Elliptical Extrapolation of Truncated 2D CT Projections Using Helgason-Ludwig Consistency Conditions G. Van Gompel1 , M. Defrise2 , D. Van Dyck1 1
2
Vision Lab, University of Antwerp, Antwerp, Belgium. Department of Nuclear Medicine, Free University of Brussels, Brussels, Belgium
ABSTRACT Image reconstruction from truncated tomographic data is an important practical problem in CT in order to reduce the X-ray dose and to improve the resolution. The main problem with the Radon Transform is that in 2D the inversion formula globally depends upon line integrals of the object function. The standard Filtered Backprojection algorithm (FBP) does not allow any type of truncation. A typical strategy is to extrapolate the truncated projections with a smooth 1D function in order to reduce the discontinuity artefacts. The lowfrequency artifact reduction however, severely depends upon the width of the extrapolation, which is unknown in practice. In this paper we develop a modified ConTraSPECT-type method for specific use on truncated 2D CT-data, when only a local area (ROI) is to be imaged. The algorithm describes the shape and structure of the region surrounding the ROI by a specific object with only few parameters, in this paper a uniform ellipse. The parameters of this ellipse are optimized by minimizing the Helgason-Ludwig consistency conditions for the sinogram completed with Radon data of the ellipse. Simulations show that the MSE of the reconstructions is reduced significantly, depending on the type of truncation.
1. INTRODUCTION The problem of reconstructing a 2D image from a complete set of 1D projections consists in computing the image from its Radon Transform, i.e. from a complete set of line integrals pθ (r), defined as: Z ∞Z ∞ pθ (s) = (Rf )(θ, s) = f (x, y)δ(s − x cos θ − y sin θ)dxdy θ ∈ [0, π], s ∈ R (1) −∞
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For CT reconstruction, the standard analytic reconstruction algorithm is (based on) the Filtered BackProjection formulation of the inverse Radon transformation. Z π f (x, y) = Qθ (x cos θ + y sin θ) dθ 0
with Qθ the Rampfiltered projections. This method is fast and accurate compared to iterative reconstruction algorithms, but has a low flexibility. It can not as easily be adapted to sinograms with non-uniform angular or radial sampling. The same remark is valid for a truncated sinogram. To reconstruct an image pixel, the FBP formulation needs a complete sinogram. Indeed, to reconstruct a specific pixel (x, y) every filtered projection is accessed once, and every point of the filtered projection is calculated from the whole corresponding measured projection. This means that no amount of truncation is tolerated for FBP when an exact reconstruction of a ROI is required. Note also that a truncated sinogram does not correspond to the full sinogram of a real object, unless for special cases such as when the original object is radially symmetrical. The FBP reconstruction of a truncated sinogram contains two types of artifacts: a high frequency artifact along Send correspondence to:
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the lines corresponding to the boundary of the measured region of the sinogram, and on the other hand a distortion of the low-frequencies over the whole ROI. In most cases, acceptable reconstructions of a ROI can be achieved for qualitative analysis. However FBP reconstruction is catastrophic when a quantitative reconstruction is needed. In this paper we consider transaxial truncation of 2D parallel projections, which is the case when the detector is too small to cover the whole object for all orientations. We consider a specific geometry where the line integrals are measured for all lines that cross a circle, which we call the Field Of View (FOV). Each point within the FOV is sampled over the full angular range [0, π]. We consider two different types of truncation: the ’interior’ and the ’truncated arms’ problem, shown in figure 1. The oval represents the object, the circle shows the FOV. In the interior problem (figure 1a) the FOV
(a) interior
(b) truncated arms
Figure 1: Representation of the two considered types of truncation does not overlap any object boundary, which means that all projections are truncated on both sides. It has been proven by Natterer [1] that the problem of reconstructing such a FOV is not uniquely solvable. Indeed, there is an infinite amount of 2D functions which are different from zero in the field of view, but for which the truncated sinogram equals zero. In the truncated arms problem (figure 1b) the FOV partly covers the boundary of the object, meaning that at least some projections are not truncated. For this type of truncation a great evolution has been achieved during the last few years. Zou et al [2], Clackdoyle et al [3] and Noo et al [4] showed how specific parts of the FOV can be reconstructed exactly with iterative methods or with an alternative formulation of the Radon inversion, the latter method not yet being validated for real data. Recall that these exact results can never be achieved using the standard FBP algorithm because its formulation does not allow any truncation. There are two strategies of extrapolating a truncated sinogram to improve the quantitative accuracy of a FBP reconstruction. The first strategy is to acquire a second, non-truncated low resolution scan and combine it with the truncated high-resolution scan. Following Grangeat and Azevedo [5], Tisson [6] showed that only few projections are needed to achieve in this way a satisfying reconstruction. However in general such additional data is not available. A typical strategy is then to reduce the high frequency artifacts corresponding to the discontinuities in the sinogram by extrapolating it with a smooth monotone function (e.g. cos2 ). This operation also reduces the low frequency distortion, though this improvement depends on the width of the extrapolating window, which in general is chosen empirically. In this paper we estimate the missing data of the Exterior Region (ER, meaning the region surrounding the FOV) using Helgason-Ludwig consistency conditions. Consistency conditions have earlier been proposed for extrapolating a truncated sinogram. Lewitt and Bates [7] noticed that these consistency conditions cannot be used in an isolated way. Since the missing ER data is not determined by the truncated sinogram, there is an infinite amount of consistent extrapolations with only the real ER Radon data yielding the aimed reconstruction. Therefore additional requirements are needed. Kudo and Saito [8] introduced 4 extra constraints in their work: nonnegativity, the measurement data, the sinogram support (assuming the object boundary is known before image reconstruction) and a predescribed reference sinogram (e.g. the most recent healthy image of the
patient). They noticed that the latter two constraints yield significantly improved results using accurate a priori information. However, in many cases such information is not available. In the PET/SPECT domain the consistency conditions were succesfully used for attenuation correction when no transmission data is available. A ConTraSPECT (Consistency Transmission SPECT) method was developped [9–12] which approximates the unknown attenuation map by a uniform ellipse of which the parameters are optimized using the HL-consistency conditions. In our paper we develop a modified ConTraSPECT type algorithm for specific use on truncated CT data when an accurate FBP reconstruction of the FOV region is aimed. We extrapolate the truncated sinogram by the Radon data of a uniform ellipse of which the 6 parameters are optimized using the Helgason-Ludwig consistency conditions. Afterwards, the Region Of Interest is reconstructed using the FBP algorithm. We offer suggestions for dealing with the discontinuity between the measured data and the extrapolation, for which the FBP algorithm is very sensitive. Furthermore we use a different cost function formulation and optimization strategy in order to avoid problems of misconvergence and instability as reported in [9] and [11].
2. METHOD 2.1. Idea The two-dimensional Radon Transform (which represents ideal CT-data) obeys a set of consistency conditions (Helgason-Ludwig) described by Z πZ ∞ sn eikθ Rf (θ, s)dsdθ = 0 (2) H(n,k) (Rf ) = 0
−∞
for integers k > n ≥ 0 and k − n even ; the CT transmission sinogram Rf (θ, s) has been defined in equation (1). A truncated sinogram T f (θ, s), which is equal to Rf (θ, s) for s ∈ [−t, t] and is zero outside this interval, is clearly not consistent in general. Here t denotes the radius of the centered FOV. A truncated sinogram contains information about the global density distribution and over the ER. We approximate the ER by a shape that can be adjusted to these properties, in this paper an ellipse is used. The parameters of this ellipse are then optimized using the Helgason-Ludwig (HL) conditions. Finally the sinogram is extrapolated with the Radon transform of the optimal ellipse and reconstructed.
2.2. Extrapolation The algorithm approximates the region surrounding the ROI by a uniform ellipse, this simple shape being chosen to avoid having to fit too many parameters. We complete the missing part of the truncated sinogram (T f (θ, s)) with the corresponding Radon data F (µ, a, b, x0 , y0 , φ, θ, s) of a uniform ellipse with parameters (µ, a, b, x0 , y0 , φ), respectively the positive uniform density, the two axes, the x- and y-position of the center and the orientation of the ellipse. This results in the extrapolated sinogram (T f (θ, s) + F (µ, a, b, x0 , y0 , φ, θ, s)), see figure 2. Remark that another shape could be used if it is expected to better match the object. The method can also be extended to a 2-ellipse approximation, which is particularly useful for objects with a local dense structure in the region surrounding the ROI. In that case, one ellipse is used to describe the boundary of the object, while the other ellipse describes the local structure. The optimization now has to be performed in a 12-parameter space, 6 parameters for each ellipse.
2.3. Optimization The next step is then to determine the optimal ellipse parameters by optimizing the consistency of the extrapolated sinogram. Define the cost function describing the amount of inconsistency of the extrapolated sinogram as X H(n,k) (T f + F ) E(µ, a, b, x0 , y0 , φ) = (3) (k,n)∈I
where I denotes the set of indices (k, n) used. To ensure that the contributions of the various terms have the same order of magnitude, we rescale the s-interval to [−2, 2].
The multiplication of the sinogram with sn in equation (2) introduces severely increasing numerical instability for increasing n. Therefore, and because only six parameters are to be optimized, the number of conditions can be limited to I = {(k, n) : n = 0, 1, 2; k = n + 2, n + 4} (4) Furthermore we notice the importance (for stability) of an analytic description of the elliptical Radon data instead of a numerical Radon calculation from the ellipse pixel image. The 6-dimensional cost function counts multiple local minima, so an analytic optimization from a random start value is not advised. Instead we use a Differential Evolution (DE) optimization which is slower but which increases the chance of finding the global minimum. DE [13] is an iterative population-based global optimization algorithm for continuous functions. Initially a random group of vectors (here 6 components) or ’parents’ is generated, each component representing a parameter of the cost function. At each iteration, every parent is combined (by cross-over) one-by-one with a set of new vectors which results in a set of trial vectors (’children’), this step referred to as the recombination stage. Each child then replaces the ’least useful’ parent (which is the parent with the largest cost function value) in the population if and only if its cost function value is lower than that of this least useful parent. There is a variety of strategies for creating the new set of vectors in the recombination stage, each of them influencing the convergence speed and misconvergence rate differently. In this work a parent is combined by cross-over with the sum of two weighted difference vectors, the first one representing the difference between the parent and the best member, the second one representing the difference between two random population members. During our experiments we noticed that the global optimum region is often wide, so only a relatively small amount of parents and iterations is needed to determine the global optimum region. The exact minimum can then be determined by a gradient based algorithm. The optimization time can be reduced significantly by downsampling the sinogram in a first step. Note that only 6 parameters are to be optimized which means that only a small amount of projections and projection pixels is needed to perform the optimization. Afterwards the parameters are fine-tuned using the full sinogram. In subsection 2.2 the possibility of a 2-ellipse extrapolation is considered. The optimization problem now consists of finding the minimum in the 12-dimensional parameter space. First results show that this space is still
s
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q Figure 2: summary of the extrapolation procedure. Upper left : measured truncated sinogram. Lower left : Radontransform of the ellipse with the region corresponding to the known measured data is set to zero. Right : the final sinogram as a result of filling up the missing region of the measured sinogram with the according data of the ellipse.
stable using the same number of conditions. The differential evolution optimization time increases because more parents are needed. The steepest descent optimization time doubles because twice as much function evaluations have to be done per iteration.
2.4. Reconstruction The extrapolating function F (µ, a, b, x0 , y0 , φ, θ, s) is determined to satisfy the HL-conditions, but in general it does not match the measured truncated projections at the boundaries of the FOV, i.e. at s = t and s = −t. To avoid amplification of these discontinuities by the FBP high pass ramp filter, we smooth the transition between T f and F using a short cos2 function.
2.5. Summary In summary our method contains following steps: 1. Complete the truncated sinogram with the Radon transform of a uniform ellipse with parameters (µ, a, b, x0 , y0 , φ), respectively the uniform density, the two axes, the x- and y-position of the center and the orientation of the ellipse. This yields an extrapolation F (µ, a, b, x0 , y0 , φ, θ, s). 2. Determine the parameters by minimizing the cost function (3), composed of the Helgason-Ludwig consistency conditions. 3. Reconstruct the completed sinogram using FBP.
3. RESULTS AND DISCUSSION 3.1. Practical Considerations In this section we test the method on three phantoms and for the ’interior’ and ’truncated arms’ types of truncation. The reconstructions are compared to those obtained with the standard cos2 -extrapolation method described in the introduction. The latter method contains one free parameter, the width of the extrapolation. Choosing this width large enough, the method is very effective in eliminating the high-frequency artifacts of the FBP reconstruction. Also the DC-shift is reduced significantly, however the latter result depends on the chosen width. In figure 3 the MSE within the FOV of the reconstruction with the cos2 method is plotted for different extrapolation widths. There is a clear minimum in this curve for which the MSE is very low, however, the width corresponding to this minimum is unknown in practice so that a guess width (based for instance on the boundary of the object if known) is used.
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Figure 3: Typical tendence of the MSE of a reconstruction for varying cos2 extrapolation widths.
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Figure 4: Comparison of the two competing 6 parameter one-ellipse results ((b) and (f), (c) and (g)) with the two-ellipse 12 parameter result ((d) and (h)) for the phantom shown in (a). The display interval of the reconstructions is reduced from [0;1.2] to [0.15;0.3] to enhance the contrast.
The optimal ellipse doesn’t necessarily describe the real boundary of the object. The reason is the ER not being uniform in general, while we assumed it can be approximated by a uniform ellipse. The resulting ellipse is therefore a trade-off between the internal structure and the object’s boundary. In case of a dense structure in the ER (which is illustrated in figure 4), there are two solutions competing for optimality: the first ellipse describing the boundary shifted towards the dense structure (sinogram (c) and corresponding reconstrution (g)), and the other one describing the dense structure ((b) and (f)). It is then preferable to choose the first solution (figure (c)), because of the catastrophic discontinuities in the solution of figure (b). Furthermore, in such cases large improvement can be achieved by using a two-ellipse optimization ((d) and (h)) with one ellipse describing the boundary and the other one describing the structure. Notice that the solution found is very close to the object that has been simulated.
(a) G22 density interval [0;0.4]
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Figure 5: Visualisation of the used phantoms
The results are shown for three different phantoms: G22, mSL (a modified Shepp-Logan phantom) and G16, respectively shown in figures 5(a), 5(b) and 5(c). Phantom G22 represents objects with a slow varying asym-
metrical density in the ER. MSL is a challenging phantom because of the dense skull surrounding the object, creating a high-contrast non-local non-uniformity in the ER. The G16 phantom represents objects with a local density in the ER. For this phantom the effect of the 12-parameter optimization is examined. The 6-parameter optimization process is performed using a 200 parent population and 40 iterations. For the 12 parameter optimization, we used 400 parents and up to 100 iterations. The 185 × 180 (pixels per projection × number of angles) sinograms are reduced to size 185 × 45 to speed up the optimization.
3.2. The interior problem The results for the interior problem for the phantoms G22 and mSL are shown in figures 6 and 7. The display interval of the reconstructions is reduced to enhance the contrast. In both figures, (a) represents the ROI of the reconstruction from the full sinogram; (b), (c) and (d) represent the reconstructions of the ROI from a truncated sinogram after using the different considered extrapolation strategies: figures (b) and (c) show the result of the cos2 -extrapolation with an empirically chosen width and the optimal width respectively, while figure (d) represents the result of the elliptical extrapolation method. In figure (e) the thin white line on the original phantom represents the optimal ellipse. Figure (f) plots the shifts (the absolute value of the difference with the exact reconstruction (a)) of a horizontal cross-section of the images shown in (b), (c) and (d). The results for phantom G16 are shown in figure 7. (a) represents the exact reconstruction from a full sinogram. (d) and (e) are reconstructions using the cos2 extrapolation with a arbitrary user chosen width and the optimal extrapolation width respectively. (f) and (g) represent the reconstructions using the elliptical extrapolation method for a 6 and 12 parameter optimization respectively. The different curves in figure (f) shows the absolute shifts of the central horizontal cross section of results (d), (e), (f) and (g) compared to the exact reconstuction. A common result for the three phantoms is that both the MSE and the absolute shift of the reconstruction with the elliptical method are significantly smaller than those of a cos2 extrapolation method with an empirically chosen width. The error (both the MSE and the cross-section through the difference image) obtained with our algorithm for the phantoms G16 and G22 is close to the error obtained with the optimal cos2 width which is a very nice result noting that the optimal cos2 width is unknown in practice. The result for the challenging mSL phantom is slightly less nice but is, as noticed earlier, still significantly better than that of the empirically chosen cos2 extrapolation.
3.3. Truncated arms problem The results for the phantoms G22, mSL and G16 are shown in figures 9, 10 and 11 respectively. The structure of the figures is the same as in the figures for the interior problem. The figures show clearly that the error (MSE and shift) of the EE-reconstruction is significantly smaller compared to that of the cos2 -reconstruction for the optimal width, even for the challenging mSL phantom. For the ‘truncated arms’ problem, exact algorithms are available [2–4] , but the accurate results obtained with the approximating elliptic extrapolation method nicely illustrate the efficiency of this approach.
4. CONCLUSIONS In this paper we present a modified ConTraSPECT method for the extrapolation of a truncated CT sinogram. This method approximates the missing data of the ER with the corresponding Radon data of an ellipse, the parameters of which are determined using the Helgason-Ludwig consistency conditions for the 2D Radon transform. The method contains suggestions for reducing the discontinuity artifacts at the transition between the measured data and the extrapolation, using a small cos2 function and by eliminating specific ellipse solutions. The modified cost function formulation and optimization strategy provide improved stability and the opportunity of optimizing a two-ellipse 12-parameter problem, even when only few Helgason-Ludwig conditions and a downsampled sinogram are used. For the interior problem, the MSE of the elliptical extrapolation are comparable to the results obtainable with the cos2 extrapolation of optimal (but unknown in practice) width. For the ‘truncated arms’ problem [2–4], exact algorithms are available, but the accurate results obtained with the elliptic extrapolation nicely illustrate the efficiency of this approach. Remark finally that the algorithm also can be used for other mathematical shapes that better describe the object.
5. ACKNOWLEDGEMENTS This work was supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen)
REFERENCES 1. F. Natterer, The mathematics of computerized tomography, SIAM, 2001. 2. Y. Zou and X. Pan, “Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT,” Physics in Medicine and Biology 49, pp. 941–959, February 2004. 3. R. Clackdoyle, F. Noo, J. Gua, and J. Roberts, “Quantitative reconstruction from truncated projections in classical tomography,” 51, pp. 2570–2578, October 2004. 4. F. Noo, C. R., and P. J.D., “A two-step Hilbert transform method for 2D image reconstruction,” Physics in Medicine and Biology 49, p. 3903, August 2004. 5. S. Azevedo, P. Rizo, and P. Grangeat, “Region-of-interest cone-beam computed tomography,” Fully 3D Image Reconstruction in Radiology and Nuclear Medicine , pp. 381–384, 1995. 6. G. Tisson, P. Scheunders, and D. Van Dyck, “ROI cone-beam CT on a circular orbit for geometric magnification using reprojection,” in IEEE Medical Imaging Conference, October 2004. 7. R. Lewitt and R. Bates, “Image reconstruction from projections: III: Projection completion methods (theory),” Optik 50, pp. 189–204, October 1977. 8. H. Kudo and T. Saito, “Sinogram Recovery with the Method of Convex Projection for Limited-Data Reconstruction in Computed Tomography.,” Optical Society of America 8, pp. 1148–1160, 1991. 9. A. Welch and R. Clack, “Toward accurate attenuation correction in SPECT without transmission measurements,” IEEE Transactions on Medical Imaging 16, pp. 532–541, October 1997. 10. A. Welch, C. Campbell, and R. Clackdoyle, “Attenuation correction in PET using consistency information,” IEEE Transactions on Nuclear Science 45, pp. 3134–3141, December 1998. 11. I. Laurette, R. Clackdoyle, A. Welch, F. Natterer, and G. T. Gullberg, “Comparison of Three Applications of ConTraSPECT,” IEEE Transactions on Nuclear Science 46, p. 6, 1999. 12. D. Gourion, D. Noll, P. Gantet, A. Celler, and J. Esquerr´e, “Attenuation Correction Using SPECT Emission Data Only.,” Nuclear Science Symposium Conference Record , pp. 1522–1526, 2001. 13. R. Storn and K. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” Journal of Global Optimization 11, pp. 341–359, 1997.
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Figure 6: Results for the G22 phantom interior prob-
Figure 7: Results for the mSL phantom interior prob-
lem. (a) reconstruction from full sinogram. (b), (c) and (d) reconstructions from truncated sinogram. The display interval of the reconstructions is reduced to [0.25;0.35] to enhance the contrast. In figure (e) the white contour represents the optimal ellipse. Figure (f) plots the horizontal cross sections through the absolute difference images of (b), (c) and (d) with (a)
lem. (a) reconstruction from full sinogram. (b), (c) and (d) reconstructions from truncated sinogram. The display interval of the reconstructions is reduced to [0.15;0.25] to enhance the contrast. In figure (e) the white contour represents the optimal ellipse. Figure (f) plots the horizontal cross sections through the absolute difference images of (b), (c) and (d) with (a)
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Figure 8: Results for the G16 phantom interior problem. (a) reconstruction from full sinogram. (d), (e), (f) and (g) reconstructions from truncated sinogram. The display interval of the reconstructions is reduced to [0.3;1] to enhance the contrast. In (b) and (c), the white contour(s) represent the optimal ellipse for the 6 parameter optimization and the 12 parameter optimization respectively. Figure (f) plots the horizontal cross sections through the absolute difference images of (d), (e), (f) and (g) with (a)
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Figure 10: Results for the mSL phantom truncated arms
problem. (a) reconstruction from full sinogram. (b), (c) and (d) reconstructions from truncated sinogram. The display interval of the reconstructions is reduced to [0.25;0.35] to enhance the contrast. In figure (e) the white contour represents the optimal ellipse. Figure (f) plots the horizontal cross sections through the absolute difference images of (b), (c) and (d) with (a)
problem. (a) reconstruction from full sinogram. (b), (c) and (d) reconstructions from truncated sinogram. The display interval of the reconstructions is reduced to [0.15;0.25] to enhance the contrast. In figure (e) the white contour represents the optimal ellipse. Figure (f) plots the horizontal cross sections through the absolute difference images of (b), (c) and (d) with (a)
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Figure 11: Results for the G16 phantom truncated arms problem. (a) reconstruction from full sinogram. (d), (e), (f) and (g) reconstructions from truncated sinogram. The display interval of the reconstructions is reduced to [0.3;1] to enhance the contrast. In (b) and (c), the white contour(s) represent the optimal ellipse for the 6 parameter optimization and the 12 parameter optimization respectively. Figure (f) plots the horizontal cross sections through the absolute difference images of (d), (e), (f) and (g) with (a)
