International Journal of Biomedical Imaging

Volume 2017, Article ID 8126019, 9 pages

https://doi.org/10.1155/2017/8126019

## Narrow-Energy-Width CT Based on Multivoltage X-Ray Image Decomposition

Shanxi Key Laboratory of Signal Capturing & Processing, North University of China, Taiyuan 030051, China

Correspondence should be addressed to Ping Chen; moc.361@2190cp

Received 29 June 2017; Accepted 17 October 2017; Published 7 November 2017

Academic Editor: Jyh-Cheng Chen

Copyright © 2017 Jiaotong Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A polychromatic X-ray beam causes the grey of the reconstructed image to depend on its position within a solid and the material being imaged. This factor makes quantitative measurements via computed tomography (CT) imaging very difficult. To obtain a narrow-energy-width reconstructed image, we propose a model to decompose multivoltage X-ray images into many narrow-energy-width X-ray images by utilizing the low frequency characteristics of X-ray scattering. It needs no change of hardware in the typical CT system. Solving the decomposition model, narrow-energy-width projections are obtained and it is used to reconstruct the image. A cylinder composed of aluminum and silicon is used in a verification experiment. Some of the reconstructed images could be regarded as real narrow-energy-width reconstructed images, which demonstrates the effectiveness of the proposed method.

#### 1. Introduction

With the development and application of advanced technology, computed tomography (CT) has changed from conventional qualitative imaging for detection to quantitative functional imaging for distinguishing and identifying different components. For instance, quantifying the composition of coal and the microstructure of mineral grain contributes to an understanding of the transformation of minerals during coal processing, which promotes the development of clean coal technologies [1]. Quantifying the three-dimensional microstructure of excipients contributes to the development and testing of new drugs [2]. Quantification of soil aggregate microstructure on abandoned cropland during vegetative succession allows determination of the retention and transport of water, gases, and nutrients in soils, thus allowing preservation of soil productivity and maintaining soil porosity and resistance to erosion [3]. In these applications, good congruity is needed between the linear attenuation coefficient and X-ray energy in the reconstructed images. In other words, the linear attenuation coefficient of the same component should be uniform, and the corresponding energy of different components’ linear attenuation coefficients should be uniform in a single reconstructed image. A higher grey value corresponds to a larger linear attenuation coefficient in one reconstructed image. It is polychromatic X-ray in the typical CT system and leads to cupping artifacts, which is that the grey of the reconstructed image depends on both the material and its position [4, 5]. So if two materials have approximately linear attenuation coefficients, their grey may overlap, which makes them difficult to distinguish. Overlapping attenuation coefficients makes quantitative imaging very challenging. The use of monochromatic radiation can eliminate cupping artifacts and accomplish a one-to-one relationship between grey values and materials [6]. But it is impractical to apply monochromatic radiation in the typical CT system [7, 8]. One feasible method is to synthesize monochromatic images using dual-energy imaging. One example is the gemstone spectral imaging (GSI) systems. It is a type of dual-energy CT and its X-ray is polychromatic [9]. In the synthesized monochromatic images, the CT numbers become more accurate, but they are still not truly monochromatic, especially at low energy [10]. Another feasible method is to obtain narrow-energy-width images, which can approximate monochromatic images. It can be accomplished through multienergy imaging, which may require an X-ray photon counting detector [11, 12]. Multienergy imaging can be seen an extension of dual-energy CT [7, 13]. The photon counting detector can count discrete photon interactions [14] and has energy selectivity. So it can improve contrast in CT and apply to the material identification [15]. The imaging system based on photon counting detector shows effectiveness in distinguishing different materials [16]. Photon counting detectors are used in nuclear medicine and spectral mammography, but they are not commercially available for CT systems [14]. Challenges remain for them since the exposure rates are insufficient when used to CT imaging [14].

The photon counting detector can obtain many narrow-energy-width images by selecting energy bands. A multienergy CT imaging method was presented based on energy spectrum filtering separation [17], which can, in theory, distinguish different components. However, the application of the multienergy CT imaging method is limited in practice because the energy spectrum can only be narrowed to maintain X-ray penetrability, and the difference between different spectra is insufficient. Another method to obtain narrow-energy-width images is to decompose the multivoltage X-ray images acquired in a typical CT system [18]. This can improve the contrast of different materials with approximately linear attenuation coefficients in reconstructed images [18]. However, these reconstructed images are not real narrow-energy-width reconstructed images, as their contrast is much larger than the theoretical value [18].

Building on previous research [18], we continued studying the decomposition approach of multivoltage X-ray images to obtain a narrow-energy-width projection with a typical CT system without changes in hardware. Herein, we present a new decomposition model based on X-ray scattering characteristics. Some reconstructed images obtained with the new decomposition model can be regarded as real narrow-energy-width reconstructed images. The remainder of this paper is organized as follows. In Section 2, the decomposition model of multivoltage X-ray images presented in [18] is introduced. In Section 3, the new decomposition model and its solution algorithm are presented. Then, in Section 4, the new method is applied to obtain the narrow-energy-width reconstructed image of a cylinder composed of aluminum and silicon. In Section 5, the discussion of innovations and shortcomings of the method are presented, along with upcoming work. Finally, our conclusions are presented.

#### 2. Previous Decomposition Model of Multivoltage X-Ray Images

The X-ray emitted from an X-ray tube is polychromatic and can be split into many narrow-energy-width bands. Therefore, a polychromatic X-ray image can be seen as the sum of many narrow-energy-width X-ray images. The X-ray imaging can be described as follows:where is initial X-ray intensity, is final X-ray intensities, means different narrow energy bands, denotes different materials, is the energy of the th narrow energy band, means the weighted coefficient of the th narrow-energy-width X-ray image, is a linear attenuation coefficient depending on the th material being traversed by the X-ray and the energy level of the th narrow-energy-width band, and the distance the X-ray traverses through the th material is denoted as [18]. is related to the incident X-ray spectrum and the detector efficiency:The weighted coefficients are unknown because the energy spectrum is unknown. The narrow-energy-width X-ray images can be get from the decomposition of multiple X-ray images with different voltages [18]. In other words, the narrow-energy-width projection can be obtained and can be used to reconstruct a narrow-energy-width CT image.

of the th pixel in the X-ray image of th voltage is denoted as , and the multivoltage X-ray imaging model is where , , , , , and is the error produced by measurement and scattering [19–21]. The th row of is the weighted coefficients of the narrow-energy-width X-ray images to constitute the X-ray image at the th voltage. The value of is the th pixel’s corresponding distance that the X-ray traversed through the th material. When several materials are uniformly mixed, they are considered one material [18]. To guarantee that the information related to narrow-energy-width X-ray images is sufficient, , , , and should satisfy the following inequality [18]:The solution is translated to a least squares optimization model asThis model can be solved with the Karush-Kuhn-Tucker (KKT) condition [18]. In the verification experiment [18], the materials with approximately linear attenuation coefficients in the reconstructed images could be significantly distinguished. However, the contrast of the materials is larger than it should be in a real narrow-energy-width reconstructed image. In other words, the reconstructed images are not real narrow-energy-width reconstructed images. This may be because scattering is not considered in model (5).

#### 3. Decomposition Model Based on X-Ray Scattering Character

During X-ray imaging, scattering is an important interference factor, especially when a flat panel detector is used [22]. Significant research on scattering is available. From [22–25], scattering is a low frequency signal related to the imaging objects. The estimated scattering is obtained by multiplying a coefficient to the low-pass filter of the original image, and the scattering suppression method is the original image minus the estimated scattering. This method was quite effective in [23, 25]. In the X-ray imaging model in [26], scattering is regarded as a constant over the entire projection and the same for all projections and depends on the object in the scan. Summarizing the aforementioned research results, scattering is a low frequency signal.

A low frequency signal indicates slow change. In other words, the difference of the neighboring sampling nodes is small; therefore, variance is used to describe this characteristic. Because scattering is related to the imaging object, different projections may result in different scattering. For this reason, the local variance of a signal sampling node is used to estimate the change rate of the sampling node. The whole scattering character is described with the sum of all local variance. The initial intensity of the X-ray beam is greater than 1, so the signal of dividing scattering by is also a low frequency signal. The decomposition model of multivoltage X-ray images can be considered:where is a parameter related to local image size (and it needs to set up first in order to solve the model) and is the local image whose center is the th pixel in the X-ray image of th voltage. For example, the local image size is , and the current pixel is the center in the 2-dimensional CT reconstruction. To reconstruct an image, the projections of many different angles are needed; then formula (6) is changed aswhere denotes different angles; means the local image, whose center is the th pixel in the th angle X-ray image of th voltage; and denotes the pixel amount in the th angle X-ray image.

Similar to [18], formula (7) can be solved by the KKT condition. The iterative formulas arewhere “” is the Hadamard product. means a matrix with every element = 1 with 1 row and columns. “” means the matrix moves left columns (right if the is negative), and the empty columns at the boundary are replaced with original columns. means a matrix with rows and columns, and only the element at the row from to in column is 1, while the others are 0. means a matrix with rows and columns, and the column iswhere the nonzero row is from to , , and the column iswhere the nonzero row is from to , , and the column iswhere the nonzero row is at to . means a matrix with rows and columns and all elements equal to 1 when the column is from to , from to , from to . All elements of other columns are 0. means a matrix with rows and columns where every column of is as ; every column of is ; every column of is .

Similar to [18], the multiplicity solution of and still exists due to putting a pair invertible matrix between and . As the eventual goal is the product* UD*, we considered that they are the same solution. Every row of is normalized according to (2). The following is the complete algorithm to solve (7):(1)initialize , , ;(2)set maximum number of iterations* niter* and a small value ;(3)** for **,(a)update according to (10);(b)normalize every row of with (c)update according to (11);(d)update according to (12);(e)compute the value of the objective function of (7), and note as* y*;(f)** if **, iteration terminates **end** **end**

The solution may be a local minimum, so the algorithmic processes must be repeated many times with different initializations. The optimal solution is selected from the many results.

#### 4. Results

A cylinder made of aluminum and silicon was used in the verification experiment because the two materials’ linear attenuation coefficients are approximate. The linear attenuation coefficients of aluminum and silicon are near-equal at approximately 60 KeV, and, from 10 to 140 KeV, their max difference is less than 13%, as shown in Figure 1. Thus, for the contrast of aluminum and silicon in the reconstructed image, the absolute value should first decrease and then increase as the voltage increases from 10 KeV. The linear attenuation coefficient was obtained from National Institute of Standards and Technology (NIST), and the values were processed using cubic spline interpolations. Some origin values of NIST and difference of the linear attenuation coefficient of aluminum and silicon are shown in Table 1. The silicon was on the outside, and the aluminum was on the inside of the cylinder. The cylinder’s diameter was 40 mm, and the aluminum’s diameter was 30 mm.