# Unsupervised 3D Motion Summarization Using Stacked Auto-Encoders
## 6 Experimental Results
### Evaluation Metrics  
As we have stated above, ground truth data have been created by experts of Greek traditional dances. These experts are affiliated with the schools of sport science of the University of Thessaloniki and University of Thessaly in Greece. The ground truth data include a set of desired key frames, as being specified by the experts. Let us denote as \(\vec{g}_{l}\) the selected key frames by the experts, with \(l=1,2,...,L\) where \(L\) is the number of representative frames as being indicated by the experts. We also symbolize as \(G\) the set containing all these selected frames, that is, \(G=\{\vec{g}_{1},\cdots,\vec{g}_{L}\}\). Let us also denote as \(\vec{\tau}_{k},k=1,2,..,K\) the extracted representative frames by any summarization algorithm and as \(R=\{\vec{\tau}_{1},\cdots,\vec{\tau}_{K}\}\) the respective set containing all \(K\) representatives extracted. Indices \(l,k\) are actually the frame instances of the ground truth key frames and the ones extracted by a summarization algorithm respectively. Thus, one objective criterion for evaluating the performance of a summarization scheme is to find, for each of the \(K\) extracted frames by an algorithm, the time instance (i.e., the frame index) of the experts' selected frame which is closest to the first one and then take the frame index difference of the ideal (experts' selected frame) and the extracted one. In other words,  
\[\hat{I}(k)=\arg\min_{for\ all\ l\in G}|l-k|\ \ \forall\vec{\tau}_{k}\in R, \tag{3}\]  
where \(\hat{I}(k)\) is the optimal frame index returned over all \(L\) selected frames in \(G\) for an examined extracted frame in \(R\), say the \(k\)-th. We should notice that different extracted key frames \(\vec{r}_{k}1,\vec{r}_{k}2\) with \(k_{1}\
eq k_{2}\) may yield the same selected frame \(\vec{g}_{l(k)}\) meaning that some of the \(L\) selected frames may not correspond to any of the \(K\) extracted key frames. Then, the absolute difference \(|\hat{I}(k)-k|\) describes how close is the \(k\)-th representative frame (by a summarization algorithm) to the closest ground truth one. In particular, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type of Dance** & **Description** & **Main Choreographic Steps** \\ \hline \multirow{4}{*}{Sitros (3-Beat)} & A Greek folklore dance in a slow rhythm performed by both women and men. & (1) Initial Posture (IP); (2) Cross Leg (CL); (3) Initial Posture (IP); (4) Left Leg Up (LLU); (5) Initial Posture (IP); (6) Right Leg Up (RLU) \\ \hline \multirow{4}{*}{Sitros (2-Beat)} & A Greek folkloric circular dance & (1) Initial Posture (IP); (2) Left Leg Back (LLB); (3) Cross Legs (CL); (4) Cross Legs (CL); (5) Cross Legs (CL); (6) Initial Posture (IP); (7) Right Leg Back (RLB); \\ \cline{1-1}  & A Greek folkloric circular dance & (1) Initial Posture (IP); (2) Cross Legs (6) Right Leg Up (RLU) \\ \hline \multirow{4}{*}{Makedonikos} & A Greek folkloric circular dance & (1) Initial Posture (IP); (2) Cross Legs (CLB); (3) Cross Legs (CL); (4) Left Leg Front (LLF); (5) Right Leg Back (RLB) \\ \cline{1-1}  & & (1) Initial Posture (IP); (2) Cross Legs (2) Backwards (CLB); (3) Cross Legs (CL); (4) Left Leg Front (LLF); (5) Right Leg Back (RLB) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A brief description of the recorded dances. Figure 7: The main choreographic steps of Sytros (3-beat) dance. \[\mu =\frac{1}{K}\sum_{k=1}^{k=K}|\hat{l}(k)-k| \tag{4}\] \[\mu_{max} =\max_{\forall k\in R}|\hat{l}(k)-k|,\]  
where \(\mu\) is the average time instance deviation among all \(K\) extracted representatives and \(\mu_{max}\) the maximum deviation (worst case) among all \(K\) extracted frames. Another criterion is to estimate how well all frames of a dance sequence can be reconstructed (represented) by the key frames. This is performed in our case by calculating the correlation coefficient of the feature vector for each frame of the dance sequence \(\vec{x}_{i},i=1,...,N\) against all representative frames \(\vec{r}_{k},k=1,...,K\).