# 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.

Page 1

Penalized-likelihood sinogram restoration for CT

artifact correction

Patrick J. La Rivière

I. INTRODUCTION

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 [1]. In current

practice, such effects are addressed by a sequence of indepen-

dent sinogram-preprocessing steps, including recursive correc-

tions for detector afterglow [2] and deconvolution-type correc-

tionsforoff-focalradiation[3], thathavethepotentialtoamplify

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

bestpossibleestimateofthelineintegralsneededforreconstruc-

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.

II. METHODS

A. Measurement model

WeassumetheCTscanproducesasetofmeasurementsymeas

i = 1,...,NY, where NYis the total number of measurements in

the scan. Each ymeas

i

is regarded as a realization of a random

variable Ymeas

i

whose statistics are described by:

?Ny

j=1

i

,

Ymeas

i

∼Gi×Poisson

∑

Ijbije−f(lj)+si

?

+Normal?di,σ2

i

?.

(1)

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

?

lj≡

Lj

µ(x)dl.

(2)

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: pjlarivi@midway.uchicago.edu).

coefficients can generally be determined empirically from mea-

surements made using a water phantom [1]. 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 [6].

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 [3]. 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-

counted for.

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

terminedfromcalibrationscansanddark-currentmeasurements,

and the scatter term could be determined by scatter-estimation

techniques [4].

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

acquisition geometry.

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

??Ymeas

?Ny

j=1

i

,

Yi

≡

i

−di

Gi

?

Ijbije−f(lj)+si+σ2

+σ2

i

G2

i

?

+

∼

Poisson

∑

i

G2

i

?

,

(3)

0-7803-8701-5/04/$20.00 (C) 2004 IEEE

Page 2

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,

that maximizes

Φ(l) ≡ L(l)−βR(l),

where L(l) is the usual Poisson likelihood function

?Ny

j=1

(4)

L(l) =

NY

∑

i=1

yilog

∑

Ijbije−f(lj)+ri

?

−

?Ny

j=1

∑

Ijbije−f(lj)+ri

?

(5)

,

and where we have defined

ri≡ si+σ2

i

G2

i

.

(6)

The roughness penalty R(l) can be expressed very generally in

the form

K

∑

k=1

given by Fessler [9], 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

relativeinfluenceofthelikelihoodandsmoothnessterms. Inthis

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.

R(l) =

ψk([Cl]k),

(7)

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 [9], 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

l(n+1)

j

=

l(n)

j

−

nj−β∑K

k=1ckjωk

c(n)

?

Cl(n)?

k

j+βvj

+

,

(8)

where

nj=

NY

∑

i=1

Ijbij˙ gi

?Ny

∑

j=1

Ijbije−f(l(n)

j)+ri

?

e−f(l(n)

j)˙f(l(n)

j)

(9)

˙ 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)

j

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)

j

j

are chosen appropriately.

III. RESULTS

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 [1]. 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 [3].

i=

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 [4]. 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

[4] 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

0-7803-8701-5/04/$20.00 (C) 2004 IEEE

Page 3

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

studies.

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

investigation.

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

Thepenaltyfunctionplaysanimportantroleinanypenalized-

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 [9] to slightly more complex weak-plate priors based on

discrete approximations of the Laplacian [13].

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

0-7803-8701-5/04/$20.00 (C) 2004 IEEE

Page 4

Center point

1234

Resolution (mm)

0

50

100

150

Standard Deviation in ROI (HU)

ATM Filter

Proposed PL

Right point

1234

Resolution (mm)

0

50

100

150

Standard Deviation in ROI (HU)

ATM Filter

Proposed PL

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

valuesofβfromthemeasurementsthemselves[16], thesewould

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 [4]. 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

forpurelyinthesinogramdomain, theircomputationalburdenis

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.

APPENDIX

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

function φ

such that finding

?

l;l(n)?

l(n+1)= argmax

l≥0φ

?

l;l(n)?

,

(10)

0-7803-8701-5/04/$20.00 (C) 2004 IEEE

Page 5

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

?Ny

j=1

?

l(n+1)?

≥ Φ

?

l(n)?

.

L(l) =

NY

∑

i=1

gi

∑

aije−f(lj)+ri

?

.

(11)

Now we make use of a multiplicative trick similar to one em-

ployed by De Pierro in the context of emission tomography and

write

y(n)

i

Ny

∑

j=1

aije−f(lj)+ri=

Ny

∑

j=1

aije−f(l(n)

j)

e−f(lj)

e−f(l(n)

j)y(n)

i

+

?

ri

y(n)

i

?

(12)

y(n)

i,

where

y(n)

i

≡

Ny

∑

j=1

aije−f(l(n)

j)+ri.

(13)

Using this expression, we can rewrite

j=1

L(l) =

NY

∑

i=1

gi

Ny

∑

aije−f(l(n)

j)

y(n)

i

e−f(lj)

e−f(l(n)

j)y(n)

i

+

?

ri

y(n)

i

?

y(n)

i

.

(14)

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:

S

?

l,l(n)?

≡

Ny

∑

j=1

Sj

?

lj,l(n)?

,

(15)

where

Sj

?

lj,l(n)?

=

Ny

∑

i=1

aije−f(l(n)

y(n)

i

j)

gi

?

e−f(lj)

e−f(l(n)

?

j)y(n)

l,l(n)?

i

?

+

ri

y(n)

i

gi

?

y(n)

i

?

.

(16)

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

l,l(n)?

lj,l(n)?

+˙Sj

S

l(n),l(n)?

= L

?

l(n)?

. The surrogate S

?

is still difficult

?

to each Sj

?

lj,l(n)?

lj−l(n)

:

Pj

?

lj,l(n)?

≡

Sj

?

l(n)

j,l(n)?

?

are chosen such that Pj

?

l(n)

j,l(n)??

j

?

(17)

−c(n)

j

2

lj−l(n)

j

?2,

where the curvatures c(n)

?

function is then given by

j

?

lj,l(n)?

≤

Sj

lj,l(n)?

, ∀lj≥ 0. The overall surrogate for the likelihood

P

?

l,l(n)?

≡

Ny

∑

j=1

Pj

?

lj,l(n)?

,

(18)

which also satisfies both P

?

A separable, quadratic surrogate for the penalty function has

been given previous by Fessler:

?

j

l,l(n)?

are chosen appropriately

≤ L(l), ∀lj ≥ 0 and

P

l(n),l(n)?

= L

?

l(n)?

if the c(n)

R

?

l,l(n)?

≡

Ny

∑

j=1

Rj

?

lj,l(n)?

,

(19)

where

Rj

?

lj,l(n)?

≡

K

∑

k=1

γkjωk1

2

?ckj

γkj

?

lj−l(n)

j

?

+[Cl]k

?2

.

(20)

Here the γkjmust satisfy γkj≥ 0 and ∑Ny

Fessler, we choose γkj=??ckj

φ

j=1γkj= 1. Following

??/ck.

P

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.

l,l(n)?

=

?

l,l(n)?

+βR

?

l,l(n)?

.

(21)

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