Comparison of FBP and Iterative Algorithms with Non-Uniform Angular Sampling

Abstract

Some people may believe that the filtered backprojection (FBP) algorithm does not work if the projection data are measured non-uniformly. Some may also believe that iterative algorithms can automatically handle the non-uniformly sampled data in the projector/backprojector. This paper claims that the FBP algorithm can effectively handle the situation where the angular sampling is not uniform. This paper compares the images that are reconstructed by both the FBP and the iterative Landweber algorithms when the angular sampling is nonuniform. When the iteration number is low, the iterative algorithms do not handle the non-uniform sampling properly. A weighting strategy is then suggested and it makes the image resolution more isotropic. In few-view tomography, the FBP and iterative algorithms both perform poorly if no other prior information is used. We have made the following observations: 1) When using an iterative algorithm, one must use early solutions due to noise amplification. 2) An early solution can have anisotropic spatial resolution if the angular sampling is not uniform. 3) The anisotropic resolution problem can be solved by introducing angle dependent weighting, which is not noise dependent. 4) The weighting is not effective when the iteration number is large. The weighting only affects the early solutions, and does not affect the converged solution. 5) When the iteration number is large, the model-mismatch errors are amplified and cause artifacts in the image. 6) The FBP algorithm is not sensitive to the model-mismatch errors, and does not have the “early solution” problems. 7) In few-view tomography, both FBP and iterative algorithms perform poorly, while the FBP algorithm gives a sharper image than the iterative algorithm does.

I. Introduction

Usual tomographic imaging systems scan an object (or a patient) by rotating the camera around the object 360° with uniform angular intervals. In some rare situations, it may be beneficial to skip some view-angles and use a non-uniform angular sampling scheme [1–6]. For instance, we can skip some view angles on purpose in order to reduce radiation dose to the patient in an X-ray CT scan. In single photon emission computed tomography (SPECT), one can skip some highly attenuated views while spend more acquisition times at the views where the attenuation is less.

The capability to account for physical phenomena and for a non-ideal imaging system is one of the advantages of iterative approaches compared with the FBP method [5]. The iterative algorithms can be applied to any irregular imaging geometry with any irregular data sampling intervals [6]. This flexibility allows researchers to automatically choose an iterative algorithm for the image reconstruction task whenever data sampling is irregular.

In fact, the filtered backprojection (FBP) algorithm can be used in many situations where the data are sampled irregularly [7]. The goal of this paper is to compare the results of the FBP and iterative algorithms when they are applied to the projection data that are acquired with non-uniform angular intervals. We will demonstrate that the FBP algorithm is more powerful than one may believe and the iterative algorithm may not live up to its reputation.

II. Methods

The FBP algorithm is derived and expressed in the continuous signal domain. The backprojector is represented as an integral over the projection view angle θ [8]:

$$f(x,y)=\underset{0}{\overset{2\pi }{\int }}q(s,\theta ){\mid }_{s=xcos\theta +ysin\theta }d\theta$$ (1)

where q(s, θ) is the filtered projection, s is the index on the detector, and (x, y) represents the reconstruction point of the image.

The discrete implementation of (1) is to replace the integral by a summation, and replace dθ by Δθ, which is normally a constant 2π/(number of views). It is straightforward to extend this regular backprojector to include the cases where the angular range is not uniformly sampled. The only change is to make the interval Δθ vary. For example, the current view angle is θ_k, and then the backprojection weighting Δθ_k at angle θ_k is given by

$$\mathrm{\Delta }{\theta }_k=\frac{{\theta }_{k+1}-{\theta }_k}{2}+\frac{{\theta }_k-{\theta }_{k-1}}{2}=\frac{{\theta }_{k+1}-{\theta }_{k-1}}{2}.$$ (2)

The derivation of the weight (2) is based on the discrete approximation of the continuous backprojection integral (1). There are many methods to approximate a continuous definite integral, such as the rectangle rule, the trapezoidal rule, the Simpson’s, and so on. We choose the rectangle rule because it is simple to implement and it provides accurate enough results.

Other than using a variable backprojection weighting factor, the revised FBP algorithm is exactly the same as the conventional FBP algorithm, which consists of the ramp-filtering step q(s, θ) = p(s, θ)*h(s) and the backprojection step (1). Here, p(s, θ) is the projection and h(s) is the convolution kernel of the ramp filter.

The FBP algorithm is a linear algorithm. In order to conduct a fair comparison study, we choose to use a linear iterative Landweber algorithm, which is a gradient-descent-type algorithm. Both of the analytical FBP algorithm and the iterative Landweber algorithm are able to model the projection data noise; they both can incorporate prior information (or Bayesian penalty) about the image [9], [10].

Without the noise-weighing and Bayesian penalty, the iterative Landweber algorithm can be expressed as

$$f^{(k+1)}=f^{(k)}+\alpha A^T(p-{Af}^{(k)}),$$ (3)

where f^(k) is the estimation of the discretized image at the kth iteration written as a column vector, p contains the discrete projections written as a column vector, A is the discrete projection matrix, and A^T is the backprojection matrix which is also the transpose matrix of A.

This paper focuses on the effect of angular sampling. Neither noise weighting nor Bayesian penalty are considered. In the Results section, we will show that it is beneficial to use (2) as the weighting function in the iterative Landweber algorithm when the angular sampling is irregular and the number of iteration is low.

Firstly, a noiseless computer simulation study is presented. The computer generated phantom is shown in Fig. 1. The phantom had an elliptical shape, with a row and a column of small circles to test the resolution in different directions. The imaging geometry was curved fan-beam with 896 detector bins (i.e., detector channels) and the bin width was 0.52 mm. The fan-beam focal length was 600 mm. The images were reconstructed in an 840x840 array. The normal number of views over 360° in a typical CT scanner is 1200; that is, the angular interval is 0.3°. The non-uniform angular sampling strategy is to have finer sampling when the X-rays travel across the longer paths [4]. A longer path attenuates the X-ray photons more severely and makes the projection noisier. Using finer sampling at the views where the X-ray photon counts are low can increase the signal-to-noise ratio.

Fig. 1.

FBP reconstruction of a simulated resolution phantom using uniform angular sampling data.

We used the angular interval of 0.3° in the angular range of 0° ± 30° and in the range of 180° ± 30°. The angular interval of 1.2° was used in the angular range of 90° ± 60° and in the range of 270° ± 60°.

Secondly, a clinical data set was used. In this study, a cadaver was scanned using a standard clinical dose, with tube voltage of 120 kV and current of 500 mA. The imaging geometry and the parameters were exactly the same as in the computer simulation mentioned above. In the above two studies, we used the 1200-view, uniform angular interval Δθ = 2π /1200 reconstructions as the gold standard.

For the comparison purposes, we also used a larger angular interval of 2.4° in the angular range of 90° ± 60° and in the range of 270° ± 60° for the two cases mentioned above. We must point out that this large angular interval of 2.4° is not recommended for clinical CT, because it produces under-sampling artifacts.

In iterative reconstruction, a high iteration number of 20000 was used to provide the near convergence solutions, and a low iteration number of 50 was used to provide an early solution.

Thirdly, few-view tomography using the clinical CT data was then attempted with both the FBP and iterative Landweber algorithms, using only 20 views over 360°. No Bayesian methods such as total variation (TV) minimization, l₁ norm minimization, or l₀ norm minimization methods were used [11–15].

The step size α in (3) was chosen inversely proportional to the number of views (NOV) in the iterative reconstruction. For the uniform angular data with NOV = 1200, α was chosen as 0.0000005. For the uniform angular data with NOV = 20, α was chosen as 0.00006. For the non-uniform angular data with NOV = 400 + 200 = 600, α was chosen as 0.000001, where 400 views with Δθ = 0.3° and 200 views with Δθ = 1.2°. For the non-uniform angular data with NOV = 400 + 100 = 500, α was chosen as 0.0000012, where 400 views with Δθ = 0.3° and 100 views with Δθ = 2.4°.

III. Results

Figs. 1–15 show the results of the computer simulation using both of the FBP and the iterative Landweber algorithms. It is observed that the FBP gold-standard uniform angular data reconstruction (Fig. 1) and the weighted FBP non-uniform angular data regular data reconstruction (Fig. 2) are of almost the same image quality by visual inspection.

Fig. 15.

Weighted iterative Landweber reconstruction of a simulated resolution phantom using non-uniform angular sampling data. Iteration number is 50. Maximum angular gap is 2.4°.

Fig. 2.

FBP reconstruction of a simulated resolution phantom using non-uniform angular sampling data with utilization of proposed weighting function.

On the other hand, when the weighting function is not used, the FBP reconstruction has severe non-uniformity artifacts (Fig. 3), if the data are non-uniformly sampled. The phantom was finely sampled when the detector was closed to the 0° and 180° positions. The phantom was sparsely sampled when the detector was closed to the 90° and 270° positions. In Fig. 3, the vertical small dots are better separated than the horizontal dots. By using a proper weighting function, the separation of the horizontal dots is improved as shown in Fig. 2. This shows the effectiveness of the proposed weighting function in the FBP algorithm.

Fig. 3.

FBP reconstruction of a simulated resolution phantom using non-uniform angular sampling data without utilization of proposed weighting function.

The iterative algorithm, in principle, does not need any weighting functions to handle the non-uniform sampling. The image obtained with un-weighted iterative algorithm (Fig. 6) and the image obtained with weighted iterative algorithm (Fig. 7) show very similar results, when the number of iteration (20000) is extremely high.

Fig. 6.

Iterative Landweber reconstruction of a simulated resolution phantom using non-uniform angular sampling data. Iteration number is 20000.

Fig. 7.

Weighted iterative Landweber reconstruction of a simulated resolution phantom using non-uniform angular sampling data. Iteration number is 20000.

However, when the iteration number is low, angular sampling affects the results. When the angular sampling is uniform, the horizontal and vertical small dots have a similar degree of separation even though they are blurred; see Fig. 5. When the angular sampling is non-uniform, the horizontal dots have much worse separation than the vertical dots; see Fig. 8.

Fig. 5.

Iterative Landweber reconstruction of a simulated resolution phantom using uniform angular sampling data. Iteration number is 50.

Fig. 8.

Iterative Landweber reconstruction of a simulated resolution phantom using non-uniform angular sampling data. Iteration number is 50.

The iteration number is fairly low for most practical image reconstruction tasks. If data sampling is non-uniform (which is uncommon), one should also pay attention to the anisotropic resolution issues, similar to the situation when the FBP algorithm is applied to the non-uniformly sampled data.

One way to make image resolution more isotropic when the iteration number is low is to apply the weighting function (2) in the iterative algorithm, and the result is shown is Fig. 9.

Fig. 9.

Weighted iterative Landweber reconstruction of a simulated resolution phantom using non-uniform angular sampling data. Iteration number is 50.

The weighting factors affect the convergent rate. Due to noise in a real-world problem, we are not interested in a converged solution. We are only interested in non-converged “early solutions.” The purpose of the weighting in Fig. 9 is to slow down the convergent rate for the densely sampled views in order to maintain isotropic image resolution. However, this weighting scheme somehow causes a faster convergent rate for the sparsely sampled views. Figs. 8 and 9 show the effect of a small number of iterations. When a large iteration number is used, Fig. 8 will look like Fig. 6 and Fig. 9 will look like Fig. 7. Resolution improves as the iteration number increases.

After we replaced the angular interval from 1.2° to 2.4° in the angular range of 90° ± 60° and in the range of 270° ± 60°, a new set of images were obtained. They are Figs. 10–15, which are the counterparts of Figs. 2, 3, and 6–9, respectively.

Fig. 10.

FBP reconstruction of a simulated resolution phantom using non-uniform angular sampling data with utilization of proposed weighting function. Maximum angular gap is 2.4°.

For the cadaver data set, the reconstructed images are shown in Figs. 16–30, and the image order and layout are the same as those in Figs. 1–15. For non-uniformly sampled data, the proposed weighting function is able to improve the image uniformity and reduce the X-shaped artifacts; see Figs. 17 and 18. Fig. 18 has severe X-shape artifacts.

Fig. 16.

FBP reconstruction of a patient torso slice using uniform angular sampling data.

Fig. 30.

Weighted iterative Landweber reconstruction of a patient torso slice using non-uniform angular sampling data. Iteration number is 50. Maximum angular gap is 2.4°.

Fig. 17.

FBP reconstruction of a patient torso slice using nonuniform angular sampling data with utilization of proposed weighting function.

Fig. 18.

FBP reconstruction of a patient torso slice using nonuniform angular sampling data without utilization of proposed weighting function.

The iterative reconstructions (Figs. 19–24 and 27–30), in principle, do not need any weight functions. When the iteration number is low, as expected, the reconstructed images (Figs. 20 and 23) are blurry. When the angular sampling is not uniform, one can observe some X-shape artifacts in Fig. 23. When iteration number is low, artifacts can appear in addition to image blurring. The X-shape artifacts are not found in Fig. 20.

Fig. 19.

Iterative Landweber reconstruction of a patient torso slice using uniform angular sampling data. Iteration number is 20000.

Fig. 24.

Weighted iterative Landweber reconstruction of a patient torso slice using non-uniform angular sampling data. Iteration number is 50.

Fig. 27.

Iterative Landweber reconstruction of a patient torso slice using non-uniform angular sampling data. Iteration number is 20000. Maximum angular gap is 2.4°.

Fig. 20.

Iterative Landweber reconstruction of a patient torso slice using uniform angular sampling data. Iteration number is 50.

Fig. 23.

Iterative Landweber reconstruction of a patient torso slice using non-uniform angular sampling data. Iteration number is 50.

The weighting function can improve the balance of all projection measurements. By using (2) as the weighting function in an iterative algorithm, the X-shape artifacts are significantly reduces and the image resolution becomes more isotropic (see Fig. 24).

When an extremely high iteration number is used, the reconstructed image can contain some high frequency structural artifacts and ripples (see Figs. 4, 6, 19, and 21). Small ripples are normal in Fig. 6 and Fig. 21, because the non-uniform data set uses fewer projections (with a larger angular gap) than the uniform data set. In our non-uniform angular sampling examples, the angular gap is larger than the clinically accepted angular gap. The main objective here is to compare the images with the weighting function and without the weighting function. The objective is not to compare the non-uniformly (and sparsely) sampled study with the uniformly (and densely) sampled study.

Fig. 4.

Iterative Landweber reconstruction of a simulated resolution phantom using uniform angular sampling data. Iteration number is 20000.

Fig. 21.

Iterative Landweber reconstruction of a patient torso slice using non-uniform angular sampling data. Iteration number is 20000.

For the uniform data set, the high iteration number itself can cause artifacts and errors. This is because the projector in an iterative algorithm does not perfectly match the projection procedure in the actual data acquisition. These model-mismatch errors propagate from one iteration to another. When the iteration number is as large as 20000, the model-mismatch errors can cause many un-expected artifacts, which we do not normally see in an FBP reconstruction.

Comparing the 1.2° angular gap with the 2.4° angular gap, we make the following observations:

At a low iteration number of 50, the weighting function is able to influence the iterative algorithm and make the image spatial resolution isotropic; see Figs. 14 and 15. At a high iteration number of 20000, the image is not yet converged due to sparse angular sampling. At this high iteration number, the weighting function has hardly any effects on the iterative algorithm and the horizontal resolution appears worse than the vertical resolution; see Figs. 12 and 13. The weighted FBP reconstruction (Fig. 10) at this situation outperforms the iterative reconstructions.

Fig. 14.

Iterative Landweber reconstruction of a simulated resolution phantom using non-uniform angular sampling data. Iteration number is 50. Maximum angular gap is 2.4°.

Fig. 12.

Iterative Landweber reconstruction of a simulated resolution phantom using non-uniform angular sampling data. Iteration number is 20000. Maximum angular gap is 2.4°.

Fig. 13.

Weighted iterative Landweber reconstruction of a simulated resolution phantom using non-uniform angular sampling data. Iteration number is 20000. Maximum angular gap is 2.4°.

Finally, we slightly touch the few-view tomography problem, where the number of views is very small, say, 20, both the FBP and the iterative Landweber algorithms perform poorly if no additional information (e.g., the piecewise constant) is used; see Figs. 31 and 32.

Fig. 31.

FBP reconstruction of a patient torso slice using few-view data. The number of views is 20.

Fig. 32.

Iterative Lanweber reconstruction of a patient torso slice using few-view data. The number of views is 20. The number of iterations is 20000.

The few-view tomography problem can be tackled by using, say, total variation minimization techniques [11–15] to enforce the piecewise constant property of the image, and this is not the focus of this paper.

We observe that when the view number is extremely high, even after iteration number of 20000, the iterative reconstruction, Fig. 32, is still very blurry, while the FBP reconstruction, Fig. 31, is a lot sharper. In few-view tomography if no other constraints are used, the iterative algorithm may require an (impractical) extremely high iteration number to reach the same image resolution as the FBP algorithm can provide.

The above examples demonstrate that the FBP algorithm is more powerful than one may believe, and the iterative algorithm sometimes dose not live up to its reputation.

IV. Discussion and Conclusions

This paper shows that when a proper weighting function is introduced in the FBP algorithm, the weighted FBP algorithm can be used in the situation where the angular sampling is not uniform.

The theoretical development of iterative algorithms has been focused on their convergence and the properties of the converged solution [16]. Since the routine clinical tasks require low number of iterations, it is important to investigate the properties of the intermediate “solutions” that an iterative algorithm can provide.

The weighting scheme in an iterative algorithm has been focused on the noise model. This paper uses examples to illustrate that weighting schemes that are not based on the noise model sometimes are helpful.

The iterative algorithms are believed to be fairly versatile. In principle, no special weighting function is necessary when the imaging geometry or data sampling scheme is irregular.

However, when the number of iterations is low, the iterative algorithm can cause some anisotropic artifacts. In the nonuniform angular sampling case, if we apply the weighting function that was derived for the FBP algorithm to the iterative algorithm, the anisotropic resolution problem is significantly relieved.

At very high iterations, the result of weighted iterative algorithm is almost the same as that of the un-weighted iterative algorithm, because weighting is only effective when the iteration number is low. The weighting function does not affect the converged image.

Ideally speaking, one would like to have the true solution by using a huge iteration number. This true solution should have isotropic resolution. However, the real data are always noisy. It is impractical to seek for convergence. Only a small number of iterations are used in practice and an early “solution” is obtained to avoid too much noise corruption. We need to live with early solutions and understand them. Noise and resolution tradeoff is a classical concern in medical imaging. Large number of iterations, leading to isotropic solutions, are unreachable in practice. Moreover, due to noise, the reconstruction is often regularized by early stopping the iterations. We showed that one must be aware of the influence of the weighting scheme (regardless of it is noise weighting or geometric weighting) on the resolution and the convergence rate. Depending on the clinical task, an image can have higher resolution and be less noisy, if we allow anisotropic resolution and non-stationary noise distribution.

If the ML-EM (maximum likelihood expectation maximization) algorithm [17], [19] were to be used for image reconstruction in the situation of non-uniform angular sampling, it could incorporate the angular weighting factor w_j in the backprojector. The revised ML-EM algorithm can be expressed as

$$f_i^{(k+1)}=\frac{f_i^{(k)}}{{\displaystyle \sum _j}{w}_j{a}_{ij}}\sum _j{w}_j{a}_{ij}\frac{p_j}{{\displaystyle \sum _n}{a}_{nj}{f}_n^{(k)}}.$$ (4)

There are not many fundamental differences between an analytical algorithm and a linear iterative algorithm. The analytical algorithms and the iterative algorithms can learn from each other.

When compared with the iterative algorithm, the strengths of the FBP algorithm are:

1. Its computation is fast and it can be used in real-time imaging applications.

2. It does not have the “early solution” problems and can provide images with isotropic resolution. One of the “early solution” problems in an iterative algorithm is anisotropic resolution.

3. It does not have the problem of the propagation of model-mismatch errors which are amplified at each iteration in an iterative algorithm.

When compared with the iterative algorithm, the weaknesses of the FBP method are:

1. It is not easy to incorporate non-quadratic penalty functions.

2. It is not easy to model imaging physics.

Noise is always an issue in imaging. The iterative algorithms are able to incorporate the noise distribution model into the algorithms as weighting factors and use the stopping rule or Bayesian terms to control noise. As shown in References [7], [9] and [13], the weighted FBP algorithm can do the same by introducing windowed ramp filter for each projection ray. The noise-weighted FBP algorithm has the similar noise performance as the iterative algorithms. The comparison results were reported in [13], [18] and [19].

As shown in Figs. 31 and 32, the proposed method does not work in few-view tomography. The reason of it is the lack of sufficient angular sampling. In other words, the angular interval is too large. The iterative algorithm (without the TV-norm or the L₁-norm constraints) does not work for few-view tomography either. It is interesting to notice that the FBP method produces a sharper image than that by the iterative method in few-view tomography. In any case, we require that the maximum angular interval be small enough for the FBP and the iterative algorithms to work properly.

Fig. 11.

FBP reconstruction of a simulated resolution phantom using non-uniform angular sampling data without utilization of proposed weighting function. Maximum angular gap is 2.4°.

Fig. 22.

Weighted iterative Landweber reconstruction of a patient torso slice using non-uniform angular sampling data. Iteration number is 20000.

Fig. 25.

FBP reconstruction of a patient torso slice using nonuniform angular sampling data with utilization of proposed weighting function. Maximum angular gap is 2.4°.

Fig. 26.

FBP reconstruction of a patient torso slice using nonuniform angular sampling data without utilization of proposed weighting function. Maximum angular gap is 2.4°.

Fig. 28.

Weighted iterative Landweber reconstruction of a patient torso slice using non-uniform angular sampling data. Iteration number is 20000. Maximum angular gap is 2.4°.

Fig. 29.

Iterative Landweber reconstruction of a patient torso slice using non-uniform angular sampling data. Iteration number is 50.