# A Noniterative Simultaneous Rigid Registration Method for Serial Sections of Biological Tissues +
## 4 Experiments
### Comparison with state-of-the-art methods  
To test the proposed algorithm's applicability to real-world scenes, we first used our noniterative simultaneous rigid registration algorithm (NSRR) to register serial section images of a zebrafish, which included 336 microscopic images imaged by scanning electron microscopy (SEM). The thickness of each section is 50 nm, the image resolution is 6144 by 6144 pixels, and the pixel size of the images is 110 nm. This problem involves different levels of rotation, translation, scaling, and nonlinear deformation. All the experiments were run on the same PC equipped with an intel i7-4790 CPU. Baseline methods' programs are publicly available in Fiji (the software can be downloaded from: [[https://imagej.net/Fiji](https://imagej.net/Fiji)]([https://imagej.net/Fiji](https://imagej.net/Fiji)) and [[http://www-o.ntust.edu.tw/](http://www-o.ntust.edu.tw/)]([http://www-o.ntust.edu.tw/](http://www-o.ntust.edu.tw/))\(\sim\)cweiwang/3 Dregistration/). We utilized the SIFT flow method [20] to obtain dense correspondences, from which we selected a sparse set of correspondences with high confidence from the foreground, as illustrated in Figure 4.  
Figure 4: An example of the correspondences we extracted (marked by red dots). We utilized the SIFT flow method to obtain dense correspondences between adjacent images; then, from those, we selected a sparse set of pairs with low matching errors. The two rows show adjacent images. Clearly, our approach can obtain reliable correspondences even under conditions that involve illumination variation, tissue removal, and spatial transformation. Figure 3: Plot of MSE versus noise ratio. The performance metric is SSIM [22], which evaluates the structural similarity between each pair of registered adjacent images; these values are summed and averaged across the entire sequence. Meanwhile, the SSIM variance along the whole sequence was calculated to evaluate the algorithms in terms of stability. We computed the total time needed to register an image pair to measure algorithm performance. We adopted the widely used state-of-the-art methods described in Section 2 as baselines, including Pairwise method [23], BUnwarpJ [10], CW_R_color [3], LeastSquare [8], and Elastic method [12]. Here, Pairwise method was applied sequentially to register adjacent section images. Table 1 lists comparisons of the results among the baselines and our method. Compared with the baseline methods, our method achieves the highest accuracy in the least amount of time. Moreover, our method is very stable, and it achieves the smallest accuracy deviations. As Figure 5 shows, our method achieves the best results and does not cause error accumulation and propagation. To further prove the effectiveness of our method, we also adopted the data acquired by FIB-SEM [24], which contains 62 serial section images of a _Drosophila_ brain whose image resolution is 6684 \(\times\) 6516 pixels with thicknesses and pixel sizes all equal to 9.15 nm. The advantage of using these data are that all the sections were imaged in situ; therefore, these data can be treated as the ground truth. It is well known that ground truth for image registrations is difficult to obtain; however, based on these data, we can construct a testing set that includes ground truth. Each section was randomly rotated and shifted, and some sections were removed to mimic the common thicknesses of serial sections. Compared with the zebrafish SEM data, these data are considerably more complex, because the image content can be replete with many textures, the matching of fine details needs to be considered. Similar to the last experiment, we compared the performance of our algorithm with all the baseline methods. We used endpoint error (EPE) to evaluate accuracy; EPE is the Euclidean distance between the registered images and the ground truth images, and it is averaged over all pixels. Here, an image is reshaped to a vector. Table 2 shows a comparison among the baseline methods and our method.