Generating Matlab-based 3D FDFD Computational Modeling by ...

Generating Matlab-based 3D FDFD Computational Modeling by ...

Qiuzhao Dong(NU), Carey Rappapport(NU) (contact: [email protected],[email protected])
This work was supported in part by CenSSIS, the Center for Subsurface Sensing and Imaging Systems, under the Engineering
Research Centers Program of the National Science Foundation (Award Number EEC-9986821)
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Inverse Problems: FDFD matrix-based Inversion
es Ak

Abstract
The FDFD electromagnetic model computes wave scattering by
directly discretizing Maxwells equations along with specifying the material
characteristics in the scattering volume. No boundary conditions are need
except for the outer grid termination absorbing boundary. We use a sparse
matrix Matlab code with loose generalized minimum residue (LGMRES)
Krylov subspace iterative method to solve the large sparse matrix equation,
along with the Perfectly Matched Layer (PML) absorbing boundary
condition. The PML conductivity profile employs the empirical optimal
value from[1-2]. This method is easily manipulated and general-geometry
oriented, it is fast comparing to other models for solving the whole 3D
computational grids.
The inverse scheme based on the forward FDFD model is also
investigated. A novel matrix-based Born approximation is used instead of
the traditional integral Born approximation. Tikhnov Regularization is
employed. The good results have been obtained based on the simulated data
from 2D FDFD TM model.
Microwave breast cancer detection is becoming a promising
technique because of the high electrical contrasts between malignant tumors
and normal tissue. This method investigates the electrical field properties of
the 3D breast model with and without tumors at different frequencies, low
frequency has big penetrating depth. The detection of tumor in 2D is
presented.

Application

Analysis:

Comparing to 2D FDFD TMz model

From the results, the skin make a big contribution for the total
electrical field; Therefore, it is important to choose suitable
surrounding medium to minimize reflection from the skin.

Uniform wet sand background with the relative permittivity
=20+1.06i at 1GHz
The step size is 0.0045m
The rectangular target with the relative permittivity =2.63+0.016i
The grid size is 89x89 with 8 PML at each side for 2D and 89x89x29
for 3D with plates located at 3-5 and 24-26 along z direction
Line source is located in the center(37,37) of the computational region
in 3D model

z=15
70
60
50
40
30
20
10

0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05

60
50

10

S4

10

60
50

-2

40

-3

30
20

70

3

60

2

50

1

40

0

30

10 20 30 40 50 60 70

0.18

-3

2D FDFD
70

0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05

60
50
40
30
20
10

R2

10 20 30 40 50 60 70

3

60

2

50

1

40

0

30

0.1
0.08

-1

20

0.06

-2

10
10 20 30 40 50 60 70

0.04

-3

0.02

2D FDFD TM model

R3

Phase of
scattering field z
component of
tumor : clearly
shows that the
higher frequency
has shorter
penetrating depth.

3Dline30
2Dline30
3Dline40
2Dline40

0.12

70

4
6

transmitters
5cm-radius round breast image
2mm-radius tumor
Receivers surrounding the
breast, 14 each side

0.14

2D FDFD

0

10

20

30

40

50

60

70

80

Single frequency data: 3GHz
Breast fat inhomogeneity
ignored

II. Tumor detection in vertical plane perpendicular to
chest wall (2D)

State of Arts

- 2D Matlab-based FDFD methods deal with complicated geometries and isotropic,
dispersive media;
- Our approach about 3D Matlab-based FDFD method is a valuable forward modeling
for layered 3D inhomogeneous, dispersive media and high frequencies in reasonable
memory and computational time.

Opportunities for Technology Transfer
- The general purpose of this research is detecting the subsurface targets according to
their EM properties. This model can be applied to the well-logging in the oil field by
the induction (or resistivity) coupling voltage. The geometry for well logging is
commonly anisotropic multi-layered & multi-faulted structure, which is suitable for
the proposed model .

freq

1.5G

2G

2.5G

3G

fat

5.2504 + 1.0792i

5.2180 + 0.9935i

5.1777 + 0.9780i

5.1307 + 0.9944i

6.4743 + 2.2212i

6.4157 + 2.0133i

-- Based on the general Maxwells equations, the wave equation is

2
2
E ( E ) ( i ) E 0

K k 2
where = 0.
-- Equipped with the popular PML (perfectly matched layer) ABC
(absorbing boundary conditions).
-- Employing the Yee cell geometry as the grid structure of finite
difference method.

49.1976+17.7194i

48.7264+15.8215i

48.1420 +15.1728i

47.4587 +15.1000i

Muscle (chest
wall)

57.6727 +21.1531i

55.1529+20.3072i

52.6478 +19.8237i

50.3367 +19.3200i

skin

37.8866+13.5757i

The applying mathematical method
The method finally leads to solving the problem of matrix equation:
Ax=B; where A is the coefficient matrix, B is the source column matrix and x
is the unknown. A is a very large sparse matrix. Therefore the problem is
suitable for the Krylove subspace iterative methods. One of them, LGMRES
(Generalized minimum residue method), is employed after optimalizing the
structure of matrix A by multiplying the assisted matrix and doing some
permutations.

7

37.5306+11.8164i

37.0961 +11.0554i

Single frequency data: 3GHz

Magnitude of
total field:
z component

36.5975 +10.7527i

Phase of total field:
z component

Breast geometry at supine
position, the breast
immersed in the media with
=2.6;
Semi-ellipsoid model for
breast terminated at the
planar chest wall
System of transmitter and
receivers surrounding the
breast
Transmitter: magnetic
dipole source with z
polarization at (4.6cm,-5.1cm,
2.5cm) .
The 2mm-radius tumor
located in (1cm, -2.0cm,2.0cm)

Breast fat inhomogeneity
ignored
chest wall has strong effect to
the detection of tumor to the
detection in cylindrical geometry.

Conclusion and Future works:
3D FDFD model is general-geometry objected and fast solver for the whole

Source (white star)

Tumor (blue)

4

Receivers surrounding the
breast (except chest wall), 44 total

6.3480 + 1.9067i

Tumor (HWC)

3D matlab-based FDFD (finite difference frequency domain) method :

3

2mm-radius tumor

- This model can be also applied to other fields such as mine detection and tumor
detection with the corresponding high and low frequencies.

3D FDFD Modeling

2

half elliptical breast image
(Rl=5.7cm; Rs=4.5cm)

The relative permittivity for different dispersive breast tissues at 4
frequencies[7-8]:

6.5219 +2.6424i

1
5

Spatial distribution of electrical
properties for the plane focused
onto the tumor

Breast Cancer Imaging

fibrograndular

transmitters

6

The comparison agrees very well to each other, the error is less than
3%.
- Scalar Helmholtz wave equation in frequency domain are well computed with
different boundary condition and inhomogeneous media in 2D ; 3D Fortran-based
FDFD modeling is time and memory consuming with simple geometries;

7

3

Plane geometry

3D FDFD metal plates

S5

2

10 20 30 40 50 60 70

-2

10

5
1

10

-1

20

k : approximation of perturbation to the background: (I+Eb-1Es), where
is the difference of square wavelamber between region with objects
and region without objects (background field).
Robustness with respect to the measurement noise.
I. Tumor detection in Cylindrical Breast Geometry (2D)

70

-1

20

A : equals (A0-1Eb), where A0 is related to the background coefficient
matrix, Eb is background E-field;

Magnitude of
scattering field z
component of
tumor : it decay
very fast due to
the high decay
rate

0.16

L2Validating
TestBEDs
L1Fundamental
Science
R1

30

10 20 30 40 50 60 70

Enviro-Civil

S2 S3

0

z=22

20

S1

1

40

z=22

30

L3

2

50

10 20 30 40 50 60 70

40

Bio-Med

3

60

10 20 30 40 50 60 70

70

Value Added to CenSSIS

70

es : measured E-scattering field data;

The breast tissues are dispersive and lossy, the penetrating depth
()=(1/())1/2 ( the place at (1/e) of breast fat surface intensity,
the relative intensity vs. depth d is (1/e)^(d/) ), for 3GHz, =1.2cm
for breast fat.

z=15
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05

Based on the Matlab-based FDFD forward model:

region computation;
Microwave breast imaging is investigated with full 3D version:
distance of transmitter and receiver to the tumor is guiding the level of
signal detection from tumor due to the penetrating length ;
Skin have important contribution to the total reflected electrical field. The
further work to minimize effect of the skin will be done.
Microwave breast tumor detection in 2D:
Tumor in Cylindrical breast geometry has a good recovery;
Chest wall has a strong effect on tumor recovery which causes a big noise.
Future plan: Extension investigation on microwave breast imaging ; 2D and
3D inverse algorithm to detect the breast tumor. More medical application in
FDFD model due to its high inhomogeniety-handling properties, Multilayer
inhomogeneous, dispersive media modeling and detection.

References
[1] J. Berenger, A Perfectly matched layer for the absorption of electromagnetic waves, J. Computat. Phys., vol. 114, pp.185-200,Oct,1994;
[2] E. Marengo, C. Rappaport and E. Miller, Optimum PML ABC Conductivity Profile in FDFD,in review IEEE Transactions on Magnetics, 35,1506-1509, (1999)
[3] S. Winton and C. Rappaport, Profiling the Perfectly Matched Layer to Improve Large Angle Performance, IEEE Transactions on Antenna and Propagation, Vol 48,No. 7,July,2000
[4] C. Rappaport, M. Kilmer, and Eric Miller, Accuracy considerations in using the PML ABC with FDFD Helmholtz equation computation, Int. J. Numer. Modeling, Vol 13, pp. 471-482,Sept. 2001.
[5] Carey M. Rappaport, Qiuzhao Dong, Emmett Bishop, A. Morgenthaler, M. Kilmer, Finite Difference Frequency Domain (FDFD) Modeling of Two Dimensional TE Wave Propagation , URSI Symposium Conference Proceedings, to
appear 2004.
[6] ) Qiuzhao Dong, He Zhan and Carey Rappaport, Efficient 3D Finite Difference Frequency-Domain Modeling of Scattering in Lossy Half-space Geometries, IEEE Antenna and Propagation conference proceedings, to appear, June 2007.
[7]C.Rappaport, E. Bishop, and P. Kosmas, Modeling FDTD wave propagation in dispersive biological tissue using a single pole Z-transform function, in IEEE Int. Engineering in Medicine and Biology Soc. Conf., Cancun, Mexico, Sept,
2003, pp.3789-3792.
[8]P. Kosmas, C. Rappaport, E. Bishop, Modeling with the FDTD Method for Microwave Breast Cancer Detection, IEEE Trans. Microwave theory Tech., vol. 52, No. 8, AUGUST 2004.

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