1

A compressive sensing approach for enhancingbreast cancer detection using a hybrid DBT / NRI

configurationRichard Obermeier and Jose Angel Martinez-Lorenzo

Abstract—This work presents a novel breast cancer imagingapproach that uses compressive sensing in a hybrid Digital BreastTomosynthesis (DBT) / Nearfield Radar Imaging (NRI) systemconfiguration. The non-homogeneous tissue distribution of thebreast, described in terms of dielectric constant and conductivity,is extracted from the DBT image, and it is used by a full-waveFinite Difference in the Frequency Domain (FDFD) method tobuild a linearized model of the non-linear NRI imaging problem.The inversion of the linear problem is solved using compressivesensing imaging techniques, which lead to a reduction on therequired number of sensing antennas and operational bandwidthwithout loss of performance.

Keywords—breast cancer detection, microwave imaging, com-pressive sensing, convex optimization.

I. INTRODUCTION

It has been shown [1] that X-ray based Digital BreastTomosynthesis (DBT) enhances the imaging capabilities ofConventional Mammography (CM) by producing volumetricimaging. Unfortunately, DBT still suffers from the limitedradiological contrast existing between cancerous and healthyfibroglandular tissue, which is of the order of 1%. At mi-crowave frequencies, the contrast between cancerous andhealthy fibroglandular tissue is of the order of 10% [2]. Forthis reason, Nearfield Radar Imaging (NRI), working at thesefrequencies, is an appealing technology to detect canceroustissue within the breast. Unfortunately, NRI by itself fails todetect cancerous tissue embedded in a random, heterogeneousmatrix of fibroglandular and fatty breast tissue.

A recent paper [3] demonstrated that it is possible to fuseDBT with NRI to achieve improved breast cancer detection.The successful detection of breast cancer using this configura-tion required a high performance NRI sensor operating with thefollowing requirements: 1) 17 transmitting/receiving antennasin a multi-static configuration, and 2) a wideband system,having a 1GHz bandwidth at a 1.5GHz center frequency.These high performance requirements make the NRI sensortoo expensive for widespread use in a clinical setting.

This paper presents preliminary 2D results of a new 3Dimaging approach for the NRI sensor [4], based on techniquesfrom compressive sensing theory. This approach utilizes amultiple monostatic configuration with fewer antennas and asmaller operating bandwidth than the NRI system presentedin [3]. The new approach reduces the cost of the NRI systemwithout loss of performance, thereby making the technologymore suitable for clinical use.

II. METHODOLOGY

A. System Configuration

The concept of a fused DBT / NRI imaging system wasfirst introduced in [3], [5], and the configuration of the 3Dimaging mechatronics device was presented in [4]. In thisapproach, the breast is placed under clinical compression, andthe DBT measurements are recorded using low-dosage X-rays.The DBT measurements are used to create a 3D reconstructionof the fat distribution in the breast. Simultaneously, a set ofNRI measurements are collected using a series of microwaveantennas, which are placed along the breast in a bolus materialin order to maximize its coupling with the breast surface.

B. The Sensing Problem

The NRI system operates using a multiple monostatic con-figuration, in which each antenna utilizes stepped-frequencywaveforms. Measurements of nf frequencies are taken byna antennas for a total of M = na · nf measurements.The relationship between the electric fields E(r, ω) and theunknown complex permittivity ε(r, ω) of the breast tissues canbe expressed as:

Es(r, ω) =

∫Gb(r, r

′, ω)k2b (r′, ω)E(r′, ω)χ(r′, ω)dr′ (1)

E(r, ω) = Eb(r, ω) +Es(r, ω) (2)

where Gb(r, r′, ω) are the Green’s functions of the background

medium, Eb(r, ω) are the incident electric fields, Es(r, ω) arethe scattered electric fields, and χ(r, ω) = ε(r,ω)−εb(r,ω)

εb(r,ω)are

the contrast variables [6]–[8].Eq. 1 is a nonlinear function of the contrast variable χ(r, ω)

and total electric field E(r, ω), and so nonlinear programmingtechniques such as the Contrast Source Inversion (CSI) al-gorithm [6]–[8] must be applied in order to recover χ(r, ω).These types of nonlinear algorithms typically require severalcalls to a forward model solver in each iteration, which makesthem computationally expensive. As a result, it is desirableto make some simplifying assumptions in order to reduce thecomputation time. This work makes two such assumptions.First, the Born approximation is applied, E(r, ω) ≈ Eb(r, ω),in order to linearize Eq. 1. Second, the complex permittivitiesare assumed to be approximately constant over the frequencyrange of the NRI system, i.e. ε(r, ω) ≈ ε(r) and εb(r, ω) ≈εb(r), so that the contrast variable is also approximately

arX

iv:1

603.

0615

1v1

[m

ath.

OC

] 1

9 M

ar 2

016

2

constant over frequency. With these two modifications, Eq. 1can be rewritten in the following form:

Es(r, ω) =

∫Gb(r, r

′, ω)k2b (r′, ω)Eb(r

′, ω)χ(r′)dr′ (3)

+ es(r, ω)

where es(r, ω) is the error introduced by the approximatingassumptions.

Eq. 3 can be discretized as y = Ax + e + η, wherex ∈ CN are the contrast variables, y ∈ CM are the measuredfields, A ∈ CM×N is the sensing matrix constructed fromthe incident fields and Green’s functions of the backgroundmedium, e ∈ CM is the error vector, and η ∈ CM isthe random noise introduced by the measurement system. Inpractice, M < N , and so this system has an infinite number ofsolutions satisfying y = Ax+e+η. When ‖e‖`2 � ‖η‖`2 , theperformance of linear inverse techniques only depends uponthe vector η. When ‖e‖`2 ∼ ‖η‖`2 , then the performanceof linear inverse techniques depends upon both e and η. Inpractice, the statistics of the measurement noise η can beestimated in order to tune the proposed inverse algorithmaccordingly. However, it is difficult to estimate e, since itrequires a-priori knowledge of the unknown contrast variablex.

Stand-alone NRI systems tend to perform poorly in breastcancer imaging applications because they lack a suitablebackground model εb(r, ω). Without any prior knowledge, NRIsystems select a homogeneous background medium that has adielectric constant derived from averaging that of low-water-content (LWC) fatty tissue and high-water-content (HWC)fibroglandular tissue. Choosing a homogeneous dielectric con-stant leads to a contrast variable x that has a significant numberof large, non-zero elements. This violates the assumptionsmade by the Born approximation and produces an error vectore with a large norm, which ultimately challenges the accurateinversion of the linear system of equations.

The hybrid DBT / NRI system utilizes the prior knowl-edge obtained from the DBT image in order to constructthe heterogeneous background model, thereby overcoming theaforementioned limitations of the homogeneous model usedby stand-alone NRI systems. In this hybrid system, the DBTimage is segmented into three types of tissue, skin, muscle(pectoralis major), and breast tissue, and it is assumed that thelatter only contains healthy tissue. Additionally, each pixel ofbreast tissue consists of p% of fatty tissue and (100 − p)%of fibroglandular tissue, and its constitutive properties εr(r, ω)and σ(r, ω) are determined from a composite model, whichconsiders the dispersive properties of the breast tissues [9].Fig. 1 displays the fat content segmented from a singleslice of a 3D DBT image. The segmented geometry is inputto an electromagnetic numerical simulation based on FiniteDifferences in the Frequency Domain (FDFD) [10] in orderto model the NRI sensing process. The FDFD is a full-wavemodel that accounts for all mutual interactions within thebreast, and is used to generate the incident fields Eb(r, ω) andthe non-uniform, cancer free Green’s functions Gb(r, r

′, ω).

Fig. 1. Fat content of breast segmented from a 3D DBT image. High intensityindicates a high percentage of fat.

III. IMAGING WITH COMPRESSIVE SENSING

Ideally, the hybrid DBT / NRI system would segment thehealthy breast tissue perfectly and would classify any cancer-ous tissue as HWC fibroglandular tissue, producing a contrastvariable x that is non-zero only at the locations of cancerouslesions. In this case, the reconstruction process becomes asparse recovery problem, which can be solved using techniquesfrom compressive sensing (CS) theory. CS theory is a relativelynovel signal processing technique, which was first introducedby Candes et al. in [11], and it has since been refined inmany other works [12]–[15]. CS establishes that sparse signalscan be recovered using far fewer measurements than requiredby the Nyquist sampling criterion. In order to apply suchprinciples, certain mathematical conditions must be satisfiedby the sensing matrix A and the reconstructed image x. Thesensing matrix must satisfy the Restricted Isometry Propertycondition, which is related to the independence of its columns,and the number of non-zero entries, Nnz , of the reconstructedimage must be much smaller than the total number of elements,N . If the two aforementioned conditions are satisfied, thenthe reconstruction of the unknown vector can be performedwith a small number of measurements by solving a norm-1optimization problem. In this work, the contrast variables xare recovered by solving the modified basis pursuit denoisingproblem:

minimizex

1

2‖Ax− y‖2`2 + λ‖x‖`1 (4)

subject to Re(diag(εb)x+ εb) ≥ 1

Im(diag(εb)x+ εb) ≥ 0

where the constraints on the variable x ensure that the solutionis physically realizable, i.e. εr ≥ 1 and σ ≥ 0. In practice, theweighting factor λ must be selected based upon the expectederror in the measurement vector y and sparsity of the vectorx. This problem can be efficiently solved using Nesterov’saccelerated gradient method for non-smooth convex functions[16], which was first applied to the basis pursuit denoising

3

problem in [14]. In this method, the norm-1 term is replacedby the smooth approximation:

gµ(x) = gµ(x1, . . . , xN ) =

N∑n=1

fµ(xn) (5)

where the Huber function fµ(·) is defined as:

fµ(x) =

{|x| |x| ≥ µ12µ |x|

2 |x| < µ(6)

The Huber function is Lipschitz continuously differentiable,so first-order techniques such as Nesterov’s method can beused to minimize the smoothed objective function 1

2‖Ax −y‖2`2 +λgµ(x). At each step in Nesterov’s method, the desiredvariable is updated using a step direction derived from the gra-dients up to and including the current iteration. This updatedpoint is then projected onto the feasible set to ensure that allof the constraints are satisfied. For the imaging problem of Eq.4, the projection operator can be expressed as the solution tothe following convex optimization problem:

minimizex

‖x− z‖2`2 (7)

subject to Re(diag(εb)x+ εb) ≥ 1

Im(diag(εb)x+ εb) ≥ 0

This problem is separable in the elements x1, . . . , xN of thevector x. For a scalar contrast x, it can be shown that thisprojection problem has the following closed form solution:

PQp(x) =Pε(εbx+ εb)− εb

εb(8)

Pε(ε) = max(Re(ε), 1) + max(Im(ε), 0) (9)

The reader is referred to the literature [14], [16] for furtherdetails on Nesterov’s method

IV. NUMERICAL RESULTS

Following the process of Section II-B, a 2D model of ahealthy breast was generated by segmenting a 2D slice froma 3D DBT image. In order to simulate data from a cancerouscase, a lesion with frequency-dependent electrical propertiesmodeled after [2] was added to the healthy breast. A 2Dversion of the FDFD code was used to generate the syntheticNRI measurements of the healthy breast, the synthetic NRImeasurements of the cancerous breast, and the sensing matrixof the healthy breast A according to Eq. 3. Note that theFDFD model accounted for the dispersive properties of boththe healthy breast tissue and the cancerous tissue; only the in-version process utilized the simplifying assumptions of SectionII-B. In the simulation, the NRI system used six transmittingand receiving antennas operating in a multiiple monostaticconfiguration. Each antenna was excited with three differentfrequencies, 500MHz, 600MHz, and 700MHz, for a total of18 measurements among the antennas.

Figure 2 displays the true contrast variable obtained whenthe fat percentage is perfectly segmented from the DBT image.In this plot, the white dots represent the antenna positions

Fig. 2. Real and imaginary parts of true contrast variable χε obtained whenthe DBT image is segmented perfectly.

and the green curves represent the breast and lesion borders.Since the fat percentage was segmented perfectly, the contrastvariable is non-zero only at the location of the cancerous le-sion. Figure 3 displays the estimated contrast variable obtainedusing the perfect fat percentage segmentation and noiselessmeasurements. The artifacts within the image are due to theerror vector es(r, ω) that is introduced to the measurementvector when the simplifying assumptions of Section II-B areapplied. Despite these artifacts, the algorithm is able to locatethe cancerous lesion. This represents a significant improvementover the phase-based results presented in [3], which utilized17 antennas in a multi-static configuration and 11 frequenciesover a 1GHz bandwidth in order image the cancerous lesion.

Fig. 3. Real and imaginary parts of reconstructed contrast variable χε

obtained when the DBT image is segmented perfectly and there is nomeasurement noise.

Figure 4 displays the true contrast variable obtained whenthe fat percentage is segmented from the DBT image with10% random error. More specifically, the fat percentage valueswere corrupted by i.i.d. random noise following a uniformdistribution, taking values between ±10% with equal prob-ability. Since the fat percentage is not segmented correctly,the true contrast variable is non-zero within the healthy tis-

4

sue. Nevertheless, the true contrast variable is approximatelycompressible, and so Eq. 4 can still be used to image thebreast. This result can be seen in Figure 5, which displaysthe estimated contrast variable obtained using the noisy fatpercentage segmentation and noiseless measurements.

Figure 6 displays the estimated contrast variable obtainedusing the noisy fat percentage segmentation and measurementswhose SNR = 49dB. It is important to note that the signalsin this SNR calculation are the electric fields scattered by theentire breast, and not just the fields scattered by the cancerouslesion. In this example, the fields scattered by the lesionare approximately 40dB lower in magnitude than the fieldsscattered by the rest of the breast, so that the “lesion signal tonoise ratio” is on the order of 10dB. Therefore, the NRI systemmust have a significant SNR to ensure that the fields producedby cancerous lesions are not overwhelmed by the noise, or itmust have antennas with higher directivity in order to improvethe SNR - the latter case may require the CS algorithm touse additional measurements. With this high SNR, the CSalgorithm is able to image the cancerous lesion with some

Fig. 4. Real and imaginary parts of true contrast variable χε obtained whenthe fat percentage is segmented from the DBT image with 10% error.

Fig. 5. Real and imaginary parts of reconstructed contrast variable χε

obtained when the fat percentage is segmented from the DBT image with10% error and there is no measurement noise.

additional artifacts compared to the noiseless case. However,when the SNR is decreased to 43dB, the algorithm is no longerable to image the lesion, as can be seen in Figure 7.

Fig. 6. Real and imaginary parts of reconstructed contrast variable χε

obtained when the fat percentage is segmented from the DBT image with10% error and and the measurement SNR = 49dB.

Fig. 7. Real and imaginary parts of reconstructed contrast variable χε

obtained when the fat percentage is segmented from the DBT image with10% error and and the measurement SNR = 43dB.

V. CONCLUSIONS

This work presents a novel approach for imaging breastcancer using compressive sensing techniques in a hybrid DBT/ NRI imaging system. Using the prior knowledge obtainedfrom the DBT reconstruction, the hybrid system creates abackground tissue distribution of the assumed healthy breast,thereby overcoming a common pitfall in stand-alone NRIimaging systems. Since cancerous lesions tend to be localizedto a relatively small region of the breast, image reconstructioncan be expressed as a sparse recovery problem, which is solvedusing techniques from compressive sensing. Numerical resultsshow that the CS imaging algorithm can localize cancerouslesions within in the breast, even when corrupted by DBTsegmentation and measurement errors.

5

ACKNOWLEDGMENT

This work has been funded by the U.S. National ScienceFoundation award number 1347454.

REFERENCES

[1] D. Kopans, S. Gavenonis, E. Halpern, and R. Moore, “Calcifications inthe breast and digital breast tomosynthesis,” The breast journal, vol. 17,no. 6, pp. 638–644, 2011.

[2] M. Lazebnik, D. Popovic, L. McCartney, C. B. Watkins, M. J. Lind-strom, J. Harter, S. Sewall, T. Ogilvie, A. Magliocco, T. M. Breslinet al., “A large-scale study of the ultrawideband microwave dielectricproperties of normal, benign and malignant breast tissues obtained fromcancer surgeries,” Physics in Medicine and Biology, vol. 52, no. 20, p.6093, 2007.

[3] J. Martinez Lorenzo, R. Obermeier, F. Quivira, C. Rappaport, R. Moore,and D. Kopans, “Fusing digital-breast-tomosynthesis and nearfield-radar-imaging information for a breast cancer detection algorithm,”in Antennas and Propagation Society International Symposium (AP-SURSI), 2013 IEEE, July 2013, pp. 2038–2039.

[4] J. H. J. J. A. M.-L. A. Molaei, A. Ghanbarzadeh Dagheyan, “Minia-turized uwb antipodal vivaldi antenna for a mechatronic breast cancerimaging system,” in IEEE AP-S International Symposium, 2015.

[5] R. Obermeier, M. Tivnan, C. Rappaport, and J. A. Martinez-Lorenzo,“3d microwave nearfield radar imaging (nri) using digital breast to-mosynthesis (dbt) for non-invasive breast cancer detection,” in URSI2014 — USNC-URSI Radio Science Meeting, July 2014.

[6] P. Van Den Berg and A. Abubakar, “Contrast source inversion method:state of art,” Journal of Electromagnetic Waves and Applications,vol. 15, no. 11, pp. 1503–1505, 2001.

[7] A. Abubakar, G. Pan, M. Li, L. Zhang, T. Habashy, and P. van den Berg,“Three-dimensional seismic full-waveform inversion using the finite-difference contrast source inversion method,” Geophysical Prospecting,vol. 59, no. 5, pp. 874–888, 2011.

[8] A. Zakaria, I. Jeffrey, and J. LoVetri, “Full-vectorial parallel finite-element contrast source inversion method,” Progress In Electromag-netics Research, vol. 142, pp. 463–483, 2013.

[9] J. Martinez Lorenzo, A. Basukoski, F. Quivira, C. Rappaport, R. Moore,and D. Kopans, “Composite models for microwave dielectric constantcharacterization of breast tissues,” in Antennas and Propagation SocietyInternational Symposium (APSURSI), 2013 IEEE, July 2013, pp. 2036–2037.

[10] C. Rappaport, A. Morgenthaler, and M. Kilmer, “FDFD modeling ofplane wave interactions with buried objects under rough surfaces,,” in2001 IEEE Antenna and Propagation Society International Symposium,2001, p. 318.

[11] E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery fromincomplete and inaccurate measurements,” Communications on pureand applied mathematics, vol. 59, no. 8, pp. 1207–1223, 2006.

[12] E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles:Exact signal reconstruction from highly incomplete frequency informa-tion,” Information Theory, IEEE Transactions on, vol. 52, no. 2, pp.489–509, 2006.

[13] D. L. Donoho, “Compressed sensing,” Information Theory, IEEE Trans-actions on, vol. 52, no. 4, pp. 1289–1306, 2006.

[14] S. Becker, J. Bobin, and E. J. Candes, “Nesta: a fast and accurate first-order method for sparse recovery,” SIAM Journal on Imaging Sciences,vol. 4, no. 1, pp. 1–39, 2011.

[15] A. Massa, P. Rocca, and G. Oliveri, “Compressive sensing inelectromagnetics-a review,” Antennas and Propagation Magazine,IEEE, vol. 57, no. 1, pp. 224–238, 2015.

[16] Y. Nesterov, “Smooth minimization of non-smooth functions,” Mathe-matical programming, vol. 103, no. 1, pp. 127–152, 2005.