% \section{Ablation Studies}
% \subsection{Weighted Landmark Pose Estimation Strategies}
% \label{appendix:weighted_landmark}
% For completeness, we describe the three weighting strategies used in the ablation study of uncertainty-aware pose estimation. All methods modify the standard weighted registration objective
% \begin{align}
%     \min_{\theta} \sum_{c \in V_0} w_c \, \bigl\| \pi(\theta, X_c) - y_c \bigr\|_2^2,
% \end{align}
% where \(X_c\) is the 3D landmark, \(y_c\) is the predicted 2D location, \(\pi(\theta, \cdot)\) denotes the projection under pose parameters \(\theta\), and \(w_c\) is the weight assigned to correspondence \(c\).

% Let \(d_c \ge 0\) denote the MC-dropout uncertainty for landmark \(c\), and let $\tilde{d}_c$ denote the deviation after replacing missing values with the median. All three weighting functions were evaluated by substituting \(w_c = w_c^{(1)},\, w_c^{(2)},\, w_c^{(3)}\) in the weighted registration objective above. These analyses are included solely as comparative baselines and are not part of the proposed method.

% \subsubsection{Inverse-Deviation Weighting}
% Landmarks with smaller uncertainty should receive higher weight. We therefore set
% \begin{align}
%     w_c^{(1)} \;=\; \frac{1}{\tilde{d}_c + \varepsilon}, \qquad w_c^{(1)} \leftarrow \frac{w_c^{(1)}}{\max_{c'} w_{c'}^{(1)}},
% \end{align}
% where \(\varepsilon\) ensures numerical stability. This produces normalized weights in \([0,1]\).

% \subsubsection{Softmax-Like Weighting}
% To avoid excessively large ratios between low- and high-uncertainty landmarks, we apply an exponential transformation:
% \begin{align}
%     w_c^{(2)} = \exp\!\bigl(-\beta \tilde{d}_c\bigr),\qquad w_c^{(2)} \leftarrow \frac{w_c^{(2)}}{\max_{c'} w_{c'}^{(2)}},
% \end{align}
% where \(\beta\) controls the sharpness of the weighting distribution. This behaves similarly to a softmax over negative uncertainty.

% \subsubsection{Rank-Based Weighting}
% Because absolute uncertainty magnitudes may not be meaningful across images, we also evaluated a rank-normalized weighting:
% \begin{align}
%     r_c = \mathrm{rank}(\tilde{d}_c)\quad\text{(0 = smallest deviation)}, \qquad w_c^{(3)} \;=\; 1 - \frac{r_c}{C - 1},
% \end{align}
% where $C$ is the number of landmarks. This linearly maps the most reliable landmark to weight $1$ and the least reliable to weight $0$.

% % \subsection{Dropout Rate}