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\title{MDPose: Real-Time Multi-Person Pose Estimation \\via Mixture Density Model\\(Supplementary Material)}

% The standard author block has changed for UAI 2023 to provide
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\author[1]{\href{mailto:<zzzlssh@snu.ac.kr>?Subject=MDPose}{Seunghyeon~Seo}{}}
\author[2]{\href{mailto:<yoojy31@webtoonscorp.com>?Subject=MDPose}{Jaeyoung~Yoo}{}}
\author[3]{\href{mailto:<hjh881120@gmail.com>?Subject=MDPose}{Jihye~Hwang}{}}
\author[1]{\href{mailto:<nojunk@snu.ac.kr>?Subject=MDPose}{Nojun~Kwak}{}}
% Add affiliations after the authors
\affil[1]{%
    Seoul National University
}
\affil[2]{%
    NAVER WEBTOON AI
}
\affil[3]{%
    LG Electronics
    }
  
  \begin{document}
  
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\section{Analysis of the distribution of mixture components}

\begin{figure}[h]
\centering
% \includegraphics[width=0.85\linewidth]{figure/distribution_vis_big.pdf}
\includegraphics[width=\linewidth]{figure/distribution_vis_big.pdf}
\caption{\textbf{Visualization of estimation results with different mixture distributions of MDPose.}}
\label{supp_fig:dist_vis}
\end{figure}

\begin{table}[h]
\caption{\textbf{Mixture model of different exponential distributions.} The Laplace is more suitable than the others for multi-person pose estimation.}
  \label{supp_tab:abl_dist}
  \centering
  \begin{tabular}{lccccccc}
    \toprule
    Dist.     & AP$^{kp}$ & AP$_{50}^{kp}$ & AP$_{75}^{kp}$ & AP$_{M}^{kp}$ & AP$_{L}^{kp}$ & Underflow R. \\ 
    \midrule
    Gaussian   & 50.5 & 79.7 & 54.0 & 41.1 & 63.8 & 0.184  \\
    Cauchy    & 50.6 & 79.6 & 54.1 & 41.4 & 63.5 & \textbf{0.0}  \\
    Laplace & \textbf{51.5}  & \textbf{80.4}  & \textbf{55.1}  & \textbf{42.0}  & \textbf{64.7} & 0.086  \\
    \bottomrule
  \end{tabular}
    % \vspace{-1mm}
\end{table}

Tab. \ref{supp_tab:abl_dist} shows the accuracy and underflow ratio of different mixture distributions.
The MDPose with Laplace mixture distribution outperforms the one with either the Gaussian or Cauchy with a noticeable gap of AP$^{kp}$.
Since the tails of Laplace and Cauchy fall off \textbf{less sharply} than the Gaussian, they are relatively free from the underflow problem.
Furthermore, as the tails of Laplace fall off \textbf{more rapidly} than the Cauchy and it has a \textbf{sharper peak}, it leads to more efficient weighting for good and bad estimations during the training process.
As demonstrated in Fig. \ref{supp_fig:dist_vis}, the Laplace mixture distribution enables more accurate localization of human keypoints than the respective mixture distributions of the Gaussian and Cauchy.

\section{Analysis of grouping randomness through visualization}

\begin{figure}[h]
\centering
\includegraphics[width=1.0\linewidth]{figure/supp_randomness_vis.pdf}
% \vspace{-5mm}
\caption{\textbf{Visualization of our MDPose with (a) non-random grouping and (b) RKG.}}
\label{supp_fig:randomness_vis}
\end{figure}

As mentioned in Sec. 4.2 in the main paper, our proposed RKG strategy enables learning of the overall joint distributions of all keypoints while the non-random grouping learns only the joint distributions of each pre-defined keypoint groups.
Fig. \ref{supp_fig:randomness_vis} shows the qualitative results of our MDPose with (a) non-random grouping and (b) RKG.
The results are obtained from the MDPose (ResNet-50~\citep{he2016deep}) with $K_g=3$, $N_g=6$ and 320x320 input size on the COCO validation set~\citep{lin2014mscoco}.
As shown in Fig. \ref{supp_fig:randomness_vis} (a), the model trained by non-random grouping has a difficulty in differentiating the left and right of limbs, due to lack of learning the overall relationship between every keypoint.
On the contrary, the MDPose trained by RKG (Fig. \ref{supp_fig:randomness_vis} (b)) shows superior performance with well-distinguished left and right of limbs.

\newpage
\section{Qualitative results}
\begin{figure}[h!]
\centering
\includegraphics[width=0.77\linewidth]{figure/supp_qualitative_results_ochuman.pdf}
% \includegraphics[width=0.9\linewidth]{figure/supp_qualitative_results_ochuman.pdf}
\caption{\textbf{Qualitative results of MDPose (ResNet-101) on OCHuman validation set, with $K_g=3$, $N_g=6$ and 896x896 input size.}}
\label{supp_fig:qual_res_ochuman}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[width=0.8\linewidth]{figure/supp_qualitative_results.pdf}
% \includegraphics[width=\linewidth]{figure/supp_qualitative_results.pdf}
\caption{\textbf{Qualitative results of MDPose (ResNet-50) on COCO validation set, with $K_g=3$, $N_g=6$ and 896x896 input size.}}
\label{supp_fig:qual_res}
\end{figure}

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