Sparse-view tomography via displacement function interpolation
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Sparse-view tomography has many applications such as in low-dose computed tomography (CT). Using under-sampled data, a perfect image is not expected. The goal of this paper is to obtain a tomographic image that is better than the naïve filtered backprojection (FBP) reconstruction that uses linear interpolation to complete the measurements. This paper proposes a method to estimate the un-measured projections by displacement function interpolation. Displacement function estimation is a non-linear procedure and the linear interpolation is performed on the displacement function (instead of, on the sinogram itself). As a result, the estimated measurements are not the linear transformation of the measured data. The proposed method is compared with the linear interpolation methods, and the proposed method shows superior performance.
KeywordsLimited data imaging Tomography Estimation
According to compressed sensing theory, sparse-view tomography may still be possible if some image domain constraints are used to compensate for the missing data [9, 10, 11, 12, 13, 14, 15]. However, this paper focuses on analytic filtered backprojection (FBP) reconstruction. Iterative reconstruction methods and machine learning methods are beyond the scope of this paper.
Sometimes the unmeasured views can be approximately estimated by interpolation methods [16, 17, 18, 19]. Recently, machine learning methods are popular and successful in sparse-view data estimation [20, 21, 22].
In this paper, we argue that interpolation with linear convolution approaches simply introduce slightly rotated images to the main image. By the “main” image, we imply the image that is reconstructed by only the measured sparse data. The results of machine learning methods depend on the training sets; therefore, the results may not be applicable to all applications. Here we propose a nonlinear method to estimate unmeasured projections by using displacement functions, which will be discussed in Section II. The linear interpolation method assumes that the unmeasured value is the average of its neighbors. On the other hand, the deformation method assumes that the unmeasured value has the same value as one of its neighbor’s value. The deformation method is nonlinear and is able to avoid or reduce the rotational interpolation artifacts. Comparison simulations will be presented in Section III, and Section IV concludes the paper.
The rotation effect of the linear interpolation method
An intuitive way to understand this phenomenon is to consider a different example that copies the available measurements at view 2 m + 1 to view 2 m, i.e., p(n, 2m) = p(n, 2m + 1). Using the extended sinogram, the reconstructed image will be the summation of two images: one is the original image and the other is a rotated version of the original image.
In general, sinogram interpolation via convolution yields an image that is a combination of the main reconstruction and some rotated versions of the main reconstruction. Similar phenomena are expected for other sinogram estimation methods that based on linear interpolation.
These two simple examples imply that in order to significantly improve the sinogram estimation, we must use some sort of nonlinearity. The idea of non-rigid deformation may be borrowed, altered, and applied to our sinogram estimation [23, 24, 25]. Another non-linear way is to use sine wave approximation . This paper proposes a displacement function interpolation method.
Use the deformation function for non-linear interpolation
The main idea of our algorithm is illustrated below. A pair of measured sinogram views is provided: p(n, m1) and p(n, m2), where n is the index along the radial direction and m1 and m2 are two angular indices. The goal is to estimate p(n, m) with m between m1 and m2.
where λ is a pre-set parameter to balance the weighting between constraints in the objective function F. We set λ = 0.01 in our implementation of (5). The purpose of the sign function is to encourage that the slopes of the deformed function p(n + u(n), m1) and the target function p(n, m2) have the same sign.
If we restrict u(n) to be integers in [−N, N] with N being a pre-set positive integer, it is efficient to evaluate the objection F with all possible u(n) values in [−N, N] and use a “min” function to determine the optimal displacement function u(n). Here, “min” is a built-in function in Matlab® to find the minimum value in an array. The motivation of using the “min” function instead of an iterative algorithm (such as the gradient decent algorithm) is to make the algorithm more efficient. The “min” function method only evaluates the deformed function 2 N + 1 times, while an iterative method evaluates the deformed function at least equal to the number of iterations, which is much greater than 2 N + 1.
Computer simulations and patient study
This proposed sinogram extension method was applied to two computer simulation cases and one real patient case. In the computer simulations, the image size was 256 × 256, and the detector size was 367. In this paper, the word “sinogram” is used in a general sense, and the “sinogram” can be parallel projections and can be fan-beam projections or in other geometries.
In computer simulations, the original under-sampled sinogram was generated analytically without noise. We can better observe the image distortion in noiseless studies. In the first computer simulation study, the original measured number of views was 60 over 360°. After sinogram extension, the number of views was increased to 180 over 360°. In the second computer simulation study, the original measured number of views was 120 over 360°. After sinogram extension, the number of views was increased to 360 over 360°. The absolute error image between the estimated sinogram and the true sinogram was calculated and reported in the next section.
In the patient study, the sinogram data was obtained by a CT scan using 500 mAs. The detector was curved, and the imaging geometry was cone-beam. The central slice of the cone-beam data was used as the fan-beam data. The data set had 896 detector bins at one view and 1200 views over 360°.
In this paper, an under-sampled data set was a subset of the original data set by using only 400 fan-beam views over 360°. After sinogram extension with displacement interpolation, there were 1200 views over 360°.
Sinogram estimation results using two-adjacent-view linear interpolation was also obtained and reported in the next section.
Rotation displacement artifacts due to sinogram linear interpolation
Linear interpolation between sinogram views is equivalent to linear combination of the images from the original sparse-view reconstruction and rotated versions of the sparse-view reconstruction. These effects are illustrated by an exaggerated sketch in Fig. 1. The rotational artifacts become more severe at locations away from the center-of-rotation in the image. The observation of these artifacts motivated the investigation of a nonlinear sinogram interpolation method.
Using function deformation for sinogram interpolation
Figure 2 shows two curves p(n, m1) and p(n, m2), one being a solid curve and the other being a broken curve. These two curves represent two sinogram measurements at view indices m1 and m2. A displacement function u(n) was estimate according to (3) so that the deformed version of one function approximately equal the other function (p(n, m2) ≈ p(n + u(n), m1)) . The displacement function is shown in Fig. 3.
Using the displacement function u(n) for sinogram interpolation was realized as follows. A missing view at the angle exactly between the two measured views can be estimated by replacing u(n) by 0.5 × u(n).
There are two pairs of small black-and-white dots in the phantom. The pair at the bottom is blurred more than the pair at the center be the estimation algorithms. We also observe that for the linear methods there is a circular region and the background noise texture is different within and outside this region.
Computer simulation sinogram estimation errors
Initial data set
Maximal absolute errors in the expanded sinogram
Sum of absolute errors in the expanded sinogram
Computer simulation FBP reconstruction errors in RMSE
Initial data set
Raw under-sampled data
Raw under-sampled data
Computer simulation iterative reconstruction errors in RMSE
Number of iterations
There are two types artifacts: the under-sampling streaking texture in the uniform areas and the blurry artifacts due to sinogram interpolation. The blurring artifacts can be easily detected by the pair of black-and-white dots at the bottom of the image. All methods perform poorly for the data set that has only 60 views.
Figure 8 shows the FBP reconstruction with 400 views. This image contains lots of streaking artifacts due to angular aliasing. For patient images, all images are displayed twice using two different display windows: [min, max] and [− 400, 400] Hounsfield units (HU).
Figure 9 shows the FBP reconstruction result from linear interpolation method. Severe rotation artifacts are observed in the image. The most severe rotation artifacts are observed at the outer regions inside the patient.
Figure 10 shows the result of proposed method that uses a non-rigid deformation technique. The rotation artifacts are no longer present. However, this image is not perfect. Compared with the gold standard shown in Fig. 7, some shadow artifacts are observed along the high contrast boundaries, and the spatial resolution is somewhat degraded.
In order to appreciate the improvements of the proposed method, a small rectangular sub region at the right part of the original image is cut out and is displayed in a larger format in Fig. 11 for images in Figs. 7, 8, 9, 10.
Patient study iterative reconstruction errors in RMSE
Number of iterations
Few-view tomography in CT is an open problem. This paper made an observation that linear convolution-based sinogram interpolation methods may produce rotational artifacts. To overcome this problem, this paper suggests a nonlinear method to estimate the unmeasured views. In this proposed method, two adjacent views in the original under-sampled sinogram are used to estimate the missing views between them. A displacement function is estimated by a non-iterative method. A fraction of the displacement function is used to estimate the missing views between the original measurements. One advantage of the proposed method is that the resultant FBP reconstruction using the estimated sinogram does not have the rotation artifacts. Our estimated sinogram is more accurate than the sinogram estimated by linear convolution-based methods, which is demonstrated by the absolution errors as shown in Tables 1, 2 and 3.
In our patient study, there are 400 views over 360° and there are 896 bins on the detector. The number of view angles is extremely small, about 1/4.5 of the value required by the Shannon’s sampling theorem. The proposed algorithm produces fewer artifacts than the linear interpolation method as demonstrated in Fig. 11.
The iterative Lanweber algorithm is also used for the under-sampled data image reconstruction. However, it requires a large number of iterations to produce high resolution images. At the 1500th iteration, the reconstructed image is still blurry.
When the number of views is extremely low, as in the computer simulation with 60 views, the proposed algorithm is not effective, and the reconstructed image is rather blurry even though the streaking artifacts are significantly reduced. It is still an open problem to effectively reconstruct an image with extremely under-sampled data.
The authors thank Raoul M.S. Joemai of Leiden University Medical Center for collecting and providing us raw clinical data.
All authors read and approved the final manuscript.
This research is partially supported by NIH grant R15EB024283.
The authors declare that they have no competing interests.
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