Dual energy CT has the ability to give more information about the test object by reconstructing the attenuation factors under different energies. These images under different energies share identical structures but different attenuation factors. By referring to the fully sampled lowenergy image, we show that it is possible to greatly reduce the sampling rate of the highenergy image in order to lower dose. To compensate the attenuation factor difference between the two modalities, we use piecewise polynomial fitting to fit the lowenergy image to the highenergy image. During the reconstruction, the result is constrained by its distance to the fitted image, and the structural information thus can be preserved. An ASDPOCSbased optimization schedule is proposed to solve the problem, and numerical simulations are taken to verify the algorithm.
Computed tomography has become an important nondestructive detection method in medicine, industry, and security. Typically CT scans the object by a single energy to reconstruct the attenuation factors in order to evaluate the density distributions inside the test object. However, some materials’ attenuation factors are close and hard to distinguish, which brings trouble for diagnosis. Since the attenuation factors are different under different Xray energies, DECT [
In DECT, the test object is scanned under different energies while keeping the object fixed. Thus, two different images, the lowenergy image
Dose has been concerned more and more recently with the increasing public awareness of the possible risks brought about by the radiation of CT scans. One of the most efficient ways to reduce dose is to reduce the sampling number. According to the concept of compressed sensing (CS) [
Although the two images share the same structure, their attenuation factors are different under the two different energies, which leads to the grey scale difference in the reconstructed images. Similar situations can be found in multimodality imaging, where the grey scale values of the images are far different from each other. Bowsher prior has been used for MR/SPECT imaging to improve the reconstruction quality of SPECT [
It has been shown that invoking reference images in CSbased reconstruction is able to improve the reconstruction quality, but grey scale value compensation remains a challenge in DECT. For example, PICCS algorithm requires the reference image and the target image to be as close as possible, but the attenuation factor may differ widely under different energies. The ratio image method, on the other hand, requires that the changes in the grey scale values are proportional so that there are less edges in the ratio image and its gradient image is sparser. However, the change of the attenuation factor under different energies is unpredictable and the conditions required for the above methods may be violated sometimes.
Here, we propose a novel CSbased method for undersampled DECT reconstruction. The wellreconstructed lowenergy image
The method is very much motivated by one of our previous works [
The paper is organized as follows. In Section
The reconstruction formula of piecewise polynomial function constrained (PPFC) method is as follows:
Before introducing PPF, we will firstly show the least squares polynomial fitting method and some of its properties. In DECT, one of the ways to sparsify
Image approximation by polynomial fitting is to solve the following equation:
Equation (
The approximation has some good properties. Firstly, loworder polynomials have the property of smoothness, which maps similar values in
In our experiments, the order
As we have stated, loworder polynomial fitting requires that the grey values of the image distribute around only a few points. Thus, the size of the image for fitting should be small for accuracy. This can be achieved by extracting patches from the original image and applying polynomial fitting on the sub images. On the other hand, the patches should be big enough to hold the structural information with them. There should also be adequate overlapping areas between adjacent patches or the difference between patches will lead to obvious blocklike artifacts in the approximated image. In our experiments, the patch size is selected as 16 by 16, and the offset of adjacent patches is 4 by 4, which leads to an overlapping size of 12 by 12.
Figure
The PPF results on noisy phantom with different patch sizes and offsets. From the top row to the bottom row, the corresponding offsets between adjacent patches are 4, 8, and 16. From the leftmost column to the rightmost column, the corresponding patch sizes are 8, 16, and 32. The grey scale value window is
When approximating
Then putting the patch back to its original position can be achieved by the transposed matrix
In (
The transpose of
Since
The transposed operator will be of future use and we will take a look at its properties here. The PPF matrix
The formula for optimization is shown in (
Using the chain derivative rule, one can easily derive the gradient of the objective function:
The initial value for the iterations is crucial for the convergence speed. For PPFC, 10 times SART is used to estimate
The sparsity of the PPFC transform
The approximation result of PPF.
Multienergy projections are used for the experiments. The spectrum of the Xray source is generated by the Monte Carlo method. Three different spectrums are used for the experiments: a 90 kVp spectrum, a 120 kVp spectrum, and a beamhardened 160 kVp spectrum. There are two different phantoms for testing, a cylinder phantom and a realistic dental phantom. The cylinder phantom is forward projected by the 90 kVp spectrum and the 160 kVp spectrum. The energy used for the dental phantom is 90 kVp and 120 kVp. The lowenergy image
The phantoms used for simulation: (a) the cylinder phantom; the base cylinder is PMMA and the inner cylinders from the biggest to the smallest are made of bone, water, and 1%, 2%, 5%, and 10% NaI solutions and (b) the dental phantom.
The cylinder phantom is first forward projected by the 90 kVp and 160 kVp spectrums with fan beam geometry to get the high and low projections
The scan geometry for the cylinder phantom.
Parameter  Value 

Source to center distance  75 cm 
Source to detector distance  105 cm 
Total width of detector  9.6 cm 
Number of detector bins  512 
Reconstruction resolution 

Reconstruction size  6.4 cm 
The quantitative estimations of the reconstruction results of the cylinder phantom.
Method  RMSE  Iteration number  Time (s) 

Noiseless PPFC 

10  31.2 
Noiseless PICCS 

100  34.3 
Noisy PPFC 

10  31.2 
Noisy PICCS 

100  34.4 
The RMSE is calculated by comparing to the noiseless result of FBP.
The iterations numbers do not include the 10 times preiterations and are enough for convergence.
The code is implemented by C++ on a machine with Intel Core i5 processor and 3 GB memory.
The reconstruction results of cylinder phantoms. The top row is the results from noiseless projections and the bottom row is the results from noisy projections. From left to right: the first column is the fully sampled
In the results of PICCS, the cylinder made of 1% NaI solution is blurred. The reason is that in the reference image
As for the results of PPFC, all the cylinders including the 1% NaI solution cylinder are well recovered. Furthermore, the results of the noisy projections have not degraded much comparing to the noiseless results. Thus, the experiments indicate that PPFC is compensating grey scale values difference and noise stable.
The dental phantom is forward projected by the 90 kVp and 120 kVp spectrums with fan beam geometry. The lowenergy image
The scan geometry for the dental phantom.
Parameter  Value 

Source to center distance  50 cm 
Source to detector distance  80 cm 
Total width of detector  30 cm 
Number of detector bins  960 
Reconstruction resolution 

Reconstruction size  19.2 cm 
The quantitative estimations of the reconstruction results of the dental phantom.
Method  RMSE  Iteration number  Time (s) 

PPFC 

20  238 
PICCS 

200  288 
The RMSE is compared to the noiseless results of FBP.
The reconstruction results of the dental phantom. From left to right: the first one is the
The experiments on the realistic phantom show that PPFC is able to reconstruct objects with complicated structures as well as the simple objects. It also shows an advantage over PICCS on the aspect of RMSE.
In this paper, we propose a CSbased method for undersampled DECT reconstruction with piecewise polynomial function constraint. The lowenergy image is reconstructed from fully sampled projections and the highenergy image is reconstructed from severely corrupted samples with the wellreconstructed lowenergy image as the reference. The proposed piecewise polynomial fitting method has good ability to compensate for the grey scale value difference between the high and lowenergy images. Under most conditions, the target image can be well approximated by the reference image using the PPF, which ensures the sparsity of the PPFC transform. The simulation results show that our method is both accurate and stable.
The drawback of the algorithm is that the piecewise polynomial fitting is still not efficient enough. However, the fitting of each patch is independent and the algorithm can be further accelerated by parallel computation. Furthermore, the algorithm has the potential to reconstruct the decomposition coefficients images in DECT, whose values are far from the values in the lowenergy image. Applying the method to a dualeffect or dualmaterial decomposition reconstruction is of future concerns.
This work was partly supported by the grants from NNSFC 10905030 and the Beijing Natural Science Foundation (research on key techniques of medical conebeam CT reconstruction from little data based on compressed sensing theory).