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A novel CT reconstruction model is proposed, and the reconstruction is completed by this kernel-based method. The reconstruction kernel can be obtained by combining the approximate inverse method with the FDK algorithm. The computation of the kernel is moderate, and the reconstruction results can be improved by introducing the compact support version of the kernel. The efficiency and the accuracy are shown in the numerical experiments.

In the last years, the approximate inverse method [

At present, image reconstruction from X-ray cone-beam projections collected along a single circular source trajectory is commonly done with the FDK reconstruction method despite of the drawback that the FDK method is inaccurate [

As the integral operators in most tomography problems are compact operators acting on suitable infinite dimensional Hilbert spaces, their inverse operators are not continuous, which would amplify the unavoidable data errors. When a reconstruction work is processing, the demand for highest possible accuracy and the necessary damping of the influence of the unavoidable data errors should be balanced [

In this paper, a novel CT reconstruction model is proposed based on the approximate inverse where the kernel of the FDK method is derived and is used to complete the reconstruction. With the classical FDK method, the coordinate of the reconstruction point is first transformed from the globe Cartesian coordinate system to the detector coordinate system. It means that the reconstruction point is combined with the measured projection directly, and, furthermore, the invariance and symmetries of the coordinate system are preserved. Then the reconstruction kernel of the FDK method can be derived by defining some operators while the simplicity is preserved. In some ways, it presents a modification for the FDK method, and it is also a novel realization of the approximate inverse method. With the FDK kernel, the reconstruction can be completed efficiently. However, imposed ring artifacts arise around the area where the density is high.

Ring artifacts mainly arise in a third-generation CT system. They are caused by errors in a single or multiple channels over an extended range of views. It can be found that even a little projection errors can cause perceivable ring artifacts in the image. An error in an isolated view is mapped to a streak by the backprojection process. If the same error is persistent over a range of views, the streaks are blended, and an arc is generated [

From the figure of the FDK kernel, it can be observed that many minute oscillations exist at the edge of the kernel, and the FDK kernel has infinite support set. In consideration of the approximate inverse method, the oscillations present the contribution errors in multiple channels. As the shift-invariant property of FDK kernel is utilized in the back-projection process, and when the projections in the whole circle are considered, these errors generate the ring artifacts. Thus, in order to eliminate the ring artifacts, we update the reconstruction model by truncating the kernel with proper radiuses to make the kernel compact. For a given radius, the values of the points far from the center are set as zero, and thus the errors are set as zero. Reconstruction results show that the compact support FDK kernel is advantageous to reduce the ring artifacts and enhance the edge clarity.

The remainder of this article is organized as follows. In Section

In this section, the approximate inverse method is briefly introduced. For more details, we refer to [

Let

The approximation of

With the auxiliary function

In [

In circular tomography, let the condition of Tuy-Kirillove be fulfilled, the inversion of the 3D Radon transform is given as

The integral operator

Let

With operators above, the inverse formula of the 3D Radon transform shown as (

As we know that the circular trajectory could not fulfill the sufficient condition, only approximate results are possible. At present, FDK reconstruction method is the most popular method for circular reconstruction. In next section, we concentrate on deriving the FDK kernel from the well-known FDK method and then proposed the compact support FDK kernel reconstruction model.

The 3D FDK reconstruction method is a heuristic development of the 2D Radon transform [

Let

Let

Define three new operators:

According to the auxiliary problem

Thus for a suitable mollifier

Then according to the approximate inverse method, the projections can be processed on the FDK kernel for reconstruction. Then we obtain the FDK kernel reconstruction model as:

The FDK kernel is computed in numerical experiments, and the shape of the original FDK kernel is shown in Figure

Kernel of the FDK method with the CT geometry in Table

With the FDK kernel reconstruction model, we reconstruct the Shepp-Logan phantom and the Turbell clock phantom in numerical experiments. Reconstruction results are shown in Figures

Original phantom and reconstruction results of the middle plane. (a) is the original phantom of the midplane. (b) is the reconstruction result with the Radon kernel. (c) is the result with the original FDK kernel model. Results with the compact FDK kernel model where the radius

As we all know, the FDK reconstruction method is a filtered back-projection reconstruction method. From the formula of the kernel as (

In our study, we employ a simple but useful method to reduce the ring artifacts. In order to eliminate the oscillations and make the kernel compact, the kernel is truncated with different radius. A truncation operator

Then projections are processed on the compact kernel for reconstruction according to the approximate inverse method and then we can denote our compact reconstruction model as follows:

Results with the compact kernel are also shown in Figures

As the kernel is a function of the reconstruction point

From the derivation of the kernel, we can see that the kernel is expressed with the coordinate system on the detector plane. And as coordinate system on the detector plane is shift invariant, the invariance and symmetries of the chose reconstruction relation operator are all directly inherited by the kernel. And with the translational and rotational invariant Gaussian function as the mollifier, the FDK kernel has the invariance below.

For the fixing point

As the position

With the above two results, the kernel of an arbitrary point

Then for the reconstruction work, we are needed to compute the kernel at the

In the experiments, the original FDK kernel reconstruction model and the compact FDK kernel reconstruction model are employed for comparison. We choose the well-known Shepp-Logan phantom and the Turbell clock phantom [

Parameters of Measurement.

Measurement parameters | |

Detector | |

Projections | 180 |

Source-Detector | ~100 cm |

Source-Object | ~87.3 cm |

Reconstruction parameters | |

Reconstruction grid | |

0.00165 cm |

Figure

Reconstruction results with compact FDK kernel model are shown in Figure

Profiles along the midline of the midplane where the dashed and solid curves represent the true and reconstructed values with the compact FDK kernel with

Original phantom and reconstruction results of the plane parallel and 20 mm apart from the midplane. (a) is the original phantom of the plane. (b) is the reconstruction result with the Radon kernel. (c) is the result with the original FDK kernel model. Results with the compact FDK kernel model where the radius

Original Turbell clock and reconstruction results. (a) is the original Turbell clock. (b) is the reconstruction result with the Radon kernel. (c) is the result with the original FDK kernel model. Results with the compact FDK kernel model where the radius

Figure

Figure

In this paper, a novel CT reconstruction model is proposed based on the approximate inverse where the kernel of the FDK method is derived and is used to complete the reconstruction. In order to eliminate the imposed ring artifacts, the kernel is truncated with proper radius. Reconstruction results show that the compact support FDK kernel reconstruction model can suppress the ring artifacts. The proposed reconstruction model preserves the simplicity of the FDK reconstruction method and also provides an alternative to realize the approximate inverse method for circular trajectory. And when the kernel of an algorithm is modified, the corresponding reconstruction formula is also modified accordingly. And this gives us another way to improve the existing reconstruction methods.

This work is supported in part by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program no. 11JK0504).