## 6 Evaluations  
Consider an observation of a scene, where geometrical components have been extracted, by applying the methods described in Section 4.1. Let \(O\) be the set of all geometrical components observed on a scene, then:  
\[O=\{O^{1},...,O^{N_{k}}\} \tag{11}\]  
where \(O^{k}\) is the subset of geometrical components of the type \(k\). In this way, observed horizontal geometrical components found on the scene are in the same subset, lets say \(O^{*}\). Consequently, it is possible to extract each one of them in the subset and then compare them to the main nodes for each furniture graph. Once the similarity between the horizontal components on the scene and the models has been calculated, all the categories with a similarity higher than a certain threshold are chosen as probable models for each horizontal component. A graph is then constructed for each horizontal component, where adjacent geometrical components are merged with it. Then, this graph is compared with the subgraphs of the probable models previously selected. The first column in Figure 14 shows some scenes from the environment, where different pieces of furniture are present. Only four images with the six types of furniture are shown. The scene in Figure 13(a) is composed of a dinning table and a chair. After the geometrical components are extracted (Figure 13(b)), two horizontal planes corresponding to the dinning table and the chair are selected. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Side** & **Front** & **Side** & **Back** & **Top** \\ \hline area & 1575.5 & 9155.5 & 1575.5 & 9155.5 & 1914 \\ Model \% & 6.74 & 39.16 & 6.74 & 39.16 & 8.19 \\ Partial View \% & 12.45 & 72.41 & & & 15.13 \\ Partial View \% & & 72.41 & 12.45 & & 15.13 \\ \hline \end{tabular}
\end{table} 
Table 2: Example of the weights for the comparison of the chest of drawers graph, based on the area of each geometrical component. Figure 13: Graph models for the chest of drawers: in (**a**), the graph for the full model, and in (**b**,**c**), graphs for partial front views, left and right, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Table** & **Leg** & **Leg** & **Leg** \\ \hline area & 8159 & 947 & 947 & 947 & 947 \\ Model \% & 68.31 & 7.92 & 7.92 & 7.92 & 7.92 \\ Partial View \% & 68.31 & 7.92 & 7.92 & 7.92 & 7.92 \\ \hline \end{tabular}
\end{table} 
Table 3: Example of the weights for the comparison of the dining table graph, based on the area of each geometrical component. A comparison of those horizontal components to each one of the main nodes of the furniture graphs is performed. This results in two probable models (table and chest of drawers) for the plane labeled as \"H0\", which actually corresponds to the table, and three probable models (chair, center table and couch) for the plane labeled \"H01\" (which corresponds to a chair). The similarities computed can be observed as \"Node Sim.\" in Table 4.  
Next, graphs are constructed for each horizontal plane (the main node) and adding its adjacent components. In this case, both graphs have only one adjacent node. Figure 15a shows the generated Graph (\"G0\") from the scene and the partial-views graphs from the selected probable models. It can be observed that \"G0\" has an adjacent leg node, so it can only be a sub-graph for the table graph since the chest of drawers graph has only adjacent vertical nodes. For \"G1\" (Figure 16a), its adjacent node is a vertical node with a higher height than the main node, so it is matched to the backrest node of the chair and of the couch (Figure 16b,d). Moreover, there is no match with the center table graph (Figure 16c). The similarity for adjacent nodes is noted in Table 4. Graph similarity is calculated with Equation (9) and shown in the last column of the table. \"G0\" is selected as a table and \"G1\" as a chair (Figure 14c). Figure 14 shows the results of applying the described procedure to different scenes with different types of furniture. The first column (Figure 14a,d,g,j) shows the point clouds from the scenes.