the fact that the measurement setup is multistatic in the horizontal plane but quasi-monostatic in the vertical direction. This property is reflected in the calculated PSF-based SR urn:x-wiley:00486604:media:rds20747:rds20747-math-0069, urn:x-wiley:00486604:media:rds20747:rds20747-math-0070, urn:x-wiley:00486604:media:rds20747:rds20747-math-0071, DFZ, and DInt, as defined by equations 22 and 23. ASR D is presented in Table 4. Figure 10 illustrates the value comparison of D, SR DFZ (circles), and DInt (crosses).

Details are in the caption following the image
Images of the point object (3D point spread function [PSF]) placed in the center of the phantom related to Figure 10a: 3-GHz, 2-mm period of the vertical array scan, object position (0, 0, 20 mm), nonlinear consideration. Sections of the 3D PSF: (a) by the plane z = 20 mm and (b) by the plane y = 0.
Table 4. Computed ASR and PSF-Based SR urn:x-wiley:00486604:media:rds20747:rds20747-math-0072, urn:x-wiley:00486604:media:rds20747:rds20747-math-0073, urn:x-wiley:00486604:media:rds20747:rds20747-math-0074, and DInt (Equations 22 and 23) Depending on the Period dz of the Vertical Array Scan
dz SR (mm)
urn:x-wiley:00486604:media:rds20747:rds20747-math-0075 urn:x-wiley:00486604:media:rds20747:rds20747-math-0076 urn:x-wiley:00486604:media:rds20747:rds20747-math-0077 DFZ DInt D
2 10.2 10.2 34.8 15.4 11.7 13.3
4 10.2 10.2 34.7 15.3 12.0 13.8
6 10.2 10.2 34.9 15.3 12.0 14.1
8 11.1 11.1 33.5 16.0 13.8 14.4
10 11.1 11.1 33.6 16.1 13.8 14.7
  • Note. Measurement setup of Figure 10: frequency 3 GHz, 13BG background.; 13BG = 1.3 butylene glycol.

Given the diffraction limit of 23 mm for the 13BG background and 12 mm for the ED phantom for 3 GHz, Table 4 shows that the horizontal SR urn:x-wiley:00486604:media:rds20747:rds20747-math-0078, urn:x-wiley:00486604:media:rds20747:rds20747-math-0079 corresponds to the super-resolution for the background and is even smaller than the RL for the body. The values of the averaged SR DFZ, DInt, and D are greater than the quarter-wavelength level (see Figure 10 and Table 4) and are related to the effect of a relatively large vertical component of SR (e.g., equation 23).

The presented 3D analysis demonstrates that the nonlinear image reconstruction can provide the super-resolution, consistent with the findings by Cui et al. (2004). The main conclusion of these authors is that the scattered propagating waves carry information about the evanescent waves inside the object, which appear because of multiple scattering. This multiscattering provides coupling and mutual transformations of evanescent and propagating waves to the contrast heterogeneity. A similar mechanism was considered by the study of Simonov (1998, 2000), which showed that the coupling between the propagating plane wave and the evanescent waves scattered in the inhomogeneous material explains its nonlocal properties.

Although we use homogeneous phantoms, the nonlinear reconstruction provides the multiscattering consideration by multiplying the Jacobian by the matrix IGC in 27 and 29. This finding explains the super-resolution effects obtained in the current and previous sections.

6 Inhomogeneous Distribution of SR in the Reconstructed Image of Phantom

For additional verification of the suggested ASR approach, we demonstrate here an example of nonlinear image reconstruction of 2D phantom (Figure 12) at frequency 3 GHz. This phantom, immersed into a matching liquid, consists of circular body with relatively big contrast and two circular inclusions with lower contrast. The body's parameters, matching liquid, and measurement setup correspond to that of section 4.1.1; the difference is only in the ability of two inclusions. The body's material is ED, the background liquid is 13BG, the inclusions material is most fatty breast type (see averaged dielectric parameters in Table 3 at 3 GHz; Shea et al., 2010). The body's diameter is 100 mm, the diameter of inclusions is 32 mm, and the gap between inclusions is 8 mm. We used the contrast variance 0.1 and the ANSR = 1.0E−4, and the spacing of array elements is 11.8 mm (compare with Figure 8b). The calculated value of the ASR in the phantom is D = 7.3 mm for the nonlinear consideration, at 3 GHz, and for the investigation domain, coincided with the phantom shape. Note that this D is close to the 8-mm gap between the inclusions.

Details are in the caption following the image
Original images of 2D phantom and an image, smoothed by the image resolution matrix Rr 32. (a) Permittivity, (b) conductivity, (c) contrast module, and (d) image of contrast module, after smoothing it by the image resolution matrix Rr, calculated in nonlinear consideration, using 6, 26, and 27.

Figure 12d demonstrates the smoothed image of the contrast, which is the product of original contrast vector C (Figure 12c with the image resolution matrix Rr 6, which corresponds to nonlinear image reconstruction. We calculate PSFs as columns of matrix Rr, and Figure 13 shows that their images in different points of the phantom have different shapes and SR. PSF in Figure 13a is located in the center point between the inclusions, having an elongated form, while PSF in Figure 13b corresponds to the point above the inclusions and has a round profile. Notice that the SR of PSFs in Figures 13c and 13d is bigger than that of PSFs in Figures 13a and 13b, due to the influence of the lower contrast of the inclusions. Actually, PSF in Figure 13c is located in the center of the inclusion, and PSF in Figure 13d corresponds to the edge of the inclusion. All images of PSFs in Figure 12 demonstrate that the evaluated ASR = 7.3 mm is sa reasonable value.

Details are in the caption following the image
Images of 2D point spread function (PSF), calculated in the nonlinear consideration inside the phantom (Figure 12). (a) PSF in the phantom's center, (b) PSF in the point (0, 20) mm, (c) PSF in the center of inclusion, and (d) PSF in the point (36, 0) mm—the edge of the inclusion.
To create reconstructed images of the phantom in Figure 12, we conduct forward and inverse scattering electromagnetic (EM) simulations, wherein the forward EM solver employs 2D volume integral equation and the image reconstruction uses the iteration Gauss-Newton (nonlinear) optimization method. We utilize the following signal measurement configuration:
  1. array diameter—150 mm
  2. spacing of array elements—11.8 mm
  3. number of array elements—40
  4. full multistatic scattering matrix
  5. testing frequency—3 GHz
The Gauss-Newton iteration method provides updating of contrast Cn at each nth iteration, using equation Cn + 1 = Cn + Nst ΔCn, where Nst is a Newton step. The contrast increment ΔCn is a solution of the following equation (see, e.g., Rubæk et al., 2007, Eqs (6)–(12)):
urn:x-wiley:00486604:media:rds20747:rds20747-math-0080(31)
For comparison, we apply the image reconstructions with two different regularization methods: normal Tikhonov and TSVD-like, which also includes additional Tikhonov regularization. For the case of TSVD-like regularization, we seek a solution in the form of the contrast vector Cr, which is projection of the original contrast C to the image space of the matrix Vr ≡ V · Ir 5 and, hence, it corresponds to parameters of ASR calculation:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0081(32)

Here urn:x-wiley:00486604:media:rds20747:rds20747-math-0082 is a smoothed image of the original contrast C. The effective rank r of J is defined by equation 12. Figure 12d demonstrates an example of such a smoothed image.

At the next step, we reduce dimensions of matrix equation 31, truncating zero columns of Jr and zero elements of vector ΔCr, n for indexes i > r; in such a way, we transform them into urn:x-wiley:00486604:media:rds20747:rds20747-math-0083 and urn:x-wiley:00486604:media:rds20747:rds20747-math-0084 variables, correspondingly. In addition, we sum γ0 with noise vector nm 3 of normally distributed random numbers with variance 1.0E−4, which is equal to the selected ANSR parameter:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0085(33)
Definition of the parameter r by equation 12 guarantees that Jacobian urn:x-wiley:00486604:media:rds20747:rds20747-math-0086 is well conditioned at urn:x-wiley:00486604:media:rds20747:rds20747-math-0087 C and equation 33 can be solved directly, without regularization. However, preliminary tests show that at first iterations, when contrast is lower and the effective rank of the Jacobian is smaller than r, additional regularization is necessary. Taking into account these facts, we utilize additional Tikhonov regularization of 33 in the form of
urn:x-wiley:00486604:media:rds20747:rds20747-math-0088(34)
Here urn:x-wiley:00486604:media:rds20747:rds20747-math-0089 is the truncated identity matrix 6 and αn is a normalized Tikhonov parameter. We used the following equation for αn:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0090(35)
where Nit is number of iteration, σ1 and σr are singular values in 5 and 10. Notice that αn = 0 at n = Nit, which provides an effective dimension of the reconstruction image equal to r.
In the case of the normal Tikhonov regularization, we use the full Jacobian J 1 rather than the truncated one urn:x-wiley:00486604:media:rds20747:rds20747-math-0091 and the following equation that also provides effective dimension r of the reconstruction image:
urn:x-wiley:00486604:media:rds20747:rds20747-math-0092(36)
where I is the identity matrix.

We use Nst = 0.3 and 200 iterations for both imaging methods. As an initial guess, we applied homogeneous distribution of contrast over the whole IZ equal to Cig = 1.3 + 0.58i with initial misfit error err(1) = ‖Cig − C2/‖C2 = 0.84 for both cases. After 200 iterations, the misfit error err(200) = 5.8E−3 for the case of TSVD-like reconstruction, and err(200) = 1.1E−2 for the Tikhonov regularization.

Figure 14 demonstrates reconstructed images (module of contrast and permittivity) for both regularizations. Notice that the images, reconstructed by these different methods, look similar, and it can be concluded that their SR is consistent with the SR of PSFs in Figure 12 and the calculated ASR = 7.3 mm.

Details are in the caption following the image
Reconstructed images for truncated singular value decomposition (TSVD)-like and for normal Tikhonov regularizations (see equations 3436). Module of contrast (a) for TSVD-like reconstruction and (b) for normal Tikhonov regularization. Permittivity (c) for TSVD-like reconstruction and (d) for normal Tikhonov regularization.

7 Conclusions

This work analyzes the SR and the super-resolution effects, utilizing a TSVD-based approach for the MT system in the linear (Born approximation) and nonlinear modeling in near-field zone, employing 2D and 3D considerations. The following conclusions are formulated.

The SR and the super-resolution effects in the MT depend on many factors, including the aperture size of the antenna, background coupling medium, operating frequency, total number of measurements, average NSR of the measured data, distance between the antennas and the object under investigation, sampling and illumination periods, and method of image reconstruction (linear or nonlinear).

This study examines the case of vertically oriented point sources arranged in a circular array of the MT system. Clearly, the limited-sized antennas provide the SR, which is greater than that for the point sources, because of the space filtering properties of the aperture.

In the Born approximation, the SR of the MT system can be smaller than the illumination or sampling periods because of the MIMO method of information gathering. For example, a super-resolution imaging can be observed even if the illumination or sampling periods exceed the half-wavelength in the background.

Investigations on the 2D linear case demonstrate that the SR depends mostly on the total number of measurements and almost does not depend on specific numbers of illumination and sampling points. This conclusion assumes that in all considered cases, the illumination and sampling points are evenly distributed in the same observation domain. The noticeable deviation from this rule occurs only for sampling or illumination numbers smaller than 4.

This work also examines the case in which the illumination period becomes considerably smaller than the half-wavelength in the matching liquid. This dependence of the SR demonstrates a weak decline (in the semilogarithmic scale) when the argument tends to 0. In the zero-spacing limit, the MT configuration transforms into the electrical capacitance tomography, in which the SR depends on the spacing of the electrodes but is not limited by the wavelength.

This work also confirms results of some previous studies, for example, Cui et al. (2004) and Gilmore et al. (2010) that the nonlinear reconstruction produces an SR that can be remarkably smaller than that for the linear consideration. Furthermore, the super-resolution effect can occur even when the system measures propagating waves only.

Acknowledgments

This work was supported by the Electronics and Telecommunications Research Institute grant funded by the Korean government (17ZR1400, Research on Beam Focusing Algorithm for Microwave Treatment). According to the AGU data policy, all simulation data of the method proposed in this paper will be available upon request.