The multidimensional phase unwrapping integral and applications to microwave tomographical image reconstruction

Review and Analysis of Tumour Detection and Image Quality Analysis in Experimental Breast Microwave Sensing

This review evaluates the methods used for image quality analysis and tumour detection in experimental breast microwave sensing (BMS), a developing technology being investigated for breast cancer detection. This article examines the methods used for image quality analysis and the estimated diagnostic performance of BMS for image-based and machine-learning tumour detection approaches. The majority of image analysis performed in BMS has been qualitative and existing quantitative image quality metrics aim to describe image contrast-other aspects of image quality have not been addressed. Image-based diagnostic sensitivities between 63 and 100% have been achieved in eleven trials, but only four articles have estimated the specificity of BMS. The estimates range from 20 to 65%, and do not demonstrate the clinical utility of the modality. Despite over two decades of research in BMS, significant challenges remain that limit the development of this modality as a clinical tool. The BMS community should utilize consistent image quality metric definitions and include image resolution, noise, and artifacts in their analyses. Future work should include more robust metrics, estimates of the diagnostic specificity of the modality, and machine-learning applications should be used with more diverse datasets and with robust methodologies to further enhance BMS as a viable clinical technique.

Addressing Multipath Signal Corruption in Microwave Tomography and the Influence on System Design and Algorithm Development

In developing a microwave tomography system, we started by examining the fundamental signal measurement challenges-i.e., how to interrogate the target while suppressing unwanted multi-path signals. Beginning with a lossy coupling bath to suppress unwanted surface waves, we have developed a robust and reliable system that is both simple and low profile. However, beyond the basic measurement configuration, the lossy coupling medium concept has also informed our choice of array antenna and imaging algorithms. The synergism of these concepts has produced a novel concept which is embodied in a system that has been successfully translated to the clinic.

Advanced Fast 3-D Electromagnetic Solver for Microwave Tomography Imaging

This paper describes a fast-forward electromagnetic solver (FFS) for the image reconstruction algorithm of our microwave tomography system. Our apparatus is a preclinical prototype of a biomedical imaging system, designed for the purpose of early breast cancer detection. It operates in the 3-6-GHz frequency band using a circular array of probe antennas immersed in a matching liquid; it produces image reconstructions of the permittivity and conductivity profiles of the breast under examination. Our reconstruction algorithm solves the electromagnetic (EM) inverse problem and takes into account the real EM properties of the probe antenna array as well as the influence of the patient's body and that of the upper metal screen sheet. This FFS algorithm is much faster than conventional EM simulation solvers. In comparison, in the same PC, the CST solver takes ~45 min, while the FFS takes ~1 s of effective simulation time for the same EM model of a numerical breast phantom.

Electrical Characterization of Glycerin: Water Mixtures: Implications for Use as a Coupling Medium in Microwave Tomography

We examine the broadband behavior of complex electrical properties of glycerin and water mixtures over the frequency range of 0.1 - 25.0 GHz, especially as they relate to using these liquids as coupling media for microwave tomographic imaging. Their combination is unique in that they are mutually miscible over the full range of concentrations which allows them to be tailored to dielectric property matching for biological tissues. While the resultant mixture properties are partially driven by differences in the inherent low frequency permittivity of each constituent, relaxation frequency shifts play a disproportionately larger role in increasing the permittivity dispersion while also dramatically increasing the effective conductivity over the frequency range of 1 to 3 GHz. For the full range of mixture ratios, the relaxation frequency shifts from 17.5 GHz for 0% glycerin to less than 0.1 GHz for 100% glycerin. Of particular interest is the fact that the conductivity stays above 1.0 S/m over the 1-3 GHz range for glycerin mixture ratios (70-90% glycerin) we use for microwave breast tomography. The high level of attenuation is critical for suppressing unwanted multipath signals. This paper presents a full characterization of these liquids along with a discussion of their benefits and limitations in the context of microwave tomography.

A Robust Wrap Reduction Algorithm for Fringe Projection Profilometry and Applications in Magnetic Resonance Imaging

In this paper, we present an effective algorithm to reduce the number of wraps in a 2D phase signal provided as input. The technique is based on an accurate estimate of the fundamental frequency of a 2D complex signal with the phase given by the input, and the removal of a dependent additive term from the phase map. Unlike existing methods based on the discrete Fourier transform (DFT), the frequency is computed by using noise-robust estimates that are not restricted to integer values. Then, to deal with the problem of a non-integer shift in the frequency domain, an equivalent operation is carried out on the original phase signal. This consists of the subtraction of a tilted plane whose slope is computed from the frequency, followed by a re-wrapping operation. The technique has been exhaustively tested on fringe projection profilometry (FPP) and magnetic resonance imaging (MRI) signals. In addition, the performance of several frequency estimation methods has been compared. The proposed methodology is particularly effective on FPP signals, showing a higher performance than the state-of-the-art wrap reduction approaches. In this context, it contributes to canceling the carrier effect at the same time as it eliminates any potential slope that affects the entire signal. Its effectiveness on other carrier-free phase signals, e.g., MRI, is limited to the case that inherent slopes are present in the phase data.

3D parallel-detection microwave tomography for clinical breast imaging

A biomedical microwave tomography system with 3D-imaging capabilities has been constructed and translated to the clinic. Updates to the hardware and reconfiguration of the electronic-network layouts in a more compartmentalized construct have streamlined system packaging. Upgrades to the data acquisition and microwave components have increased data-acquisition speeds and improved system performance. By incorporating analog-to-digital boards that accommodate the linear amplification and dynamic-range coverage our system requires, a complete set of data (for a fixed array position at a single frequency) is now acquired in 5.8 s. Replacement of key components (e.g., switches and power dividers) by devices with improved operational bandwidths has enhanced system response over a wider frequency range. High-integrity, low-power signals are routinely measured down to -130 dBm for frequencies ranging from 500 to 2300 MHz. Adequate inter-channel isolation has been maintained, and a dynamic range >110 dB has been achieved for the full operating frequency range (500-2900 MHz). For our primary band of interest, the associated measurement deviations are less than 0.33% and 0.5° for signal amplitude and phase values, respectively. A modified monopole antenna array (composed of two interwoven eight-element sub-arrays), in conjunction with an updated motion-control system capable of independently moving the sub-arrays to various in-plane and cross-plane positions within the illumination chamber, has been configured in the new design for full volumetric data acquisition. Signal-to-noise ratios (SNRs) are more than adequate for all transmit/receive antenna pairs over the full frequency range and for the variety of in-plane and cross-plane configurations. For proximal receivers, in-plane SNRs greater than 80 dB are observed up to 2900 MHz, while cross-plane SNRs greater than 80 dB are seen for 6 cm sub-array spacing (for frequencies up to 1500 MHz). We demonstrate accurate recovery of 3D dielectric property distributions for breast-like phantoms with tumor inclusions utilizing both the in-plane and new cross-plane data.

Efficient reconstruction of dielectric objects based on integral equation approach with Gauss-Newton minimization

Reconstruction of unknown objects by microwave illumination requires efficient inversion for measured electromagnetic scattering data. In the integral equation approach for reconstructing dielectric objects based on the Born iterative method or its variations, the volume integral equations are involved because the imaging domain is fully inhomogeneous. When solving the forward scattering integral equation, the Nyström method is used because the traditional method of moments may be inconvenient due to the inhomogeneity of the imaging domain. The benefits of the Nyström method include the simple implementation without using any basis and testing functions and low requirement on geometrical discretization. When solving the inverse scattering integral equation, the Gauss-Newton minimization approach with a line search method (LSM) and multiplicative regularization method (MRM) is employed. The LSM can optimize the search of step size in each iteration, whereas the MRM may reduce the number of numerical experiments for choosing the regularization parameter. Numerical examples for reconstructing typical dielectric objects under limited observation angles are presented to illustrate the inversion approach.

Improving image-quality of interference fringes of out-of-plane vibration using temporal speckle pattern interferometry and standard deviation for piezoelectric plates

Interferometry provides a high degree of accuracy in the measurement of sub-micrometer deformations; however, the noise associated with experimental measurement undermines the integrity of interference fringes. This study proposes the use of standard deviation in the temporal domain to improve the image quality of patterns obtained from temporal speckle pattern interferometry. The proposed method combines the advantages of both mean and subtractive methods to remove background noise and ambient disturbance simultaneously, resulting in high-resolution images of excellent quality. The out-of-plane vibration of a thin piezoelectric plate is the main focus of this study, providing information useful to the development of energy harvesters. First, ten resonant states were measured using the proposed method, and both mode shape and resonant frequency were investigated. We then rebuilt the phase distribution of the first resonant mode based on the clear interference patterns obtained using the proposed method. This revealed instantaneous deformations in the dynamic characteristics of the resonant state. The proposed method also provides a frequency-sweeping function, facilitating its practical application in the precise measurement of resonant frequency. In addition, the mode shapes and resonant frequencies obtained using the proposed method were recorded and compared with results obtained using finite element method and laser Doppler vibrometery, which demonstrated close agreement.

Importance of phase unwrapping for the reconstruction of microwave tomographic images

Microwave image reconstruction is typically based on a regularized least-square minimization of either the complex-valued field difference between recorded and modeled data or the logarithmic transformation of these field differences. Prior work has shown anecdotally that the latter outperforms the former in limited surveys of simulated and experimental phantom results. In this paper, we provide a theoretical explanation of these empirical findings by developing closed form solutions for the field and the inverted electromagnetic property parameters in one dimension which reveal the dependency of the estimated permittivity and conductivity on the absolute (unwrapped) phase of the measured signal at the receivers relative to the source transmission. The analysis predicts the poor performance of complex-valued field minimization as target size and/or frequency and electromagnetic contrast increase. Such poor performance is avoided by logarithmic transformation and preservation of absolute measured signal phase. Two-dimensional experiments based on both synthetic and clinical data are used to confirm these findings. Robustness of the logarithmic transformation to variation in the initial guess of the reconstructed target properties is also shown. The results are generalizable to three dimensions and indicate that the minimization form with logarithmic transformation offers image reconstruction performance characteristics that are much more desirable for medial microwave imaging applications relative to minimizing differences in complex-valued field quantities.

Importance of phase unwrapping for the reconstruction of microwave tomographic images

Microwave image reconstruction is typically based on a regularized least-square minimization of either the complex-valued field difference between recorded and modeled data or the logarithmic transformation of these field differences. Prior work has shown anecdotally that the latter outperforms the former in limited surveys of simulated and experimental phantom results. In this paper, we provide a theoretical explanation of these empirical findings by developing closed form solutions for the field and the inverted electromagnetic property parameters in one dimension which reveal the dependency of the estimated permittivity and conductivity on the absolute (unwrapped) phase of the measured signal at the receivers relative to the source transmission. The analysis predicts the poor performance of complex-valued field minimization as target size and/or frequency and electromagnetic contrast increase. Such poor performance is avoided by logarithmic transformation and preservation of absolute measured signal phase. Two-dimensional experiments based on both synthetic and clinical data are used to confirm these findings. Robustness of the logarithmic transformation to variation in the initial guess of the reconstructed target properties is also shown. The results are generalizable to three dimensions and indicate that the minimization form with logarithmic transformation offers image reconstruction performance characteristics that are much more desirable for medial microwave imaging applications relative to minimizing differences in complex-valued field quantities.

Phase local approximation (PhaseLa) technique for phase unwrap from noisy data

The local polynomial approximation (LPA) is a nonparametric regression technique with pointwise estimation in a sliding window. We apply the LPA of the argument of cos and sin in order to estimate the absolute phase from noisy wrapped phase data. Using the intersection of confidence interval (HCI) algorithm, the window size is selected as adaptive pointwise varying. This adaptation gives the phase estimate with the accuracy close to optimal in the mean squared sense. For calculations, we use a Gauss-Newton recursive procedure initiated by the phase estimates obtained for the neighboring points. It enables tracking properties of the algorithm and its ability to go beyond the principal interval [-pi, pi] and to reconstruct the absolute phase from wrapped phase observations even when the magnitude of the phase difference takes quite large values. The algorithm demonstrates a very good accuracy of the phase reconstruction which on many occasion overcomes the accuracy of the state-of-the-art algorithms developed for noisy phase unwrap. The theoretical analysis produced for the accuracy of the pointwise estimates is used for justification of the HCI adaptation algorithm.

Log transformation benefits parameter estimation in microwave tomographic imaging

Microwave tomographic imaging falls under a broad category of nonlinear parameter estimation methods when a Gauss-Newton iterative reconstruction technique is used. A fundamental requirement in using these approaches is evaluating the appropriateness of the regression model. While there have been numerous investigations of regularization techniques to improve overall image quality, few, if any, studies have explored the underlying statistical properties of the model itself. The ordinary least squares (OLS) approach is used most often, but there are other options such as the weighted least squares (WLS), maximum likelihood (ML), and maximum a posteriori (MAP) that may be more appropriate. In addition, a number of variance stabilizing transformations can be applied to make the inversion intrinsically more linear. In this paper, a statistical analysis is performed of the properties of the residual errors from the reconstructed images utilizing actual measured data and it is demonstrated that the OLS algorithm with a log transformation (OLSlog) is clearly advantageous relative to the more commonly used OLS approach by itself. In addition, several high contrast imaging experiments are performed, which demonstrate that different subsets of data are emphasized in each method and may contribute to the overall image quality differences.

Initial clinical experience with microwave breast imaging in women with normal mammography

RATIONALE AND OBJECTIVES

We have developed a microwave tomography system for experimental breast imaging.

MATERIALS AND METHODS

In this article, we illustrate a strategy for optimizing the coupling liquid for the antenna array based on in vivo measurement data. We present representative phantom experiments to illustrate the imaging system's ability to recover accurate property distributions over the range of dielectric properties expected to be encountered clinically. To demonstrate clinical feasibility and assess the microwave properties of the normal breast in vivo, we summarize our initial experience with microwave breast exams of 43 women with negative mammography according to the Breast Imaging Reporting and Data System (BI-RADS 1).

RESULTS

The clinical results show a high degree of bilateral symmetry in the whole breast average microwave properties. Focal assessments of microwave properties are associated with breast tissue composition evaluated through radiographic density categorization verified through magnetic resonance image correlation in selected cases. Specifically, both whole-breast average and local microwave properties increase with increasing radiographic density, in which the latter exhibits a more substantial rise.

CONCLUSION

These findings support our hypothesis that water content variations in the breast play an influential role in dictating the overall dielectric property distributions and indicate that the microwave properties in the breast are more heterogeneous than previously believed based on ex vivo property measurements reported in the literature.