Blind Beam-Hardening Correction from Poisson Measurements

Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Acronyms

Blind Beam-Hardening Correction fromPoisson Measurements�

Aleksandar Dogandžić

Electrical and Computer Engineering

Iowa State University

� joint work with Renliang Gu, Ph.D. student

supported by

1 / 57

Terminology and Notation I

“�” is the elementwise version of “�”;

B1 spline means B-spline of order 1,For a vector a D Œa1; : : : ; aN �

T 2 RN , definenonnegativity indicator function

IŒ0;C1/.a/ ,

(0; a � 0

C1; o.w.I

elementwise logarithm�lnı.a/

D ln ai ; 8i:

2 / 57

Terminology and Notation II

ιL.s/ is the Laplace transform of ι.�/:

ιL.s/ ,Z

ι.�/e�s� d�;

Laplace transform with vector argument:

bLı.s/ D bLı

ˇ266664s1s2:::

377775

266664bL.s1/

bL.s2/:::

bL.sN /

377775 :

3 / 57

X-ray CT

An X-ray computedtomography (CT) scanconsists of multipleprojections with the beamintensity measured bymultiple detectors.

Figure 1: Fan-beam CT system.

4 / 57

Introduction to CT Imaging (Parallel Beam)

A detector array is deployed parallel to the t axis and rotatesagainst the X-ray source collecting projections. Sinogram isthe set of collected projections as a function of angle atwhich they are taken.

x-ray source

sinogram6 / 57

Exponential Law of AbsorptionThe fraction dI=I of plane-wave intensity lost intraversing an infinitesimal thickness d` at Carte-sian coordinates .x; y/ is proportional to d`:

D � �.x; y; "/™attenuation

d` D � �."/˛.x; y/šseparable

where " is photon energy and

�."/ � 0 is the mass attenuationfunction of the material and

˛.x; y/ � 0 is the density map of theinspected object.

(κ, α)

To obtain the intensity decrease along a straight-line path` D `.x; y/, integrate along ` and over ". The underlyingmeasurement model is nonlinear.

8 / 57

Polychromatic X-ray CT Model

Incident energy I in spreads along photon energy " withdensity �."/:Z

�."/ d" D I in:

Noiseless energy measurementobtained upon traversing a straightline ` D `.x; y/ through an objectcomposed of a single material:

Iout D

l�."/ exp

��."/

˛.x; y/ d`�

(κ, α)

9 / 57

Linear Reconstruction Artifacts

Figure 3: FBP reconstruction of an industrial object.

Note the cupping and streaking artifacts of the linear filteredbackprojection (FBP) reconstruction, applied to ln Iout.

10 / 57

Problem Formulation and Goal

Assume that both

o the incident spectrum �."/ of X-ray source and

o mass attenuation function �."/ of the object

are unknown.

Goal: Estimate the density map ˛.x; y/.

11 / 57

Polychromatic X-ray CT Model UsingMass-Attenuation Spectrum

Mass attenuation �."/ and incident spectrum density �."/ areboth functions of ".Idea. Write the model as integrals of �rather than ":

I in D

Zι.�/ d� D ιL.0/

Iout D

Zι.�/ exp

��

˛.x; y/ d`�

D ιL�Z

˛.x; y/ d`�:

R Need to estimate one function, ι.�/,rather than two, �."/ and �."/!

κ(ε)

ι(ε)

∆κj

∆εj

12 / 57

Mass-Attenuation Spectrum

κ(ε) κ

ι(ε)ι(κ)

∆κj ∆κj

∆εj

Figure 4: Relation between mass attenuation �, incident spectrum�, photon energy ", and mass attenuation spectrum ι.�/.

13 / 57

Noiseless Polychromatic X-ray CT Model UsingMass-Attenuation Spectrum ι.�/: Summary

I in D ιL.0/

Iout D ιL�Z

˛.x; y/ d`� (κ, α)

14 / 57

Mass-Attenuation Spectrum and LinearizationFunction

For s > 0, the function

ιL.s/ D

ι.�/e�s� d�

is an invertible decreasing function of s.

.ιL/�1 converts the noiseless measurement

Iout D ιL�Z

˛.x; y/ d`�

into a linear noiseless “measurement”R

` ˛.x; y/ d`.

15 / 57

Mass-Attenuation Spectrum and LinearizationFunction

The .ιL/�1 ı exp.��/ mapping corresponds to the linearizationfunction in (Herman 1979) and converts ln Iout into a linearnoiseless “measurement”

R` ˛.x; y/ d`.

0 2 4 6 8 10 12 14 16

Polychromaticprojections

Monochromatic projections

− ln ιL(·)

The standard FBP method fits a linear model to log ofthe energy measurementso for monochromatic X-ray source, �ln Iout is an affine

function ofR

`˛.x; y/ d`. 16 / 57

Basis-function expansion of mass-attenuationspectrum ι.�/ D b.�/I

š.�/

š.�/b.�/I

Figure 5: B1-spline expansion ι.�/ D b.�/I, where the B1-splinebasis is b.�/“

D�b1.�/; b2.�/; : : : ; bJ .�/

�. ι.�/ � 0 implies I � 0.

17 / 57

B1-Spline Basis Functions

Select spline knots from a growing geometric series withcommon ratio

which yields the B1-spline basis functions:

bj .�/ D

˚� � qj �1�0

.q � 1/qj �1�0; qj �1�0 � � < qj �0

�� C qj C1�0

.q � 1/qj �0; qj �0 � � < qj C1�0

0; otherwise

where the j th basis function can be obtained by q-scalingthe .j � 1/th basis function:

bj .�/ D bj C1.q�/:

18 / 57

Comments

The geometric-series knots have a wide span, from �0 toqJ C1�0, and compensate larger � with a “geometrically”wider integral range, which results in an effectiveapproximation of the noiseless measurements.

The common ratio q determines the resolution of theB1-spline approximation.

In summary, the following three tuning constants:

.q; �0; J /

define our B1-spline basis functions b.�/.

19 / 57

Discretization of ˛.x; y/

˛ � 0 is a p � 1 vectorrepresenting the 2D image thatwe wish to reconstruct and

� � 0 is a p � 1 vector of knownweights quantifying how mucheach element of ˛ contributesto the X-ray attenuation on thestraight-line path `.

detector array

b X-ray source

R` ˛.x; y/ d` � �T˛.

20 / 57

Shift Ambiguity of the Mass-Attenuation Spectrum

By notingbj .�/ D bj C1.q�/:

and the �-scaling property of the Laplace transform,

bj .q�/L!

�we conclude that selecting q times narrower basis functions

b.q�/ D�b0.�/; b1.�/; : : : ; bJ �1.�/

�and q times larger density map and spectral parameters (q˛and qI) yields the same mean output photon energy.

21 / 57

Mean output photon energy:

Iout.˛;IIb.�// D bLı.ˆ˛/I

Iout.q˛; qIIb.q�// D1

�ˆq˛

The right-hand sides are identical!

22 / 57

Shift Ambiguity of the Mass-Attenuation Spectrum

Iout�˛; Œ0; I2; : : : ; IJ �

D Iout�q˛; qŒI2; : : : ; IJ ; 0�

This shift ambiguity of the mass-attenuation spectrumallows us to rearrange leading or trailing zeros in themass-attenuation coefficient vector I and position thecentral nonzero part of I.

23 / 57

Multiple Measurements and Projection Matrix

Denote by N the total number of measurements from allprojections collected at the detector array.For the nth measurement, define its discretized lineintegral as �T

n ˛.Stacking all N such integrals into a vector yields

ˆ’monochromaticprojection of ˛

�projectionmatrix

266664�T

�T2:::

377775N �p

24 / 57

Noiseless Measurement Model

The N � 1 vector of noiseless measurements is

Iout.˛;I/ D bLı.ˆ˛/˜output

basis-functionmatrix

bLı.s/ D

266664bL.s1/

bL.s2/:::

bL.sN /

377775and s D ˆ˛ is the monochromatic projection.

25 / 57

Noise Model

Assume noisy Poisson-distributed measurementsE D .En/

NnD1, with the negative log-likelihood (NLL)

functionL.ˆ˛;I/

L.s;I/ D 1T�bLı.s/I � E

�� ET

�bLı.s/I

�� lnı E

The Poisson model is a good approximation for the moreprecise compound-Poisson distribution (Xu and Tsui2014; Lasio et al. 2007).

26 / 57

NLL of I

Define A D bLı.ˆ˛/.The NLL of I for known ˛ reduces to the NLL for Poissongeneralized linear model (GLM)‡ with identity link and designmatrix A:

LA.I/ D 1T .AI � E/ � ET�lnı.AI/ � lnı E

‡See (McCullagh and Nelder 1989) for introduction to GLMs.27 / 57

NLL of ˛

The NLL of ˛ for fixed ι.�/ is also a Poisson GLM:

Lι.˛/ D 1T�ιLı.ˆ˛/ � E

�� ET

�ιLı.ˆ˛/

�� lnı E

owith the link function equal to the inverse of ιL.�/. Since ι.�/

is known, we do not need its basis-function expansion.

R Lι.˛/ is convex, under conditions that we established in (G.and D. 2016)!

28 / 57

Regularized NLL Minimization

min˛;I

L.ˆ˛;I/C ur.˛/C IŒ0;C1/.I/“,f .˛;I/

where u > 0 is a scalar tuning constant and r.˛/ is thedensity-map regularization term:

r.˛/ D k .˛/k1 C IŒ0;C1/.˛/:

Here, .˛/ D� i .˛/

iD1, where

i .˛/ D

qPj 2Ni

.˛i � j /2 is the gradient map,

Ni is the index set of neighbors of ˛i in an appropriate2D arrangement.

29 / 57

TV Sparsifying Transform

p pixels

$grad. map i .˛/

# significant coeffs � p

30 / 57

Goal and Minimization Approach

Goal: Estimate the density-map and mass-attenuationspectrum parameters

.˛;I/

by minimizing the penalized NLL f .˛;I/.Approach: A block coordinate-descent that uses

Nesterov’s proximal-gradient (NPG) (Nesterov 2013) and

limited-memory Broyden-Fletcher-Goldfarb-Shanno withbox constraints (L-BFGS-B) (Byrd et al. 1995)

methods to update estimates of the density map andmass-attenuation spectrum parameters.We refer to this iteration as NPG-BFGS algorithm.

31 / 57

Iteration i of NPG-BFGS

Alternate between 1 and 2 :

1 Update ˛ using NPG. Set the mass-attenuationspectrum ι.�/ D b.�/I.i�1/ and descend the objectivefunction f

�˛;I.i�1/

�D Lι.˛/C ur.˛/ by applying an

NPG step for ˛, which yields ˛.i/:

f�˛.i/;I.i�1/

�� f

�˛.i�1/;I.i�1/

2 Update I using BFGS. Set A D bLı�ˆ˛.i/

�and

minimize f�˛.i/;I

I.i/D arg min

I�0LA

�using the inner L-BFGS-B iteration.

32 / 57

1 Update ˛: NPG Step

Set ι.�/ D b.�/I.i�1/ and compute the new iterate of ˛ as

˛.i/D arg min

�˛ � x

.i/�T

.i/�

2ˇ.i/

˛ � x.i/

2C ur.˛/

where ˇ.i/ > 0 is a step size,

D ˛.i�1/C

�.i�1/�1�.i/

�˛.i�1/

� ˛.i�2/�

Nesterov accel.

� .i/D1

�� .i�1/

�and the minimization is computed using an inner iterationthat employs the total-variation (TV)-based denoisingmethod in (Beck and Teboulle 2009, Sec. IV).

33 / 57

Comments

The optimization task in Step 1 is a proximal-gradient(PG) step:

˛.i/D proxˇ .i/ur

.i/� ˇ.i/

.i/��

If we do not apply the Nesterov’s acceleration and useonly the PG step to update the density-map iterates ˛,i.e., x.i/ D ˛.i�1/, then the corresponding iteration is thePG-BFGS algorithm;

We select the step size ˇ.i/ adaptively to account forvarying local Lipschitz constants of the objectivefunction and restart the Nesterov acceleration by� .i/ D 0 when the objective function f .˛;I/ is notdecreasing (O‘Donoghue and Candès 2015).

34 / 57

2 Update I: L-BFGS-B

Broyden-Fletcher-Goldfarb-Shanno (BFGS) iteration usedto update iterates of I is a state-of-the-artquasi-Newton method (Thisted 1989, Sec. 4.3.3.4).

box constraints in the L-BFGS-B variant impose thenonnegativity

I � 0:

35 / 57

Numerical Examples

B1-spline constants set to satisfy

J D 20; # basis functions

qJD 103; span

�0qd0:5.J C1/e

D 1; centering

36 / 57

Real X-ray CT Example I

360 equi-spaced fan-beamprojections with 1° spacing,

X-ray source to rotation centeris 3492� detector size,

measurement array size of 694elements,

projection matrix ˆconstructed directly on GPU,

detector array

bX-ray source

D rotate

imaginarydetector array

yielding a nonlinear estimation problem with N D 694 � 360measurements and an 512 � 512 image to reconstruct.Implementation available at github.com/isucsp/imgRecSrc.

Real data provided by Joe Gray, CNDE. Thanks!37 / 57

(a) FBP (b) NPG-BFGS (u D 10�5)

Figure 6: Real X-ray CT: Full projections.

38 / 57

Comments I

Our reconstruction eliminates

the streaking artifacts across the air around the object,

the cupping artifacts with high intensity along theborder.

The regularization constant u has been tuned for goodreconstruction performance.

39 / 57

Comments II

The slight non-uniformity of the reconstructed densitymap in Fig. 6b may be due to

detector saturation that leads to measurementtruncation,scattering,noise-model mismatch, orthe bowtie filter applied to the X-ray source.

We leave further verification of causes and potentialcorrection of this problem to future work and note thatthis issue does not occur in the simulated-dataexamples that we constructed.

40 / 57

Inverse Linearization Function Estimate

0 1 2 3 4 5 6 7 8 9

NPG-BFGSFBP

fitted − ln[bL(·)I

Figure 7: The polychromatic measurements as function of themonochromatic projections and its corresponding fitted curve.

Observe the biased residual for FBP, the unbiased residualfor NPG-BFGS and its increasing variance.

41 / 57

Real X-ray CT Example II

360 and 120 equi-spaced fan-beam projections,

X-ray source to rotation center is 8696 times of a singledetector size,

measurement array size of 1380 elements,

projection matrix ˆ constructed on GPU with fullcircular mask.

yielding a nonlinear estimation problem withN D 1380 � 360 measurements and an 1024 � 1024 imageto reconstruct.We employ same convergence constants as in the previousexample.

42 / 57

Figure 8: Reconstructions from 360 fan-beam projections with 1°spacing.

43 / 57

Figure 9: Estimated ˛ and �ln�bL.�/I

�from 360 fan-beam

projections.

44 / 57

Inverse Linearization Function Estimate

−0.5

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

NPG-BFGSFBP

fitted − ln[bL(·)I

Figure 10: The polychromatic measurements as function of themonochromatic projections and its corresponding fitted curve.

45 / 57

10−1

0 1000 2000

,I)−

Number of iterations

PG-BFGSNPG-BFGS

Figure 11: The objective function as a function of iteration index.

46 / 57

Figure 12: Reconstructions from 120 fan-beam projections with3° spacing.

Observe aliasing artifacts in the FBP reconstruction.47 / 57

(a) 360 projections (b) 120 projections

Figure 13: NPG-BFGS (u D 10�5) reconstructions from fan-beamprojections.

The reconstructed density maps are uniform, except thedefect region.

48 / 57

Biconvexity, KL Property, and Convergence

Under certain condition,o L.ˆ˛;I/ is biconvex with respect to ˛ and I.o our objective function f .˛;I/ satisfies the

Kurdyka-Łojasiewicz (KL) inequality.

The above facts can be used to establish the localconvergence for alternating proximal minimizationmethods (Attouch et al. 2010; Xu and Yin 2013)

o e.g., PG-BFGS,o not NPG-BFGS.

See (G. and D. 2016; G. and D. 2015).

49 / 57

Conclusion

Developed a blind method for sparse density-map imagereconstruction from polychromatic X-ray CT measurementsin Poisson noise.

50 / 57

Future Work

Apply to more real X-ray CT data sets to test andpotentially improve our proposed algorithm,

Generalize our polychromatic signal model to handlemultiple materials and develop correspondingreconstruction schemes.

51 / 57

Publications

R. G. and A. D., “Blind X-ray CT image reconstructionfrom polychromatic Poisson measurements,” IEEETrans. Comput. Imag., vol. 2, no. 2, pp. 150–165,2016. doi: 10.1109/tci.2016.2523431.

R. G. and A. D. (Sep. 2015), Polychromatic X-ray CTimage reconstruction and mass-attenuation spectrumestimation, arXiv: 1509.02193 [stat.ME].

52 / 57

References I

H. Attouch, J. Bolte, P. Redont, and A. Soubeyran, “Proximal alternatingminimization and projection methods for nonconvex problems: An approachbased on the Kurdyka-Łojasiewicz inequality,” Math. Oper. Res., vol. 35,no. 2, pp. 438–457, May 2010.

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrainedtotal variation image denoising and deblurring problems,” IEEE Trans. ImageProcess., vol. 18, no. 11, pp. 2419–2434, 2009.

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm forbound constrained optimization,” SIAM J. Sci. Comput., vol. 16, no. 5,pp. 1190–1208, 1995.

R. G. and A. D. (Sep. 2015), Polychromatic X-ray CT image reconstructionand mass-attenuation spectrum estimation, arXiv: 1509.02193 [stat.ME].

R. G. and A. D., “Blind X-ray CT image reconstruction from polychromaticPoisson measurements,” IEEE Trans. Comput. Imag., vol. 2, no. 2,pp. 150–165, 2016.

G. T. Herman, “Correction for beam hardening in computed tomography,”Phys. Med. Biol., vol. 24, no. 1, pp. 81–106, 1979.

References II

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging.New York: IEEE Press, 1988.

G. M. Lasio, B. R. Whiting, and J. F. Williamson, “Statistical reconstructionfor X-ray computed tomography using energy-integrating detectors,” Phys.Med. Biol., vol. 52, no. 8, p. 2247, 2007.

P. McCullagh and J. Nelder, Generalized Linear Models, 2nd ed. New York:Chapman & Hall, 1989.

Y. Nesterov, “Gradient methods for minimizing composite functions,” Math.Program., Ser. B, vol. 140, no. 1, pp. 125–161, 2013.

B. O‘Donoghue and E. Candès, “Adaptive restart for accelerated gradientschemes,” Found. Comput. Math., vol. 15, no. 3, pp. 715–732, 2015.

R. A. Thisted, Elements of Statistical Computing. New York: Chapman &Hall, 1989.

J. Xu and B. M. Tsui, “Quantifying the importance of the statisticalassumption in statistical X-ray CT image reconstruction,” IEEE Trans. Med.Imag., vol. 33, no. 1, pp. 61–73, 2014.

References III

Y. Xu and W. Yin, “A block coordinate descent method for regularizedmulticonvex optimization with applications to nonnegative tensorfactorization and completion,” SIAM J. Imag. Sci., vol. 6, no. 3,pp. 1758–1789, 2013.

Glossary

BFGS Broyden-Fletcher-Goldfarb-Shanno. 33

CT computed tomography. 4, 49

FBP filtered backprojection. 8, 14, 39, 45, 56

GLM generalized linear model. 25, 26

KL Kurdyka-Łojasiewicz. 47

L-BFGS-B limited-memoryBroyden-Fletcher-Goldfarb-Shanno with boxconstraints. 29, 30

NLL negative log-likelihood. 24–26, 29

NPG Nesterov’s proximal-gradient. 29, 30, 39

PG proximal-gradient. 32

TV total-variation. 28, 3155 / 57

Polychromatic X-ray CT Model via Mass Attenuation

For invertible �."/§, define its inverse as ".�/. Then,

I in D

Zι.�/ d�; Iout D

Zι.�/ exp

��

˛.x; y/ d`�

whereι.�/ , �.".�//j"0.�/j

is the mass attenuation spectrum and the function ".�/ isdifferentiable with derivative

"0.�/ Dd".�/

§Assumed for simplicity, extends easily to arbitrary �."/.

Initialization and Convergence Criteria I

Initialize the density-map iterates as follows:

˛.�1/D yFBP; ˛.0/

D 0; � .0/D 0

where yFBP is the standard FBP reconstruction (Kak andSlaney 1988, Ch. 3).

Convergence criterion

ı.i/ , ˛.i/

� ˛.i�1/

2< �

where � > 0 is the convergence threshold.

Initialization and Convergence Criteria II

Inner-loop convergence criteria

˛.i;k/� ˛.i;k�1/

2< �˛ı

.i�1/ˇLA

�I.i;k/

�� LA

�I.i;k�1/

�ˇ� �Iı

where ı.i/L D

�˛.i/;I.i�1/

�� L

�˛.i�1/;I.i�1/

�ˇ,

k are the inner-iteration indices, and

the convergence tuning constants �˛ 2 .0; 1/ and�I 2 .0; 1/ are chosen to trade off the accuracy andspeed of the inner iterations and provide sufficientlyaccurate solutions by these iterations.

Blind Beam-Hardening Correction from Poisson Measurements

Engineering

Transcript of Blind Beam-Hardening Correction from Poisson Measurements

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