I will describe a sparse image reconstruction method for Poisson-distributed polychromatic X-ray computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. To obtain a parsimonious mean measurement-model parameterization, we first rewrite the measurement equation by changing the integral variable from photon energy to mass attenuation, which allows us to combine the variations brought by the unknown incident spectrum and mass attenuation into a single unknown mass-attenuation spectrum function; the resulting measurement equation has the Laplace integral form. The mass-attenuation spectrum is then expanded into basis functions using a B-spline basis of order one. We develop a block coordinate-descent algorithm for minimization of a penalized Poisson negative log-likelihood (NLL) cost function, where penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the density map image. This algorithm alternates between a Nesterov’s proximal-gradient (NPG) step for estimating the density map image and a limited-memory Broyden–Fletcher–Goldfarb–Shanno with box constraints (L- BFGS-B) step for estimating the incident-spectrum parameters. To accelerate convergence of the density-map NPG steps, we apply a step-size selection scheme that accounts for varying local Lipschitz constants of the objective function. I will discuss the biconvexity of the penalized NLL function and outline preliminary results on convergence of PG-BFGS schemes. Finally, I will present real X-ray CT reconstruction examples that demonstrate the performance of the proposed scheme.
Blind Beam-Hardening Correction from Poisson Measurements
1. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Blind Polychromatic X-ray CT Reconstruction
from Poisson MeasurementsŽ
Renliang Gu and Aleksandar Dogandžić
Electrical and Computer Engineering
Iowa State University
Ž presented by Aleksandar Dogandžić
supported by
1 / 42
2. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
References
R. G. and A. D., “Blind polychromatic X-ray CT
reconstruction from Poisson measurements,” in Proc.
IEEE Int. Conf. Acoust., Speech, Signal Process.,
Shanghai, China, Mar. 2016, pp. 898–902.
R. G. and A. D., “Blind X-ray CT image reconstruction
from polychromatic Poisson measurements,” IEEE
Trans. Comput. Imag., vol. 2, no. 2, pp. 150–165,
2016. DOI: 10.1109/TCI.2016.2523431.
2 / 42
3. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Terminology and Notation I
“ ” is the elementwise version of “ ”;
B1 spline means B-spline of order 1,
For a vector a D .an/ 2 RN , define
nonnegativity indicator function
IŒ0;C1/.a/ ,
(
0; a 0
C1; o.w.
I
elementwise logarithm
Œlnı an D ln an; 8n:
3 / 42
4. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Terminology and Notation II
ιL.s/ is the Laplace transform of ι.Ä/:
ιL
.s/ ,
Z
ι.Ä/e sÄ
dÄ;
Laplace transform with vector argument:
bL
ı.s/ D bL
ı
ˇ2
6
6
6
6
4
s1
s2
:::
sN
3
7
7
7
7
5
D
2
6
6
6
6
4
bL
.s1/
bL
.s2/
:::
bL
.sN /
3
7
7
7
7
5
:
4 / 42
5. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
X-ray CT
An X-ray computed
tomography (CT) scan
consists of multiple
projections with the beam
intensity measured by
multiple detectors.
Figure 1: Fan-beam CT system.
5 / 42
6. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Third Generation X-ray CT
(a) medical CT
Extremely fast and accurate
For
con
The
feat
con
blad
to r
By v
requ
GE R
When compared with conventional cone beam CT, highly collimated fa
quality CT slice result, due to the reduction of image artifacts caused b
Robotic sample
manipulation
Turbine blade
GE Jupiter linear
detector array
X-ray fan beam
Blade rotation
X-ray projection
imagesAcquisition of
slice projections
ISOVOLT Titan
450 kV X-ray source
In
m
on
tu(b) GE’s system for CT turbine-blade
inspection
6 / 42
7. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Introduction to CT Imaging (Parallel Beam)
A detector array is deployed parallel to the t axis and rotates
against the X-ray source collecting projections. Sinogram is
the set of collected projections as a function of angle at
which they are taken.
x-ray source
Â
t
Image
Â
t
sinogram
7 / 42
8. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Iterative Reconstruction
exactly the inverse of backprojection. It is per-
formed by summing up the intensities along
the potential ray paths for all projections
through the estimated image. The set of
projections (or sinogram) generated from
the estimated image then is compared with
and for this reason have been more slowly
adopted in the clinical setting. However, as
computer speeds continue to improve, and
with a combination of computer acceleration
techniques (e.g., parallel processors), and
intelligent coding (e.g., exploiting symmetries
FIGURE 16-17 Schematic illustration of the steps in iterative reconstruction. An initial image estimate is made and
projections that would have been recorded from the initial estimate then are calculated by forward projection. The
calculated forward projection profiles for the estimated image are compared to the profiles actually recorded from the
object and the difference is used to modify the estimated image to provide a closer match. The process is repeated until
Object
⍺(x,y)
CT system
Measured
projection data
sinogram p(r,φ)
Image
estimate
ɑ *(x,y)
Calculated
projection
data p(r,φ)
Compare
converged?
Yes
No
Update
image
estimate
Forward
projection
Reconstructed
image
from (Cherry et al. 2012)
8 / 42
9. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Exponential Law of Absorption
The fraction dI=I of plane-wave intensity lost in
traversing an infinitesimal thickness d` at Carte-
sian coordinates .x; y/ is proportional to d`:
dI
I
D .x; y; "/™
attenuation
d` D Ä."/˛.x; y/š
separable
d`
where " is photon energy and
Ä."/ 0 is the mass attenuation
function of the material and
˛.x; y/ 0 is the density map of the
inspected object.
(κ, α)
Iin
Iout
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10. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Exponential Law of Absorption
The fraction dI=I of plane-wave intensity lost in
traversing an infinitesimal thickness d` at Carte-
sian coordinates .x; y/ is proportional to d`:
dI
I
D .x; y; "/™
attenuation
d` D Ä."/˛.x; y/š
separable
d`
where " is photon energy and
Ä."/ 0 is the mass attenuation
function of the material and
˛.x; y/ 0 is the density map of the
inspected object.
(κ, α)
Iin
Iout
To obtain the intensity decrease along a straight-line path
` D `.x; y/, integrate along ` and over ". The underlying
measurement model is nonlinear.
9 / 42
11. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Polychromatic X-ray CT Model
Incident energy Iin spreads along photon energy " with
density Ã."/:
Z
Ã."/ d" D Iin
:
Noiseless energy measurement
obtained upon traversing a straight
line ` D `.x; y/ through an object
composed of a single material:
Iout
D
l
Ã."/ exp
Ä
Ä."/
Z
`
˛.x; y/ d` d":
(κ, α)
Iin
Iout
10 / 42
12. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Problem with Linear-Model Based Reconstructions
(a) (b)
Figure 3: (a) FBP and (b) linear-model reconstructions from
polychromatic X-ray CT measurements.
11 / 42
13. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Problem Formulation and Goal
Assume that both
o the incident spectrum Ã."/ of X-ray source and
o mass attenuation function Ä."/ of the object
are unknown.
12 / 42
14. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Problem Formulation and Goal
Assume that both
o the incident spectrum Ã."/ of X-ray source and
o mass attenuation function Ä."/ of the object
are unknown.
Goal: Estimate the density map ˛.x; y/.
12 / 42
15. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Polychromatic X-ray CT Model Using
Mass-Attenuation Spectrum
Mass attenuation Ä."/ and incident spectrum density Ã."/ are
both functions of ".
κ(ε)
ι(ε)
0
0
ε
ε
13 / 42
16. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Polychromatic X-ray CT Model Using
Mass-Attenuation Spectrum
Mass attenuation Ä."/ and incident spectrum density Ã."/ are
both functions of ".
Idea. Write the model as integrals of Ä
rather than ":
Iin
D
Z
ι.Ä/ dÄ D ιL
.0/
Iout
D
Z
ι.Ä/ exp
Ä
Ä
Z
`
˛.x; y/ d` dÄ
D ιL
ÂZ
`
˛.x; y/ d`
Ã
:
4 Need to estimate one function, ι.Ä/,
rather than two, Ã."/ and Ä."/!
κ(ε)
ι(ε)
∆κj
∆εj
0
0
ε
ε
13 / 42
17. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Mass-Attenuation Spectrum
κ(ε) κ
ι(ε)
ι(κ)
∆κj ∆κj
∆εj
0
0
0ε
ε
Figure 4: Relation between mass attenuation Ä, incident spectrum
Ã, photon energy ", and mass attenuation spectrum ι.Ä/.
14 / 42
18. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Noiseless Polychromatic X-ray CT Model Using
Mass-Attenuation Spectrum ι.Ä/: Summary
Iin
D ιL
.0/
Iout
D ιL
ÂZ
`
˛.x; y/ d`
Ã
(κ, α)
Iin
Iout
15 / 42
19. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Mass-Attenuation Spectrum and Linearization
Function
For s > 0, the function
ιL
.s/ D
Z C1
0
ι.Ä/e sÄ
dÄ
is an invertible decreasing function of s.
16 / 42
20. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Mass-Attenuation Spectrum and Linearization
Function
For s > 0, the function
ιL
.s/ D
Z C1
0
ι.Ä/e sÄ
dÄ
is an invertible decreasing function of s.
.ιL/ 1 converts the noiseless measurement
Iout
D ιL
ÂZ
`
˛.x; y/ d`
Ã
into a linear noiseless “measurement”
R
` ˛.x; y/ d`.
16 / 42
21. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Mass-Attenuation Spectrum and Linearization
Function
The .ιL/ 1 ı exp. / mapping corresponds to the linearization
function in (Herman 1979) and converts ln Iout into a linear
noiseless “measurement”
R
` ˛.x; y/ d`.
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10 12 14 16
Polychromaticprojections
Monochromatic projections
− ln ιL
(·)
17 / 42
22. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Basis-function expansion of mass-attenuation
spectrum ι.Ä/ D b.Ä/I
š.Ä/
Ä
š.Ä/
b.Ä/I
Figure 5: B1-spline expansion ι.Ä/ D b.Ä/I, where the B1-spline
basis is b.Ä/
“
1 J
D b1.Ä/; b2.Ä/; : : : ; bJ .Ä/ . ι.Ä/ 0 implies I 0.
18 / 42
23. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Discretization of ˛.x; y/
˛ 0 is a p 1 vector
representing the 2D image that
we wish to reconstruct and
0 is a p 1 vector of known
weights quantifying how much
each element of ˛ contributes
to the X-ray attenuation on the
straight-line path `.
detector array
X-ray source
ϕi
αi
R
` ˛.x; y/ d` T
˛.
19 / 42
24. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Multiple Measurements and Projection Matrix
Denote by N the total number of measurements from all
projections collected at the detector array.
For the nth measurement, define its discretized line
integral as T
n ˛.
Stacking all N such integrals into a vector yields
ˆ˛’
monochromatic
projection of ˛
where
ˆ
projection
matrix
D
2
6
6
6
6
4
T
1
T
2
:::
T
N
3
7
7
7
7
5
N p
:
20 / 42
25. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Noiseless Measurement Model
The N 1 vector of noiseless measurements is
Iout
.˛; I/ D bL
ı.ˆ˛/
˜
output
basis-function
matrix
I
where
bL
ı.s/ D
2
6
6
6
6
4
bL
.s1/
bL
.s2/
:::
bL
.sN /
3
7
7
7
7
5
and s D ˆ˛ is the monochromatic projection.
21 / 42
26. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Noise Model
Assume noisy Poisson-distributed measurements
E D .En/N
nD1, with the negative log-likelihood (NLL)
function
L.ˆ˛; I/
where
L.s; I/ D 1T
ŒbL
ı.s/I E ET
˚
lnıŒbL
ı.s/I lnı E
«
:
The Poisson model is a good approximation for the more
precise compound-Poisson distribution (Xu and Tsui
2014; Lasio et al. 2007).
22 / 42
27. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
NLL of I
Define A D bL
ı.ˆ˛/.
The NLL of I for known ˛ reduces to the NLL for Poisson
generalized linear model (GLM)‡ with identity link and design
matrix A:
LA.I/ D 1T
.AI E/ ET
lnı.AI/ lnı E :
‡See (McCullagh and Nelder 1989) for introduction to GLMs.
23 / 42
28. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
NLL of ˛
The NLL of ˛ for fixed ι.Ä/ is also a Poisson GLM:
Lι.˛/ D 1T
ιL
ı.ˆ˛/ E ET
˚
lnı ιL
ı.ˆ˛/ lnı E
«
with the link function equal to the inverse of ιL. /. Since ι.Ä/
is known, we do not need its basis-function expansion.
24 / 42
29. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
NLL of ˛
The NLL of ˛ for fixed ι.Ä/ is also a Poisson GLM:
Lι.˛/ D 1T
ιL
ı.ˆ˛/ E ET
˚
lnı ιL
ı.ˆ˛/ lnı E
«
with the link function equal to the inverse of ιL. /. Since ι.Ä/
is known, we do not need its basis-function expansion.
4 Lι.˛/ is convex, under conditions that we established in (G.
and D. 2016b)!
24 / 42
30. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Penalized NLL
objective
function
f .˛; I/ D L.˛; I/ C u k‰H
˛k1 C IR
p
C
.˛/ C IRJ
C
.I/
25 / 42
31. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Penalized NLL
objective
function
f .˛; I/ D L.˛; I/ C u k‰H
˛k1 C IR
p
C
.˛/ C IRJ
C
.I/
NLL
25 / 42
32. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Penalized NLL
objective
function
f .˛; I/ D L.˛; I/ C u k‰H
˛k1 C IR
p
C
.˛/ C IRJ
C
.I/
NLL
penalty term
u > 0 is a scalar tuning constant
we select gradient-map sparsifying transform k‰H
˛k1
25 / 42
33. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
TV Sparsifying Transform
p pixels
$
grad. map
ˇ
ˇŒ‰H
˛i
ˇ
ˇ
# significant coeffs p
Total-variation (TV) regularization.
26 / 42
34. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Goal and Minimization Approach
Goal: Estimate the density-map and mass-attenuation
spectrum parameters
.˛; I/
by minimizing the penalized NLL f .˛; I/.
Approach: A block coordinate-descent that uses
Nesterov’s proximal-gradient (NPG) (Nesterov 1983) and
limited-memory Broyden-Fletcher-Goldfarb-Shanno with
box constraints (L-BFGS-B) (Byrd et al. 1995; Zhu et al.
1997)
methods to update estimates of the density map and
mass-attenuation spectrum parameters.
We refer to this iteration as NPG-BFGS algorithm.
27 / 42
35. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Numerical Examples
B1-spline constants set to satisfy
J D 20; # basis functions
qJ
D 103
; span
Ä0qd0:5.J C1/e
D 1; centering
28 / 42
36. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Real X-ray CT Example I
360 equi-spaced fan-beam
projections with 1° spacing,
X-ray source to rotation center
is 3492 detector size,
measurement array size of 694
elements,
projection matrix ˆ
constructed directly on GPU,
x
y
detector array
X-ray source
D rotate
imaginary
detector array
yielding a nonlinear estimation problem with N D 694 360
measurements and an 512 512 image to reconstruct.
Implementation available at github.com/isucsp/imgRecSrc.
Real data provided by Joe Gray, CNDE. Thanks!
29 / 42
37. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
(a) FBP (b) NPG-BFGS (u D 10 5
)
Figure 6: Real X-ray CT: Full projections.
30 / 42
38. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Comments
Our reconstruction eliminates
the streaking artifacts across the air around the object,
the cupping artifacts with high intensity along the
border.
The regularization constant u has been tuned for good
reconstruction performance.
31 / 42
39. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Inverse Linearization Function Estimate
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4 5 6 7 8 9
Polychromaticprojections
Monochromatic projections
NPG-BFGS
FBP
fitted − ln
[
bL
(·)I
]
Figure 7: The polychromatic measurements as function of the
monochromatic projections and its corresponding fitted curve.
Observe the biased residual for FBP, the unbiased residual
for NPG-BFGS and its increasing variance.
32 / 42
40. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
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 single
detector size,
measurement array size of 1380 elements,
projection matrix ˆ constructed on GPU with full
circular mask.
yielding a nonlinear estimation problem with
N D 1380 360 measurements and an 1024 1024 image
to reconstruct.
We employ same convergence constants as in the previous
example.
33 / 42
41. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
(a) FBP (b) NPG-BFGS (u D 10 5
)
Figure 8: Reconstructions from 360 fan-beam projections with 1°
spacing.
34 / 42
43. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Inverse Linearization Function Estimate
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Polychromaticprojections
Monochromatic projections
NPG-BFGS
FBP
fitted − ln
[
bL
(·)I
]
Figure 10: The polychromatic measurements as function of the
monochromatic projections and its corresponding fitted curve.
36 / 42
44. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
(a) FBP (b) NPG-BFGS (u D 10 5
)
Figure 11: Reconstructions from 120 fan-beam projections with
3° spacing.
Observe aliasing artifacts in the FBP reconstruction.
37 / 42
45. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
(a) 360 projections (b) 120 projections
Figure 12: NPG-BFGS (u D 10 5
) reconstructions from fan-beam
projections.
The reconstructed density maps are uniform, except the
defect region.
38 / 42
46. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Simulated X-ray CT Example
10−1
100
101
102
103
40 80 120 160 200 240 280 320 360
RSE/%
Number of projections
FBP
linearized FBP
NPG-BFGS0
linearized BPDN
NPG-BFGS
NPG (known ι(κ))
Figure 13: Average relative square errors (RSEs) as functions of
the number of projections.
39 / 42
47. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
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 local
convergence for alternating proximal minimization
methods (Attouch et al. 2010; Xu and Yin 2013)
o e.g., PG-BFGS,
o not NPG-BFGS.
See (G. and D. 2016b; G. and D. 2015).
40 / 42
48. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Conclusion
Developed a blind method for sparse density-map image
reconstruction from polychromatic X-ray CT measurements
in Poisson noise.
41 / 42
49. Background Mass-Attenuation Spectrum Parameter Estimation Numerical Examples Conclusion References
Future Work
Generalize our polychromatic signal model to handle multiple
materials and develop corresponding reconstruction
schemes.
42 / 42
50. References I
H. Attouch, J. Bolte, P. Redont, and A. Soubeyran,
“Proximal alternating minimization and projection
methods for nonconvex problems: An approach based
on the Kurdyka-Łojasiewicz inequality,” Math. Oper.
Res., vol. 35, no. 2, pp. 438–457, May 2010 (cit. on
p. 47).
R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited
memory algorithm for bound constrained optimization,”
SIAM J. Sci. Comput., vol. 16, no. 5, pp. 1190–1208,
1995 (cit. on p. 34).
S. R. Cherry, J. A. Sorenson, and M. E. Phelps, 4th ed.
Philadelphia, PA: W. B. Saunders, 2012, pp. 493–523
(cit. on p. 8).
51. References II
R. G. and A. D. (Sep. 2015), Polychromatic X-ray CT
image reconstruction and mass-attenuation spectrum
estimation, arXiv: 1509.02193 [stat.ME] (cit. on p. 47).
R. G. and A. D., “Blind polychromatic X-ray CT
reconstruction from Poisson measurements,” in Proc.
IEEE Int. Conf. Acoust., Speech, Signal Process.,
Shanghai, China, Mar. 2016, pp. 898–902 (cit. on
p. 2).
R. G. and A. D., “Blind X-ray CT image reconstruction
from polychromatic Poisson measurements,” IEEE
Trans. Comput. Imag., vol. 2, no. 2, pp. 150–165,
2016 (cit. on pp. 2, 28, 29, 47).
G. T. Herman, “Correction for beam hardening in
computed tomography,” Phys. Med. Biol., vol. 24,
no. 1, pp. 81–106, 1979 (cit. on p. 21).
52. References III
G. M. Lasio, B. R. Whiting, and J. F. Williamson,
“Statistical reconstruction for X-ray computed
tomography using energy-integrating detectors,” Phys.
Med. Biol., vol. 52, no. 8, p. 2247, 2007 (cit. on
p. 26).
P. McCullagh and J. Nelder, Generalized Linear Models,
2nd ed. New York: Chapman & Hall, 1989 (cit. on
p. 27).
Y. Nesterov, “A method of solving a convex
programming problem with convergence rate O.1=k2
/,”
Sov. Math. Dokl., vol. 27, no. 2, pp. 372–376, 1983
(cit. on p. 34).
53. References IV
J. Xu and B. M. W. Tsui, “Quantifying the importance of
the statistical assumption in statistical X-ray CT image
reconstruction,” IEEE Trans. Med. Imag., vol. 33,
no. 1, pp. 61–73, 2014 (cit. on p. 26).
Y. Xu and W. Yin, “A block coordinate descent method
for regularized multiconvex optimization with
applications to nonnegative tensor factorization and
completion,” SIAM J. Imag. Sci., vol. 6, no. 3,
pp. 1758–1789, 2013 (cit. on p. 47).
C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm
778: L-BFGS-B: Fortran subroutines for large-scale
bound-constrained optimization,” ACM Trans. Math.
Softw., vol. 23, no. 4, pp. 550–560, Dec. 1997
(cit. on p. 34).