# Detail-Preserving Shape Unfolding
## 6 Integration with User Control  
While our algorithm works well on a large number of models, it is difficult to generate highly uniform canonical forms for models with very different poses. In Figure 8, (a)-left and (b)-left are two poses of the same person. After mesh unfolding, their legs (a)-right and (b)-right have different poses. To solve this problem, an extra item is added in Equation (3) to consider the information provided by users. \[ \begin{split}\mathcal{L}(\mathbf{P}^{\prime})&=- \sum_{i}\sum_{j\in\mathcal{N}_{i}}k_{ij}\|\mathbf{p}_{i}^{\prime}-\mathbf{p}_ {j}^{\prime}\|^{2}\\ &+\beta\sum_{i}\sum_{j\in\mathcal{N}_{i}}w_{ij}\|\mathbf{p}_{i}^ {\prime}-\mathbf{p}_{j}^{\prime}-\mathbf{R}_{i}(\mathbf{p}_{i}-\mathbf{p}_{ j})\|^{2}\\ &+\gamma\sum_{(i,j)\in\mathcal{U}}u_{ij}\|\mathbf{p}_{i}^{\prime }-\mathbf{p}_{j}^{\prime}-\mathbf{t}_{ij}\|^{2}\end{split}  \tag{11}\]  
where, \(\gamma\) is a parameter to balance the user information, \(u_{ij}\) is the weight for vertex pairs, which is also set to 1, \(\mathcal{U}\) is the control set of vertex pairs and \(\mathbf{t}_{ij}\) is the control vector specified by user. Similar to the derivation process of Section 3.2, we can obtain the following linear system  
\[(\beta\mathbf{V}_{2}+\gamma\mathbf{V}_{3}-\mathbf{V}_{1})\mathbf{P}^{\prime} =\beta\mathbf{H}+\gamma\mathbf{K}\,, \tag{12}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & **Alien** & **Garilla** & **Snake** & **Woman** \\ \hline Num. of Vertices & 4062 & 5091 & 4229 & 4823 \\ Time of DPMU & 687.05 & 190.12 & 1137.23 & 147.39 \\ Time of ours & 84.12 & 157.01 & 83.25 & 115.57 \\ Time of ours (parallel) & 30.87 & 54.37 & 31.02 & 46.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Running time of examples in Figure 7 (Unit: second). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Ori** & **LSMDS**[11] & **SCF**[28] & **DPMU**[24] & **Ours** \\ \hline ORT & 48.2 & 91.3 & 89.5 & 95.1 & 94.6 \\ CMBOF & - & 99.2 & 99.2 & 99.5 & 99.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Retrieval accuracy for different methods on the dataset [47]. where  
\[\mathbf{V}_{3} =\sum_{(i,j)\in\mathcal{C}}(\mathbf{e}_{i}-\mathbf{e}_{j})^{T}( \mathbf{e}_{i}-\mathbf{e}_{j})\] \[\mathbf{K} =\sum_{(i,j)\in\mathcal{C}}(\mathbf{e}_{i}-\mathbf{e}_{j})^{T} \mathbf{t}_{ij} \tag{13}\]  
We find that parameter \(\gamma\) has a similar role to \(\beta\) in experiments, so we set \(\gamma=\beta\) in our algorithm with user control. In Figure 8, an example is given to show how to specify vertex pairs and control vectors by users. As shown in Figure 8c-left, we first pick one point pair \(\mathbf{p}_{i}\) and \(\mathbf{p}_{j}\) on the feet (red nodes), then specify their direction \(\mathbf{t}_{ij}\) and the length after unfolding with the following strategy. As the intrinsic symmetry plane of this model is perpendicular to z-axis, that is \((0,0,1)\), we restrict \(\mathbf{t}_{ij}\) to be parallel to \((0,0,1)\) and the length after unfolding to be \(||\mathbf{p}_{i}-\mathbf{p}_{j}||/2\). The result after solving Formulation (11) is shown in Figure 8c-right which almost has the same pose as the one in Figure 8a-right. For the Octopus model shown in Figure 9, our method can give a satisfactory result ((a)-middle). However, its tentacles do not in a plane when viewed from another view ((a)-right). Our algorithm with user control can be used to further improve the quality of the canonical form. We first pick one point at the end of each antenna (red points in (b)-left). Then the directions and lengths between the adjacent antennas are restricted to approximate a regular octagon. The resting results are shown in Figure 9b. We can clearly see that all tentacles are almost in the same plane. Figure 8: Different poses of the same model ((a)-left and (b)-left) results in different canonical forms ((a)-right and (b)-right). With user control (red point pair in (c)-left), the quality of the canonical form can be largely improved ((c)-right).