Robust Multifrequency Inversion in Terahertz TomographyKe Chen and David Castanon ECE Department, Boston University

Motivation• Problem: The developed joint multifrequency inversion technique out-

performs single-frequency inversion, but requires complete and accuratespectral information

• Goal: Robust reconstruction approaches which do not assume exact priorinformation of spectra

Research to realityThis work provides a general framework for robust joint image formationusing multiple frequencies for improved reconstruction and sequent recogni-tion. The algorithms are applicable to Terahertz(THz) pulsed imaging whichhas attracted much attention for its ability to reveal unique spectral charac-teristics of chemicals in THz range and thus to fingerprint explosives.

Background

Object function:o(r, f)= n(r, f)2−1

Measurement:us(r)=u(r)−u0(r)

• Fourier Diffraction Theorem[1] relates scattering with object spectrum un-der the Born approximation u(r)≈u0(r)

Us(ωx) =ieilωy

2ωyO(ωx, ωy − k0), ωy ,

√k20 − ω2

x

• Model: {fm} in THz radiation spectrum

ym = Ψmxm, m = 1, . . . ,M

ym ∈ CKm Fourier transform(FT) of THz field measurement

xm ∈ CN Estimate of object function at fm

Ψm Nonuniform FT operator with fast approximation NUFFT[2] Tm ≈ Ψm

Joint inversion with E

JM: min(x,s)‖y − T x‖

2

+ α21‖DMx‖

2

WM+ ϕ(s, γ)

where x = [x1 , . . . ,xM]T , T , diag[Tm], Wn := W⊗ In, derivative operatorD

• Data-fidelity term in frequency domain, smooth penalty in spatial domain• Jointly construct boundaries s ∈ RN : Invariant across {fm}• Mumford-Shah[3]: ϕ(s, γ)=γ2‖∇s‖2 + 1

γ2 ‖s‖2

Spatially varying weighting: W , diag[(1− [s]i)

2]

Methodologies

Joint inversion with EJMP: min

(ν,s)‖y − T H ν‖

2

+ α21‖DJν‖

2

WJ+ ϕ(s, γ)

where ν=[ν1,...,ν

J]T ,H=(E1···EJ )⊗IN

• Given exact spectral prior Ej ∈ CM , j = 1, . . . , J

• Abundance of the j-th material νj ∈ RN , [νj ]i∈ [0, 1]

• Linear transform[4] x = Hν

Case 1. With partial E

RJMP1: min(x,s)‖y − T x‖

2

+ α21‖D(M+J)x‖

2

W(M+J)+ α2

2‖xc‖1

1+ ϕ(s, γ)

where xc=[xc1, . . . ,x

cM ]T, x=[ν,xc]T , T =(T H, T )

• Augment xcm∈ CN to compensate prior incomplete

• `1norm penalty on xc: ‖x‖11≈∑Ni=1

([x]2i + β

)1/2Case 2. With uncertain E• Uncertainty modelling: Ej−∆j ≤ ej ≤ Ej+∆j

• Inverse with worst-case prior: H = (e1 · · · eJ)⊗ IN

RJMP2: min(ν,s)

ψ(ν, s) + α21‖DJν‖

2

WJ+ ϕ(s, γ)

ψ(ν, s) , maxH‖y − T H ν‖

2

• NP-hard solving ψ: possible solution ψj(ν, s) at Hj , j=1, . . . , 22J

• Smoothing approximation[5]: ψp(ν, s) , λ−1log(∑

j exp(λψj(ν, s))

Experiment

Target and spectrum prior

An 81×81 phantom consisting of 3 explo-sives and air as background was generatedwith known spectral prior[6]. We simulatedTHz incident fields at 19 projection angles,and collected complex amplitude of the scat-tering at 4 frequencies for reconstruction.Each method was evaluated under variouslevels of Gaussian noise.

Noiseless SNR30 SNR20 SNR10

0.05

0

0.01

0.02

0.03

0.04 JM

JMP

1.8THz

wc-JMP

RJMP2

RJMP1

Noiseless SNR30 SNR20 SNR10

0.05

0

0.01

0.02

0.03

0.04JM

JMP

2.0THz

wc-JMP

RJMP2RJMP1

Noiseless SNR30 SNR20 SNR10

0.04

0

0.01

0.02

0.03JM

JMP

2.4THz

wc-JMP

RJMP2RJMP1

Noiseless SNR30 SNR20 SNR10

0.04

0

0.01

0.02

0.03 JM

JMP

3.0THz

wc-JMP

RJMP2

RJMP1

Exp1. ReconstructionRJMP1 is given only spec-tra of RDX. wc-JMP andRJMP2 are given the worst-case spectral priors under5% uncertainty.

Noiseless SNR30 SNR20 SNR10 SNR5

0.09

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08 JM

RJMP1

JMP

RDX

Noiseless SNR30 SNR20 SNR10 SNR5

0.09

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

JM

RJMP1JMP

PETN

Noiseless SNR30 SNR20 SNR10 SNR5

0.09

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

JM

RJMP1

JMP

Metabel

Noiseless SNR30 SNR20 SNR10 SNR5

0.01

0

0.001

0.002

0.004

0.006

0.008

JM

Air

Air

JMP

Exp2. Recognitionpixel-wise labeling j at r:JM:minimizer of Euclideandistance from x to Ej .JMP: maximizer νj .RJMP1: maximizer νj forsmall xc

Discussion

• JMP: force spectral and spatial consistency during inversion; sensitive tothe completeness and accuracy of spectral priors

• RJMP1: encourage sparse representation using given spectral priors

• RJMP2: : minimax optimization techniques

• Future work: handle uncertainties in accuracy and completeness at thesame time

References

ALERT

Three-Level Strategic Approach

Level 3: Grand

Challenges

Level 2

Enabling

Technology:

Testbeds

Level 1

Fundamental

Science

C1: Ultra-Reliable

Screening

C3: Unequivocal Pre-

& Post-Blast

Mitigation

C2: >50 meter

Stand-Off Discovery

and Assessment

C4: Rapid & Thorough

Preparedness and

Response

F1: Explosives

Characterization

F2: Explosives

Detection Sensors

F3: Explosives

Detection Sensor

Systems

F4: Blast

Mitigation

T1: Multimodal

Suicide Bomber

Detection

T2: Trace

Explosives

Detection/Screening

T3: VBIED

Signatures

Testbed T5: Roaming Ex-

plosives Detection

Sensor Suite

T4: Multisensor

Luggage

Scanner

Fieldable

Products

Teaming with

National Labs

& Industry

Research

Drivers

Research

Barriers

Proof-of-Principle

Tests of Research

This Work

[1] A. Kak, M. Slaney, Principles of Computerized Tomographic Imaging, 2001.[2] J. Fessler, B. Sutton, IEEE Tr. Sig. Proc. vol. 51, no. 2, pp. 560-574, 2003.[3] R. Weisenseel, Ph.D. diss., Boston University, 2004.[4] A. Li and et al. Appl. Opt. vol. 44, no. 10, pp.1948-1956, 2005.[5] E. Polak and et al. J Optimiz Theory App vol. 119, no. 3, 2003.[6] C. Baker and et al. Proc. of IEEE vol.95, no. 8, pp1559-1565, 2007.

This work was supported in part by Gordon-CenSSIS, the Bernard M. Gordon Center for Subsurface Sensing andImaging Systems, under the Engineering Research Centers Program of the National Science Foundation (Award NumberEEC-9986821), and the U.S. Department of Homeland Security (award number 2008-ST-061-ED0001).

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