## 7 Design Space Exploration
### Pre-Selection of Data Formats and Representations  
To pre-select the data formats and reduce the size of the design space, we use a top-down approach. At the beginning, we decided to explore only standardized data formats that are also implemented in the new commercial architectures, such as GPUs, and TPUs. In this way, we focused on floating-point and fixed-point representations.  
For interventional CT applications, the goal is to minimize latency while maintaining high accuracy for having a real-time reconstruction. For this reason, in the second step, from the pre-selected data representations, we considered only formats with data widths in the range from 16 to 32. The values of this range are related to the data widths of raw sensor data and reconstructed data. In fact, raw sensor data are usually represented with short format (16-bit data width), and reconstructed data with float format (16-bit data width). In addition, we considered the double format (64-bit data width), which we used as a reference point in the image quality analysis. Therefore, for the floating-point representation, we limited our study to three different encodings: _half, single_ and _double_ precision.  
By contrast, regarding the fixed-point representation with a fixed-rounding configuration, if all possible formats in the range from 16 to 32 bits are considered, 408 configuration formats are possible. Therefore, we considered only the upper bound and lower bound configurations, which are 16 and 32 bits. For these two data widths, there are 16 plus 32 possible configurations as fixed-point representation. Therefore, to reduce our DSE from these 48 to the desired 2 configurations, we analyzed the raw sensor data and selected one configuration for 32-bit fixed-point and one for 16 bit fixed point. The raw sensor data are represented as 16-bit unsigned, so we configured the 32-bit fixed-point with I and F both equal to 16 bits. In this way, the 32-bit fixed-point does not approximate any values in the _sensor data conversion stage._  
Due to the fractional part of the fixed-point representation, the 16-bit fixed point cannot contain the 16-bit unsigned, without approximation. Therefore, to find the best configuration, we started from the 16-bit fixed-point configuration that has 8 bits for integer and fractional, and we estimated the MSE. The MSE is the mean squared difference between a reference value and an approximated value [43]. This is often used to measure the image quality between two images [44]. To find other configurations, we analyzed the multiplication factor of Formula (7), which can be approximated with a shift of the dot in the number. So, we removed the multiplication and shifted the dot by decreasing/increasing the integer and fractional parts, respectively, to reach the lowest MSE. In this way, we reached the best configuration of 16-bit fixed-point with 4-bit and 12-bit for the integer and fraction parts, respectively.  
With this methodology, we pre-selected four data representations for the DSE, which are half-precision and single-precision floating-point, and 16-bit and 32-bit fixed point, where I and F are equal to 16 and 16 bits, and equal to 4 and 12 bits, respectively.