To investigate the local micro/nanoscale region in a large scale sample, an image reconstruction method for nanometer computed tomography (nanoCT) was proposed in this paper. In the algorithm, wavelets were used to localize the filteredbackprojection (FBP) algorithm because of its spacefrequency localization property. After the implementation of the algorithm, two simulation local reconstruction experiments were performed to confirm its effectiveness. Three evaluation criteria were used in the experiments to judge the quality of the reconstructed images. The experimental results showed that the algorithm proposed in this paper performed best because (1) the quality of its results had improved 20%–30% compared to the results of FBP and 10%–30% compared to the results of another wavelet algorithm; (2) the new algorithm was stable under different circumstances. Besides, an actual reconstruction experiment was performed using real projection data that had been collected in a CT experiment. Twodimensional (2D) and threedimensional (3D) images of the sample were reconstructed. The microstructure of the sample could be clearly observed in the reconstructed images. Since much attention has been directed towards the nanoCT technique to investigate the microstructure of materials, this new waveletbased local tomography algorithm could be considered as a meaningful effort.
With the rapid development of material science, advanced experiment and investigation technique have been developed and used to investigate the micro/nanoscale morphology and mechanical properties of samples. It is well known that the mechanical properties are determined by the material itself and its internal microstructure. It has been found that the mechanical properties of materials changed greatly as their internal structure varies from macroscale to micro/nanoscale. So it is very important to find the technique that is capable of investigating the micro/nanoscale structure and mechanical properties. The major investigation techniques included electron microscope and ultrasonic inspection. But considering that the structure to be investigated was extremely tiny and the sample was sometimes placed in external field (including stress, heat and electric), those techniques couldnot complete the investigation under these circumstance. But the CT technique was a good choice because of its nondestructive and noncontacted characteristics. More importantly, 2D and 3D images of the internal microstructure of materials could be obtained by CT and the images could achieve micro/nanoscale spatial resolution. Taking the nanoCT device in Beijing Synchrotron Radiation Facility as an example, its spatial resolution has reached 30 nm/pixel.
However, the number of pixels is limited by the size of the Xray acquisition lens. If the spatial resolution was improved, the view field decreased. Once again taking the nanoCT device as an example, there were 1024 pixels along the horizontal direction. So the transverse size of the view field was about 30 um. If the size of a sample exceeded that number, the sample could not be correctly reconstructed using conventional image reconstruction algorithm (i.e., the FBP). On the other hand, fabricating tiny samples was usually a challenge and the samples may be too small to represent the real structure of the material. Thus, how to complete exact tomography under the condition of relative large sample and small view field has become a problem that needs urgent solution. The problem could also be described as local reconstruction in which only the projections of a small local region were acquired and used. The corresponding image method was referred to as local reconstruction algorithm. During the past years, several different local reconstruction algorithms have been proposed by Smith [
Wavelets have received much attention in the past few years. As wavelets are designed to have many good characteristics, it is possible to apply wavelet theory to image reconstruction field. The research on that aspect has been developed rapidly. Some local tomography algorithms have been proposed in recent years [
In this paper, a new waveletbased reconstruction algorithm was proposed and implemented. At first, the reason why the FBP could not complete local reconstruction was discussed. Then a new local reconstruction algorithm based on wavelet was proposed and implemented. The key to this algorithm is that many wavelets have the spacefrequency localization property. The difference between this new method and another wavelet reconstruction algorithm was also discussed. Next, simulation experiments were performed to test the algorithm. The results of the new algorithm were presented in this paper, together with the results of other algorithms. Besides, the new algorithm was applied to an actual CT experiment and the reconstructed images of local area were also presented.
The most commonly used image reconstruction algorithm is the FBP. However, it is not competent to reconstruct a local region with only local projection data. The reason is that the ideal ramp filtering function in the algorithm is truncated by a spectral window in frequency domain and the windowed ramp filter is unbounded in space domain. So the key point of local reconstruction is to find a window function which ensures that the windowed ramp filter is bounded both in frequency and space domain. A possible alternative is the wavelet due to its timefrequency localization property. So the waveletbased local reconstruction algorithm was proposed. More details about it will be given below.
At the beginning, the terminology and definitions required in the subsequent discussions will be briefly introduced. In this paper, the following notations are used. The
The goal of CT is to reconstruct an image from a set of its lineintegral projections. In the 2D case, the projection of
Conversely, function
The filtering step can be written as
And the backprojection step is
Because the ideal ramp filter
Generally, the Hamming window is chosen to be
The truncated ramp filter by Hamming window. (a) Space domain. (b) Frequency domain.
Considering the equivalent relation that
For the reasons outlined above, the functions which are compactly supported and have several vanishing moments are wanted. This second condition will ensure that the functions remain compactly supported after the differential and Hilbert transform process and will ensure that a reconstruction from local data can be accomplished. Wavelets are generally designed with many vanishing moments. So they can be used as the window function in local reconstruction algorithm. In this section, a waveletbased reconstruction algorithm is presented. The key point of the algorithm is that each projection is filtered by four different wavelet filters, respectively. The filtering formula comes from (
Many kinds of wavelet have been designed by researchers so far. For example, Daubechies, Coiflets, Symlets are commonly used in signal and image processing. Most kinds of wavelet come from a corresponding unique function called the scaling function (see Figure
An example of wavelets. (a) Coiflets. (b) Scaling function of Coiflets.
As wavelet and scaling functions are both compactly supported and have many vanishing moments, the functions
Besides, the filtering process in (
It is obvious that the filter at a certain angle is the multiplication of the ideal ramp filter and the slice of the Fourier transform of the 2D wavelet at the same angle:
These angledepended filters are simplified into
Spectrogram of (a) ramped scaling function and (b) ramped wavelet function. In comparison, the ideal ramp filter was also curved in red line.
In the following, the new waveletbased reconstruction algorithm is summarized in four steps.
For each projection data at angle
The results that have been obtained in step (I) are filtered as the formulas in (
Filtered projections are back projected using (
The four coefficients are composed into the reconstruction image by 2D inverse wavelet transform.
However, it’s worth mentioning that another wavelet algorithm has been proposed in [
It can be derived that (
The projection of the SheppLogan head phantom at
In this section, two different phantom images were introduced to the local reconstruction experiment. The reconstruction algorithms included FBP and wavelet reconstruction algorithm proposed in this paper and in [
The model images used in the experiment were the stacked particles phantom and the SheppLogan head phantom (see Figure
Two phantom images. (a) Stacked particles phantom. (b) SheppLogan head phantom.
In the following were three criteria to objectively evaluate the differences between original and reconstructed images.
The reconstructed images of the local area in stacked particles phantom were shown in Figure
Criteria values for the local reconstruction of the stacked particles phantom.
Algorithm 

Relative change of 

Relative change of 

Relative change of 

FBP  0.6446  —  0.7782  —  0.7896  — 
Method in [ 
0.4347 

0.5295 

0.9047 

Method in this paper  0.3918 

0.4749 

0.9222 

Criteria values for the local reconstruction of the SheppLogan head phantom.
Algorithm 

Relative change of 

Relative change of 

Relative change of 

FBP  1.8632  —  2.6380  —  0.7095  — 
Method in [ 
1.7999 

2.5667 

0.8116 

Method in this paper  1.5042 

1.8156 

0.9796 

Reconstructed images of the local area in stacked particles phantom. (a) Local area in the phantom. (b) Local reconstruction using FBP. (c) Using wavelet method in [
Reconstructed images of the local area in SheppLogan head phantom. (a) Local area in the phantom. (b) Local reconstruction using FBP. (c) Using wavelet method in [
From the figures and tables, the following could be found.
In previous section, the effectiveness of this new wavelet method was validated by two phantom images. In this section the method was applied to practical data collected in a CT experiment. Figure
(a) Reconstruction by FBP using global data. (b) Reconstruction by FBP using local data. (c) Reconstruction by the new algorithm using local data. (d) and (e) were the amplifications of (b) and (c), respectively.
Based on the properties of wavelet, a local reconstruction algorithm has been proposed and implemented. It has been observed that for some wavelet bases with many vanishing moments, the scaling and wavelet functions have essentially the same support after differentiation and Hilbert transform. According to this fact, a local reconstruction scheme has been developed to reconstruct a local region of a crosssection of a sample with essentially local data. Some experiments have been performed and the results confirmed the effectiveness of this new algorithm. Because this algorithm is able to reconstruct local small region in a large sample that exceeds the view field, it may contribute to make more kinds of materials appropriate for CT investigation whose spatial resolution can achieve micro/nanoscale. However, there are still some unsolved problems. For example, when using different wavelet functions, the results of the algorithm were different. But how to select a suitable wavelet function for a certain situation is still no solution.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was supported by the National Nature Science Foundation of China under Contract Nos. 11272305, 11172290, 10902108, the National Basic Research Program of China (973 Program, Grant No. 2012CB937504). The authors greatly acknowledge Jian Fang, Yu Xiao and Wenchao Liu at USTC and Honglan Xie, Biao Deng and Yanan Fu at SSRF for their valuable contribution to this work.