1. Introduction

 Computed Tomography(CT) is an important and most common modality in medical imaging. In CT examinations there is trade off between radiation dose and image quality. If radiation dose is decreased, the noise will unavoidably increase degrading the diagnostic value of the CT image and if the radiation dose is increased, the associated risk of cancer also increases especially in paediatric applications.^[1] The main source of noise in CT is quantum noise. It results from statistical fluctuations of X-ray quanta reaching the detector. X-ray quantum noise is originated from a signal-dependent Poisson distributed noise source. The variance of the Poisson noise increases linearly with signal amplitude, i.e., local X-ray exposure. The observed quantum noise decreases with increasing X-ray exposure. The goals of improving the image quality of medical X-ray image are to remove quantum noise from low dose X-ray images to reduce X-ray exposure to patients and to enhance the edges in diagnostic regions of the images to discern them clearly. Many different approaches for noise suppression in CT images have been investigated, for example Trilateral Filter and Bilateral Filter(BF).^[2,3,4] The key idea of these filters is that for a pixel to influence another pixel, it should not only occupy a nearby location but also have a similar value. But, the performances of these filters depend on the parameters. It is hard to get an optimal parameter. Furthermore, several filtering methods for noise reduction in CT images have been proposed. In those filters, the Guided filter generates the filtering output by considering the content of a guidance image. Guided filter can perform as an edge preserving smoothing operator like the BF, but has better result near the edges. But, the output depends on the threshold value to differentiate between an edge and flat area.^[5,6]

 The hybrid filter for removing quantum noise from images was proposed. The proposed filter consists of the two types of BF, a Neural Edge Enhancer(NEE), and a Neural Network(NN). The BF should not only occupy a nearby location but also have a similar value. The BF takes into account the difference in value with the neighbors to preserve edges while smoothing. Signal-dependent Gaussian noise as a model of quantum noise was used. The generated noisy image has a data like impulse noise when a standard deviation of quantum noise generator is large. In case of a data like impulse noise, the BF may need to mollify the input image before use. The median-filtered version of BF is used to mollify these kinds of impulse data. NEE is used for enhancing the desired edges clearly from noisy images. The NN in a proposed filter acts like a fusion operator and attempts to construct an enhanced output image by combining the information from several sources. The fundamental superiority of the proposed filter over other filters is that it efficiently removes quantum noise from medical images while successfully preserving edges and fine details in the original image. The image quality measures like the root mean square error(RMSE) and improvement in signal to noise ratio(ISNR) are used for performance evaluation when having a target image. The measures like a mean to standard deviation ratio(MSR) and contrast to noise ratio(CNR) are used for performance evaluation in case of not having a target image. Based on these all image quality measures, the proposed filter shows better results than the BF and Guided filter. Specially, the proposed filter shows best results in severe noise environments.

2. Proposed Hybrid Filter

2.1. Bilateral filter

 The strength of Gaussian filter influence depends only on the spatial distance. As a result, image edges are blurred because pixels across discontinuities are averaged together. The BF is also defined as a weighted average of nearby pixels. The difference is that the BF takes into account the difference in value with the neighbors to preserve edges while smoothing. The BF is defined by:

Where normalization factor W_p ensures pixel weights sum to 1.0. Equation (1) is a normalized weighted average where G_σS is a spatial Gaussian weighting, G_σr is a range Gaussian weighting. As the range parameter σ_r increases, the BF gradually approximates Gaussian convolution more closely because the range Gaussian G_σr widens and flattens. Increasing the spatial parameter σ_s smooths larger features.^[3] Quantum noise originates from a signal-dependent, Poisson-distributed noise source. The variance of the Poisson noise increases linearly with the signal amplitude. The Poisson-distributed noise can be approximated by a Gaussian when the number of quanta is relatively large. So, the signal-dependent Gaussian noise as a model of quantum noise was used. Assuming that noise is quantum noise, a noisy image can be represented by

 where f (x,y) is a noiseless image, N(σ) is white Gaussian noise when its standard deviation σ is , and K_N is a parameter determining the amount of noise.^[7] The standard deviation of quantum noise generator is large when the gray level of input image is high. So, the generated noisy image has a data like impulse noise. To mollify this kind of noisy images, compute the range Gaussian weights on a median-filtered version of the image. If M describes median filtering, this gives:

2.2. Neural Edge Enhancer

 A Neural Edge Enhancer(NEE) for enhancing the desired edges clearly from noisy images was proposed by Dr. Suzuki.^[7] The NEE consists of a modified multilayer NN, which can directly handle input gray levels and output edge magnitudes. The pixel values in an input region R_S are normalized and then input to the NEE. Although the most common use of an NN is a classifier that determines whether a certain pixel belongs to the class, such as an edge or a background, the output of the NEE is not a class, but the estimate for the edge magnitude, represented by

 Where G_M is a normalization factor and NN(∙) is the output of the modified multilayer NN. The multilayer NN as a nonlinear kernel is designed by training such that the input images are converted to a map of the desired edge magnitudes. The NEE is trained with a set of noisy input images and the teaching images, including the desired edge magnitudes, by adjusting the weights between layers. The error to be minimized by training is defined by

where p is a training pixel number,  is the pth training pixel in the teaching images,  is the pth training pixel in the output images, and P is the number of training pixels. The number of units in the input and hidden layers of the NN are 25 and 20. The NEE is trained by the modified back propagation algorithm.^[8] After training, the NEE will output the desired edge magnitudes.

2.3. Proposed Hybrid Filter

 Figure 1 shows the structure of the proposed hybrid filter. The proposed filter consists of the BF, median version BF, NEE, and NN. The local noise level estimate is needed to get an optimal range parameter σ_r for satisfying result in BF. But, it is hard to get it. In this paper, 5 different range parameters were used. So, each BF has 5 different kernels. The NN utilizes the informations from the two types of BF, NEE and the noisy input image to compute the output of the system, which is equal to the restored value of the noisy input pixel. The number of units in the input and hidden layers of the NN are 12 and 20. The internal parameters of NN are iteratively optimized so that its output converges to the output of the ideal noise filter which completely removes the noise from its input image. This is represented by the original training image. The training is easily accomplished by using simple artificial images that can be generated in a computer. The NN is trained by the back propagation algorithm.

Figure 1. Block diagram of the proposed hybrid filter.

 Recently, in the field of signal processing, nonlinear filters based on multilayer NN have been studied.^[9-11] A multilayer NN is employed as a convolution kernel. The NN can acquire the function of various linear and nonlinear filters through training.

 I developed a hybrid filter based on a multilayer NN. The NN in a proposed filter consists of a multilayer NN in which the activation functions of the units in the input, hidden, and output layers are an identity function, a sigmoid function, and a linear function. I adopted a linear function instead of an ordinarily used sigmoid one as the activation function of the unit in the output layer because the characteristics of the NN become better in the applications to continuous mapping issues such as image processing.^[12] The inputs to the NN are a noisy pixel and the outputs of BF, median version BF, and NEE. The architecture of the NN is shown in Figure. 2. The output of the NN is represented by

Figure 2. Architecture of the neural network in proposed filter.

 where x and y are the indices of spatial coordinates, I_x,y denotes the input vector, NN(I _x,y) is the output of the multilayer NN, and G_M is a normalization factor. The error to be minimized by training is defined as

 where p is a pattern number,  is a pth pattern in the teaching image, f^p is a pth pattern in the output image, and P is the number of patterns. By using the back propagation algorithm, it is expected that the NN would have the function to convert the input image to the desired teaching image, by presenting the filtered outputs of the noisy input image together with the teaching noiseless image, details of which are clear, and that the NN will be able to remove quantum noise from images while preserving the edge and image details.

 The NN acts like a fusion operator and attempts to construct an enhanced output image by combining several informations.

3. Experimental Results

The proposed hybrid filter has been applied to large number of medical images for performance evaluation and comparison with other filters. 

The performances of the noise reduction methods were evaluated. In order to quantify how much noise is suppressed, there are experiments in the two kinds of environments : in cases of having target images and not having target images. The RMSE and ISNR are computed when having target images. The input images are low dose images and the target(teaching) images are high dose images to compute the RMSE and ISNR. The low dose images were synthesized from the high dose images by the quantum noise generator. The output images of the proposed filter, BF and Guided filter under the K_N=1.2 are shown in Figure. 3. In Figure. 3, there are some noises in the results of Guided filter. The noises are reduced effectively in the outputs of BF and proposed filter. But, there are little bit noises in the BF. The edges become sharper in proposed filter. In this paper, I didn't insert the figures of K_N=3.0 because of limited pages.

Figure 3. The output images of BF, Guided filter, and the proposed filter on the head, abdomen and phantom images. Low dose images (1st row), BF (2nd row), Guided filter (3rd row), and proposed filter (4th row).

 The image quality is evaluated quantitatively by the RMSE and ISNR. Table 1 and 2 show the RMSE and ISNR values of the proposed filter and other filters under different noise environments. The RMSE and ISNR of the proposed filter are better than those of the BF and Guided filter. Specially, the proposed filter shows the best results in severe noise environment( K_N=3.0) by visual way and image quality measures than other filters.

Table 1. RMSE and ISNR values for different filters in K_N=1.2.

Table 2. RMSE and ISNR values for different filters in K_N=3.0.

 All these filters are applied to real low dose images in the second experiment. There are no target images(high dose images) in these experiments. The input images are real low dose images. So, the mean to standard deviation ratio(MSR) and contrast to noise ratio(CNR) are used to evaluate the performance of the filters. Increase in MSR and CNR is preferable. The MSR in a desired region of interest (DROI) is computed as

where μ_d and σ_d are the mean and the standard deviation computed in the DROI. The CNR is defined as 

 where  μ_u and σ_u are the mean and the standard deviation computed in an undesired region of interest(UROI) such as a window or background. Both MSR and CNR measurements are proportional to the medical image quality.^[13]

The real low dose image and output images of filters are shown in Figure 4. Real low dose image on the abdomen of a mouse is used. There are some noises in the result of Guided filter. The noises are reduced effectively in the outputs of BF and a proposed filter. But, the proposed filter is a little bit better than the BF with respect to noise reduction. The edges become sharper and the fine details are more preserved in proposed filter. Table 3 shows the MSR and CNR values of the proposed filter and other filters. The proposed filter shows the best results than other filters with respect to the MSR and CNR values.

Figure 4. The real low dose image on the abdomen of mouse and output images of filters. (a) Low dose image, (b) BF, (c) Guided filter, (d) proposed filter.

Table 3. MSR and CNR values for different filters and low dose image.

4. Conclusion

 In this paper, the hybrid filter for removing quantum noise from CT images was proposed. The fundamental superiority of the proposed filter over other filters is that it efficiently removes quantum noise from medical images while successfully preserving edges and fine details in the original image. The proposed filter is combined by two types of BF, a NEE, and a NN. The NN acts like a fusion operator and attempts to construct an enhanced output image by combining the informations from several sources. The image quality measures like RMSE and ISNR are used for performance evaluation when having a target image. The MSR and CNR measures are used in case of not having a target image. It is seen that visually the proposed filter has better results than other filters. Based on these image quality measures, the proposed filter shows better results than BF and guided filter. Specially, the proposed filter shows best results in severe noise environment by visual way and these image quality measures. Our future research needs to perform the mathmatical analysis on the statistical properties of the quantum noise.