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Page 1: Beam Hardening Correction via Mass Attenuation …isucsp.github.io/imgRecSrc/pdf/icassp2013poster.pdfBeam Hardening Correction via Mass Attenuation Discretization Renliang Gu and Aleksandar

Beam Hardening Correction via Mass Attenuation DiscretizationRenliang Gu and Aleksandar Dogandžić {renliang, ald}@iastate.edu

Department of Electrical and Computer Engineering

Background

(µ, α)

I in

Iout

According to the Lambert-Beer’s law [Jenkins and White 1957], the fraction dI/I of plane wave intensitylost in traversing an infinitesimal thickness dℓ at Cartesian coordinates (x, y) is proportional to dℓ, so that

dII

= −µ(ε)α(x, y) dℓ.

where▶ µ(ε) is the mass attenuation coefficient of the material (in cm2

g ),which depends only on the photon energy ε,

▶ α(x, y) is the density map of the inspected object (in gcm3 ).

Therefore, a monochromatic X-ray signal at photon energy ε attenuatesexponentially as it penetrates an object composed of a single material:

Iout = I in exp[−µ(ε)

∫ℓ

α(x, y) dℓ].

where Iout and I in are the emergent and incident X-ray signal energies, respectively.However, X-rays generated by vacuum tubes are not monochromatic. To describe the polychromatic X-raysource, assume that its incident intensity I in spreads along photon energy ε following the density ι(ε), i.e.,∫

ι(ε) dε = I in. (1a)

Then, the noiseless measurement collected by an energy integral detector upon traversing a straight lineℓ = ℓ(x, y) is

Iout =

∫ι(ε) exp

[− µ(ε)

∫ℓ

α(x, y) dℓ]

dε. (1b)

Polychromatic X-ray CT Model via Mass Attenuation

bb

µ(ε)

ι(ε)

∆µj

∆εj

0

0

ε

ε

Figure 1: Mass attenuation andincident spectrum as functions ofphoton energy ε.

Assumption:▶ Both incident spectrum ι(ε) and mass attenuation function µ(ε) of

the object are unknown.Objective:

▶ Estimate the density map α(x, y).Based on the fact that mass attenuation µ(ε) and incident spectrumdensity ι(ε) are both functions of ε (see Fig. 1), our idea is to:

▶ write the model as integrals of µ rather than ε;▶ estimate ι(µ) rather than ι(ε) and µ(ε).

For invertible µ(ε), define its inverse as ε(µ) and rewrite (1a) and (1b) as

I in =

∫ι(µ) dµ, Iout =

∫ι(µ) exp

[−µ

∫ℓ

α(x, y) dℓ]

dµ (2)

whereι(µ) ≜ ι(ε(µ))|ε′(µ)|

and ε(µ) is differentiable with derivative ε′(µ) = dε(µ)dµ . Invertibility of µ(ε) is assumed for simplicity: easy

to extend (2) to arbitrary µ(ε).

Discretizations over Mass Attenuationp

I in

IoutX-ray path ℓ

ϕi

αj

Figure 2: Discretization overspace:

∫ℓ α(x, y) dℓ ≈ ϕTα.

Discretize (2) in the mass attenuation and spatial domains using J mass attenuation bins and p pixels:

I in =J∑

j=1

Ij and Iout =J∑

j=1

Ij e−µj ϕTα (3)

where▶ α is a p × 1 column vector representing the 2-D image α(x, y) that

we wish to reconstruct,▶ ϕ is a p × 1 vector of weights quantifying how much each element ofα contributes to the X-ray attenuation on the straight-line path ℓ,

▶ µ0 < µ1 < · · · < µJ are known discretization points along the µ axis,∆µj = µj − µj−1, and

▶ Ij = ι(µj)∆µj ≈ ι(εj)∆εj.The mass attenuation coefficient (MAC) discretization is facilitated by the following facts:▶ µ of almost all materials at any energy level are within the range 10−2 cm2/g to 104 cm2/g,▶ the energy level of an X-ray scan is usually selected so that the function µ(ε) is as flat as possible.

Choose discretization points {µj}Jj=1 with a sufficiently wide range to cover µ(supp(ι(ε))):

µj = µ0qj, j = 1, 2, . . . , J. (4)Identifiability:▶ The value of µ0 in (4) can be arbitrary,▶ if I1 = 0, Iout ((α, [I1, I2, . . . IJ]

T)) ≡ Iout ((αq, [I2, I3, . . . IJ, 0]T)),

▶ if IJ = 0, Iout ((α, [I1, I2, . . . IJ]T)) ≡ Iout ((α/q, [0, I1, I2, . . . IJ−1]

T)).Hence, if the range of {µj}J

j=1 is sufficiently large to allow for zero edge elements of I , then the recoveryof α will be correct up to a scale of common ratio q.

Measurement Model, Assumptions, and Constraints

Figure 3: Projections at angles θ1 and θ2.

An X-ray CT scan consists of multiple projections with the beam intensity measured by multiple detectors,see Fig. 3. Model a vector z of N log-scale noisy measurements as

z = f(θ) + n = −ln◦[A(α)I] + nwhere ln◦(x) denotes elementwise logarithm,

▶ I = [I1, I2, . . . , IJ]T,

▶[A(α)

]i,j = exp(−ΦT

(i)αµj),▶ Φ = [Φ(1)Φ(2) · · ·Φ(N)]

T is the Radon transform matrix for ourimaging system, and

▶ n is additive white Gaussian noise.Our goal: estimate the image and incident energy density parameters

θ =(α,I

).

Assumptions:▶ Object’s shadow covered by the receiver array;▶ Known upper bound I in

MAX on incident energy I in: I in =∑J

j=1 Ij = 1TI ≤ I inMAX;

▶ ι(µ) and α(x, y) are nonnegative for all ε, x and y: I ⪰ 0 and α ⪰ 0.Notation: y ⪰ 0 denotes that all elements of a vector y are nonnegative.

Parameter EstimationpTo leave some margin for the noise and discretization effects, we relax the nonnegative signal constraintα ⪰ 0 and propose the following penalized least-squares (LS) objective function:

Lν,t(θ) =1

2∥z − f(θ)∥22 +

ν

2∥(−α)+∥22 + t

[− 1T

J×1 ln◦(I)− ln(I in

MAX − 1TJ×1I

)](5)

energy of negative pixels log barrier penaltywhere ν and t are scalar tuning constants for the signal nonnegativity and sparsity penalty terms.Notation: (x)+ keeps positive elements of x intact and sets the rest to zero.Minimization Algorithm: Define the gradient vectors gα,ν(θ), gI,t(θ) and Hessian matrices Hα,ν(θ),HI,t(θ) of the objective function (5) with respect to α and I , respectively. Descend (5) by alternatingbetween (i) and (ii):

(i) the Polak-Ribière nonlinear conjugate-gradientstep for α [Shewchuk 1994, Sec. 14.1] where Iis fixed and set to I(i);

α(i+1) = α(i) − sαgTα,ν(θ

(i))d(i)

d(i)THα,ν(θ(i))d(i)d

(i)

wheree(i) = gα,ν(θ

(i))− gα,ν(θ(i−1))

β(i) = max{0,

gTα,ν(θ

(i))e(i)

∥gα,ν(θ(i−1))∥22

}d(i) = gα,ν(θ

(i)) + β(i)d(i−1);

(ii) ▶ if1N×1 − z + f

(θ̃(i))

⪰ 0N×1 (6)holds, apply the Newton step for I :

I(i+1) = I(i) − sI[HI,t(θ̃

(i))]−1

gI,t(θ̃(i))

whereθ̃(i)

= (α(i+1),I(i));

▶ otherwise, i.e., if (6) does not hold, keepthe old I :

I(i+1) = I(i).

Note: (6) ensures HI,t(θ̃(i)) ≥ 0.

Numerical ExampleSimulation example based on a binary 1024× 1024 image in Fig. 5a obtained by thresholding the pixel valuesof a reconstruction in [Qiu and Dogandžić 2012, Fig. 5(b)].

▶ Inspected object, assumed to be made of iron, contains irregularly shaped inclusions; mass-attenuationfunction µ(ϵ) for iron extracted from the NIST database.

▶ Polychromatic sinogram simulated using photon-energy discretization with 130 equi-spaced discretizationpoints over the range 20 keV to 150 keV that approximates well the support of ι(ε).

10−1

100

101

102

20 60 100 140

µ(ε)

ι(ε)

ε/keV

µ(ε)/cm

2g−

1

0

0.1

0.2

0.3

ι(ε)

Figure 4: Simulated mass attenuationand incident X-ray spectrum asfunctions of the photon energy ε.

▶ 1024-element measurement array employed.▶ Radon transform Φ constructed using nonuniform fast Fourier

transform (NUFFT) with full circular mask [Dogandžić et al. 2011].▶ 180 equi-spaced parallel-beam projections with 1° spacing.▶ Performance metric is the relative square error (RSE) of an estimateα̂ of the signal coefficient vector:

RSE{α̂} = 1−(

α̂Tα

∥α̂∥2∥α∥2

)2

.

We compare▶ the traditional filtered backprojection (FBP) method, and▶ our MAC reconstruction upon convergence of the iteration (i)–(ii).

Figs. 5f and 5g show the histograms of the residuals z − Φα̂FBP and z − f(θ(+∞)).

0

0.2

0.4

0.6

0.8

1

(a) The ‘ground truth’

0

0.5

1

1.5

RSE=6.89 %

(b) FBP

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6RSE=0.54 %

(c) MAC

0

1

2

0 200 400 600 800 1000

FBPMAC

true

Pixel Position

Reconstructed

Pixel

Value

(d) The 500th row profile

0

1

2

0 200 400 600 800 1000

FBPMAC

true

Pixel Position

Reconstructed

Pixel

Value

(e) The 700th row profile

0

2

4

6

8

10

12

0 0.2 0.40−0.2−0.4

×103

Residual

(f) FBP

0

5

10

15

20

25

0 0.5 1.00−0.5−1.0

×103

×10−3Residual

(g) MAC

Figure 5: (a)–(c) The true image and FBP and MAC reconstructions, (d)–(e) corresponding 500th and 700th row profiles, and(f)–(g) residual histograms from the FBP and MAC reconstructions.

Future Workp

▶ Generalize the proposed MAC discretization to handle multiple materials,▶ Incorporate signal sparsity and develop an active set approach for estimating incident energy density

parameters I [Gu and Dogandžić 2013],▶ Iteratively refine the selection of the mass attenuation discretization points {µj}J

j=1 based on the obtainedestimates of I .

Referencesp

Aleksandar Dogandžić, Renliang Gu, and Kun Qiu. “Mask iterative hard thresholding algorithms forsparse image reconstruction of objects with known contour.” Proc. Asilomar Conf. Signals, Syst.Comput. Pacific Grove, CA, Nov. 2011, pp. 2111–2116.Renliang Gu and Aleksandar Dogandžić. Sparse signal reconstruction from polychromatic X-ray CTmeasurements via mass attenuation coefficient discretization. Tech. rep. NSF/IU. Ames, IA: CNDE,Iowa State Univ., Mar. 2013.Francis A Jenkins and Harvey E White. Fundamentals of Optics. 3rd ed. New York: McGraw-Hill, 1957.Kun Qiu and Aleksandar Dogandžić. “Sparse signal reconstruction via ECME hard thresholding.” IEEETrans. Signal Process. 60 (Sept. 2012), pp. 4551–4569.J. R. Shewchuk. An introduction to the conjugate gradient method without the agonizing pain.Tech. rep. CMU-CS-94-125. Pittsburgh, PA: Carnegie Mellon Univ., 1994.G. Van Gompel, K. Van Slambrouck, M. Defrise, K.J. Batenburg, J. de Mey, J. Sijbers, and J. Nuyts.“Iterative correction of beam hardening artifacts in CT.” Med. Phys. 38 (2011), S36–S49.

Acknowledgment ICASSP 2013General ChairsRabab Ward

University of British ColumbiaLi Deng

Microsoft

Technical Program ChairsVikram Krishnamurthy

University of British ColumbiaKostas Plataniotis

University of Toronto

Finance ChairJane Wang

University of British Columbia

Special Sessions ChairsXiaodong He

Microsoft Wu Chou

Huawei

Tutorials ChairKhaled El-Maleh

Qualcomm Technologies Inc.

Local Arrangement ChairPanos Nasiopoulos

University of British Columbia

Social Program ChairRabab Ward

University of British Columbia

Webmaster ChairMehrdad Fatourechi

University of British Columbia

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Arizona State UniversityMichel Sarkis

Qualcomm Technologies Inc.

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University of VictoriaVicky Zhao

University of Alberta

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MicrosoftWu Chou

HuaweiHank Liao

Google

Entrepreneurial RelationshipTon Kalker

Huawei

Conference ManagementBillene Mercer

Conference Management Services, Inc.

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