Package inspection using inverse diffraction

Package inspection using inverse diffraction Alastair D. McAulay Lehigh University, ECE Department, Bethlehem, PA, USA. ABSTRACT More efficient cost-effective hand-held methods of inspecting packages without opening them are in demand for security. Recent new work in TeraHertz sources,1 millimeter waves, presents new possibilities. Millimeter waves pass through cardboard and styrofoam, common packing materials, and also pass through most materials except those with high conductivity like metals which block light and are easily spotted. Estimating refractive index along the path of the beam through the package from observations of the beam passing out of the package provides the necessary information to inspect the package and is a nonlinear problem. So we use a generalized linear inverse technique that we first developed for finding oil by reflection in geophysics.2 The computation assumes parallel slices in the packet of homogeneous material for which the refractive index is estimated. A beam is propagated through this model in a forward computation. The output is compared with the actual observations for the package and an update computed for the refractive indices. The loop is repeated until convergence. The approach can be modified for a reflection system or to include estimation of absorption. Keywords: Package inspection Industrial inspection Diffraction Inverse diffraction Inverse problems Inverse scattering Terahertz light Signal Processing

1. INTRODUCTION Concern has increased in recent years over the shipment of packages with dangerous contents such as bombs, pistols, chemical explosives, amd hazardous chemicals. It is generally too expensive to open all packages at sources and destinations. Current machines for checking contents without opening packages, such as X-Ray machines are expensive and/or may not provide adequate 3-D viewing of the contents. We describe a novel approach that provides 3-D scans cost effectively without opening packages. The approach uses recently developed 100GHz sources which have 3mm wavelength—between optics and microwaves.1 A wavelength of 3mm passes through materials with particles less than 3mm in size such as cardboard and styrofoam, the usual packing materials for packages. Experiments are in progress to use 3mm waves for people inspection at airports because of its ability to penetrate through clothing. In section 2 we describe the approach in detail. Equations for the approach are presented in section 3. This is ongoing research for which we are currently performing computer simulations.

2. PRINCIPLES FOR PROPOSED UNOPENED PACKAGE INSPECTION A handheld scanner passes a short pulse from a collimated beam at 100GHz through the package to be inspected onto a sensor array, figure 1. The sensor is connected to a laptop computer for displaying the properties of the material along the path of the beam through the package. For the purpose of computation, the region through which the beam passed in the package is sliced into vertical slices, each considered to have different refractive index, absorption and reflection coefficients at their front surface. Each vertical slice is itself homogeneous. The thickness of the slices dk are selected for constant travel time T to simplify the computations3,4,5 . dk =

cT nk

(1)

After passing through the package, the light pulse is diffracted through air onto a detector. A computer algorithm uses the entry and string of exit pulses at the display caused by reverberation in the layer stack Further author information: E-mail: [email protected], Telephone: 1 610 758-6079, www.eecs.lehigh.edu/˜ amcaulay. Optics and Photonics for Information Processing II, SPIE 7072-16, Aug., 2008 Optics and Photonics for Information Processing II, edited by Abdul Ahad Sami Awwal, Khan M. Iftekharuddin, Bahram Javidi, Proc. of SPIE Vol. 7072, 70720H, (2008) 0277-786X/08/$18 · doi: 10.1117/12.796109 Proc. of SPIE Vol. 7072 70720H-1 2008 SPIE Digital Library -- Subscriber Archive Copy

dk

Light source

Detector

Layers in package Figure 1. Proposed system for inspection of unopened packages Initial refractive index nk

Output time sequence

Forward computation Compute J Update nk nk(m+1) = nk(m) + nk(m)

Deconvolve pulse out Difference x

Inverse computation. Solve for from J nk = x

nk

nk Is residual small? End

Figure 2. Block diagram for inverse computation for inspection of unopened packages

to estimate the properties of each of the vertical slices through which the beam passed. An output display shows functions for reflection coefficients, refractive index and (if included) absorption along the package in the direction of light propagation. Reflectivity and refractive index are not independent, one can be computed from the other (equation (2)). Sudden changes in reflectivity between vertical slices can reveal metallic reflective materials, sudden changes in refractive index indicate changes in material density affecting the speed of light such as between solids and clothes, and changes in absorption tends to indicate certain liquids and conductive materials. By moving the handheld device across the package, cross-sectional shapes of objects inside the package may be observed. Further, the handheld device can be used from the sides or back of the package to further define the contents. As an alternative approach, the region of the package can be discretized in the transverse, x and y direction, as well as the z direction, and an array of sensors used. A fixed configuration rather than a hand-held device can also be used.

3. COMPUTATION FOR PROPOSED UNOPENED PACKAGE INSPECTION We use a method similar to one we used previously in geophysics6,2 for estimating the parameters in a plane layer model of the earth for prospecting for oil. In our case we subdivide the package into vertical slices to form a vertically arranged stack of plane layers, figure 1. As the estimation of parameters for layers is nonlinear, we iterate in a loop as shown in figure 2. The forward computation computes the field reaching the sensor when the layer parameters are known. The inverse computation then finds values for updating the layer parameters so as to bring the output closer to that observed at the sensor.

Proc. of SPIE Vol. 7072 70720H-2

kth interface

+

E r

t

1 (a)

1

(b)

+

E

Figure 3. Reflection and transmission of TE mode at an interface: (a) entering from left and (b) entering from right

k

E

z

E

Ek

k

r’

t’

(k+1)th interface

d k

x

k+1

+

k

E k+1 nk

Figure 4. Propagation across a single layer

3.1 Forward computation We use the matrix method7,8 ; we have used it previously for leaking fields in integrated optics9 , and curved dielectric layered fields10 . We have also used it for determing the values of refractive index in a few layers when illuminating the waveguide at an angle and observing the angle of output beams for each mode as used in the prism coupling method in optical integrated circuits. Figure 3 shows the reflection and transmission for waves entering from the left of an interface (a) and waves entering from the right of the interface (b). Figure 4 shows propagation of right and left fields across a single layer, the kth layer of width dk . The reflection coefficients for the TE waveguide mode, having the electric field E perpendicular to the plane of incidence, entering from left and right of the kth interface respectively are given by nk − nk−1 and rk = −rk (2) rk = nk + nk−1 The transmission coefficients for the TE waveguide mode from above and below the kth interface respectively are given by: 2nk 2nk−1 tk = and tk = (3) nk + nk−1 nk + nk−1 We derive an equation for the right and left fields entering the kth interface in figure 4 in terms of those entering the k + 1th interface. In other words, the field just to the left of an interface to that just to the left of the following interface. Inspection of figures 3 and 4 provides Ek+

Ek− or

−tk Ek+ −rk Ek+

Therefore

−tk −rk

0 1

= tk Ek+ + rk Ek−

= tk Ek− + rk Ek+

+Ek−

Ek+ Ek−

(4)

−Ek+ +

= =

=

−1 rk 0 tk

rk Ek− tk Ek−

Ek+ Ek−

(5)

leading to

Ek+ Ek−

1 = −tk

1 rk

0 −tk

−1 rk 0 tk

Ek+ Ek−

1 = −tk


−1 −rk

−rk rk rk − tk tk

Ek+ Ek−

(6)

d2

d1 E1+

E1

n0

+

E1

E

-

E2-

E2

E3-

+

E2-

E3

+

E2

1

dK

n1

+

E 3+

EK-

EK

EK+1

E 3-

EK

+

EK-

EK+1

n3

n2

+

n K-1

+

nK

+ E K+1

E K+1 n0

Figure 5. Layer stack for inspection of unopened packages

where we used equation (2), rk = −rk . So, more simply + 1 Ek 1 = Ek− tk rk

rk 1

Ek+ Ek−

(7)

From figure 4, in passing √ from kth layer to the k + 1th layer or from k + 1th layer to the kth there is a unit delay represented by the Z transform operator, √ + = ZEk+ Ek+1 √ − Ek− = ZEk+1 (8) or

Ek+ Ek−

=

√ 1/ Z 0

√0 Z

+ Ek+1 − Ek+1

Substitute equation (9) into equation(7) √ + + 1 −1 rk Ek+1 1/ Z √0 Ek = − 0 tk Ek− Ek+1 0 Z tk √ + √ 1 Ek+1 1/ √Z rk√ Z = − Ek+1 Z tk rk / Z From equation (10) we name the matrix for propagating through the kth layer from right to left as √ √ 1 1 1/ √Z rk√ Z 1 rk Z = √ Sk = Z Z tk rk / Z Ztk rk

(9)

(10)

(11)

Figure 5 shows a vertical stack of layers for a package. Using equations (11) and (9), the field at the right of the stack can be written: +‘ +‘ E1 EK K S = Π (12) −‘ k=1 k E1−‘ EK where ΠK k=1 is the product of two-by-two matrices from k = 1 to k = K. There is a final interface at the right that can be accounted for by assuming an additional K + 1 layer with zero width dK+1 = 0 for which the scattering matrix, equation (11), reduces to 1 rK SK+1 = (13) rK 1


Then

+‘ EK −‘ EK

=

1 tK

1 rK

rK 1

+‘ EK+1 −‘ EK+1

= SK

+ EK − EK

We can define the product result for two-by-two matrices for the whole plane layer stack in a matrix M. + + +‘ EK M11 M12 E1 EK K Π S = S = K+1 k=1 k − M21 M22 E− EK E1−‘

(14)

(15)

In equation (15) there are four unknowns, the right and left fields at the top of the stack of layers and the right and left fields at the output at the right of the stack, but only two equations. We assume that there are no obstacles in the vicinity of the output of the layer stack. Therefore no field couples back into the right side − = 0. Then of the layer stack, or EK+1

E1+‘ E1−‘

=

M11 M21

M12 M22

+‘ EK 0

(16)

or the output sequence of the layer stack relates to the input, Q=

+ EK −1 = M11 E1+‘

(17)

where the series M11 is a single term of matrix M, defined in equation (15).

4. INVERSE PROBLEM OF COMPUTING PARAMETERS FOR LAYERS Following the generalized linear inverse approach2 of figure 2 we compute a Jacobian (or sensitivity) matrix that shows the sensivity of the output, that is the effect on output Qp at time sample p due to a change nk in the kth layer refractive index. ∂Qp Jp,k = (18) ∂nk From equation (17), Qp results from the product of two-by-two matrices for each layer. From equation (2), for the kth layer, only nk and nk−1 are involved, the other matrices in the two-by-two product chain remaining unchanged.2 In this case an element of the Jacobian matrix can be computed from ∂(M M ) j−1 j j−2 ΠK+1 S Jp,j = Πk=1 Sk k k=j+1 ∂nj ∂Mj ∂Mj−1 j−2 S = Πk=1 Sk Mj−1 + Mj ΠK+1 k k=j+1 ∂nj ∂nj ∂M j ΠK+1 = Πj−1 k=j+1 Sk k=1 Sk ∂nj ∂M j−1 K+1 + Πj−2 S S (19) Π k k k=j k=1 ∂nj From equation (19) the four product terms are partial computations for the forward computation. So the Jacobian is most easily computed at the same time as the forward computation, figure 2. The output of the forward computation series Qp as a function of time is now compared with that measured at the sensor Q after deconvolving the pulse out if necessary. Hence the difference vector ∆y p = Qp − Qp is computed. The difference vector ∆y is now used with the Jacobian matrix Jp,j to compute an update vector ∆n for the refractive indices for the layers by solving the linear equations ∆y = J∆n


(20)

The refractive indices are now updated for the next (m + 1)th iteration of the generalized inverse algorithm, figure 2, using, (21) nm+1 = nm + ∆nk (m) We repeat loops of forward and inverse computation until the rms of the update vector converges to a lower limit. When the Jacobian is sparse we use a conjugate gradient algorithm.5

5. CONCLUSION We describe a generalized linear inverse alogrithm that we adapted for inspection of unopended packages and that we originally developed for oil exploration.2 A 100GHz beam passes through cardboard and styrofoam and hence through the unopened package onto a sensor. The computation assumes parallel slices in the package of homogeneous material for which the refractive index is estimated. A beam is propagated through this model in a forward computation. The output is compared with the actual observations for the package and an update computed for the refractive indices. The loop is repeated until convergence. The same method can be modified for reflection only2 that would allow a hand scanner, looking like a bar-code reader to view the inside of the package from one side. Also, absorption can be estimated at the same time by allowing a complex refractive index.6 We plan to perform computer simulations and also investigate faster methods when no absorption is present that make use of the recursive Levinson-Durbin algorithm for solving equations with Toeplitz matrices.

ACKNOWLEDGMENTS I would like to thank the Electrical and Computer Engineering Department, the Center for Optical Technologies, and the Center for Advanced Materials and Nanotechnology at Lehigh University for their support.

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