A TSVD Analysis of the Impact of Polarization on Microwave Breast Imaging using an Enclosed Array of Miniaturized Patch Antennas

Abstract

Microwave breast imaging performance is fundamentally dependent on the quality of information contained within the scattering data. We apply a truncated singular-value decomposition (TSVD) method to evaluate the information contained in a simulated scattering scenario wherein a compact, shielded array of miniaturized patch antennas surrounds an anatomically realistic numerical breast phantom. In particular, we investigate the impact of different antenna orientations (and thus polarizations), namely two array configurations with uniform antenna orientations and one mixed-orientation array configuration. The latter case is of interest because it may offer greater flexibility in antenna and array design. The results of this analysis indicate that mixed-polarization configurations do not degrade information quality compared to uniform-polarization configurations and in fact may enhance imaging performance, and thus represent viable design options for microwave breast imaging systems.

I. Introduction

Microwave inverse scattering has been extensively investigated for a variety of breast imaging applications. The dielectric properties of breast tissue have been well characterized in the microwave frequency range [1] and suggest both anatomical and functional imaging potential at microwave frequencies. Microwave medical imaging is low cost compared to MRI and produces three-dimensional tomographic images, in contrast to the two-dimensional images produced by conventional X-ray mammography. Thus it has the potential to provide a safe, effective tool for periodic breast imaging.

The quality of the microwave image depends on the quality of the scattering data recorded by the antenna array surrounding the object [2]. One method of analyzing the quality of scattering data in this type of setting is based on the truncated singular value decomposition (TSVD) of the scattered field data. The utility of this method was previously demonstrated for microwave breast imaging using an open array of idealized dipole antennas [3]. In this paper we apply the technique to an enclosed/shielded array of miniaturized patch antennas recently proposed for microwave breast imaging [4] to address the following question: What is the impact of array topology and antenna orientation (and thus polarization) on the quality of the measured data?

A variety of polarization arrangements have been proposed for use in microwave imaging systems. For example, the use of cross-polarized antennas [5], [6] has been examined as a means of increasing sensitivity in radar-based breast cancer screening systems. Among breast imaging arrays designed for microwave inverse scattering, some use vertically oriented monopole antennas [7] while others use horizontally oriented monopoles [8]. Yet another [9] reports experimental results for imaging simple objects with a dual-polarized array.

A comparison of an array of vertical and horizontal orientations [10] revealed superior performance with the horizontal configuration, with the caveat that the improvement may have been related to the radiation pattern of the monopoles rather than the polarization itself. In contrast, a previous TSVD analysis [3] compared vertically and horizontally oriented dipole antennas and showed a slight improvement in noise performance for the vertical polarization configuration. Thus there is a lack of consensus about monopole/dipole open-array configurations for microwave inverse scattering. Additionally, there has been no prior investigation of polarization effects in inverse scattering systems with complex scatterers such as the breast.

A systematic study of the impact of polarization on microwave imaging systems is particularly important for establishing the feasibility of new antenna designs with multiple polarizations. For example, a quad-band miniaturized antenna operating in the 1-3 GHz range has recently been proposed [11]. Quad-band performance with broadside radiation patterns in each band is achieved by exploiting the TM₁₀₀, TM₃₀₀, TM₅₀₀, and TM₀₁₀ modes of the patch antenna. The TM₀₁₀ mode is cross-polarized relative to the other modes. As a result, an array employing these quad-band antennas will have all antennas polarization matched within each band but the array overall will make use of horizontal polarization in some bands and vertical polarization in others. While introducing multiple resonant bands and miniaturized antennas helps reduce the ill-posedness of the microwave inverse scattering problem, the impact of introducing additional polarizations has not been quantified.

In this paper we employ simpler dual-band slot-loaded patch antennas [12] to investigate whether horizontal or vertical polarization yields better performance for the same operating frequency. We also examine the performance of an array with mixed polarizations within each frequency band.

II. Analysis Method and Numerical Test Platform

The following explanation of the TSVD method is based on [3] which applied it to an open array of dipole antennas. Microwave inverse scattering is governed by the following electric field integral equation:

$${\overrightarrow{E}}^s(\overrightarrow{r},{\overrightarrow{r}}_{\mathrm{s}})={\int }_V{\bar{G}}^b(\overrightarrow{r},{\overrightarrow{r}}^{\prime }){\overrightarrow{E}}^t({\overrightarrow{r}}^{\prime },{\overrightarrow{r}}_{\mathrm{s}})(k^2\left({\overrightarrow{r}}^{\prime }\right)-k_b^2\left({\overrightarrow{r}}^{\prime }\right))d{\overrightarrow{r}}^{\prime }$$ (1)

where ${\overrightarrow{E}}^s(\overrightarrow{r},{\overrightarrow{r}}_{\mathrm{s}})$ is the scattered electric field measured at position $\overrightarrow{r}$ due to a source at position ${\overrightarrow{r}}_{\mathrm{s}};{\overrightarrow{E}}^t$ is the total electric field inside the imaging domain, $V;k^2-k_b^2$ is the unknown contrast between the object dielectric profile, k, and the known background profile, k_b; and Ḡ^b is the background Green’s function. Inverting this equation in an imaging scenario requires approximating ${\overrightarrow{E}}^t$ since it depends on the unknown contrast. However, in the computational environment of this analysis we can compute the total fields for the exact dielectric profile everywhere inside the domain. This allows us to evaluate the performance of different array designs independent of a particular imaging algorithm and any degradation due to the approximation of ${\overrightarrow{E}}^t$ [3].

Applying (1) to multiple {r, r_s} rsg pairs and discretizing the domain enables us to rewrite (1) as a system of linear equations: b = Ax. We efficiently account for the dispersive nature of biological tissue by expressing x in terms of the Debye parameters ε_s, Δε, and σ_s (assuming that a single, fixed Debye relaxation time constant, τ , is suitable for all materials in the imaging domain). For a system with M measurements and N unknowns, the scattering matrix A is M × N, the unknown contrast x is N × 1 and the scattered field data vector b is M × 1. Note that a measurement in this case constitutes the signal from a single transmit-receive antenna pair at a particular frequency. Because the number of measurements M is in general significantly smaller than the number of unknowns N, one cannot simply invert this linear system. Instead the singular value decomposition (SVD) of A is taken: A = UΣV^H. This allows for the construction of the pseudo-inverse of A: A⁺ = VΣ⁺U^H. A truncated version of A⁺ is used to compute an estimate of the object, x̂:

$$\widehat{\mathrm{x}}=\sum _{i=1}^{i_n}\left(\frac{{\mathbf{u}}_i^H\mathbf{b}}{{\sigma }_i}\right){\mathbf{v}}_i$$ (2)

where i_n is the truncation index. This truncation index can be chosen to be as small as 1 or as large as N . The vector x̂ contains the estimated dielectric properties at each voxel in the imaging domain. We refer to this three-dimensional estimate as the TSVD reconstruction. We evaluate the quality of an estimate x̂ for a given truncation index i_n by correlating the estimate and exact object profile x^* using a normalized inner product [13]:

$$f\left(\widehat{\mathbf{x}}\right)=\frac{\langle {\mathbf{x}}^{\ast },\widehat{\mathbf{x}}\rangle }{\Vert {\mathbf{x}}^{\ast }{\Vert }_2\Vert \widehat{\mathbf{x}}{\Vert }_2}$$ (3)

The quality of the scattering data for a given array is evaluated by plotting the fidelity metric given in (3) as a function of truncation index.

Simulations were performed using the finite-difference time-domain method (FDTD) to obtain scattered field data at the antennas, total field data over the imaging volume, and background Green’s function for each antenna array configuration under investigation. We used two MRI-derived numerical breast phantoms developed in [14] as our test bed. The two phantoms represent American College of Radiology (ACR) Class 2 and Class 3 breast densities: scattered fibroglandular and heterogeneously dense, respectively. The phantoms are modeled on a 2-mm grid. The total fields computed throughout the imaging volume are recorded for use in (2) at 2-mm intervals.

III. Array Configurations

We investigated three different enclosed arrays of 36 dual-band slot-loaded patch antennas. The dimensions of the patch antennas are given in [4]; the operating frequencies are the resonant frequencies of the TM₁₀₀ and TM₃₀₀ modes: 1.6 GHz and 3.02 GHz, respectively. Each array consists of six panels arranged to form a box with the antennas radiating into the interior. This array design is similar to that of [4]: we use the same dimensions for the enclosed array, but we use more antennas and different antenna layouts on the side panels. In all cases the bottom panel of the enclosed array is bare substrate backed with solid copper, the side panels are solid copper-backed substrate with patch antennas etched on the inside face, and the top panel is copper with a hole cut in the center to allow the breast to extend into the array from the top. We define the z axis to be the vertical direction; in this case the vertical side panels are parallel to either the x-z or y-z planes. These four side panels each have nine antennas arranged in a 3×3 configuration. In each array, the four side panels have an identical antenna configuration. The three different configurations are shown in Fig. 1. We investigated two configurations with uniform polarization: vertical (Fig. 1(a)) and horizontal (Fig. 1(b)). We also investigated a mixed polarization configuration (Fig. 1(c)).

Fig. 1.

Three configurations for the four side panels of the enclosed array: (a) vertical (ẑ), (b) horizontal (x̂ or ŷ depending on the orientation of the side panel), and (c) mixed polarization. The antennas are slot-loaded patches on ground-plane-backed substrate [4]. All four side panels in a given array have the same configuration.

The vertical configuration has the largest number of co-polarized antennas (36), as all antennas are polarized in the ẑ direction. The horizontal configuration has 18 x̂-polarized and 18 ŷ-polarized antennas, and the mixed-polarized configuration has 8 x̂-polarized, 8 ŷ-polarized and 20 ẑ-polarized antennas. Comparing the performance of the layouts shown in Fig. 1(a) and Fig. 1(b) addresses the question of whether uniform vertical or uniform horizontal polarization yields better performance. Evaluating the performance of the layout shown in Fig. 1(c) further addresses the question of whether employing additional polarizations is a viable strategy for increasing the number of resonant bands, as proposed in [11].

IV. Results and Discussion

Fig. 2 shows the fidelity results for the three array configurations and the two numerical phantoms. One important feature is the value of the peak fidelity. The ideal fidelity curve would have a peak of unity, corresponding to a perfect reconstruction at that truncation index. Another important feature of these fidelity plots is the optimal truncation index, defined as the index at which the peak fidelity occurs. The optimal truncation index gives insight into the resolution available from the data, as higher-indexed singular vectors generally correspond to higher-spatial-frequency basis functions. Comparing both the peak fidelity and optimal truncation index for different scattered-field data sets suggests the relative imaging performance available from the data.

Fig. 2.

Fidelity of the estimated profile versus TSVD truncation index for two numerical breast phantoms: (a) class 2 (scattered fibroglandular) and (b) class 3 (heterogeneously dense). Results are shown for three array configurations: vertical, horizontal, and mixed polarization (see Fig. 1).

For the Class 2 phantom (Fig. 2(a)), we observe that the mixed polarization layout achieves a slightly higher peak fidelity than both the vertical and horizontal layouts. The peak for the mixed array also occurs at a higher truncation index than the vertical array and at approximately the same index as the horizontal array, indicating potentially superior or equivalent resolution, respectively. For the Class 3 phantom (Fig. 2(b)), the mixed polarization arrangement again reaches a slightly higher peak fidelity than both the vertical and horizontal configurations. The peak fidelity also occurs at a higher truncation index than the vertical layout and is comparable to the peak location for the horizontal array.

The relative performance of the completely vertical and completely horizontal arrays changes slightly between the two different phantoms. Specifically, for the Class 2 phantom the horizontal layout reaches a higher peak fidelity at a higher truncation index than does the vertical layout. For the Class 3 phantom, the vertical layout reaches a higher peak fidelity while the horizontal layout achieves its peak at a higher truncation index. The mixed polarization configuration consistently performs slightly better than the other two. The results for the uniform configurations indicate that while the horizontal or vertical polarization may perform better for a given patient, the differences are slight and not consistent between different breast tissue environments. Thus there is no reason to avoid introducing additional polarizations if doing so will yield other benefits such as additional resonant frequencies.

Fig. 3 shows the TSVD reconstructions of the interior of the breast phantoms, compared against the exact phantom profiles. The reconstructions shown are the Δε components of x̂ computed from (2) using the truncation index that yields the highest fidelity. While three Debye parameters were reconstructed (ε_s, Δε; and σ_s), the results for all three parameters are highly correlated so we only present Δε. We present coronal cross-sections at 1 cm intervals in the vertical direction, with the base of the breast at the top of the figure and the anterior end at bottom. The low-permittivity regions (dark blue in the color version) correspond to adipose tissue and the higher permittivity regions (light blue through red in the color version) correspond to fibroglandular tissue.

Fig. 3.

Coronal cross sections through the exact and reconstructed profiles using the vertical, horizontal and mixed-polarization array configurations. Left: Class 2 (scattered fibroglandular) and Right: Class 3 (heterogeneously dense) phantom. The Debye parameter Δε is shown. The other Debye parameters track the Δε distribution very closely. Adjacent cross sections are separated vertically by 1 cm.

The reconstruction underestimates the magnitude of Δε; for both classes of phantom, but more severely for the Class 2 phantom.

The Class 2 reconstruction in Fig. 3 shows the mixed array performing slightly better, particularly in the middle cross sections. All three arrays reconstruct the regions of higher Δε in the correct locations, and all three exhibit similar underestimation in the reconstructed value. The Class 3 reconstruction in Fig. 3 exhibits a more accurate Δε magnitude, again with the mixed array performing slightly better, particularly in the middle cross sections.

V. Conclusion

The results of the TSVD analysis indicate that the horizontal and vertical array configurations perform similarly overall; for a given phantom, one polarization gives slightly better results than the other, but the polarization that achieves better performance changes from object to object. This suggests that there is no compelling reason to favor one polarization over the other and points to the possibility of using antennas that have different polarizations in different bands such as those in [11]. Furthermore, the array configuration with mixed polarization within each band performs at least as well as – if not better than – those with uniform polarization. These results indicate that polarization heterogeneity may be introduced as needed to achieve other design goals, such as reducing mutual coupling or adding resonant bands, yielding greater flexibility in designing near-field microwave medical arrays.