Penalized-likelihood sinogram restoration for CT artifact correction
ABSTRACT In CT sinogram preprocessing, the best possible estimate of the line integrals needed for image reconstruction from the set of noisy, degraded detector measurements is reported. A general imaging model relating the degraded measurements to the ideal sinogram and the estimation of the ideal line integrals by iteratively maximizing an appropriate penalized statistical likelihood function are discussed. Image reconstruction is then performed by the use of existing non-iterative approaches.
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ABSTRACT: We have compared the performance of two different penalty choices for a penalized-likelihood sinogram-restoration strategy we have been developing. One is a quadratic penalty we have employed previously and the other is a new median-based penalty. We compared the approaches to a noniterative adaptive filter that loosely but not explicitly models data statistics. We found that the two approaches produced similar resolution-variance tradeoffs to each other and that they outperformed the adaptive filter in the low-dose regime, which suggests that the particular choice of penalty in our approach may be less important than the fact that we are explicitly modeling data statistics at all. Since the quadratic penalty allows for derivation of an algorithm that is guaranteed to monotonically increase the penalized-likelihood objective function, we find it to be preferable to the median-based penalty.International Journal of Biomedical Imaging 01/2006; 2006:41380.
Penalized-likelihood sinogram restoration for CT
Patrick J. La Rivière
CT measurement data are degraded by a number of physi-
cal factors—including noise, scatter, detector afterglow, beam
hardening, and off-focal radiation—that will produce artifacts in
reconstructed images unless properly corrected . In current
practice, such effects are addressed by a sequence of indepen-
dent sinogram-preprocessing steps, including recursive correc-
tions for detector afterglow  and deconvolution-type correc-
noise. Noise itself is generally mitigated through apodization of
the reconstruction kernel, which is equivalent to shift-invariant
smoothing and thus effectively ignores the measurement statis-
tics, although in very high-noise situations, adaptive filtering
methods that loosely model data statistics are applied [4,5].
In this work we formulate CT sinogram preprocessing as a
statistical restoration problem in which the goal is to obtain the
tion from the set of noisy, degraded detector measurements. We
present a general imaging model relating the degraded measure-
ments to the ideal sinogram and propose to estimate the ideal
line integrals by iteratively maximizing an appropriate penal-
ized statistical likelihood function. Image reconstruction can
then proceed by use of existing, non-iterative approaches.
A. Measurement model
i = 1,...,NY, where NYis the total number of measurements in
the scan. Each ymeas
is regarded as a realization of a random
whose statistics are described by:
The first term here represents the effects of photon-counting
statistics. Ijis the incident x-ray intensity along the jthmeasure-
ment line, which we denote Lj; ljis the line integral through the
attenuation map µ(x) along that measurement line, given by
The effects of beam hardening are modeled by replacing the
ideal line integrals ljin the exponent with lower transformed
values f(lj). The function f(x) depends on the particular spec-
trum of x-rays used in a given scanner and, at least in one case,
has been shown to be well-approximated as a polynomial whose
Patrick J. La Rivière is with the Department of Radiology, The University of
Chicago, Chicago, IL 60637 (email: firstname.lastname@example.org).
coefficients can generally be determined empirically from mea-
surements made using a water phantom . This model is ac-
curate when the attenuating tissue is composed primarily of soft
tissues and relatively small amounts of bone. When significant
amounts of bone as well as soft tissue are present in the imaged
slice, such as in the head, it is not possible to account for beam
hardening purely in a sinogram preprocessing algorithm .
Here the ithmeasurement is seen to receive contributions not
just from the ithattenuation line (along which liis defined) but
from other attenuation lines j ?= i with weight bij. This models,
for example, the effects of off-focal radiation, which, at least
in the two-dimensional step-and-shoot case, results in contribu-
tions from other points in the sinogram lying on a locus that
can be determined by the geometry of the x-ray tube’s off-focal
halo, the x-ray collimation, and the detector geometry . The
weights bijcould also model the source and detector response
functions, to the extent that these can be modeled as local blur-
rings of ideal exponentiated attenuation line integrals. Detector
speed, afterglow, and crosstalk effects could also be included
in the bij, although if these effects are significant, they intro-
duce noise correlations among measurements that should be ac-
The final two factors in the first term are si, which represents
the scattered radiation contributing to the measurement, and Gi,
a gain factor reflecting the conversion of detected x-rays to the
current measured by the data-acquisition system.
The second term models detector dark current, of mean di,
and electronic readout noise, of variance σ2
Gi, di, σ2
and the scatter term could be determined by scatter-estimation
i. We assume that
i, and siare known. The first three quantities can be de-
B. Objective function
Our goal is to estimate the undegraded attenuation line in-
tegrals li, i = 1,...,NY, from the set of measurements ymeas
i = 1,...,NY. The estimated line integrals can then be input to
a standard analytic reconstruction algorithm appropriate to the
We will estimate the set of line integrals needed for recon-
struction by maximizing a penalized-likelihood objective func-
tion. The sum of a Poisson and a Gaussian density does not
have a tractable likelihood function, so we define new adjusted
measurements yithat are realizations of random variables
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where [x]+is x for positive x and zero otherwise. This Poisson
approximation to the distributions of theYimatches the first two
moments of their true distributions and has been used previously
by Snyder et al. in the context of image restoration for charge-
coupled devices [7,8].
Expressing the set of unknown attenuation line integrals as a
vector l, with elements li, i = 1,...,NY, and the set of adjusted
measurements as a vector y, with elements yi, i = 1,...,NY, we
then seek to find an estimateˆl, subject to positivity constraints,
Φ(l) ≡ L(l)−βR(l),
where L(l) is the usual Poisson likelihood function
and where we have defined
The roughness penalty R(l) can be expressed very generally in
given by Fessler , where C is a matrix and ψka potential
function that assigns a cost to the K combinations of attenuation
line integral values represented by the matrix product Cl. The
smoothing or regularization parameter β in Eq. 4 determines the
work, we employ a quadratic penalty ψk(t) = ωkt2/2 applied to
the simple difference of a sinogram sample with its horizontal
and vertical neighbors.
C. Maximization algorithm
The objective function we seek to maximize is not guaranteed
to be concave or even necessarily unimodal. In such situations,
an effective strategy is to employ an iterative algorithm that is
guaranteed to increase the penalized likelihood at each iteration
and to initialize it with a good starting guess. The hope is that
the starting guess would be close enough to the solution corre-
sponding to the true global maximum that the monotonic itera-
tive algorithm would converge to this solution. We have derived
such an algorithm by making use of the optimization transfer
principal , in which at each iteration one defines a surrogate
to the likelihood function, such that the vector of line integrals
maximizing this surrogate is guaranteed to have a higher penal-
ized likelihood than the previous vector estimate. Deatils are in
Appendix A. This gives rise to an update of the form
˙ gi(x) =yi
are the curvatures of the parabaloidal surrogates employed as
part of the optimization transfer process.
This update is guaranteed not to decrease the penalized like-
lihood so long as the curvatures c(n)
It does not appear possible to derive a closed-form expression
for curvatures c(n)
that guarantee monotonicity. While it would
be possible to solve numerically for such values at each itera-
tion, that may add unacceptably to the computational burden. In
practice, we make use of a precomputed curvature that does not
guarantee monotonicity, but that we have only rarely observed
to take a non-monotonic step.
x−1, vj≡ ∑K
k=1|cki|ckωk, ck≡ ∑NY
i=1|cki|, and the c(n)
are chosen appropriately.
A. Physical model employed
We made use of the measurement model given in Eq. 1. To
simulate beam hardening, we modeled the function f(l) = l −
0.01l2, which approximates the observed reduction in estimated
line integral value . We chose Gi= 0.25 pA/quanta and σ2
10.0 pA2for all i. We did not simulate the effects of scatter.
To simulate the effects of off-focal radiation we convolved
the exponentiated, beam-hardened line integrals with a sharply
peaked kernel with low tails approximately 13 channels wide.
Although the kernel was effectively one-dimensional, it was ap-
plied along diagonal lines in sinogram space to approximate the
true geometry of off-focal radiation .
B. Artifact removal
In Figure 1, we show the results of applying the approach to
a numerical, cylindrical water phantom comprising a plexiglas
shell and a water interior containing three small circular voids.
The top row involves reconstruction from noiseless data and the
bottom row reconstruction from data to which noise have been
added based on an exposure level Ij= 2.5×105for all j. On the
left are images reconstructed directly from the degraded data by
FFBP without correction for the physical degradations. On the
right are images reconstructed by FFBP after correction by the
proposed approach. It is apparent that the proposed approach
can remove the artifacts introduced by these degradations with-
out introducing additional artifacts.
C. Resolution-noise performance
In order to compare the noise-reduction properties of the pro-
posed approach to those of existing approaches, we made use
of a numerical ellipse phantom similar to the physical phantom
employed by Hsieh . This numerical phantom, shown in Fig.
2, comprises a water-equivalent background ellipse (0 HU) and
three circular higher-attenuation inserts (900 HU).
We included the effects of off-focal radiation and beam hard-
ening as described above. We processed the sinograms by use
of two different methods: the proposed approach and an exist-
ing approach combining Hsieh’s adaptive trimmed mean filter
 with deconvolution for off-focal effects.
We characterized resolution by analyzing the edge broaden-
ing of the central and right high-attenuation inserts along cen-
tral vertical profiles through these structures, as these profiles
are most strongly affected by the heavier smoothing of the noisy
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Fig. 1. Results of applying the sinogram restoration approach to noiseless (up-
per row) and noisy (bottom row) data corrupted by beam hardening and
off-focal radiation effects. In the uncorrected images on the left, a cupping
artifact characteristic of beam hardening is evident as is a bright halo near
the edge of the phantom caused by off-focal effects. These are seen to dis-
appear in the corrected images.
Fig. 2. Illustration of the numerical ellipse phantom used for resolution-noise
projections through the long dimension of the phantom. We as-
sumed that the effective broadening kernel was described by a
Gaussian of standard deviation σb, which gives rise to an edge
spread function (ESF) having the form of an error function pa-
rameterized by σb. This functional form matched the observed
ESFs very closely. We use the full-width half-maximum of the
fit Gaussian broadening kernel, given by 2.35σb, as our measure
of resolution. In order to obtain an accurate fit, we performed
targeted reconstructions of the central and right high-attenuation
inserts with 0.25 mm pixel size from 10 different noise realiza-
tions at exposure level 2.5×105. We then averaged the recon-
structions together in order to obtain relatively low-noise pro-
files on which to perform the fitting.
We characterized noise at the two locations by calculating the
standard deviation of the pixel values in circular regions of inter-
est (ROIs) of diameter 16.0 mm placed immediately to the right
of, but not overlapping, the central and right high-attenuation
inserts. We employ the average ROI standard deviation over the
10 noise realizations as our measure of noise at each of the two
positions. We computed these noise and resolution measures for
all of the reconstructed ellipse phantom images and plot them
versus each other for different values of the smoothing parame-
ter of ATM filter length in Fig. 3. It can be seen that over most
resolution ranges, and in particular the high-resolution regime
(1-2 mm) likely to be of practical interest, the proposed PL ap-
proach can achieve lower noise levels at any given resolution
than can either the ATM approach or the previously proposed
spline-based penalized-likelihood approach, with the advantage
holding at both the center and periphery of the phantom.
IV. DISCUSSION AND CONCLUSIONS
A. Choice of noise model
We assumed in Eq. 1 that CT measurement noise is dom-
inated by photon-counting statistics described by a standard
Poisson distribution. While this assumption appears to be rea-
sonable in practice [10,11], it is, in fact, based on an approxima-
tion to the true statistics of the signal, which have been shown
to follow a compound Poisson distribution [10–12]. The com-
pound Poisson distribution arises because a CT detector is not a
true photon-counting detector, like those employed in emission
tomography, but is rather an integrating detector that generates a
signal proportional to the total energy deposited in the detector
by the polychromatic photons.
There is no exact, closed-form likelihood for a compound
Poisson distribution, although Elbakri and Fessler have de-
rived saddle-point approximations that allow the likelihood to
be computed efficiently [10,11]. The extension of the penalized-
likelihood sinogram restoration approach described here to that
potentially more accurate statistical model is a topic of ongoing
In Eq. 3 we approximated the sum of a Poisson and Gaussian
distribution by a Poisson distribution. While it is more common
to approximate such a sum by a Gaussian distribution, particu-
larly at moderate-to-high count levels, as noted, this Poisson ap-
proximation has been used before in similar circumstances [7,8]
as it matches the the first two moments of the true distribution
and thus appropriately models the relative variance of each mea-
surement. Gaussian approximations to the Poisson likelihood
are sometimes used in emission tomography problems because
they result in objective functions that are quadratic in the vector
of unknowns and thus easier to maximize. However, because of
the exponential involved in the CT imaging equation, the use of
a Gaussian approximation would not lead to a quadratic objec-
tive function in this case and thus does not necessarily provide
any significant computational advantage.
B. Choice of penalty function
likelihood estimation problem. A variety of penalty functions
have been explored in the context of fully iterative statistical re-
construction, ranging from simple differences of nearest neigh-
bors  to slightly more complex weak-plate priors based on
discrete approximations of the Laplacian .
While some of these penalty functions can be applied equally
well in the context of the line integral estimation strategy we are
proposing, there are some important differences in this case that
call for a deeper examination of penalty choices. Unlike a two-
or three-dimensional image where the dimensions are all equiv-
alent, there are potentially significant differences among the di-
mensions in a CT sinogram. For example, in a two-dimensional
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Standard Deviation in ROI (HU)
Standard Deviation in ROI (HU)
Fig. 3. Resolution-noise tradeoffs for exposure 2.0×105at the center and right circular inserts in the ellipse phantom for the ATM filter (labeled “ATM Filter”),
and the penalized-likelihood approach discussed here (labeled “Proposed PL”).
sinogram, one of the dimensions is a spatial dimension extend-
ing along the detector array but the other is a projection angle.
It may be desirable to employ an asymmetric penalty function
that enforces smoothness more strongly in the detector array di-
rection than in the projection angle direction. In multi-slice CT,
there is a second dimension to the detector array, but the detector
elements may not be square and so again it might be desirable to
consider an asymmetric penalty function in that case. Future re-
search will examine the effect of different choices, particularly
on CT resolution uniformity and isotropy. We will also consider
the design of data-dependent and space-variant penalty func-
tions that seek to achieve specific resolution properties, such as
uniformity and isotropy, in reconstructed images [14,15].
C. Choice of smoothing parameter
The choice of the regularization parameter β that controls
the tradeoff between goodness of fit to the measurements and
smoothness of the resulting estimates has a significant influence
on the estimated attenuation line integrals and thus on the ap-
pearance of the reconstructed images. While there are some au-
tomatic and statistically principled ways of determining optimal
add unacceptably to the computational load of the procedure.
One more practical approach would be to characterize as thor-
oughly as possible the effect of β on image resolution for vari-
ous measured count levels and distributions. This could be done
both empirically, through analyzing reconstructions of numeri-
cal phantoms made using a range of count levels and smoothing
parameters, and also analytically. The goal would be to tabulate
appropriate choices of β for achieving a desired resolution at a
given count level.
Another, perhaps more appealing, possibility is to deliber-
ately undersmooth the measurements by choosing a relatively
low value of β and then to apply further post-filtering of the pro-
jections prior to reconstruction. Even a small degree of smooth-
ing using the penalized-likelihood framework would likely re-
duce the variability of the very noisiest measurements, which
tend to have an out-sized influence on noise properties and es-
pecially on the presence of noise-induced streak artifacts in re-
constructed images . The subsequent post-filtering could be
done by use of the standard proprietary reconstruction kernels
available on all commercial CT scanners, thus leading to images
whose overall resolution properties are somewhat standardized
and familiar to clinicians.
D. Prospects for fully iterative CT reconstruction
Naturally, a more principled statistical approach would frame
the problem as one of estimating the attenuation map of interest,
and not just its line integrals, from the noisy, degraded mea-
surements. While such fully iterative approaches to CT recon-
struction are potentially very powerful as they can model effects
such as multi-tissue beam hardening that cannot be accounted
extremely high due to the need for multiple backprojections and
reprojections. This burden will only grow when faced with the
need for 3D projection and backprojection in conebeam scan-
ners. The proposed sinogram restoration approach thus rep-
resents a middle road between full-blown iterative reconstruc-
tion and the current non-statistical methods, potentially offering
many of the benefits that come from accurately modeling the
data statistics at a substantially lower computational cost.
We seek to derive an algorithm for generating a sequence of
estimates l(n)that monotonically increase Φ(l) defined in Eq. 4.
We do so by making use of the optimization-transfer principle,
in which at each iteration we define a new, surrogate objective
such that finding
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necessarily implies Φ
In order to construct such surrogate functions for Φ(l), we
consider the likelihood and penalty terms separately. In the like-
lihood term, given in Eq. 5, we simplify notation by defining
aij≡ Ijbijand gi(x) ≡ yilogx−x, which allows us to write
Now we make use of a multiplicative trick similar to one em-
ployed by De Pierro in the context of emission tomography and
Using this expression, we can rewrite
Since g(x) is concave on (0,∞) and, by construction, the sum of
the Ny+1 coefficients inside the square brackets is 1, we can in-
voke Jensen’s inequality to obtain a surrogate to the likelihood:
This surrogate satisfies both S
to maximize when coupled to the surrogate we will derive for
the penalty function. For easier maximization, we instead con-
struct a paraboloidal surrogate Pj
≤ L(l), ∀lj≥ 0 and
. The surrogate S
is still difficult
to each Sj
are chosen such that Pj
where the curvatures c(n)
function is then given by
, ∀lj≥ 0. The overall surrogate for the likelihood
which also satisfies both P
A separable, quadratic surrogate for the penalty function has
been given previous by Fessler:
are chosen appropriately
≤ L(l), ∀lj ≥ 0 and
if the c(n)
Here the γkjmust satisfy γkj≥ 0 and ∑Ny
Fessler, we choose γkj=??ckj
j=1γkj= 1. Following
So the overall surrogate is given by
Because of the quadratic nature of the surrogates, the maximiza-
tion required by Eq. 10 is straightforward to calculate simply by
zeroing the first derivative of the surrogate. Doing so, and solv-
ing for ljyields the update of Eq. 8.
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