# FPGA Implementation of ECT Digital System for Imaging Conductive Materials
## 3 Image Reconstruction Algorithm  
An appropriate image reconstruction algorithm is applied to solve the inverse problem. The input of the inverse problem is the capacitance measurements while the result is a sequence of tomographic images describing the conductivity distribution inside the area of interest. Typically, a PC is used for implementing this algorithm where the data is processed and the final images are displayed. The relationship between the capacitance measurements (\(C\)) and the materials distribution (\(H\)) is highly nonlinear. However, this relation can be approximated as the following [14]:  
\[C_{m\times 1}=S_{m\times n}\times H_{n\times 1}, \tag{1}\]  
where \(C\) is a vector of \(m\) normalized measurements related to a certain conductive distribution, vector \(H\) has \(n\) normalized pixel values indicating the metal distribution, and \(S\) a normalized \(m\times n\) matrix called sensitivity matrix. The values in \(H\) are either zero or one, where zero means no metal exists and one for metal. Each row in the sensitivity matrix represents how the capacitance measurements of each sensor change by altering each pixel from metal to no metal, therefore the \(S\) is also called a capacitance coefficient. In the case of six sensors, \(m\) equals 6, and the image is divided into \(16\times 16\) pixels, so \(n\) is 256. The sensitivity matrix contains six sets of capacitance coefficients, and each one has 256 values related to each individual pixel in the image.  
Figure 2: Wireless electrical capacitance tomography system.  
The discrete form of the image reconstruction algorithm computes \(H\) by using the values of \(C\), while \(S\) is kept constant for simplicity [27]. One of the initial reconstruction algorithms is Linear Back Projection (LBP) which applies a transpose of the sensitivity matrix instead of its inverse to calculate grey levels of the image pixels. Thus, to calculate \(H\), the transpose of \(S\) is multiplied by \(C\):  
\[H=S^{T}\times C. \tag{2}\]  
Usually, applying LBP as a single step solution for the inverse problem creates blurred images since the problem is highly nonlinear. Therefore, to improve the quality of the reconstructed images, the inverse problem has to be solved iteratively. Typically, a modified version of the LBP algorithm that solves Equation (2) iteratively is common and called the Iterative Linear Back Projection (ILBP) algorithm [28]. The image vector is calculated as shown in Equation (3):  
\[H_{k+1}=H_{k}+\alpha S^{T}(SH_{k}-C_{m}), \tag{3}\]  
where vector \(C_{m}\) contains the measured capacitance, \(SH_{k}\) is the solution of the forward problem, \(\alpha\) and \(k\) are the relaxation factor and the iteration number, respectively. Figure 3 presents a general nonlinear reconstruction algorithm. One of the most effective stages of the inverse problem is the solution of the forward problem which was applied in the iterative loop and significantly affects the speed and resolution of reconstructed images.  
The ILBP algorithm can be summarized in the following steps:  
1. An initial image is obtained by the LBP algorithm Equation (2).
2. The forward problem Equation (1) is solved to calculate a vector of capacitance measurements.  
Figure 3: Nonlinear reconstruction algorithm in ECT.  
Differences between the calculated and the actual measurements is multiplied by \(S^{T}\) to calculate pixels' errors.
* The difference between the previous image and the pixels' errors represents the new image.
* The termination is reached when the difference in step 3 reaches a certain acceptable value.  
Typically, these algorithms involve a large number of matrix operations, therefore implementing it on a parallel processing platform rather than a serial PC is crucial. For example, the ILBP algorithm implemented on a 2.53 GHz-i5 PC with 4 GB RAM generates a 32 \(\times\) 32 image in more than 1.5 s.