Statistical Projection Completion in X-ray CT Using Consistency Conditions

1528 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 8, AUGUST 2010

Statistical Projection Completion in X-rayCT Using Consistency Conditions

Jingyan Xu*, Katsuyuki Taguchi, and Benjamin M. W. Tsui

Abstract—Projection data incompleteness arises in many sit-uations relevant to X-ray computed tomography (CT) imaging.We propose a penalized maximum likelihood statistical sinogramrestoration approach that incorporates the Helgason–Ludwig(HL) consistency conditions to accommodate projection dataincompleteness. Image reconstruction is performed by the fil-tered-backprojection (FBP) in a second step. In our problemformulation, the objective function consists of the log-likelihoodof the X-ray CT data and a penalty term; the HL condition posesa linear constraint on the restored sinogram and can be imple-mented efficiently via fast Fourier transform (FFT) and inverseFFT. We derive an iterative algorithm that increases the objectivefunction monotonically. The proposed algorithm is applied to bothcomputer simulated data and real patient data. We study differentfactors in the problem formulation that affect the properties ofthe final FBP reconstructed images, including the data truncationlevel, the amount of prior knowledge on the object support, as wellas different approximations of the statistical distribution of theavailable projection data. We also compare its performance withan analytical truncation artifacts reduction method. The proposedmethod greatly improves both the accuracy and the precision ofthe reconstructed images within the scan field-of-view, and to acertain extent recovers the truncated peripheral region of theobject. The proposed method may also be applied in areas such aslimited angle tomography, metal artifacts reduction, and sparsesampling imaging.

Index Terms—Computed tomography (CT) artifacts, fanbeam computed tomography, image reconstruction, incompleteprojection.

I. INTRODUCTION

D IFFERENT kinds of projection data incompleteness arisein various situations in X-ray computed tomography

(CT). When the patient body extends outside of the scannerfield-of-view (SFOV), the projection data will be truncated inthe transverse direction. Radiation dose concerns may also leadto reduced scan schemes so that a small region-of-interest isfully enclosed in the SFOV while other organs are truncated. Ifnot properly treated, data truncation produces edge and cupping

Manuscript received November 09, 2009; revised March 01, 2010; acceptedApril 01, 2010. First published May 03, 2010; current version published August04, 2010. This work was supported in part by a research contract with SiemensHealthcare. Asterisk indicates corresponding author.

*J. Xu is with the Division of Medical Imaging Physics, Department of Ra-diology, Johns Hopkins University School of Medicine, Baltimore, MD 21287USA (e-mail: [email protected]).

K. Taguchi and B. M. W. Tsui are with the Division of Medical ImagingPhysics, Department of Radiology, Johns Hopkins University Schoolof Medicine, Baltimore, MD 21287 USA (e-mail: [email protected];[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMI.2010.2048335

artifacts near the edge of SFOV that obscure the underlyinganatomy and make diagnosis difficult [1].

In iterative image reconstruction algorithms that use repeatedforward and backward projections, projection incompletenesscan be included as part of the system matrix modeling. Two dif-ferent approaches for treating the missing projection rays, ig-nore or estimate, were proposed in [2] and [3] that resulted inthe same fixed point in their respective maximum-likelihood es-timation algorithms. The observation was that, if the reconstruc-tion volume is big enough to cover the scanned object [3], [4],then projection incompleteness is taken care of by the systemgeometry modeling, thus artifacts caused by data incomplete-ness can be reduced.

Despite their capability of dealing with truncation andmany other artifacts, the high computational cost of iterativereconstruction methods nowadays prevents their routine appli-cation in clinical X-ray CT practice. The algorithm-of-choiceon commercial CT scanners is still the filtered-backprojec-tion (FBP) type analytical reconstruction algorithms. If thetruncated projection data are directly fed to these algorithms,the abrupt change of projection values at the edge of SFOVcauses significant artifacts in the reconstructed images [1]. Aprojection data completion method is a preprocessing step forobtaining a complete projection data, so that subsequent FBPreconstruction can have reduced artifacts within the SFOV, andextend the reconstruction field-of-view (RFOV) beyond theSFOV [5], [6].

It is well known that the sinogram of a physical functioncannot be arbitrary and satisfies the so-called consistencyconditions [7], [8], which essentially characterize the range ofthe Radon transform. In the interest of this work, we dividethe different projection data completion methods into twocategories. The first category takes an extrapolation approachwithout explicit use of consistency conditions, such as the min-imal value extension method in [9] and [10] and the symmetricmirroring method in [6]. The second category [5], [11]–[14]makes use of consistency conditions. Among these, [5], [11]use the first order consistency condition (constant body mass)to constrain the data extrapolation procedure. In [12] a new fanbeam data consistency condition was derived and the missingprojection values were filled in point-by-point in an iterativemanner. There are pros and cons in the two categories. Sincethe consistency conditions are most conveniently employedin the parallel beam geometry, fan-to-parallel rebinning isusually needed for the second category of methods, while inthe first category the projection extrapolation or data extensioncan be performed in the native geometry. However withoutincorporating the consistency conditions, extrapolation tends

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XU et al.: STATISTICAL PROJECTION COMPLETION IN X-RAY CT USING CONSISTENCY CONDITIONS 1529

to be ad-hoc and usually relies on local continuity conditionsat the edge of the SFOV. They may also require variousthresholds settings that may not be straightforward when theimaging condition changes, e.g., high dose versus low doseapplications.

Two approaches [13], [14] in the second category explicitlyuse the Helgason–Ludwig [7], [8] consistency condition1 andexpand the Radon transform in terms of its basis functions [15,Ch. IV and equation (3.13)]. Their methods therefore incorpo-rate not just one or two consistency conditions as in [5], [6],but theoretically an infinite number of such constraints [14].Our proposed method builds on these previous works and ex-tends the methodology to data completion problems in X-rayCT imaging.

We develop a statistical sinogram restoration method forX-ray CT that incorporates 1) the noise statistics of the X-rayCT data, and 2) the HL consistency condition to deal withprojection data incompleteness. The recovered sinogram is sub-sequently reconstructed by the analytical FBP type algorithms.We compare our results with direct FBP reconstruction with nocorrection and after projection completion using the symmetricmirroring method [6], an ad-hoc data extrapolation approachthat tries to recover the truncated projection values from thoseavailable data in a certain mirroring manner. Our comparisondemonstrates both improved accuracy and precision of theobject attenuation coefficients within the SFOV.

Though all make explicit use of the HL condition and are for-mulated as a sinogram restoration approach, there are signif-icant differences among [13], [14], and our method, in termsof 1) the problem formulation, 2) algorithm derivation, and 3)procedures to enforce the HL condition. In [13], the completesinogram was determined by satisfying some prescribed desir-able properties, e.g., consistency, known sinogram support, anda priori knowledge of a reference sinogram, that are formulatedas convex sets. A POCS algorithm was used to find the solutioniteratively. In [14] a different set of basis functions were usedand the HL condition was cast as hard constraints by forcingsome basis expansion coefficients of the sinogram to be zero. Intheir iterative scheme, these hard constraints were incorporatedinto a penalized least squares objective function by introducingLagrange multipliers. A primal-dual type algorithm was imple-mented to find the optimizer [14].

We consider the noise statistics of the X-ray CT projectionand formulate the sinogram restoration problem in the penalizedmaximum-likelihood framework. By borrowing the linear rep-resentation from [13], the HL condition is again posed as a linearcondition on the solution sinogram. This formulation makes thealgorithm derivation easier without using Lagrange multipliers.In different X-ray CT applications, the projection data can ei-ther be in the pre-log operation format, or in the after-log op-eration format. We look at two separate statistical assumptionsof the X-ray projection data for the two situations, a Poissonmodel and a Gaussian model. We derive an iterative algorithmthat monotonically increases the objective function by using theoptimization transfer principle [16], [17]. Moreover, enforcing

1We abbreviate the Helgason-Ludwig consistency condition as the HL con-dition in the rest of the paper for convenience.

TABLE I2-D RADON TRANSFORM NOTATIONS

the HL condition is achieved by fast Fourier transform (FFT)and inverse FFT (IFFT).

The organization of this paper is as follows. For complete-ness, in Section II we first introduce the HL condition followingthe exposition of [13] and [14] and explain how the sinogramseries expansion can be calculated via FFT. In Section III wepresent our sinogram restoration problem with explicit incor-poration of the HL condition and derive an iterative algorithmthat monotonically increases a penalized likelihood function.Reconstruction results using computer simulated data and realpatient acquisitions are presented and analyzed in Sections IVand VI. We also compare results from our proposed approachwith that from the symmetric mirroring method [6]. We con-clude this paper and discuss possible extensions of this work inSection VII. In Table I below we assemble some notations from[15] related to the 2-D Radon transform.

II. BACKGROUND

The HL condition is part of our sinogram restoration problemformulation. In this section, we first introduce the HL conditionof the 2-D Radon transform, which states that some of the seriesexpansion coefficients of the 2-D Radon transform must be zerofor it to be physically possible. For the purpose of efficient algo-rithm implementation, we then explain how the series expansioncoefficients and the 2-D Radon transform are related by Fouriertransform pair.

A. Helgason–Ludwig Consistency Conditions

With occasional exceptions, our notations here are almostidentical to those in [13]. The interested readers are referredto [13], [14] for more information. We denote by the2-D Radon transform that maps the Hilbert space to

, . Any physically consistentsinogram can then be written as a series expansion usingthe basis functions of

(1)

where and . The complex constantsare the expansion coefficients. The function is the th

order Chebyshev polynomial of the second kind. Let, the inner product of and is defined

as follows:

(2)


The functions form an orthogonalbasis of . The HL consistency conditiondictates that not all expansion coefficients can be arbitrary.In fact, the HL condition is embodied in the following require-ment on [13]:

(3)

Let , then some of its expansion coeffi-cients violate the HL condition (3). We use the notationto denote the operation of zeroing out these “inconsistent” co-efficients; the resulting sinogram obtained by pluggingthe “rectified” expansion coefficients into (1) satisfies

(4)such that .

B. Sinogram Series Expansion Using FFT

If we define

(5)

then (1) can be rewritten as

(6)

In other words, are the coefficients of the Fourier se-ries expansion of with respect to . The conversion of

therefore can be implemented via 1D FFTalong the view direction. The transformation betweenand in (5) can be calculated by a discrete sine transform[18], [19]. The key is to utilize the following definition of theChebyshev polynomial :

and realize that by resampling , at ,we then have

(7)

To complete as a periodic function in , we may define

(8)

The coefficients can then be obtained from a discretesine transform of , a resampled version of . Forthe convenience of our algorithm description, we use thephrase “sinogram expansion” to indicate the calculation of

; and “sinogram synthesis” the reverse direc-tion. Combining (6), (7), and (8), sinogram expansion can

be achieved via fast Fourier transform (FFT), first along thechannel direction , then along the view direction . Thereverse direction, sinogram synthesis, can be achieved via in-verse FFT. For , its expansion coefficients

necessarily satisfy the HL condition in (3). Conversely,if some of the conditions in (3) are violated, the synthesized“sinogram” from such ’s can not be consistent with anyphysical objects.

III. METHODS

In this section, we formulate the sinogram restorationproblem in the discrete sinogram space incorporating the HLconsistency conditions. To connect with the continuous spacedescription of the HL condition and the sinogram series ex-pansion in Section II, we first discretize the series expansionof Section II and obtain in Section III-A a discrete, linearoperator form describing the HL consistency condition, basedon which we present our problem definition in Section III-B.In Section III-C we provide a flowchart of the iterative sino-gram restoration algorithm, the derivation of which is givenin Appendix A. Our problem formulation and algorithm im-plementation distinguish two index sets, and , thecombination of which forms the index set of a complete sino-gram, the output of our sinogram completion algorithm. InSection III-E, we use an example to describe the relationshipbetween a truncated sinogram and the index sets , ,and explain how they can be determined or estimated prior tosinogram restoration.

A. An Operator Form of the HL Condition

Using in (7) as the Fourier series coefficients, we de-fine a “resampled” version of the sinogram along thechannel direction as the following:

(9)

Our projection completion procedure is applied to a discretizedversion of rather than . The convenience ofworking with is mainly computational as and

are related by the Fourier transform pair. Note thatis a periodic function in both and ; only half of it, definedon corresponds to a resampled versionof the sinogram .

To formulate our sinogram restoration problem, in the rest ofthe paper we use a discrete linear operator form for (9) as

(10)

The right-hand side vector is a concatenation of the basis co-efficients which includes both the “consis-tent” and the “inconsistent” parts; the th column of the ma-trix is the vector ,

, and , listed in a lexicograph-ical order. The non-subscripted variablerepresents the discretized version of . For each ,


for a pair of specific and . With propernormalization, it is easy to see from (9) that , theidentity operator.

Let , then a consistent sinogramthat minimizes (4) can be equivalently

obtained from and , respectively the cosine-re-sampled versions of and at , byminimizing the following function:

(11)

instead. Both involve zeroing-out the “inconsistent” coefficientsthat violates the HL condition (3). The solution to (11) can

be described concisely by , where representssinogram expansion using FFT, and sinogram synthesis byinverse FFT.

Unfortunately, minimizing (4) or (11) does not provide a sat-isfying solution to the projection completion problem. The in-complete projection data are usually padded with zerosto fill in the missing region. The solution to (4) and (11) is aclosest match, in their respective (weighted) least squares sense,to regardless of whether the match takes place in the mea-sured region of , or over regions that we artificially padwith zeros. A more satisfying solution should be one such thatthe good match only takes place in those “trusted” region, overthe missing region the solution should assume some con-sistent values but not seek agreement with those padded zeros.Next we formulate a statistical sinogram restoration problemand define the “good” match, the “trust” region, and consistencyin a precise manner.

B. Statistical Projection Completion Using ConsistencyConditions

Following [3], we denote the index set of an ideal (complete)set of measurements by , the available2 but incomplete set ofmeasurements , and their difference, the missing portion by

.We model the transmission CT data as Possion distributed

random variables with mean , , where isthe air scan intensity at the detector cell , and is the unknownline integral of attenuation coefficients from the X-ray source tothe th detector cell. The measured (incomplete) data avail-able for . Our sinogram restoration algorithm is thenformulated as the following:

(12a)

(12b)

(12c)

The objective function in (12) consists of two terms. Thefirst term in (12a) represents the Poisson log-likelihood of the

2Here, “available” means either measured, or not measured but assumed tobe zero, i.e., outside of the projection support region. See Section III-E.

available incomplete data. The second term (12b) represents asmoothness penalty defined for all . The constantis a penalization weighting that controls the overall strength ofsmoothing. In this work, we use the quadratic penalty function[20] that penalizes abrupt changes in

(13)

where consists of the neighborhood pixels of the pixel .For a nonboundary pixel , we penalize the horizontal (channeldirection) and vertical (view direction) differences in the sino-gram similar to [20], i.e., a neighborhood of four pixels of equalweights. The constraints in (12c) requires that the restored sino-gram , must obey the HL condition. Subject to thisconstraint, the objective function seeks to maximize the penal-ized data log-likelihood when the data are available; when theyare not, the sinogram restoration relies on the local smoothnesspenalty. The unknowns in our problem formulation (12) are ,

, the known input data are , , .As written in (12c), the “effective” unknowns are those free

basis coefficients s that are allowed to be nonzero underthe HL consistency condition. We may partition the vector as

, the free parts and the a priorily zero parts (for

consistent sinograms) , then . If we similarlypartition , then the expression in (12c) is equiv-alent to . The unknowns of our sinogram restorationproblem are the ’s. The expression in (12c) emphasizes thatthe computation of can be achieved via FFT, which neces-sarily involves but does not affect the result of .

If we treat the constraint in (12c) as a generic linear operatorrather than the HL condition, the problem formulation of (12) issimilar to a fully iterative reconstruction problem in transmis-sion CT imaging [21]. Some considerations are called for if weattempt to follow the derivation of a “reconstruction” algorithmfor as that in [21] and [22]. First, in the problem statement of(12), both the unknowns and the “system” matrix are madeof complex numbers, whereas in a reconstruction problem theseare all nonnegative real numbers. Second, a sequential updatingscheme such as the one in [21] may not fully exploit the partic-ular structure of embodying the HL condition. An efficientimplementation of the “projection” and “backprojection” op-erator (in the language of fully iterative reconstruction tech-niques) is critical for our sinogram restoration algorithm. In theAppendix we derive an iterative algorithm for updating andhence of the problem in (12) that uses FFT and IFFT we de-scribed in Section II-B to perform the forward and backwardprojection operations and , respectively. Starting from acertain consistent sinogram , the update equa-tions for can be symbolically expressed as follows:

(14a)

(14b)

where is a column vector and is a negative-defi-nite diagonal matrix. Their exact expressions are derived inAppendix A. The constant is a fixed relaxation parameter,

.


C. Reconstruction From Incomplete Projections

Combining the contents of the previous sections, a flowchartof our proposed statistical projection completion algorithm canbe described as follows.

1) Zero-pad channels so that the .2) Fan-to-parallel beam rebinning.3) Cosine transform along the channel direction.4) Initialization.

5) Do {• Calculate , and .• Update according to (14).} Until convergence criterion is met.

6) Arc-cosine transform along the channel direction.7) Parallel beam FBP reconstruction.To extend the RFOV beyond the SFOV, we first zero-pad a

number of channels to the projection data matrix; this step mayrequire a priori knowledge of the size and shape of the scannedobject. Secondly, we perform fan beam to parallel beam rebin-ning for application of the HL condition. The third step is toimplement the cosine re-sampling operation in (7) so that en-forcing the HL condition can be implemented by FFT and IFFTas explained in Section III-A. A natural initialization schemeof is to fill it with the direct estimate of the line integralswhere they are available, i.e., and zero elsewhere, i.e.,

. However such an initial estimate is not a consistentsinogram, therefore some rectification is needed as indicated inStep 4. From this point the iteration starts (Step 5). We apply thepositivity constraint after sinogram synthesis to ensure .This helps to reduce the numerical error accumulation in the re-peated FFTs.

The stopping criterion we used in this work is either a max-imum of 2000 iterations or , the sum of theabsolute difference between consecutive updates is less than 1,whichever is met first. From the at the last iteration, we per-form arc-cosine transform along the channel direction to comeback to the sinogram domain, and apply FBP for parallel beamgeometry to obtain the final reconstructed image.

D. Weighted Least Squares Formulation

In a clinical environment, the raw data we can obtain from anX-ray CT scanner are preprocessed line integrals. An alternativeproblem formulation is the following weighted least squares ob-jective function subject to the HL consistency condition:

(15)

where ’s are the acquired line integral data, the least squaresweighting term can be taken as the variance of . If

, and is Poisson with mean , then canbe approximated by [23]. An update equation for andcan be obtained similar to Section III-B. Please see Appendix B.

Fig. 1. The truncated projection data (after log operation) and its relationshipwith� and� .� is obtained from the forward projection of an objectsupport estimate. A loose object support estimate is used in this example.

Both the Poisson model (12) and the Gaussian model (15)for a statistical formulation of low dose X-ray CT imaging havebeen proposed in the literature [24], [25]. They are both approx-imate since the detection and the preprocessing of raw data ina clinical scanner is complicated and therefore approximationis necessary. One of our simulation studies will be to comparethe properties of the final FBP reconstructed images using thetwo models in (12) and (15), in order to empirically evaluatethe effects of different approximations in our particular problemsetting.

E. Obtaining and

As a first step of the sinogram completion procedure, the trun-cated projection is padded with zeros to enlarge the RFOV (Sec-tion III-C, flowchart). The index sets and are definedon the padded sinogram and their estimates can be obtainedfrom the forward projection of the object boundary (support)information (Fig. 1). If truncation is mild, the direct FBP imageusing the truncated projection data can provide an estimate ofthe object support; otherwise some a priori knowledge of the ob-ject size may be necessary. This object support is then forwardprojected, the parts that project to the extend projection regionare marked as , and the other regions, i.e., everything mea-sured plus parts of the extended region where the object supportsdoes not project to, . In other words, isthe complete sinogram that can be applied directly to FBP. If

, the corresponding is not acquired (missing). If, the corresponding is either measured but noisy, or

not measured but is assured to be zero from an estimated objectsupport.

A question then naturally arises as how much the objectboundary estimate affects the reconstructed images withinthe SFOV and within the RFOV. To answer the question, westudy the performance of the proposed sinogram completionalgorithm using three object support estimates, i.e., an exactobject boundary, a tight estimate, and a loose estimate in oursimulation studies.


Fig. 2. The phantom definition. The numbers indicate the attenuation values�cm � in different organs. The long axis of the body ellipse is 30 cm. Thisphantom is placed at (2.5, 2.5) cm from the iso-center. The dashed circles markthe SFOV at three truncation levels.

IV. COMPUTER SIMULATIONS AND EVALUATION METHODS

We investigated three factors that may affect the performanceof our proposed algorithm: 1) the severeness of truncation, 2) theaccuracy of the object support estimation, and 3) the regular-ization weight and different statistical approximations of theprojection data.

We used a digital phantom that mimics the human torsoanatomy (Fig. 2) and simulated parallel beam projection datafirst without truncation. To avoid sampling symmetry, thisphantom was placed tilted and offset (by 2.5 cm in both and

direction) from the iso-center of a simulated SFOV of 50 cm.The maximum line integral is around 9 along the long-axisof the phantom. A high density pin was inserted in the lungregion to measure the resolution properties in the FBP recon-structed images using different projection completion methods.A selected number of boundary channels in the projectionimage were then set to zero [Fig. 3(a)] simulating transversetruncation. The line integrals of attenuation coefficients wereexponentiated and scaled by the uniform air scan intensity of

, , for different noise levels. Poisson perturbationwas then introduced to projection data based on the mean trans-mitted counts in each detector cell. Noise-free projection datawith the high density pin present, and Poisson distributed noisydata without the high density pin were separately simulated.Twenty independent noise realizations were generated in orderto calculate the mean profiles and standard deviations. In thenext three subsections, we describe the simulation studies andthe evaluation methods for each of the above three factors.

A. Dependence on the Degree of Data Truncation

At each air scan intensity level , , and , we variedthe number of cutoff channels and compared the performanceof our proposed sinogram completion method with the sym-metric mirroring method [6]. The complete projection data con-sisted of 729 channels and 512 views over 180 acquisition.Fig. 3 shows one example of the truncated projection data (lineintegrals) at cm and the corresponding

Fig. 3. (a) The truncated projection data (after log operation). The vertical axisis the acquisition angle in degrees, the horizontal axis is the channel index.(b) The corresponding map of� (dark) and� (white). The cutoff channelnumber is � � �� cm� in this example.

Fig. 4. Three object support estimates: the exact object support, a tight convexsupport, and a loose rectangle object support.

trust region map with (dark) and (white) derived fromthe exact object boundary.

Since our simulation was performed in the parallel geometryand data truncation was artificial, we skipped steps 1 and 2 inthe flowchart of Section III-C. Both the truncated projection data[the scaled, exponentiated version of Fig. 3(a)] and the trust re-gion map [Fig. 3(b)] went through cosine-transform along thechannel direction, and followed the steps in the flowchart untilthe final reconstructed images were obtained. The implementa-tion of symmetric mirroring method and the related parametersettings followed the description in [6]. For each truncation case

cm of cm , andcm , we plotted the mean profile, from

20 noise realizations at the air scan , across the long axis ofthe reconstructed images obtained from different methods.

B. Dependence on the Accuracy of the Object SupportEstimate

We also studied the performance of our proposed methodwhen different object support estimates were used. In additionto the exact object support (see also Fig. 3), we also used a tightconvex object support, and a loose rectangle support estimateshown in Fig. 4. The object support estimate was forward pro-jected to generate the trust region and . Similar to Sec-tion IV-A, we plotted the mean profile across the long axis overthe reconstructed images using our proposed method but withdifferent estimation of the trust region maps. We evaluated thereconstructed images within the SFOV and the extended RFOVseparately.


C. Dependence on the Regularization Weight and DifferentStatistical Assumptions

If we compare the problem formulation in (12) or (15) withthe flowchart in Section III-C, we notice that the sinogramrestoration algorithm is applied to the cosine-transformedprojection data, not the projection data themselves. This canbe considered a side effect of resorting to FFT for convertingbetween and the basis expansion coefficients . The im-plementation in the flowchart however makes the sinogramrestoration performance statistically suboptimal, for either thePoisson or the Gaussian model. To evaluate the noise propertiesof the proposed method, we use simulation studies to comparethe noise-resolution tradeoffs of the two sinogram restorationmethods with that of the symmetric mirroring method. A bettersolution is discussed at the end of the paper.

As the weighting factor in (12) and (15) varies, the noiseand resolution properties in the final FBP reconstructed imagetrace out a noise-resolution tradeoff curve. To calculate a resolu-tion measure, we used zoomed FBP reconstruction from noise-free projection data (with the same truncation level as thenoisy projection data, and going through the same projectioncompletion procedure). The resolution was then measured asthe full-width-at-half-maximum of a 2-D Gaussian fitted to theimage of the pin. The noise was calculated as the coefficientof variation (COV) in a nearby region-of-interest (ROI) of thepin location. For this study, we fixed the air scan intensity tobe , and used the tight convex mask as the sinogram sup-port. We varied the regularization weight from toand compared the noise-resolution tradeoff curves in the FBPreconstructed images using projection data obtained from (12)and (15), respectively. In the weighted least squares formula-tion, the pixel-dependent was taken to be , the inverseof the transmission data. As a reference, we also plotted thetradeoff curve for the symmetric mirroring + FBP method. Thereconstructed images were postsmoothed by Gaussian-shapedfilters. The FWHM of the postsmoothing Gaussian filter servesthe same purpose as in the statistical formulations.

V. COMPUTER SIMULATION RESULTS

Shown in Figs. 5–10 are the results from the computer sim-ulations. Fig. 5 is a plot of the objective function in (12) as afunction of iteration numbers. We notice the monotonic increaseof the objective function as the iteration continues. At the lateriterations ( 50), the increase in the objective function per itera-tion is small compared with the earlier iterations. The completesinogram takes more iterations to converge than the apparentplateau of the objective function. Fig. 6 shows some sample re-construction images at different truncation levels using five dif-ferent processing methods. The air scan intensity to produce thesample images is the lowest in our simulations at . Figs. 7–9are the mean profiles of the five reconstruction methods at thetruncation level of , 200, and 250, respectively. Fig. 10compares the noise-resolution tradeoff of two statistical sino-gram restoration methods with that of the symmetric mirroringmethod. The simulated air scan intensity is and the trunca-tion level .

Fig. 5. A sample objective function plot of the formulation (12) for the caseof � � �� and air scan level �� . The inlet zoomed in on the change in theobjective function at later iterations. The objective function increase is smallcompared to the objective function value itself, i.e., smaller than 1e-5. This ex-plains the vertical axis labels of the inlet.

A. Sample Reconstruction Images

From Fig. 6 we observe that when the truncated projectiondata are directly applied to the FBP algorithm, the reconstructedimages show significant positive bias, the bright band artifacts,near the edge of the SFOV. This bias is greatly reduced by ap-plying either the symmetric mirroring method or our proposedsinogram restoration method. Using our proposed method, someremaining truncation artifacts can still be seen in the most severetruncation case (Fig. 6, red arrows). The differencesbetween the different support estimates mainly show up outsidethe SFOV. The exact support estimate helps to delineate the ob-ject boundary (Fig. 6, green arrows).

B. Dependence on the Degree of Data Truncation

From the mean profile plots (Figs. 7–9), we notice that withinthe SFOV our proposed method using any of the three objectsupport estimates has smaller bias than the symmetric mirroringmethod. The same observation is made at the three differenttruncation levels. Unlike the symmetric mirroring method whichuses ad-hoc data extrapolation, the proposed method uses theHL condition and tries to estimate the sinogram based on theavailable data and prior object support information. It is ex-pected that the proposed method should provide more accuratereconstruction within the SFOV.

For the most severe truncation case , some re-maining edge artifacts (profile discontinuity) can be seen usingthe proposed method (Fig. 9(a) and (b), black arrows), whilethere is smaller discontinuity around the edges using the sym-metric mirroring method. This observation agrees with the vi-sual impression from Fig. 6 (third column, red arrows). For theother two truncation cases ( , 200), the proposed methodwith the exact object support estimate has smaller errors thanthe symmetric mirroring at the edge of truncation in Fig. 7 andFig. 8(a), and comparable errors with symmetric mirroring inFig. 8(b). It appears that the remaining edge artifact using ourproposed method is related to the truncation level in the avail-able data. We have used the same parameter settings (stopping


Fig. 6. Results of the computer simulation study. The different columns correspond to different number � of truncated channels, and different rows differentprocessing methods. 1st row: no correction. 2nd row: symmetric mirroring. 3rd-5th rows: our proposed method (12) with different support estimates. The air scanintensity to produce these sample images is �� . The display window for all images is [0.15, 0.25] cm .

Fig. 7. The mean profile calculated from 20 noise realizations at � � �� nearthe edge of data truncation (right side of SFOV). The black arrow points to theedge of data truncation.

criterion and the smoothing parameter ) to produce all thereconstruction results. Finer parameter tuning in our proposedmethod should be examined.

C. Dependence on the Accuracy of the Object SupportEstimate

The results from the different object support estimates do notdiffer much within the SFOV, except when the truncation is se-vere the results from the loose rectangle support showlarger bias [Fig. 9(b)] than the exact object support and the tightconvex support.

The advantage of using the exact object support is mainly inthe extended RFOV (Fig. 6, green arrows). There is a gradualimprovement in object boundary delineation when the sinogramrestoration is applied with the loose rectangle support estimate,the convex support estimate, and the exact support; and all threehave smaller mean profile error than the symmetric mirroringmethod. The reconstructed object boundary in the symmetricmirroring method is determined indirectly from the number ofextended channels [6], an input parameter based on sub-jective judgment. On the other hand, the object support estimateenters our proposed method as part of the problem formulation.The reconstruction results are then more intuitive; the better theobject support estimate is, the better delineation of the objectboundary can be obtained.

D. Dependence on the Regularization Weight and DifferentStatistical Assumptions

Fig. 10 shows the noise-resolution tradeoff curves in the FBPreconstructed images after projection completion using the sym-metric mirroring method, the Poisson assumption (12), and theweighted least squares formulation (15). Compared with the an-alytic formulation, i.e., the symmetric mirroring method, thereis a significant improvement in terms of noise resolution tradeoffusing either of the statistical formulations. The Poisson assump-tion performs marginally better than the weighted least squaresmodel at the simulated noise level.

We do notice that for the two sinogram restoration methods,the variation in the final (rather large) FWHM is relatively smalldespite change in from to . Two possible causes


Fig. 8. Similar to Fig. 7 but at � � ��. Since both sides of the phantom are truncated, we use two arrowheads to indicate the edges of data truncation. (a) and(b) are near the left and right side of the SFOV in the region near data truncation.

Fig. 9. Similar to Fig. 8 but at � � ��.

Fig. 10. The noise-resolution tradeoffs at the pin location.

may play a role here. The effect of is related to its relativemagnitude with respect to the air scan and the object at-tenuation properties (attenuation line integrals from 0–9). Forpaths through the object that are not severely attenuated, the rel-ative magnitude of is small. The rather large FWHM could

be attributed to the channel-wise cosine and arc-cosine inter-polation in the sinogram restoration procedure (cf. flowchart inSection III). Note that these interpolations are performed onlyonce at the beginning and the end of the sinogram restorationprocedure, not repeatedly at each iteration.

VI. PATIENT DATA

We studied the performance of the proposed method using adata set from an obese patient whose body size extended outsideof the SFOV. The original fan-beam projection has 672 (chan-nels) 1052 (views) per 360 acquisition covering a SFOV of50 cm diameter. We compared three processing methods: 1) di-rect FBP of the truncated projection data, 2) FBP after projectioncompletion using the symmetric mirroring method, and 3) afterapplying our proposed method.

The symmetric mirroring method was performed in the nativefan beam geometry and fan beam FBP reconstruction was ap-plied afterwards. To apply our proposed sinogram completionalgorithm, we performed fan to parallel rebinning with inter-polation; the rebinned parallel data has 673 channels and 1024views per 180 acquisition. We padded 200 zero channels oneither end of the rebinned parallel data to extend the FOV to


Fig. 11. (a) Truncated projection data (line-integrals) of the patient scan afterfan beam to parallel beam rebinning. The horizontal axis is the channel index,the vertical axis is the acquisition angle in degrees. (b) The trust region mapshowing � (dark) and � (white) for the patient data.

Fig. 12. Reconstructed images (a) without correction, (b) with symmetric mir-roring method, and (c) with our proposed method. (d) Comparison of percentbias. The display window is [0.14, 0.21] cm .

79.7 cm [Fig. 11(a)]. From the truncated projection data in Fig.11(a), we obtained the trust region map of Fig. 11(b) by vi-sually identifying those truncated channels and marked themoff as banded . The left band has a width of 90 channels,and the right band 60 channels. The two numbers are chosenpurely based on visual judgment. Using the trust region map ofFig. 11(b), the sinogram completion algorithm then further con-fines the patient size to be channels or 61cm. Since the raw data were (scaled) line-integrals (after log-op-eration), we exponentiated the line-integrals and then scaledthem by an arbitrary air scan intensity3 of . Similar to theprocessing of the simulation data, the (exponentiated) truncatedprojection and the trust region map were processed followingthe steps in the flow chart of Section III-C until the final recon-structed images were obtained.

3Scaling by an arbitrary air scan intensity may cause statistical model mis-match. We address this problem further in Section VII.

Fig. 12 shows the reconstructed images from the three pro-cessing methods. To obtain a quantitative measure, we calculatethe mean attenuation value within the three ROI’s (Fig. 12(a),green regions) located inside the SFOV. The middle ROI isfar from the truncation artifacts, therefore is used a referencefor calculating the percent bias within ROI-1 and ROI-2 thatare near the truncation artifacts. The bias values are charted inFig. 12(d). Using either the symmetric mirroring method or theproposed sinogram completion method, the bright boundary ar-tifacts and the positive bias in the peripheral region within theSFOV are significantly reduced. The proposed method producessmaller bias than the symmetric mirroring method.

VII. DISCUSSION AND CONCLUSION

There are two main ideas we would like to convey in thispaper. The first is that the consistency conditions can be in-corporated in a sinogram restoration algorithm for transmissionX-ray CT to accommodate projection data incompleteness. Thesecond is that such an algorithm can be implemented in a simple,efficient, and numerically stable manner.

In our simulation studies, we used the parallel beam ge-ometry and generated Poisson distributed transmission CTdata and applied the algorithms in Section III. By varying thenumber of truncated channels, we studied the performance ofthe algorithm at different levels of data truncation. We alsocompared the noise-resolution tradeoff in the final FBP recon-structed images using the proposed method with the symmetricmirroring method. The proposed sinogram restoration approachgreatly improved both the accuracy and the precision of thereconstructed images within the SFOV. The trust region mapor the object support estimate does not affect the reconstructedimages within the SFOV significantly, however a more accurateobject support estimate can better delineate the object boundaryin the extended RFOV. The three object support estimatesthat we have in Fig. 4 are all reasonably accurate. In a realimaging environment, some a priori knowledge of the objectsize may be necessary to obtain the sinogram support estimate.We point out that this is not an extra requirement for applyingour method. Some knowledge of object size is needed forother truncation artifacts reduction algorithms as well, e.g., thezero-padding part for the symmetric mirroring method. Evenin a fully iterative image reconstruction method, a prerequisitefor the iteration to handle truncation “naturally” is to have areconstruction matrix large enough to fully contain the object.If the object size is under estimated, then artifacts will occur atthe edge of the SFOV and the prescribed RFOV.

At the three simulated data truncation levels ( , 200,and 250), not all projection views are truncated. Analytical re-construction algorithms have been developed [26] and [27] toproduce exact reconstruction in the region where the SFOV in-tersects with the object. A crucial element in these theoreticaldevelopments is the use of object space information such as ob-ject support. The proposed sinogram restoration approach worksexclusively in the sinogram domain. Though a priori sinogramdomain knowledge can be incorporated, we do recognize thatthe sinogram domain constraints (e.g., sinogram support) can bemuch weaker than the object space constraints (e.g., object sup-port). Therefore we must caution that theoretical exactness (zero


bias) using the proposed sinogram restoration method may notbe guaranteed; this is in addition to the bias introduced throughthe regularization factor . However, a marked difference fromthe approaches in [26], [27] is that we also seek to recover be-yond the SFOV. Our computer simulations obtained not onlyexcellent results within the SFOV, but made regions beyond theSFOV visible. Moreover, the statistical formulation makes ourapproach less prone to data noise, a common weakness of manyanalytical reconstruction algorithms.

We also applied our projection completion algorithm to realpatient data. The positive bias in the peripheral regions of thepatient was greatly reduced, but within the extended RFOV theobject boundary was not as well recovered as in the simulationstudies. Since the acquired raw patient data were scaled line-in-tegrals, we needed to exponentiate the patient data first to obtainthe transmission CT data. An arbitrary, constant air scan inten-sity was used since the relevant information was unavailable tous. This approach has been used in other works though its va-lidity might be arguable. In a clinical CT scanner, the applicationof bow-tie filters results in a nonconstant air scan intensity pro-file. The mismatch between this and our assumed constant airscan intensity may quantitatively affect the noise properties ofthe final reconstructed image. A better way may be to estimatethe air scan intensity from the variance of the blank scan (no ob-ject) projection values [25], [28], and use the estimated air scaninstead of an arbitrary constant to scale the exponentiated lineintegrals.

For the iterative algorithm that we proposed, the computa-tional efficiency depends on 1) the number of iterations and 2)the computations per iteration. In terms of iteration numbers,the sinogram restoration algorithm now on the average runsto 1000 iterations. It is often found that fully iterative algo-rithms run to thousands of iterations (counting subset updates)[29], [30]. The computational advantage of the proposed methodmainly comes from the application of the FFT instead of theregular forward/backward projection operators. The paper [31]demonstrated the computational savings of using FFT methodfor forward/backward projector calculations. For typical imagesize, [31] showed more than 50% computational savings; andthis advantage increases with image size.

There are some unaddressed questions in this paper. Ourevaluation studies are still preliminary. We have not performedquantitative comparisons with other projection completionalgorithms such as [5] and [9], nor have we studied the role andeffect of the smoothing parameter . In principle, both the HLconsistency condition (by its finite number of terms) andthe quadratic penalty regulate the solution. For the resultsreported in this paper, the number of the expansion termswas simply determined from the channel numbers and the viewnumbers. It is interesting to investigate the interplay betweenthe parameter and the Fourier transform length (# of s)and their effects on the final reconstructed images. Furtherresearch may also look into how the completed sinograms,hence the reconstructed images from them, evolve as a functionof iteration numbers.

If the phantom simulations were performed in the native fanbeam geometry, the interpolation from fan beam to parallelbeam geometry will change the probability distribution of the

rebinned data, and invalidate the assumption that the rebinneddata are Poisson distributed random variables. We may alterna-tively construct the following function:

(16)

The unknowns are defined on a parallel beam grid, and themeasurements are defined on the native fan beamgrid. Here may be regarded as a rebinning matrixthat takes care of the parallel to fan rebinning. The optimizationproblem is then to maximize the objective function in (16) sub-ject to the constraint that satisfies the HL condition. It will beinteresting to compare this alternative model with our approachpresented here in terms of image noise and resolution.

There are other possible extensions of this work. Since ourproblem formulation is completely general, the projection in-completeness is not confined to channel-wise data truncation,though these are the cases we studied in the simulation and ex-periments. Other types of data incompleteness, such as limitedangle data acquisition and sparse sampling are also interestingapplication areas of our proposed method.

APPENDIX ADERIVATION OF (14)

We denote by the objective function in (12). Sincefor , we have

It is straightforward to verify that if, and . According to [21], is a

legitimate surrogate function of the objective function of (12)if we restrict the step size of such that . Bycompletion of squares, we can rewrite the surrogate function

in the following form:

where

is the total number of neighborhood pixels ( in thiswork), and

(17)


Now plugging in the constraint equation with the un-derstanding that only in are the effective un-knowns, we seek such that

(18)

is maximized. In (18) we define, a diagonal matrix with the

th element , , and a column vector withelements , . The following optimality conditionof (18):

(19)

Suppose is the identity operator, then the solution to(19) is simply by the orthogonality conditionof the operator . In general, the solution of (19)

(20a)

(20b)

but the matrix inversion in (20b) may be costly to compute ateach iteration. Instead we derive a second surrogate function of

in (18) to replace the diagonal matrix by a scaledidentity matrix. The derivation is based on the observation that,for a diagonal positive definite matrix, ,and an arbitrary real number

(21)

where is the minimal diagonal element of . From (21),it is easy to verify that is a (trivial) surrogate (at thepoint ) for minimizing . We use the same trick torewrite as follows:

(22)

(23)

where , , . The inequalityin (23) defines a second surrogate function of whosesolution is

(24)

The above derivation requires that the change in is boundedfrom below by 1.5. Since , we have .With proper normalization of , the maximumchange in can be verified and ensured by the maximumchange in , which in turn, if necessary, can be adjusted by

replacing the relaxation parameter in (24) by some, .

APPENDIX BWEIGHTED LEAST SQUARES PROJECTION

COMPLETION ALGORITHM

The algorithm derivation is analogous to the Poisson distribu-tion case. Since the objective function in (15) is quadratic, wemay skip the first quadratic surrogate function. By comparingterms, it is straightforward to verify that

The rest of the derivation is exactly the same as in Appendix Awith the above and that define and .

ACKNOWLEDGMENT

The authors would like to thank E. Fishman, M.D., B. Mudge,and E. Avril for their assistance with patient data acquisition.The authors would also like to thank K. Stierstorfer, Ph.D. atSiemens Healthcare for his help with Siemens scanner dataformat.

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Statistical Projection Completion in X-ray CT Using Consistency Conditions

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