Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References Acronyms Blind Beam-Hardening Correction from Poisson Measurements Aleksandar Dogandžić Electrical and Computer Engineering Iowa State University joint work with Renliang Gu, Ph.D. student supported by 1 / 57
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Blind Beam-Hardening Correction from Poisson Measurements
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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.
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`:
dII
D � �.x; y; "/™attenuation
d` D � �."/˛.x; y/šseparable
d`
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)
I in
Iout
To obtain the intensity decrease along a straight-line path` D `.x; y/, integrate along ` and over ". The underlyingmeasurement model is nonlinear.
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.
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.
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
Set ι.�/ D b.�/I.i�1/ and compute the new iterate of ˛ as
˛.i/D arg min
˛
�˛ � x
.i/�T
rLι
�x
.i/�
C1
2ˇ.i/
˛ � x.i/
2
2C ur.˛/
where ˇ.i/ > 0 is a step size,
x.i/
D ˛.i�1/C
�.i�1/�1�.i/
�˛.i�1/
� ˛.i�2/�
Nesterov accel.
� .i/D1
2
�1C
q1C 4
�� .i�1/
�2
�and the minimization is computed using an inner iterationthat employs the total-variation (TV)-based denoisingmethod in (Beck and Teboulle 2009, Sec. IV).
The optimization task in Step 1 is a proximal-gradient(PG) step:
˛.i/D proxˇ .i/ur
�x
.i/� ˇ.i/
rLι
�x
.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).
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,
x
y
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
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.
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.
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.
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
���
Z`
˛.x; y/ d`�
d�
whereι.�/ , �.".�//j"0.�/j
is the mass attenuation spectrum and the function ".�/ isdifferentiable with derivative
"0.�/ Dd".�/
d�
back
§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< �
˛.i/
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
.i/L
where ı.i/L D
ˇL
�˛.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.