# Metal Artifact Reduction in X-ray CT via Ray Profile Correction
## 1 Introduction  
X-ray computed tomography (CT) is a leading cross-sectional imaging technique lauded for its high image resolution and rapid speed of acquisition. It is based on the differing photoelectric X-ray absorption properties of tissue and assumes that the X-rays are attenuated exponentially according to the Beer-Lambert law:  
\[I_{E}(L)\approx I_{0}(E)\exp\left\{-\int_{0}^{L}\mu(\mathbf{s}+\lambda\cdot\mathbf{r}; E)d\mathbf{\lambda}\right\}. \tag{1}\]  
In this equation, at some energy level \(E\), \(I_{0}(E)\) is the number (or intensity) of incident photons and \(\mu(\mathbf{x};E)\) is the linear attenuation coefficient at location \(\mathbf{x}\). Then, \(I_{E}(L)\) is the number (or intensity) of detected photons along a linear ray described as a line equation, \(\mathbf{s}+\lambda\cdot\mathbf{r}\), and its length is \(L\). In this equation, \(\mathbf{s}\) is the position of the X-ray source, \(\mathbf{r}\) is the directional vector of the ray, and \(\lambda\) is a step size. From this, the Radon transform [1] is derived by taking the logarithm of Equation (1) such that  
\[M(L;E)=-\ln\frac{I_{E}(L)}{I_{0}(E)}\approx\int_{0}^{L}\mu(\mathbf{s}+\lambda\cdot \mathbf{r};E)d\lambda, \tag{2}\]  
This expression forms the underlying principle of most existing CT reconstruction algorithms. It assumes that the attenuation responses along a ray are commutative and accumulate in an additive fashion. The approximation as formulated in Equation (2) only holds when the X-ray source is monochromatic. When it comes to (more widely employed) polychromatic X-ray sources, the polychromatic projection data can be experimentally calculated [2] as  
\[P(L;E)=-\ln\frac{\int I_{0}(E)\exp\left\{-\int_{0}^{L}\mu(\mathbf{s}+\lambda\cdot\mathbf{ r};E)d\mathbf{\lambda}\right\}dE}{\int I_{0}(E)dE} \tag{3}\]  
The integration over the X-ray energy spectrum in Equation (3) breaks the linearity between attenuation and penetration length during the reconstruction procedure assumed in Equation (2). Furthermore, low-energy photons are attenuated more easily than high-energy ones (see, for example, [3]), and, as a result, the X-ray beam is _hardened_ as it passes through the material, shifting the energy spectrum toward higher energy. This becomes a particular problem when CT scanning is conducted for patients bearing metallic implants, which have dramatically increased attenuation properties for lower energies. The selective photon absorption not only increases the amount of dose absorbed by the patient, it also amplifies the X-ray beam's hardness. A direct effect is that more photons are detected than expected. A failure to consider the non-linear behavior of the beam hardening effect in Equation (2) results in dark streaks along the lines of greatest attenuation [4,5]. The high pass filter used in a Filtered-back projection (FBP) reconstruction algorithm [6] further exaggerates the differences between adjacent detector elements where one has received a hardened beam and the other one has not. This unintended contrast produces bright streaks in other directions. As a consequence, by ways of these adverse mechanisms, metal artifacts obscure information about anatomical structures, making it difficult for radiologists to correctly interpret the affected CT images. There have been extensive efforts in developing metal artifact reduction (MAR) algorithms to compensate the approximation errors in Equation (2) caused by implanted metals or high density objects. These efforts can be largely divided into two types of approaches: iterative reconstruction and sinogram correction. The iterative reconstruction methods adapt existing X-ray CT systems by incorporating one or more of the following types of a-priori knowledge: (1) low-level information (such as tissue classified image or metal region classified sinogram) of the images to be reconstructed [7, 8, 9, 10]; (2) the X-ray spectrum of the source [11]; (3) the attenuation functions of the base materials [12, 13, 14]; and (4) the composition of the metal components [15].