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Basis-Image Reconstruction Directly from Sparse-View Data in Spectral CT Buxin Chen, Student member, IEEE, Zheng Zhang, Xiao Han, Student member, IEEE, Emil Sidky, Member, IEEE, and Xiaochuan Pan, Fellow, IEEE
Abstract—In the work, we investigate the feasibility of using sparse data from spectral CT systems with multiple data sets. We develop an optimization-based reconstruction method that may avoid the limitations of standard data-based decomposition methods in spectral CT, such as overlapping rays and dense angular sampling. We generated data from two distinct diagnostic range kVp spectra based on two non-overlapping cone-beam CT scanning geometries. Realistic noise level is added to the data. In specific, we employ the constrained total-variation minimization reconstruction program with beam-hardening-corrected adaptivesteepest-descent-projection-onto-convex-sets algorithm and apply to the full and sparse data for directly reconstructing basis images. Results indicate that images reconstructed from sparse data can be visually comparable to those from full data, with both free of beam-hardening artifacts, demonstrating the feasibility of the sparse data acquisition in spectral CT with multiple data sets for potentially reducing radiation dose.
I. I NTRODUCTION PECTRAL CT exploits the underlying spectral properties of materials, using multiple data sets with different incident spectra. Its advantages include improving material contrast and reducing beam-hardening (BH) effect [1], [2]. One of the methods to use these data sets is data-based decomposition combined with standard analytic-based reconstruction method [3]–[6]. However, such method often requires overlapping rays from different spectra measurements [7] and dense angular sampling. We develop an optimizationbased reconstruction method that integrates the decomposition into the reconstruction, which can help avoid aforementioned limitations, for spectral CT systems with multiple data sets [8]. In the work, we investigate the feasibility of the proposed method for spectral CT image reconstruction from sparse-view data. A motivation of the work is to reduce radiation dose, as multiple data sets usually come with multiple radiation exposure. For example, with two scans using two kVp spectra, a straightforward way to keep the dose level similar as using a single kVp scan is to reduce the number of views to half in each scan. Preliminary simulation studies with sparse-view data are conducted with two distinct diagnostic range kVp
S
Manuscript received November 18, 2014. This work was supported in part by NIH R01 Grant Nos. CA120540, CA158446, and EB000225. The contents of this article are solely the responsibility of the authors and do not necessarily represent the official views of the National Institutes of Health. B. Chen, Z. Zhang, X. Han, and E. Silky are with the Department of Radiology, The University of Chicago, Chicago, IL 60637 USA (e-mail: {bxchen, zhangzh, xiaohan, sidky}@uchicago.edu) X. Pan is with the Departments of Radiology and Radiation & Cellular Oncology, The University of Chicago, Chicago, IL 60637 USA (e-mail:
[email protected])
978-1-4799-6097-2/14/$31.00 ©2014 IEEE
spectra, realistic scanning geometry and noise level. A numerical phantom of four materials is used for data simulation. II. M ATERIALS AND M ETHODS A. Data Simulation We consider multispectral CT data acquired from an energyintegrating cone-beam CT system. The source-to-detector and source-to-rotation distances are 150 cm and 100 cm, respectively. A 46 cm × 30 cm flat panel detector with 0.06 cm pixel size is assumed. To obtain the spectral information, two sets of data are simulated with two distinct spectra, namely, 100 kVp and 140 kVp. Each set consists 637 angular projections evenly distributed over 2π. To simulate realistic conditions in existing spectral CT systems, e.g., fast kVp switching or dual source, we assume that rays from the two sets of data do not overlap, but interlace in between each other. For the work, we focus on the central slice including 765 effective detector bins. We refer to the full data as the two sets of sinogram containing 765 × 637 entries each, and the sparse data as the data extracted in half from each set of the full data. A numerical abdominal phantom is used. It consists of four materials - adipose tissue, tissue, water, and bone. A two-basis decomposition model is used (see Fig. 1) for phantom spectral responses, in combination with energy-integrating model for the detector response. Considering the multichromatic X-ray spectrum, the measured data is generated as [8] i h P R R 2 ES(E) exp − t=1 µt (E) L ct (~r)dl dE R . (1) gm = −ln ES(E)dE
Poisson noise is simulated and the noise level is made equivalent to 1 × 106 photons per detector bin, or 2.8 × 108 photons per cm2 , in the air scan. B. Optimization-Based Reconstruction 1) Linearized Data Model: We can linearize (1) by assuming a small, perturbative non-linear beam-hardening (BH) term. In specific, it can be separated as gm = g + gBH , or Z 2 X µ ¯t ct (~r)dl, (2) g = gm − gBH = t=1
L
where gBH is the non-linear beam-hardening term defined as h P i R R 2 ES(E) exp − t=1 ∆µt (E) L ct (~r)dl dE R gBH ≡ ln , ES(E)dE (3)
2
and µ ¯t ≡
Z
S(E)µt (E)dE & ∆µt (E) = µt (E) − µ ¯t .
(4)
Consequently, a linearized data model is formed as ~ = HC, ~ G
(5)
~ ⊺ = (~g l ⊺ , ~g h ⊺ ), C ~⊺ = where G (⊺ indicates transpose), and l l µ ¯l2 Al µ ¯1 A . (6) H= h h µ ¯1 A µ ¯h2 Ah (~c1⊺ , ~c2⊺ )
µ ¯l,h 1,2 can be calculated from (4) using low (100 kVp) and high (140 kVp) energy spectra weighting the linear attenuation coefficients of two basis materials. Al,h describes two discrete X-ray transforms that correspond to the scanning geometries using low and high energy spectrum, respectively. Notably, these two projection matrices are different because the rays do not overlap. ~g l,h are the model data vectors defined in (2), and ~c1,2 are image vectors representing the two basis images. 2) Reconstruction Program and Algorithm: In the work, we formulate a constrained total variation (TV)-minimization program [8], [9] as (~c1⊺ , ~c2⊺ )
= argmin(k~c1 kT V + k~c2 kT V ) ~ < ǫ and c1i , c2i > 0, s.t. D(G)
(7)
where c1i and c2i denotes the image value on the ith pixel of basis images ~c1 and ~c2 , and
2 ~ = ~ −G ~ D(G)
HC
! 2 h 2
X X
s s s s = µ ¯t ~ct − ~gm + ~gBH ,
A ·
s=l
t=1
2
is the l2 -norm of data vector difference between the model ~ BH ). ~ and the BH-corrected measured data (G ~m − G data HC ǫ is a positive parameter accounting for the inconsistencies, including noise and residual BH terms that are not corrected ~ BH . for by the estimation of G s ~ As D(G) becomes non-convex with the addition of ~gBH , we use an approach previously developed [8] to solve the reconstruction program numerically. The approach, based on the adaptive-steepest-descent-projection-onto-convex-sets s (ASD-POCS) algorithm [9], includes notably estimating ~gBH from ~c1,2 in the current iteration by using (3) and subtracting them from the measured data to correct for the BH effect. In ~ < ǫ by iteration other words, we update the constraint D(G) and render it into a convex form after each update. III. R ESULTS A. Inverse crime - consistent data We first verify the numerical convergence, as well as the code implementation, of the algorithm by conducting inverse crime studies. The consistent data refer to those as a result of the linear model combined with the BH term, as described in (1). The optimization program is set up as ǫ = 1 × 10−5 . The reconstructed basis images are shown in Fig. 1. The results are visually identical to the true numerical phantom used to
(a) adipose [0.5, 1.1]
(b) bone [0.0, 0.6]
Fig. 1: Basis images reconstructed from consistent full data using BH-corrected ASD-POCS algorithm. The results are visually identical to the truth.
generate the data. (Hence, we do not show the true phantom images here.) We report that the data distance has reached 3.2 × 10−6 while the normalized image distance to truth is 1.2 × 10−4 . B. Noisy data 1) Basis images: We present the reconstructed basis images from full data and half-view sparse data in Fig. 2. For the results from the full data with 637 views, the bone image (Fig. 2b) is visually similar to the inverse rime result. A close inspection reveals that there is still some artifacts, or cross-talk, around the vertebral bones. For the adipose image (Fig. 2a), the cross-talk is more obvious, especially around the vertebral bones. The liver edge is not as sharp as the adipose/water edge, possibly due to the fact that adipose being the basis. The visual quality of basis images is overall maintained going from full to half-view data. No typically BH artifacts around bones are present. However, a slightly elevated level of cross-talk can be observed. 2) Composite images: In addition to basis images, we present composite images, which are linearly combined from basis images at a monochromatic level, side-by-side with FBP reconstruction from a single kVp scan data. The 100 kVp scan data is used for FBP reconstruction. The monochromatic level for the composite image is thus chosen at the average energy of the 100 kVp spectrum, namely, 54 keV. Results from both full data and half-view sparse data are shown in Fig. 3. In the FBP images (Fig. 3b and 3d), conspicuous shading artifacts are observed, especially connecting neighboring bony structures. These shading artifacts typically result from the beam-hardening effect and are pronounced in the image since no correction or post-processing is applied to the data. On the other hand, the BH-corrected ASD-POCS composite images (Fig. 3a and 3c) are almost BH-free. The structures all have sharp edges and clear contrast. Further, the reduction of view to half has little impact on the image visualization, except elevated noise level. IV. C ONCLUSIONS AND D ISCUSSIONS In the work, we have investigated the feasibility of using sparse data in spectral CT with multiple data sets for the
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(a) adipose, v637
(b) bone, v637
(c) adipose, v319
(d) adipose, v319
Fig. 2: Reconstructed basis images from noisy data using BHcorrected ASD-POCS. Cross-talk between two basis images are observed, especially around vertebral bones. Image visual quality is maintained as the number of views is reduced to half.
(a) composite image at monochro- (b) FBP reconstruction matic level of 54keV, v637 100kVp data, v637
from
(c) composite image at monochro- (d) FBP reconstruction matic level of 54keV, v319 100kVp data, v319
from
Fig. 3: Monochromatic composite images (left column) and FBP images from a single kVp scan (right column). FBP images display conspicuous shading artifacts around bony structures, while composite images from BH-corrected ASDPOCS basis images are almost BH-free.
purpose of reducing radiation dose. Simulation studies were carried out using two diagnostic range kV spectra with conebeam CT geometry. The two sets of sinogram data interlace in between each other and have non-overlapping rays that are inconsistent for data-based decomposition method. Realistic noise level is also added to the simulation data. The image results indicate that the beam-hardening correction method, based on estimating the BH term and having it corrected for in the measured data, is effective in combination with the ASDPOCS algorithm. In addition, reducing the views to half does not impact the image visual quality and BH effect correction. The monochromatic composite images display no BH artifacts and sharp structure edges, even subject to added noise in the data. The basis images are naturally decomposed in the reconstruction. However, there are still cross-talk between the two basis images, with the presence of noise, and the structure edges are not as sharp as in the composite images. Such indications implies that the BH correction is less impacted than the decomposition by the presence of noise. Further studies will be carried out to investigate the robustness of the proposed BH-corrected ASD-POCS algorithm against other challenging yet realistic scanning conditions in spectral CT. R EFERENCES [1] A. Coleman and M. Sinclair, “A beam-hardening correction using dualenergy computed tomography,” Phys. Med. Biol., vol. 30, no. 11, p. 1251, 1985. [2] P. M. Joseph and C. Ruth, “A method for simultaneous correction of spectrum hardening artifacts in ct images containing both bone and iodine,” Med. Phys., vol. 24, no. 10, pp. 1629–1634, 1997. [3] R. E. Alvarez and A. Macovski, “Energy-selective reconstructions in xray computerised tomography,” Phys. Med. Biol., vol. 21, no. 5, p. 733, 1976. [4] W. H. Marshall Jr, R. E. Alvarez, and A. Macovski, “Initial results with prereconstruction dual-energy computed tomography (predect).” Radiology, vol. 140, no. 2, pp. 421–430, 1981. [5] J. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. B¨aumer, C. Herrmann, R. Steadman, G. Zeitler et al., “Experimental feasibility of multi-energy photon-counting k-edge imaging in pre-clinical computed tomography,” Phys. Med. Biol., vol. 53, no. 15, p. 4031, 2008. [6] Y. Zou and M. D. Silver, “Analysis of fast kv-switching in dual energy ct using a pre-reconstruction decomposition technique,” in Medical Imaging. International Society for Optics and Photonics, 2008, pp. 691 313– 691 313. [7] C. Maaß, M. Baer, and M. Kachelrieß, “Image-based dual energy ct using optimized precorrection functions: A practical new approach of material decomposition in image domain,” Med. Phys., vol. 36, no. 8, pp. 3818– 3829, 2009. [8] X. Pan, B. Chen, Z. Zhang, E. Pearson, E. Sidky, and X. Han, “Optimization-based reconstruction exploiting spectral information in ct,” in The Third International Conference on Image Formation in X-Ray Computed Tomography, pp. 228–232. [9] E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol., vol. 53, pp. 4777–4807, 2008.
