We present a new method to estimate noise for a single-slice sinogram of low-dose CT based on the homogenous patches centered at a special pixel, called center point, which has the smallest variance among all sinogram pixels. The homogenous patch, composed by homogenous points, is formed by the points similar to the center point using similarity sorting, similarity decreasing searching, and variance analysis in a very large neighborhood (VLN) to avoid manual selection of parameter for similarity measures.Homogenous pixels in the VLN allow us find the largest number of samples, who have the highest similarities to the center point, for noise estimation, and the noise level can be estimated according to unbiased estimation.Experimental results show that for the simulated noisy sinograms, the method proposed in this paper can obtain satisfied noise estimation results, especially for sinograms with relatively serious noises.
With continued technology advancement and wider applications, use of computed Tomography (CT) is increasing. However radiation exposure and associated risk of cancer for patients receiving CT examination have been an increasing concern in recent years. Thus, minimizing X-ray exposure to patients has been one of the major efforts in the CT field [
A simple and cost-effective means to achieve low-dose CT applications is to lower X-ray tube current (mA) as low as achievable. However, dose reduction generally leads to an increased level of noise in the measured projection data (sinogram) and the subsequent reconstructed images.
Sinograms acquired from low-dose CT are corrupted by many factors, including Poisson noise, logarithmic transformation of scaled measurements, and prereconstruction corrections for system calibration [
One common model for noise estimation is Gaussian distribution with variance depending on the sinogram [
Another model is Poisson distribution based on quantum noise for CT which is due to the limited number of photons collected by the detector [
More accurate noise model, compound Poisson model (CPM), which takes into account both the polychromatic X-ray beam and energy integration, has been investigated in [
Based on the above models, dose-reduction simulation using synthetic-noise generators, enables ethical studies of low-dose procedures to improve the quality of reconstruction images. Dose-reduction simulation has been reported using Poisson or Gaussian noise models [
In this paper, we focus on how to estimate parameters for added signal independent Gaussian noise (SIGN) on sinogram. Although it does not coincide noise models introduced in this section, it is still a valuable model in investigating the properties of the noise for sinogram and the relations between common noise models for sinogram which will be discussed in Section
Noise estimation for a single image is an important and difficult problem in image denoising for complex structures of images and is studied in the early 90s for the last century [
Recently, noise estimation from a single image both for Gaussian and Poisson noise becomes a hot spot in computer vision [
In this paper, we insist that noise estimation should be performed in a local homogenous patch and present a method, which does not need presegment or pose complex prior. Moreover, in order to get more reliable estimated results and to improve the overestimate when noise levels are low, motivated by nonlocal means and some most recent results [
In the very large neighborhood, the homogenous patches are determined by finding similar points to the center point through similarity sorting, decreasing similarity searching and variance analysis. The noise parameters will be estimated on this homogenous patch using unbiased estimate. Since very large neighborhood provides more reliable estimation, proposed method can get satisfied noise estimation results.
The remainder of this paper is arranged as the noise models will be discussed in Section
For clinical X-ray system, its detected that X-ray intensity follows a compound Poisson distribution [
In this section, we will introduce signal-independent Gaussian noise (SIGN), Poisson noise, and signal-dependent Gaussian noise, as well as their relations and the reason for addressing SIGN.
SIGN is a common noise for imaging system. Poisson noise and signal-dependent Gaussian noise can be converted to SIGN using scale transforms which will be discussed in Section
Let the original projection data be
The photon noise is due to the limited number of photons collected by the detector [
Thus we have
Both its mean value and variance are
Gaussian distributions of ployenergetic systems were assumed based on limited theorem for high-flux levels and followed many repeated experiments in [
The most common conclusion for the relation between Poisson distribution and Gaussian distribution is that the photon count will obey Gaussian distribution for the case with large incident intensity and Poisson distribution with feeble intensity [
As a statistical method for treating nonnormally distributed or signal dependent noise, scale transformations are widely-used to stabilize the variance [
Armando Manduca et al. also indicate that the Anscombe transform can convert Poisson distribution data to data with an approximately normal distribution with a constant variance. Thus we can use
Since both Poisson noise and signal-dependent Gaussian noise can be converted to SIGN, the noise estimation for low-dose CT can start from SIGN to focus on our new method itself.
Following the SIGN model discussed in Section
Noise estimation for low-dose CT projection data is an estimation problem. Using the terms in Section
However, only one noisy observation
Suppose we have a statistical model parameterized by
An estimator is said to be unbiased if its bias is equal to zero for parameter
In theory, the sample estimators shown in (
In order to measure similarity between two different points
Generally, we can predefine a threshold
Since only samples with moderate or large size can approximate the population distribution well, we must find enough samples to estimate the noise correctly. Moreover, iid assumption requires to share statistical information among similar points of noisy sinogram. Both requirements coincide with the start point of existing methods. That is, noise estimation should be performed at smoother versions of the images to suppress the influence of the complex structures of images.
However, estimation noise using smoother versions of the images has high computation burden and overestimate noise levels [
Just as discussed in this section, the key objective is to find enough similar points for noise estimation. Unlike existing global methods, which estimate noise levels using whole images, the proposed method try to estimate noise levels in a more local way to reduce computation burden and increase flexibility for estimation.
In order to accomplish the above objectives (enough samples with a more local structure), VLN is used to find similar points (samples). Moreover, the VLN should be put in a suitable position to ensure the enough samples can be found. That is, VLN should be put in a homogenous region. Since the center of VLN can determine the position of VLN, how to put VLN in a suitable position can be converted to how to locate the center point in a homogenous region.
The noise level of low-dose CT is low. In this situation, only variance is enough for describing the local homogeneity roughly. That is, large variance relates to a square near singularities while small variance relates to a homogenous square. Therefore, by comparing variances of squares with fixed size for all pixels in sinogram, the center point is defined as the pixel with the smallest variance among all pixels.
After determining the center point, we propose a new method for noise estimation based on similarity sorting, similarity decreasing searching, and variance analysis. Its main start point is from how to avoid threshold setting in finding similar points to the center point since the threshold selection depends on the noise level which makes it a “chicken and egg” problem.
Similarity sorting provides a similarity decreasing sequence (SDS). Thus the samples must be formed by the up at the front points in the SDS since points with smaller similarities to the center point maybe the outliers for the estimation.
In order to find the largest number samples with the highest similarities in the VLN, similarity decreasing searching combining variance analysis is used. That is, the samples are formed by adding 50 points each time according to the order of the SDS. Thus it ensures that each addition adds the points with the highest similarities in residual SDS.
Moreover, in order to find the largest number samples used in estimation, we must find when the outliers are added. It can be achieved by variance analysis for each addition. When the variance for an addition becomes large suddenly, it means some outliers are added. Therefore, the real samples are the last before this addition. The detailed algorithm will be introduced in Section
In this section, we will give the framework for noise estimation. Just as shown in Figure Find the center point: It is the key step to locate the VLN, which should be put in a homogenous region. Thus the center point is defined as the point with the smallest local variance in all sinogram pixels. Determine the homogenous patch: The homogenous patch is composed by similar points to the center point (samples), and these similar points are searched using similarity sorting, similarity decreasing searching and variance analysis. Estimate noise: The parameters are estimated using unbiased estimator by the samples on the homogenous patch.
The flow chart for the proposed method.
By these steps, the largest number of samples with the highest similarities can be obtained. These samples form a very large homogenous patch for noise estimation. Thus even using simple unbiased estimate, satisfied results can be obtained. The details for these three steps will be given in the remainder of this section.
The center pixel should locate at the center of a homogenous region. However, in noise, finding a large
Motivated by [
Since the variance can describe how far the numbers lie from the mean, small variances will relate to homogenous squares while large variances will relate to squares near singularities. Thus the center point is chosen as the pixel with the smallest variance in a
By this way, we can find a point centered at a homogenous square and then extend it to an irregular homogenous patch.
After finding the center point
Each similar point is found by computing the similarity between itself and
The main advantage for this scheme is that it avoids threshold setting in finding similar points. Since the threshold should be set according to different noise level, noise estimation and threshold setting become a “chickens and eggs” question.
The most straight motivation for the proposed method is that the small variance indicates only the homogenous samples, while the large variance indicates not only the homogenous samples but also outliers. Thus the variances can be considered as a sign for outliers. It reminds us that if we sort the points according to the their similarities to
Thus we can add samples according to the fact that samples with bigger similarities will be added earlier, and compute the variance after each addition. When the added variance become big suddenly, it means that there are some outliers that are mixed in estimation. Thus the real noise level is estimated using the last before this addition.
In summary, the steps for finding similar points are as follows. Compute the similarity between the center point Sort the similarities to form a similarity descending sequence (SDS). Add 50 samples each time to form collections of samples and then estimate the standard deviations (SDs) of the collection of samples to form an SDs sequence. Compute the variance difference using the The estimated variance of the noise is
In this section, we will compare our method to some well-known noise estimation methods. Just as discussed in Section
Five test images are used in this paper: a thorax phantom acquired from a GE HiSpeed multislice CT scanner (see Figure
Test images. (a) Thorax phantom data acquired at 120 kVp, 150 mA. (b) A phantom data produced by Matlab. (c) Modified Shepp-Logan head phantom whose size is
For Figure
The relative projection data for the test images.
The scanning parameters for Figures
In addition, two projection data for Figures
The SIG noises are added on the projection data in Figure
Noise estimation is not a trivial task in image processing because of the complex structures for images. Some researchers suggest that the noise levels should be estimated by filtering images previously to suppress the image structures [
This method suppresses image structures using wavelets [
This method is proposed in [
In this section, three methods, proposed method (PM), wavelets, and Fast estimation (FE), are compared. In order to compare these three methods on a fair stage, the parameters used in these methods are fixed for all noise levels and all test images. According to this standard, the method proposed recently in [
The wavelet used for wavelets is Symlets with support 4. In summary, the parameters for PM are the squares used for finding the center point which are
Firstly, in order to show the role of variance analysis for the SDs sequence, we will use two Figures, Figures
Variance analysis to estimate the noise level.
Figure | Figure | |||||||||||
Real SD | 5 | 10 | 15 | 5 | 10 | 15 | ||||||
NO. | ESD | Diff | ESD | Diff | ESD | Diff | ESD | Diff | ESD | Diff | ESD | Diff |
21 | 4.93 | 0.26 | 9.82 | 1.07 | 14.67 | 0.61 | 4.95 | 0.52 | 9.65 | −0.26 | 14.42 | 3.75 |
22 | 4.94 | 0.10 | 9.86 | 0.78 | 14.78 | 3.00 | 4.96 | 0.15 | 9.69 | 0.90 | 14.45 | 0.95 |
23 | 4.91 | 9.82 | 14.74 | 4.98 | 0.19 | 9.75 | 1.20 | 14.50 | 1.53 | |||
24 | 4.88 | 9.75 | 14.62 | 4.98 | 0.19 | 9.75 | 1.20 | 14.52 | 0.34 | |||
25 | 0.22 | 1.19 | 2.63 | 5.13 | 0.80 | 9.82 | 0.53 | 14.68 | 4.80 | |||
26 | 6.61 | 10.76 | 15.34 | 5.20 | 0.69 | 9.94 | 2.42 | 14.77 | 2.42 | |||
27 | 5.25 | 0.51 | 10.02 | 1.52 | 14.87 | 3.17 | ||||||
28 | 0.54 | 0.42 | 14.89 | 0.53 | ||||||||
29 | 5.94 | 10.35 | 2.73 | |||||||||
30 | 15.32 |
According to the steps in Subsection
The final estimated SDs can be get through four steps (see Section
Estimated noise levels for the phantoms.
Figure | (a) | (b) | (c) | ||||||
Real SD | PM | W [ | FE [ | PM | W [ | FE [ | PM | W [ | FE [ |
5 | 5.57* | 5.31 | 4.90 | 4.91 | 6.11* | 5.05 | 4.98 | 5.40* | |
6 | 6.51* | 6.30 | 5.88 | 5.89 | 7.10* | 6.03 | 5.93 | 6.39* | |
7 | 7.45* | 7.28 | 6.86 | 6.87 | 8.10* | 6.89 | 7.38* | ||
8 | 8.40* | 8.27 | 7.84 | 7.85 | 9.10* | 7.85 | 8.38* | ||
9 | 9.37* | 9.26 | 8.82 | 8.83 | 10.08* | 9.02 | 8.82 | 9.37* | |
10 | 10.35* | 9.77 | 10.25 | 9.81 | 9.81 | 11.08* | 10.01 | 9.79 | 10.37* |
11 | 11.33* | 10.75 | 11.24 | 10.79 | 10.80 | 12.07* | 10.76 | 11.37* | |
12 | 12.26 | 11.72 | 12.23 | 11.77 | 11.79 | 13.07* | 11.75 | 12.37* | |
13 | 13.24 | 12.67 | 13.22 | 12.75 | 12.77 | 14.06* | 12.73 | 13.37* | |
14 | 14.24 | 13.64* | 14.21 | 13.73 | 13.75 | 15.06* | 13.69 | 14.37* | |
15 | 15.41 | 14.60 | 14.71 | 14.74 | 16.06* | 14.69 | 15.38* |
Estimated noise levels for the real projection data.
Figure | (c) | (d) | ||||
Real SD | PM | W [ | FE [ | PM | W [ | FE [ |
5 | 5.30 | 6.12* | 5.07 | 5.83* | ||
6 | 6.23 | 6.18 | 7.08* | 6.21 | 6.80* | |
7 | 7.17 | 7.18 | 8.04* | 7.11 | 7.77* | |
8 | 8.12 | 8.16 | 9.01* | 8.06 | 8.75* | |
9 | 9.14 | 10.00* | 9.04 | 8.99 | 9.73* | |
10 | 10.14 | 10.97* | 10.00 | 9.99 | 10.71* | |
11 | 11.25 | 11.95* | 11.09 | 11.70* | ||
12 | 12.15 | 12.12 | 12.93* | 12.08 | 11.95 | 12.69* |
13 | 13.10 | 13.11 | 13.92* | 12.93 | 13.68* | |
14 | 14.11 | 14.91* | 13.91 | 14.67* | ||
15 | 15.09 | 15.11 | 15.90* | 14.92 | 15.66* |
Table
From Table
Table
From Table
It should be indicated that the estimated results using “PM” on (a) are not satisfied especially in low noises. It reminds us, that there are also future works which can be done for improving the estimated results for the proposed method which will be discussed in Section
In this paper, we propose a new method to estimate noise for sinograms of low-dose CT. The proposed method can obtain estimated results both for phantoms and real projection data, especially in relatively serious noises, which demonstrate its potential for noise estimation of sinograms of low-dose CT.
Based on the similarity sorting and variance analysis in a very large neighborhood whose scale is
In addition, avoiding convolution for suppressing image structures and relatively homogenous local structure makes the proposed method also be easily generalized to the more complex noises, such as, Possion noise and Gaussian compound noise. Thus the proposed method is also a promising method for real sinograms of low-dose CT.
Although this paper proposes a new powerful method for simulated sinogram noise estimation, it may be improved as follows. How to find a center point in a large homogenous patch to ensure that there are enough points to obtain reliable estimation. We try to use multiresolution method to solve this problem. How to determine the number of the similar points according to the size of the samples and farther variance analysis. How to generalize the framework to more complex noise estimation, such as Poisson noise or Poisson and Gaussian compound noise.
This paper is supported by the National Natural Science Foundation of China (nos. 60873102, 60873264, and 61070214), National Key Basic Research Program Project of China (nos. 2010CB732501, 2011CB302801/2011CB302802), China Postdoctoral Science Foundation of China (no. 20070410843), and Open Foundation of Visual Computing and Virtual Reality Key Laboratory Of Sichuan Province (no. J2010N03).