Blind Beam-Hardening Correction from Poisson Measurements

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

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

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

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

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

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

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

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

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
9 / 42

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

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

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

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

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

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

Polychromatic X-ray CT Model Using
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

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

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

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

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

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

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

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

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

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

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

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

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

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

Penalized NLL
objective
function
f .˛; I/ D L.˛; I/ C u k‰H
˛k1 C IR
p
C
.˛/ C IRJ
C
.I/
25 / 42

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

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

TV Sparsifying Transform
p pixels
$
grad. map
ˇ
ˇŒ‰H
˛i
ˇ
ˇ
# significant coeffs p
Total-variation (TV) regularization.
26 / 42

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

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

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

(a) FBP (b) NPG-BFGS (u D 10 5
)
Figure 6: Real X-ray CT: Full projections.
30 / 42

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

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
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

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

)
Figure 8: Reconstructions from 360 fan-beam projections with 1°
spacing.
34 / 42

Figure 9: Estimated ˛ and ln bL
. /I from 360 fan-beam
projections.
35 / 42
0
0.5
1
1.5
2
2.5
3
3.5
0 1 2 3 4 5
Itr=0
poly.proj.−lnImea
mono. proj. ϕT
α
data
ﬁtted − ln ιL
(·)

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
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

)
Figure 11: Reconstructions from 120 fan-beam projections with
3° spacing.
Observe aliasing artifacts in the FBP reconstruction.
37 / 42

(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

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

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

Conclusion
Developed a blind method for sparse density-map image
reconstruction from polychromatic X-ray CT measurements
in Poisson noise.
41 / 42

Future Work
Generalize our polychromatic signal model to handle multiple
materials and develop corresponding reconstruction
schemes.
42 / 42

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).

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).

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).

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).

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