## 3 Results and Discussion  
The overall contribution of this work is showing how a two-stage architecture can improve the accuracy of landmark detection for a coordinate regression network. Figure 5 shows the distribution of errors that occur for all input images (indicated by the number of samples in the vertical axis) for when only a single region (i.e., the entire image) is fed into the second stage versus nine regions. As can be seen, the results for nine regions had a distribution that was shifted towards the left, indicating a higher accuracy as compared to the distribution of one region, which was further shown by their average accuracy of 1.75 pixels and 7.36 pixels, respectively. Therefore, utilizing the two-stage architecture is necessary to optimize the accuracy of this landmark detection. To further optimize the performance of this two-stage architecture, we varied the size of the square region array from \(n=3\) (9 regions) to \(n=10\) (100 regions). Figure 6 shows the accuracy of the region selection network for their training, validation, and test sets, for each of the characterized region sizes. Training and validation sets are the last sets provided by the 5-fold validation. As can be seen, with an increase in the number of regions, the accuracy of detection for the first stage decreased. The incorrect predictions were usually occurring when the marker band was located at the border of the region, such that it was overlapping with another region (Figure 7). In this example of the figure, the model was predicting from 25 possible regions, and predicted correctly in the figure on the left (Figure 7A) but incorrectly in the figure on the right (Figure 7B). To state a simple example, in the case of 100 regions, the model misclassified 11 (out of 680) images. In brief, almost all the predicted regions touched the catheter tip, even if they did not point to the correct region. Figure 4: Schematic visualization of dual network. Red box shows the region of interest and red arrow points the marker. Figure 5: Distribution of errors in landmark detection for 1 region and 9 regions. Figure 6: Graph of accuracy of region selection for various numbers of regions. Figure 7: Example of incorrect (**A**) and correct (**B**) selection of region containing target. In the localizer network, the expectation is to predict the exact location of the center of the marker band within the target region. Figure 8 presents the outcomes obtained from the network. This chart indicates the averages of errors between the ground truth coordinate and the predicted landmark coordinate by the model. This error is reported separately for the training, validation, test, and fixed-test results. The \"fixed-test\" results remove all the incorrectly selected regions to demonstrate the highest accuracy the second model can achieve. The error is reported based on the 512 \(\times\) 512 pixel size versions of the original images. The test dataset contained the average of the results of both the correctly and incorrectly selected regions. Overall, the data convey two overall trends: (i) the larger the region number, the less accurately the region selection performs, and (ii) the larger the region number, the more accurately the localizer network performs. Given these two trends, it is seen that either \(n=5\) or \(n=7\) provides the optimal results, with \(n=7\) providing the highest accuracy for the fixed-test set. These conclusions are further supported by the plots in Figure A1, which show the distribution of error for all points; it can be seen that the majority of images were accurately predicted, with a minimal number of large outliers. As we plotted these errors with their ground truth coordinates of the image (Figure A2), we did not see any obvious trends due to spatial positioning within the image. The inference time to predict landmark position in this model (both networks consecutively) is 4 milliseconds on average. The frame rate of the acquisition system is typically 15 frames per second (67 ms).