# On-the-Fly Machine Learning for Improving Image Resolution in Tomography
## 4 Results
### Simulations  
First, we investigated the performance of the proposed on-the-fly machine learning technique on simulated tomographic data. **Simulation phantom** A foam ball simulation phantom was generated by removing 90,000 randomly-placed non-overlapping spheres from a large sphere made of one material. The foam ball has diameter 1. All other dimensions in the simulation are relative to this unit length. The radius of the random spheres ranges between 0.0025 and 0.2. The central slice of this phantom is displayed in Figure 6.  
**Data simulation** Using the foam phantom, three projection datasets were computed. For the first projection dataset, which has the entire foam ball in the field of view, the source-object distance is equal to the source-detector distance, yielding a magnification factor of \(\alpha_{\text{low}}=1\). In practice, having an equal source-object and source-detector distance is not possible, since the detector and object would share the same physical space. In simulations, however, this is both possible and natural, since it results in a minimal voxel size that is equal to the detector pixel size. The second and third projection dataset were acquired at a magnification factor of \(\alpha_{\text{high}}=4\). To simulate how our method copes with optical phenomena, another set of projections was created by post-processing the projections using a Gaussian blur with standard deviation of two pixels. The Gaussian blur convolves the projection images with a 2D filter defined by \(g(x)=\exp\left(-\left\|x\right\|^{2}/\sigma^{2}\right)/\sqrt{2\pi\sigma^{2}}\), where \(\sigma\) is the chosen standard deviation. We refer to the blurred data as _optics-limited_, and to the non-blurred data as _pixel-limited_. The projections were carried out using the GPU-accelerated _cone_balls_ software package, which we have made available as an open source package [39]. This package analytically computes the linear cone beam projection of solid spheres of constant density. For each dataset, 1500 projections were acquired over 360 degrees on a virtual detector with \(1000\times 1000\) pixels. For each detector pixel, four rays were cast through the phantom, and their projection values were averaged. **Processing** The reconstruction and training of the foam phantom was performed as described before in Section 4. The details are summarized in Table 2. Methods A and B are evaluated on the upper ROI reconstruction, and compared below for both the pixel-limited and optics-limited case. Figure 6: (**a**) A projection image of the foam ball; (**b**) a magnified projection image of the central region of interest (ROI); (**c**) a magnified projection image of the upper ROI; (**d**) the central cross-sectional slice of the phantom. **Evaluation** For the simulated data, the MSE and SSIM metrics are computed between the network outputs and the original phantom data, rather than the high-resolution reconstruction. As the SSIM metric is designed to operate on images with a fixed intensity interval, we clip the network outputs such that all images have the same minimum and maximum intensity as the phantom data. Similarly, the images displayed in Figure 7 are clipped to the range \([0,1]\). The MSE metrics are calculated on the unclipped data. The quantitative results for the foam phantom are given in Table 3. Both variants of method A significantly outperform cubic up-sampling on both MSE and SSIM metrics. On the pixel-limited dataset, the SSIM score of method B is lower than the methods A but higher than cubic up-sampling. The fine scale features in the pixel-limited high-resolution image can simply not be represented on the coarser grid that method B operates on. On the optics-limited dataset, on the other hand, the SSIM score of method B is comparable to the SSIM scores of methods A, and is significantly higher than that of cubic up-sampling. In Figure 7, the methods are visually compared on both the pixel-limited dataset and the optics-limited dataset.