MISSING DATA ESTIMATION FOR FULLY 3D SPIRAL CT IMAGE RECONSTRUCTIONDaniel B. Keesinga, Joseph A. O’Sullivanb, David G. Polittec, Bruce R. Whitingc, and Donald L. Snyderb

aDept. of Biomedical Engineering, bDept. of Electrical and Systems Engineering,cMallinckrodt Institute of Radiology, Washington University, St. Louis, MO, USA

AbstractReconstruction algorithms that are not set up to handle in-

complete datasets can lead to artifacts in the reconstructed

images because the assumptions regarding the size of the

image space and/or data space are violated. In this study,

two recently developed geometry-independent methods1

are applied to fully 3D multi-slice spiral CT image recon-

struction. Using simulated and clinical datasets, we dem-

onstrate the effectiveness of the missing data approaches

in improving the quality of slices that have experienced

truncation in either the transverse or longitudinal direction.

• When the support of an object lies partially outside the field of view (FOV) of a CT scanner, artifacts may arise in the reconstructed image due to undersampling.

• Most reconstruction algorithms implicitly assume the entire object is confined to the FOV, but if this is not the case, excessively large attenuation values may be recon-structed inside the boundary of the FOV.

• The reconstruction algorithm is unaware that the mea-sured data has been affected by the object's attenuation outside the FOV, so the image in the FOV is recon-structed such that the projections through it match the measured data.

Forward projection

Image update

Back-projection)(ˆ

)( xc k )(ˆ )1( xc k+

)(yd Back-projection

)(ˆ )( xb k

)(xb

Patient

Image

)(xc CT scan

We assume the photons arrive at the detector elements accord-

ing to a Poisson counting process, d(y), with mean value

where x = image voxel index, y = source-detector pair index,Io(y) = incident photon intensity, h(y|x) = projector kernel (mm),and c(x) = 3D truth image (mm-1).

Existing Analytical Methods

where:

AM algorithm (O’Sullivan and Benac9) for monoenergetic model

Problem Statement Statistical Data Model

Data and Image Spaces

Alternating Minimization

Transverse Truncation

Missing Data Extension

Longitudinal Truncation

Conclusions & Future Work

Acknowledgments

References

q(y:c(k))

This work was supported in part by a National Science Foundation Graduate Research Fellowship, by the National Cancer Institute under research grant R01CA75371 (J. F. Williamson, P. I.), and by the National Center for Super-computing Applications under grant ASC060030, which utilized the SGI Altix. We appreciate insightful comments by J. F. Williamson of Virginia Commonwealth University.

1. D. L. Snyder, J. A. O’Sullivan, R. J. Murphy, D. G. Politte, B. R. Whiting, and J. F. Williamson, “Image reconstruc-tion for transmission tomography when projection data are incomplete,” Phys. Med. Biol. 51, pp. 5603–5619, 2006.

2. S. Schaller, F. Noo, F. Sauer, K. C. Tam, G. Lauritsch, and T. Flohr, “Exact radon rebinning algorithm for the long object problem in helical cone-beam CT,” IEEE Trans. Med. Imag. 19, pp. 361–375, 2000.

3. M. Defrise, F. Noo, and H. Kudo, “A solution to the long-object problem in helical cone-beam tomography,” Phys. Med. Biol. 45, pp. 623–643, 2000.

4. Y. Zou and X. Pan, “Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT,” Phys. Med. Biol. 49, pp. 941–959, 2004.

5. J. Hsieh, E. Chao, J. Thibault, B. Grekowicz, A. Horst, S. McOlash, and T. J. Myers, “A novel reconstruction algo-rithm to extend the CT scan field-of-view,” Med. Phys. 31, pp. 2385–2391, 2004.

6. K. Sourbelle, M. Kachelriess, and W. A. Kalender, “Reconstruction from truncated projections in CT using adaptive detruncation,” Eur. Radiol. 15, pp. 1008–1014, 2005.

7. G. L. Zeng, G. T. Gullberg, P. E. Christian, and D. Gagnon, “Cone-beam iterative reconstruction of a segment of a long object,” IEEE Trans. Nucl. Sci. 49, pp. 37–41, 2002.

8. P. J. La Rivière, “Monotonic iterative reconstruction algorithms for targeted reconstruction in emission and trans-mission computed tomography,” in IEEE Nuclear Science Symposium/Medical Imaging Conference, 2006.

9. J. A. O’Sullivan and J. Benac, “Alternating minimization algorithms for transmission tomography,” IEEE Trans. Med. Imag., to appear.

10. W.P. Segars, Development of a new dynamic NURBS-based cardiac-torso (NCAT) phantom. PhD thesis, The Uni-versity of North Carolina, May 2001.

Experiment 1 (NCAT phantom10): Truncate the data from 30 detector elements on each side of sinogram (in all detector rows). Perform unregularized AM image recon-struction of 128x128x84 volume using specified methods.

Experiment 4 (NCAT phantom): Perform unregularized AM reconstruction of 128x128x84 volume using specified methods. End slice initialization was done by replacing end slices after first iteration with the nearest fully-sampled slice. From left to right, the percentage of complete data for slices 76-81 was: 81.7%, 64.5%, 49.6%, 34.1%, 16.9%, and 4.5%.

Experiment 2 (NCAT phantom): Same experiment as above, except perform regularized AM image reconstruc-tion of 128x128x84 volume using specified methods on noiseless and noisy data. A log cosh potential function was used to penalize differences between neighbors.

Experiment 3 (clinical abdominal scan): Truncate the data from 125 detector elements on each side of sino-gram (in all detector rows). Perform unregularized AM image reconstruction of 512x512x176 volume.

Corresponding author email: keesing@wustl.edu

Existing Statistical Methods

Long object methods: Among existing methods, Schaller

et al.2 developed an exact rebinning method called the PHI-

method, while Defrise et al.,3 and Zou and Pan,4 have made

use of differentiated backprojection along PI-line segments.

Transverse truncation methods: Among the extended

FOV reconstruction methods, Hsieh et al.5 and Sourbelle et

al.6 extrapolate the missing data in each projection using 2D

parallel beam consistency conditions (e.g., constant area

under projection curve in each view) and other constraints.

100 101 102104

105

106

107

Iteration

I−divergence No missing data approach

Method 1Method 2Complete data available

(a) Truth image (slices 70 and 74 shown)

(e) I-divergence vs. iteration

(a) Truth image

(b) Method 1 without end slice initialization

(c) Method 1 with end slice initialization

(d) Method 2 with end slice initialization

(b) Reconstruction without a missing data approach

(c) Method 1 reconstruc-tion using noiseless data

(a) Method 1 reconstruc-tion using noiseless data

(b) Method 2 reconstruc-tion using noiseless data

(c) Method 1 reconstruc-tion using noisy data

(a) Method 2 reconstruc-tion after 39 iterations

with 145 ordered subsets

(b) Complete data recon-struction after 39 iterations with 145 ordered subsets

(d) Method 2 reconstruc-tion using noisy data

(d) Method 2 reconstruc-tion using noiseless data

All NCAT reconstruc-tions shown after 100 full iterations with 73

ordered subsets.

ˆ

q(y : ĉ(k)) = Io(y) exp

−

x∈Xh(y|x)ĉ(k)(x)

b̂(k)(x) =

y∈Y

h(y|x)q(y : ĉ(k))

b(x) =

y∈Yh(y|x)d(y)

ĉ(k+1)(x) = max

ĉ(k)(x)− 1

Z(x)ln

b(x)

b̂(k)(x)

, 0

Z(x) = normalizing factor

Long object methods: Zeng et al.7 published an iterative

method that is similar in principle to the first approach of

Snyder et al.1 (with one major exception being the choice

of reconstruction algorithm). Rays that pass through both

the ROI slices and outer slices are not used.

Transverse truncation methods: La Rivière described a

joint estimation procedure that iteratively updates pixels

and projections within the FOV.8 An initial estimate of the

projections outside the FOV is obtained from a FBP recon-

struction inside the FOV, and then subtracting its reprojec-

tion from the measured projections.

• A feasibility study was conducted to apply two recently de-veloped geometry-independent methods to fully 3D multi-slice spiral CT image reconstruction.

• The reconstructions using transversely truncated datasets demonstrate that it is possible to reconstruct the image inside the FOV quite accurately without many iterations.

• Outside the transverse FOV, some potentially minor arti-facts are present. These artifacts diminish with increased numbers of iterations, leading to long runtimes.

• Methods 1 and 2 addressed the long object problem com-parably well when the end slices were initialized.

• It is possible that using some form of non-smoothing regu-larization, such as a prior based on consistency condi-tions, may significantly improve the convergence rate.

The image space must be large enough to fully support the

backprojections shown above, and all voxels in the image

space must be updated to guarantee monotonic convergence.

From Snyder et al.,1 the backprojections used in the image

update step are different from the above algorithm as follows:

b(x) =

y∈Yinc

h(y|x)d(y) +

y∈Ymiss

h(y|x)q(y : ĉ(k))

y∈Yinc

h(y|x)q(y : ĉ(k)) +

y∈Ymiss

h(y|x)q(y : ĉ(k))b̂(k)(x) =

Method 1: ignore missing data, i.e.,Method 2: estimate missing data, i.e.,

Ymiss = ∅Ymiss = ∅

q(y : c) = E[d(y)] = Io(y) exp

−

x∈Xh(y|x)c(x)

,

(a) Transverse view: the complete image space X circumscribes the transverse support of the patient.

(b) Longitudinal view: the complete image space X is denoted by the gray slices.

detectors (in one row)

patient

extended FOV

x-ray source

scanner FOV

detectors (in one row)

patientextended FOV

x-ray source

scanner FOV

Yinc Ym

issY m

iss

patient ROI

detectorrows

x-ray source

bed