# CUDA and OpenMp Implementation of Boolean Matrix Product with Applications in Visual SLAM
## 5 Ultrametric-PAM and Ultrametric-FABMAP
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Compactness** & **Compactness** & **Dunn Index** &
\begin{tabular}{c} **Dunn Index** \\ **Ratio** \\ \end{tabular} \\ \hline \(B^{1}\) & 0.456225532 & 1 & 0.01110 & 1 \\ \(B^{2}\) & 0.499903924 & 1.095738596 & 0.042091 & 3.7914944142 \\ \(B^{7}\) & 0.60707534 & 1.330647448 & 0.169928 & 15.3067877 \\ \(B^{35}\) & 0.286708931 & 0.628436839 & 1.7971900 & 161.88682261 \\ \hline \end{tabular}
\end{table} 
Table 2: Compactness, Donn index as we increase ultrametricity. Spiral dataset. Figure 4: Compactness, Donn index and Normalized mutual information. Spiral dataset. Figure 5: As the ultrametricity of the distance matrix of the Iris data set increases, clusters are better separated. Figure 8: Compactness, Donn index and Sum of Squared errors. Seed dataset. Figure 6: Compactness, Donn index and Sum of Squared errors. Iris dataset. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \begin{tabular}{c} **Compactness** \\ **Avg.** \\ \end{tabular} & \begin{tabular}{c} **Compactness** \\ **Ratio Avg.** \\ \end{tabular} & \
\begin{tabular}{c} **Dunn Index** \\ **Ratio** \\ \end{tabular} &
\begin{tabular}{c} **Dunn Index** \\ **Ratio** \\ \end{tabular} \\ \hline \(B^{1}\) & 0.258152 & 1 & 0.09880 & 1 \\ \(B^{2}\) & 0.287839 & 1.114996 & 0.14560 & 1.47364530 \\ \(B^{3}\) & 0.317521 & 1.229975 & 0.138675 & 1.4034885859 \\ \(B^{7}\) & 0.34202 & 1.324878 & 0.46684 & 4.7248466 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Compactness, Donn index as we increase ultrametricity. Iris dataset. Figure 7: As the ultrametriciy of the Seed dataset increases, clusters become more distinct. The ultrametricity of data sets Iris, Seeds, Spiral, Jain, Yeast, and Breast cancer was then assessed using [PERSON]'s measure, as discussed in [13]. In [PERSON]'s measure, smaller numbers represent higher ultrametricity. Using the measure defined in Section 4, we also calculated the degree of ultrametricity for the aforementioned datasets. Figure 9 on the following page compares the proposed measure to [PERSON]'s measure. We see that the two measures are pointing in the same direction and are linearly correlated with a Pearson correlation of 0.9528. For a better demonstration of the relationship, a blue regression line of degree one is also plotted. To have a preview of the datasets used in the Ultrametric-FABMAP Figure 10 is plotted. Table 5 displays the recall and accuracy results of the Ultrametric-FABMAP, Ultrametric-BoW, regular FABMAP2, and BoW algorithms. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & **Method** & **Acc** & **Recall** \\ \hline Lip6 & Ultrametric & \%68.11 & \%66 \\ Indoor & Bow & \%57 & \%87 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Accuracy and Recall for Lip6 indoor dataset considerin TUM sequence 16 as training data. Figure 10: Example of datasets used in Ultrametric-FABMAP. We run experiments on real-world-size datasets Lip6 indoor (publicly available at: [[https://animatlab.lip6.fr/AngeliVideosEn](https://animatlab.lip6.fr/AngeliVideosEn)]([https://animatlab.lip6.fr/AngeliVideosEn](https://animatlab.lip6.fr/AngeliVideosEn)), accessed on 2 March 2019) and Newer College (publicly available at: [[https://ori.ox.ac.uk/publications/datasets/](https://ori.ox.ac.uk/publications/datasets/)]([https://ori.ox.ac.uk/publications/datasets/](https://ori.ox.ac.uk/publications/datasets/)), accessed on 20 March 2022), which is a video (which we converted into 200 image sequences), and sequence 16 Monocular Visual Odometry collected at Technical University of Munich (TUM) (publicly available at: [[https://vision.in.tum.de/data/datasets/mono-dataset](https://vision.in.tum.de/data/datasets/mono-dataset)]([https://vision.in.tum.de/data/datasets/mono-dataset](https://vision.in.tum.de/data/datasets/mono-dataset)), accessed on 20 March 2020)). Sequence 16 of TUM is simple loop within a corridor. In this case, we use Algorithm 4 on a set of specific train/test datasets.