Microwave Near-field Imaging in Real TimeNatalia K. Nikolovanikolova@ieee.org

September 27, 2018

McMaster UniversityDepartment of Electrical and Computer EngineeringElectromagnetic Vision (EMVi) Research Laboratory

Hamilton, ON CANADA

Workshop on Qualitative and Quantitative Approaches to Inverse Scattering ProblemsInstitute for Mathematical Sciences, National University of Singapore

2

APPLICATIONS OF MICROWAVE NEAR-FIELD IMAGING

• penetration into optically obscured objects (clothing, walls, luggage, living tissue…) the lower the frequency the better the penetration frequency bands from 500 MHz into the mm-wave bands (≤300 GHz)

• compact relatively cheap electronics esp. in the low-GHz range

• diverse suite of image reconstruction methods

VARIOUS APPLICATIONS

whole body scanners nondestructive testing through-wall imaging

medical imaging underground radar

3

APPLICATIONS: LUGGAGE INSPECTION, NDT[Ghasr et al., “Wideband microwave camera for real-time 3-D imaging,” IEEE Trans. AP, 2017]

16×16 array

20 GHz to 30 GHz frequency range

video

[https://youtu.be/RE-PPXmtTeA]

Prof. Zoughi’s team at Missouri University of Science & Technology

4

APPLICATIONS: WHOLE BODY SCANNERS[Sheen et al., “Near-field three-dimensional radar imaging techniques and applications,” Applied Optics 2010]

40 GHz to 60 GHz (U band) cylindrical scan

10 GHz to 20 GHz polarimetric cylindrical scan

Pacific Northwest National Laboratory, Washington, USA

40 GHz to 60 GHz (U band) cylindrical scan

5

APPLICATIONS: MEDICAL IMAGING[Song et al., “Detectability of breast tumor by a hand-held impulse-radar detector: performance evaluation and pilot clinical study,” Nature Sci. Reports 2017]

Prof. Kikkawa’s team at Hiroshima University, Japan

6

MICROWAVE NEAR-FIELD IMAGING: COMMERCIAL GROWTH

• mm-wave whole-body imagers for airport security inspection ( > 30 GHz)

• through-wall and through-floor infrastructure inspection for contractors and home inspectors (UWB, 3 GHz to 8 GHz)

• numerous underground radar applications: detection of pipes, cables, tunnels, etc. ( < 3 GHz)

[video credit: https://walabot.com/diy]

[video credit: https://www.youtube.com/watch?v=_YZEa1hiGO0]

7

OUTLINE

specifics of near-field microwave imaging in

• data acquisition

• forward models

• inversion strategies (real-time)

examples and comparisons of methods

8

DATA ACQUISITION: SCANNING

Principle: N-D imaging (N=2,3) needs abundant and diverse N-D data sets

spatial scans – 1D (linear) or 2D (surface)• illuminate target from various angles• collect scattered signals at various angles/distances• acquisition surfaces – planar, cylindrical, spherical• mechanical vs. electronic scanning

bksk

sV

aS

Tx

RxRx

Rx

Rx

Rx

RxRx

mechanical scan

electronic switching

speed low HIGHcomplexity LOW highflexibility in adjusting scan parameters

GREAT limited

frequency/temporal sweeps

polarization diversity

9

DATA ACQUISITION: SPATIAL SAMPLING

spatial data diversity• each sample must add independent information – improves uniqueness• linearly dependent data may lead to ill-posed inversion problems• over-sampling: pros and cons

minmax , ,

4sinx yλζ ζ ζ

α∆ ≤ ∆ ≈ ≡

xz

aperture edges

A,maxα

scatteringobject

AΩantennabeam

A,max Amin( , / 2)α α= Ω

• effective near-field wavelengthsmay be shorter than b2 / kλ π=

eff,min max,

2

x ykπλ =

2D( , ) ( , )x yS k k S x y=

• staying below but close to the maximum spatial sampling step ensures diversity

10

DATA ACQUISITION: FREQUENCY SAMPLING

frequency data diversity in frequency-sweep measurements

• stay below but close to the maximum frequency sampling step• ensures that back-scattered signals (after IFFT) from all targets

at distances ≤ Rmax do not overlap

bmax

max max

12 4

vf fT R

∆ ≤ ∆ = ≈

OUT

TxRx

maxR

antennaarray radar

1s

is

Ns

1is +

to SPU

11

DATA ACQUISITION: TEMPORAL SAMPLING

• temporal data diversity in time-domain measurements

stay below but close to the maximum time sampling step ensures that all frequency components of the pulsed signals are fully used (Nyquist)

minmax

max

12 2

Tt tf

∆ ≤ ∆ ≈ =

s

0

known

tot

constant

sc

s

inc (2

)i (( ))ri

i Vkik k d

a aS ε εω ′′∆ ′= ⋅′∫∫∫ E rr rE r

12

FORWARD MODEL OF SCATTERING: S-PARAMETER DATA EQUATION[Nikolova et al., APS-URSI 2016][Beaverstone et al., IEEE Trans. MTT, 2017]

Green’s vectorfunction

data

• scattering from penetrable objects (isotropic scatterer)

complexpermittivitycontrast

total internal field due to Tx antenna

tot ( ) :k ′E ris

-th portk

ks

-th port

i

ka

ib

ˆ ku

ˆ iu

Tx

Rx

OUT

sV

inc ( ) :i ′E r incident internal field due to Rx antenna if it were to transmit

p, 1, ,i k N=

total number of experiments: 2pN

reciprocity: 2p p( ) / 2N N+

total internal field

,b( ) ( ) ( )r r rε ε ε′ ′ ′∆ = −r r r

13

FORWARD MODEL: BORN’S LINEARIZING APPROXIMATION

• total field is generally unknown and depends on contrast: data equation is nonlinear in the unknown contrast

• Born’s approximation of the total internal field linearizes the data equation:

( ) nctot i; ( )) (k kε ′≈′ ′∆E Er rr

( )s

tosc n t0 i ci ( )2

( ) ; ( )iik Vi kr rkS d

a aωε ε ε′ ′ ′∆ ∆′ ′= ⋅∫∫∫ r rr rE Er

Green’s function

data contrast(to be found) total internal field

( )tot ; ( )k ε′ ′∆E r r

s

inc inc

resolved

s

nt kernel(assume known)

c ( ) ( ) ( )ik V ir kS dε ′⇒ ≈ ′∆ ′ ′⋅∫∫∫ E r E r rr

• main challenge of near-field imaging with linear inversion methods: incident-field distributions in antennas’ near zone are difficult to model

14

• object is in the near-field region of antennas

2A,max

far2Dr D

λ< ≈

any one of the conditions below implies near-field (short-range) imaging

• distance from object to antennas ≤ object’s size OUT,maxr D≤

• distance from object to antennas ≤ wavelength

r λ≤

• implication: resolvent kernel depends on incident fields which do not conform to analytical free-space far-zone propagation models, e.g.,

biinc ˆ( ) ( , )

k reGr

θ ϕ−

′E r p

Goran M Djuknic - US Patent 6657596, commons.wikimedia.org/w/index.php?curid=20417988

not valid

FORWARD MODEL: FAR-ZONE vs. NEAR-ZONE ANTENNA FIELDS

15

LINEARIZED FORWARD MODEL: TIME DOMAIN

• Born’s linearizing approximation is applied in the same way

time-domain (pulsed-radar waveforms)

s Rx Tx

inc i

,T

scR R

( ; ;T

ncxx x

)x ( )( , ; ) ( )

V ts u dt hκ

′′′∗ ′′ ≈ ∫∫∫

r r rrr r r

data(known)

contrast(unknown) impulse response of

Rx incident field(assumed known)

2b

( ) r

vεκ ∆′ =r

[Nikolova, Introduction to Microwave Imaging, 2017]

Tx incident field(assumed known)

frequency-domain (S-parameters)

s

inc insc c( )( ) ( )ik V r kiS dε ′⋅′ ′′≈ ∆∫∫∫ rE E rr r

16

LINEARIZED FORWARD MODEL: BORN vs. RYTOV DATA APPROXIMATION

• Born scattered-field data approximation

• Rytov scattered-field data approximation

sa

sc tot inc inc inca ( )

data calibration step

( ) ( ) ( ) ( )ik ik ik r i kVSS S S S dε

∈ ′ ′ ′ ′∈ ≈ − ≈ ∆ ⋅ ∫∫∫r

r r E r E r r

sa

totsc inc inc inc

a inc

data calibration step

( ) ln ( ) ( ) ( )ikik ik r i kV

ik S

SS S S dS

ε∈

′ ′ ′ ′∈ ≈ ≈ ∆ ⋅

∫∫∫

r

r r E r E r r

[Tajik et al., JPIER- B, 2017][Shumakov et al., Trans. MTT, 2018]

17

FORWARD MODEL: APPROXIMATIONS

total internal field approximation (Born)• limitations on both size and contrast of the scatterer

2 2 2s b s( ) 1,a k k V− ∈r r

bksV

s ( ')k r s ( ')k r

a

• if OUT violates limit: image contains artifacts which reflect differences between and rather than contrasttot

Tx ( )′E rincTx ( )′E r

[Nikolova, Introduction to Microwave Imaging, 2017]

data approximation

s b a2 ( ) ,a k k Sπ− < ∈r r• Born – limitation on both size and contrast:

• Born – neglects multiple scattering & mutual coupling between antennas and OUT

• Rytov – limitation on contrast only: ( )2 2 2s ab b/ 1, k k k S− < ∈r

• expect trouble in areas of strong multiple scattering & mutual coupling

18

LINEAR INVERSION METHODS: THE ENGINES OF REAL-TIME IMAGING

s

incs incc ( ( )) ( )ik V kriS dε ′′ ′≈ ⋅ ′∆∫∫∫ E r rE rr

principle: linearize forward model & solve resulting linear system of equations

data(known)

contrast(unknown)

normalized incident fields(assumed known)

• contrast and compare with nonlinear inversion methods principle: solve nonlinear forward-model equations for BOTH contrast andtotal field using nonlinear optimization and/or iterative methods

( )s

tincsc to( ( )) ; ( )r rki iVkS dε ε′ ⋅′ ′ ∆ ′≈ ′∆∫∫∫ r rrE rr E

data(known)

contrast(unknown)

Green's function(assumed known)

total internal field(unknown)

Maxwell’s equations

19

RECONSTRUCTION WITH FREQUENCY-SWEEP DATA: HOLOGRAPHY

• reflected signals: S11, S22• transmitted signals: S21, S12

type of response

number of values Sik(xꞌ,yꞌ)

co-pol X-X 4xNω

co-pol Y-Y 4xNω

cross-pol X-Y 4xNω

cross-pol Y-X 4xNω

TOTAL 16xNω = NT

number of responses acquired at each position

x

y

D

z

co-pol

cross-pol

#1#2

OUT

holography refers to reconstruction methods that use both the magnitude and phase of the scattered waves recorded at a surface to produce a 3D image in a single inversion step

Tx

Rx

for a pair of antennas

20

HOLOGRAPHY: RESOLVENT KERNEL

s

inc inc,Rx ,Tx ( , , ; , , ; )

sc ( , , ,)( , , ; ) 1, , Tx y zV r x y zS x y z dx dy dz Nx y zξ ξ ξ ω

ω ξε′ ′ ′

′ ′ ′ ′ ′ ′≈ =⋅∆ ∫∫∫ E E

approximate (Born) resolvent kernel ( ; ; )ξ ω′r r

• assume kernel is translationally invariant in x and y (background is uniform or layered)

0,

Then ( , , ; , , ; )( , ; ; )

x y z x y zx x y y z

ξ

ξ

ωω

′ ′ ′ =′ ′ ′− −

0,Let ( , ; ; )( , , ;0,0, ; )

x y zx y z z

ξ

ξ

ωω

′ ′ ′′ ′ ′≡

s

sc0,( , , ; ) ( , , ) ( , ; ; )rV

S x y z x y z x x y y z dx dy dzξ ξω ε ω′ ′ ′ ′ ′ ′ ′ ′ ′⇒ ≈ ∆ − −∫∫∫

antennas at aperture center

x

z(0,0)

Tx 0z = Rxz D=

( , )x y

inc0,Tx ( , , )x y z′ ′ ′E

inc0,Tx ( , , )x x y y z′ ′ ′− −E

21

HOLOGRAPHY: RESOLVENT KERNEL, examples

• examples: analytical kernels used with far-zone reflection data 2 2 2

bi2 ( ) ( ) ( ')inc incRx Rx Rx( , , ; , , ; ) k x x y y z zx y z x y z eξ ω ′ ′− − + − + −′ ′ ′ = ⋅E E

2 2 2bi2 ( ) ( ) ( ')

Rx 2 2 2( , , ; , , ; )( ) ( ) ( ')

k x x y y z zex y z x y zx x y y z zξ ω

′ ′− − + − + −

′ ′ ′′ ′− + − + −

inc inc,Rx ,Tx( )ξ ξ=E E

plane waves:

spherical waves:

(2) 2 20 (2 ), ( ) ( ) ,H k x x y y z z constρ ρ ′ ′ ′= − + − = =cylindrical waves:

• far-field analytical kernels do not work well with near-field data

• kernels computed from simulated incident-field distributions suffer from modeling errors [Amineh et al, Trans. AP, 2011]

• near-field kernels are best determined through measuring the system PSF[Savelyev&Yarovoy, EuRAD 2012][Amineh et al., IEEE Trans. Instr.&Meas., 2015]

22

HOLOGRAPHY: MEASURING THE POINT-SPREAD FUNCTION (PSF)

• PSF is the system response to a point scatterer

scattering probe

• relating PSF to kernel

0 , ( , ,PSF0,

; )PSF ( , , ; , ;0 )

z x y zx y zzξ

ξ ωω

′ ≡′

Let

probe at centerof plane z const′=

( )0, 0 , ,sp sp( , ; ; ) PSF ( , , ; ) /z rx y z x y zξ ξω ω ε′′ = − − ∆ ΩThen

• PSF-based kernels enable quantitative imaging in real time

23

DATA EQUATION OF HOLOGRAPHY IN TERMS OF PSF

sc0 ,

,sp sp

2D convolution

1( , , ; ) PSF ( ;( , , ,) )zr z y x

rS x y z x x y yy d y dzx z x dξ ξ ωεωε

′′ ′ ′

′ ′ ′ ′ ′≈ ⋅ − −∆ Ω

′ ′ ′∆∫ ∫∫

• in real space

• in Fourier (or k) space

sc0 ,

,sp sp( , ; ; () PSF ;, ( ,; ) )x zx y xy yzr

Fx yS k k z kk k dz zk ξξ ω ωε

′′

∆ ′′ ′∆ ′≈ ⋅

∆ Ω ∫

2DFT ( , , )r x y zε ′ ′ ′∆

• system of equations to solve at each spectral position

( )( )0 ,

1( ) PSF ( )( ; )

zNmmzn

nf zS ξξ ′

=

′≈ ∑κ κκ

1, ,1, , T

m NN

ωξ==

( , )x yk k=κ

,sp sp( ; ) ( ; )n

n nr

x Fyz zf zε′ ′ ′∆ ∆ ∆

= ⋅∆ Ω

′ ′κ κ

vΩ

24

ADVANTAGES OF SOLVING IN FOURIER SPACE: DIVIDE AND CONQUER

( )( )0 ,

1( ; )( ) PSF ( )

zNmmzij ij

nij nfS z ξξ ′

=

′≈ ∑κ κκ

1, ,1, , T

m NN

ωξ==

( , )1, , ; 1, ,

ij x yx y

i k j ki N j N

= ∆ ∆= =κ

[ ] [ 1] [ 1]( ) ( ) ( )T z z Tij N N N ij N ij N Nω ω× × ×⋅ =A κ f κ d κ

(e.g. 60 5)T zN N Nω × ×

• we solve (Nx⸱Ny) such systems (on the order of 104 to 105)

• the size of each system is small:

• typical execution times: 2 to 3 seconds on a laptop using Matlab

• solution is orders of magnitude faster than solving in real space: 6 7

v v7 8

where 10 to 1010 to 10

D x y z

D x y T

N N N N N NN N N N Nω

× ==

25

FINAL STEP: BACK TO REAL SPACE

,sp sp 12D( , , ) ( ; ) , 1, ,r

r n n zn

x y z f z n Nx y zε

ε −∆ Ω′ ′ ′ ′∆ = =

′ ′ ′∆ ∆ ∆κ

• at each plane along range (z' = const)

,b( , , ) ( , , )r n r r nx y z x y zε ε ε′ ′ ′ ′ ′ ′= + ∆

5mm250MHz

x yf

∆ = ∆ =∆ =

26

EXAMPLE: METALLIC TARGETS IN AIR

X-band (WR90) open-end waveguides ( 6.56GHz)cf ≈

f (GHz) λ (mm) Dfar (mm)3 100 12.5

8.2 37 3420 15 83

[photo credit: Justin McCombe]

[Amineh et al., Trans. Instr.&Meas., 2015]

SP

27

METALLIC TARGETS IN AIR – RESULTS WITH SIMULATED KERNELS[Amineh et al., Trans. Instr.&Meas., 2015]

28

METALLIC TARGETS IN AIR – RESULTS WITH MEASURED KERNELS (PSF)

expected spatial resolution

depth:10 mmzδ ≈

lateral:, 4 mmx yδ ≈

[Amineh et al., Trans. Instr.&Meas., 2015]

29

EXAMPLE: IMAGING TISSUE

[photo credit: Daniel Tajik]Tissue Color

HighlightRelative Permittivity Averaged over 3 to 8 GHz

Chicken Wing NABone 21 10iSkin 13 6iMuscle 45 23iPeanut Butter & Jam 7 3i

Carbon Rubber 10 3i

[Tajik et al., JPIER-B 2017]

3mm100MHz

[3,8] GHz

x yf

f

∆ = ∆ =∆ =∈

a layer in the OUT

30

EXAMPLE: IMAGING TISSUE

[photo credit: Daniel Tajik]Tissue Color

HighlightRelative Permittivity Averaged over 3 to 8 GHz

Chicken Wing NABone 21 10iSkin 13 6iMuscle 45 23iPeanut Butter & Jam 7 3i

Carbon Rubber 10 3i

[Tajik et al., JPIER-B 2017]

3mm100MHz

[3,8] GHz

x yf

f

∆ = ∆ =∆ =∈

a layer in the OUT

31

EXAMPLE: ACQUIRING THE PSF[Tajik et al., JPIER-B 2017]

[photo credit: Daniel Tajik]

calibration object with small scattering probe at center: ,SP 18 i0, radius 5 mm, height 10 mmrε ≈ −

32

EXAMPLE: IMAGING TISSUE[Tajik et al., JPIER-B 2017][ EuCAP 2018]

rε ′ rε ′′

33

QUALITATIVE IMAGING WITH SENSITIVITY MAPS[Tu et al., Inv. Problems, 2015]• reconstruction formula

a

sc

incinc tot

( , )1( , )

( , )( )

( )

TN

S

S

SD S S d d

ξ

ξξ ξω ω

ξω

ωω

ε

∗

∈=

≈−

∂ ′ = − ⋅ ′∂

∑∫ ∫∫r r

r

rr r

r

• sensitivity map: 3D image of Fréchet derivative of ℓ2 norm of the differences of all total and incident responses

a

2tot inc21

( ) [ ( )] 0.5 ( , ) [ , ; ( )]TN

SD F S S d dξ ξω

ξε ω ω ε ω

∈=

′ ′ ′= = −∑∫ ∫∫rr r r r r rJ

( )( )

( )( )

Re ( )( )

Im ( )( )

mm

mm

dFDd

dFDd

ε

ε

′ =′ ′

′ = −′′ ′

rr

rr

→ indicates where contrast in ε' exists

→ indicates where contrast in ε" exists

adjoint sensitivity formula using simulated incident fields

34

SENSITIVITY MAPS USING MEASURED PSFs: THE NEAR-FIELD CASE

tot incinc inc sc,sp

sp sp

( , ) ( , )( , ) ( , ) PSF ( , ; )( ) ( ) ( )

S SS S ξ ξξ ξ ξω ωω ω ωε ε ε ε

′−∂ ∆≈ ≈ =

′ ′ ′∂ ∆ ∆ ∆

r rr r r rr r r

• from simulated incident fields to measured PSFs

scattering probe at center of z' plane

sc sc,0PSF ( , , , ; , , ) PSF ( , , , ; )x y z x y z x x y y z zξ ξω ω′ ′ ′ ′ ′ ′= − −

scattering probe at r'

uniform background along x and y

35

• sensitivity reconstruction formula with PSFs: scattered-power maps (SPM)

a

sc scsp ,0

1( ) ( ) ( , ) PSF ( , ; )

TN

SD M S d dξ ξω

ξε ω ω ω

∗

∈=

′ ′ ′−∆ ⋅ = = ⋅ ∑∫ ∫∫rr r r r r r

• SPM reconstruction formula with planar scanning

( )a

s( )

1

sc c( , , , ) PSF ,( , , ) , , ;T

S

NmM S x y z x x y y z dx y z dz dx yξω

ξξ ωω ω

=

∗ ′ ′ ′⋅ − − ′ ′ ′ = ∑ ∫∫ ∫

sc sccross-correlation of response and PSF in ( , )S x yξ ξ

[Shumakov et al., IEEE Trans. MTT, 2018][Nikolova, Introduction to Microwave Imaging, 2017]

SENSITIVITY MAPS USING MEASURED PSFs: THE NEAR-FIELD CASE – 2

• reconstruction is practically instantaneous – no systems of equations are solved

• effort shifted to near-field system calibration – measuring PSFs or simulating incident fields to compute response sensitivity inc ( , ) / ( )Sξ ω ε ′∂ ∂r r

36

SCATTERED-POWER MAPS: SIMULATION EXAMPLE (Altair FEKO)

simulation of data acquisitionsimulation of PSF acquisition

sample PSF: S11 at 4 GHz MAG/PHASE

r,b 1.0ε =r,sp 1.1ε = r,b 1.0ε =

r,OUT 1.2ε =

min

max

3 GHz16 GHz

1 GHz

fff

==

∆ =

37

SCATTERED-POWER MAPS: SIMULATION EXAMPLE (F SHAPE)

• blurring typical for cross-correlation methods – diffraction limit, limited number of responses

38

QUANTITATIVE REAL-TIME IMAGING WITH SPM

• requires measured PSFs – they scale accurately with the scattering-probe contrast

• reconstruction solves the linear problem

sr sp@

r,sp sp(( ) ( ))1

VM M dε

ε ′′′ ′ ′′=Ω

′∆∆

′∫∫∫ rr r rr

qualitative image (SPM) of OUT

qualitative image (SPM) of scattering probe

unknown contrast

• real-time solution via inversion in Fourier space (similar to holography)

sp@(0,0, )r,sp sp

convolution in ( , )

r1( , , ) ( , , )( , , ) z

z y x

x y

M x y z M x x y y zx y z dx dy dzε

ε ′′′′ ′′ ′′

′ ′ ′ ′ ′′ ′ ′′ ′ ′′ ′′ ′′= ⋅ − −∆ Ω

′′ ′′ ′′∆∫ ∫ ∫

[Shumakov et al., IEEE Trans. MTT, 2018][Tu et al., Inv. Problems, 2015]

39

QUANTITATIVE REAL-TIME IMAGING WITH SPM: FOURIER SPACE SOLUTION

vsp@(0,0, )

1r,sp sp( , , ) ( , , ), 1, ,( , , )

z

q

N

x y p z x y zx qq

y pM k k z Mf k kk z p Nk zε =

Ω= ⋅ =∆ Ω ∑

• small square system of equations to solve at each spectral position ( , )x yk k=κ( ) ( ) ( )=κ κ κM x m

( ) 1( , ) ( , )z

TNf z f z = κ κ κx

( ) 1( , ) ( , )z

TNM z M z = κ κ κm

1

1

sp@(0,0, ) 1 sp@(0,0, ) 1

( )

sp@(0,0, ) sp@(0,0, )

( , ) ( , )

( , ) ( , )

Nz

z N zz

z z

z N z N

M z M z

M z M z

=

κ

κ κ

κ κM

• final step: back to (x,y) space

,sp sp 12D

v( , , ) ( ; ) , 1, ,r

r n n zx y z f z n Nε

ε −∆ Ω′ ′ ′ ′∆ = =

Ωκ

40

QUANTITATIVE SPM: SIMULATION EXAMPLE (F SHAPE)

Re rε Im rε

[Nikolova, Introduction to Microwave Imaging, 2017]

41

QUANTITATIVE SPM: EXPERIMENTAL EXAMPLE

5 cm thick carbon-rubber sample

2nd layer

4th layer

sheet with scattering probe

,b 10 i5rε ≈ −

Re rε

Im rε

[Shumakov et al., IEEE Trans. MTT, 2018]

42

TIME DOMAIN FORWARD MODEL WITH PSF

4th layer

• linearized time-domain resolvent kernel

s

sc inc incRx Tx

( ; , ), (( ))

V ts t h h w dκ

′ ′′ ′≈ ∗′ ∗ ∫∫∫

r rr r r

incTx( )u ′′

inc incRx Rx Tx Tx( ; , ) ( ; , ) ( ; , ) ( )t h t h t w t′ ′ ′ ′′⇒ = ∗ ∗r r r r r r

• resolvent kernel and PSF all over again: measure response with each antenna pair and position of a point scatterer at

Rx Tx

inc inc0 Rx Tx sp sp Rx Tx ( 0; , , )

sp sp 0 Rx Tx

PSF ( , , )( , , )

tt h h w

tκκ

′= ′′= Ω ∗ ∗

= Ωr r r

r rr r

[Nikolova, Introduction to Microwave Imaging, 2017]

0′ =r

2b

( ) r

vεκ ∆′ =r

43

TIME DOMAIN FORWARD MODEL WITH PSF – 2

• assuming uniform background: ( )Rx Tx 0 Rx Tx( , ; ) , ; ( )t t t′ ′≈ − ∆r r r r r r

0′ ≠r

• example: analytical far-zone kernelTx Rx 2

Rx Tx 0 Rx Tx 0b b

( , ; ) PSF ( , , ) /t t t t rv v

δ′′ ′ − −

− − −

rr r r r

r r r r

some reference time

implied in r

44

DELAY AND SUM (DAS): THE CROSS-CORRELATION EXPLANATION

• cross-correlation – a measure of signal similarity

a

s a

,01

,0 ,01

( , ) PSF ( , , ) ( , )

( ) PSF ( , , ) PSF ( , , )

T

T

N

S

N

V S

X t t s t d

t t d d

ξ ξξ

ξ ξξ

κ

∈=

′′∈ ∈=

′ ′= ⊗

′′ ′ ′′ ′′⋅ ⊗

∑∫∫

∑∫∫∫ ∫∫

r

r r

r r r r r

r r r r r r r

steering filter for antenna pair at r toward voxel r'

• with large number of responses, X(r',t) ~ κ(r') as autocorrelation term dominates r" integral

a,0 ,0

1autocorrelation term: ( ) PSF ( , , ) PSF ( , , )

TN

St t dξ ξ

ξκ

∈=

′ ′ ′⋅ ⊗∑∫∫rr r r r r r

a,0 ,0

1cross-correlation terms: ( ) PSF ( , , ) PSF ( , , )

TN

St t dξ ξ

ξκ

∈=

′′ ′ ′′⋅ ⊗∑∫∫rr r r r r r

signal processing

45

DAS: SIGNAL-FLOW SCHEMATIC

• image generation: plot the “intensity” distribution max 20

( ) ( , )T

tI X t dt

=′ ′= ∫r r

46

DAS: CONCEPTUAL EXAMPLE

47

DAS: CONCEPTUAL EXAMPLE – 2

( , )X t ′r 2( ) ( , )t

I X t dt′ ′= ∫r r

plot image

48

CONCLUDING REMARKS

• we have just grazed the surface of an extensive subject

• real-time microwave imaging is rapidly growing and developing

hardware – antennas & RF/radar electronics

calibration methods

inversion methods

49

50

IMAGE SPATIAL RESOLUTION – WHAT TO EXPECT

• lateral (or cross-range) resolutioneff,min

, max,4 2x y

x ykλ πδ ≥ =

[Nikolova, Introduction to Microwave Imaging, 2017]

• depth (or range) resolution

eff,min bmax2 2zz

vBk

λ πδ ≥ = ≈

• wide viewing angles are critically important: wide-beam antennas, large scanned apertures

51

APPLICATIONS: NONDESTRUCTIVE TESTING[Sheen et al., “Near-field three-dimensional radar imaging techniques and applications,” Applied Optics 2010]

X-BAND (8 TO 12 GHZ) SCANNER WITH 16×16 ELECTRONICALLY SWITCHED ARRAY: ABSORBER INSPECTION

16×16 array

Pacific Northwest National Laboratory, Washington, USA

52

APPLICATIONS: THROUGH-WALL IMAGING[Depatla et al., “Robotic through-wall imaging,” IEEE A&P Mag. 2017]

Prof. Mostofi’s team at the University of California Santa Barbara

https://www.youtube.com/watch?v=THu3ZvAHI9A

53

LIMITATIONS OF BORN’S APPROXIMATION: TOTAL INTERNAL FIELD

• limitations on both the size and the contrast of the scatterer

2 2 2s b s( ) 1,a k k V− ∈r r

bksV

s ( ')k r s ( ')k r

a

• if OUT violates the limits: images contain artifacts which reflect differences between and rather than contrast

totTx ( )′E r

incTx ( )′E r

EXAMPLE: TOTAL VS. INCIDENT INTERNAL FIELD

0 10 20 30 40 50 60

10

15

20

25

30

mag

nitu

de

0 10 20 30 40 50 60

position inside scatterer (mm)

-200

-100

0

100

200

phas

e (d

eg)

r = 1.0

r = 1.1

r = 4.0

60 MM THICK DIELECTRIC SLAB IN AIR

[Nikolova, Introduction to Microwave Imaging, 2017]

1 GHzf =

1 GHzf =

54

NUMERICAL ASPECTS OF THE SOLUTION IN FOURIER SPACE

( )( )0 ,

1( ; )( ) PSF ( )

zNmmzij ij n ij

nf zS ξξ κ κκ ′

=

≈ ′∑

1, ,1, , T

m NN

ωξ==

( , )1, , ; 1, ,

ij x yx y

i k j ki N j Nκ = ∆ ∆= =

( ) ( ) ( )ij ij ijκ κ κ⋅ =A f d

(1) ( )1 1 1

( ) ( ) ( ) , ( ) ( ) ( ) , 1, ,T T

TT NT T Tij ij ij ij ij ij TN N N N

S S Nω

ω ωξ ξ ξκ κ κ κ κ κ ξ

× × = = = d d d d

1 1( ) [ ( ; ) ( ; )]z zT

ij ij ij N Nf z f zκ κ κ ×′ ′=f

the data vector:

the contrast vector:

the system (PSF) matrix:

solve (Nx⸱Ny) such systemsof size: T zN N Nω ×

1

1

(1) (1)0 , 0 ,1

( ) ( )0 , 0 ,

( ) PSF ( ) PSF ( )( ) where ( )

( ) PSF ( ) PSF ( )

Nz

TNz

z zij ij ij

ij ijN NN ij z zij ijω ω

ξ ξ

ξ

ξ ξ

κ κ κκ κ

κ κ κ

′ ′

′ ′

= =

AA A

A

55

SYNTHETIC FOCUSING – 2

56

THE FORWARD MODEL OF HOLOGRAPHY

s

inc inc,Rx ,Tx ( , , ;

sc

data, , ; )

( , , 1; , ,, , ,)) (r x y Tz xV y zx dx dyS x dz Nzy z yξ ξ ωξε ξω

′ ′ ′ ′ ′′ ′ ′≈ = ′⋅∆∫∫∫ E E

• the S-parameter forward model

• ξ (response type) replaces (i,k)

response1 S11

2 S22

3 S12 = S21

ξ ( , )i k(1,1)(2, 2)

(1,2) or (2,1)3TN =

TYPICAL RESPONSE TYPES

57

THE FORWARD MODEL OF HOLOGRAPHY: PLANAR SCAN

s

inc inc,Rx ,Tx ( , , ;

sc

data, , ; )

( , , , 1, ,,; , )) (r x y Tz xV y zx dx dyS x dz Nzy z yξ ξ ξ ω

ε ξω′ ′ ′

′ ′′ ′ ′≈ = ′⋅∆∫∫∫ E E

• the S-parameter forward model

• during scan, the Tx & Rx antennas in each ξ-th experiment are fixed wrt each other:if rRx is known then rTx is known, e.g., Rx Tx( , , ) and ( , , )x y z x y z D≡ ≡ −r r

position of Tx/Rx antenna pair

x

y

D

z

co-pol

cross-pol

#1#2

OUTz z=