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\begin{document}

\title[Article Title]{Supplementary Material for:
Frequency-Aware Gradient Correction for Unified Facial Landmark Detection}

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%% GivenName	-> \fnm{Joergen W.}
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\author[1]{\fnm{Shun} \sur{Ren}}\email{renshun@ctgu.edu.cn}
\author[1]{\fnm{Qingjia} \sur{Li}}\email{1990760379@qq.com}
\author[1]{\fnm{Hang} \sur{Sun}}\email{sunhang0418@whu.edu.cn}
\author[4]{\fnm{Xu} \sur{Wu}}\email{N2309016A@e.ntu.edu.sg}
\author[2]{\fnm{Beihang} \sur{Song}}\email{bhs605906@whu.edu.cn}
\author*[3,4]{\fnm{Jun} \sur{Wan}}\email{junwan2014@whu.edu.cn}


\affil[1]{\orgdiv{College of Computer and Information Technology}, \orgname{China Three Gorges University}, \orgaddress{\city{Yichang}, \postcode{443002}, \state{Hubei}, \country{China}}}
\affil[2]{\orgdiv{National Institute of Natural Hazards}, \orgname{Ministry of Emergency Management of China,}, \orgaddress{\city{Beijing}, \postcode{100085},  \country{China}}}
\affil[3]{\orgdiv{School of Information Engineering}, \orgname{Zhongnan University of Economics and Law}, \orgaddress{\city{Wuhan}, \postcode{430073}, \state{Hubei}, \country{China}}}
\affil[4]{\orgdiv{School of Computer Science and Engineering}, \orgname{Nanyang Technological University}, \orgaddress{\city{Singapore}, \postcode{639798}, \country{Singapore}}}
% \affil[5]{\orgdiv{School of Information Science and Engineering}, \orgname{Yunnan University}, \orgaddress{\city{Kunming}, \postcode{650091}, \state{Yunnan}, \country{China}}}

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\section{Unified Facial Landmark Detection}\label{sec1}

\begin{figure}[H]
\begin{center}
	\includegraphics[width=1.0\linewidth]{Unified Facial Landmark.png}
	\end{center}
	\vspace{0em}
	\caption{We propose a unified landmark index that integrates original landmarks from four popular datasets into 124 standardized points for consistent facial landmark detection.}
	\label{map}
	\vspace{0em}    
\end{figure}

Unified Facial Landmark Detection (UFLD) jointly trains a single model across multiple datasets with distinct landmark definitions and annotation densities, aiming to accurately predict dataset-specific landmarks within a unified representation space.

\textbf{Unified Landmark Index:} To unify landmark definitions across datasets (300W, WFLW, COFW, AFLW), we construct a 124-point unified set $U$. Each dataset's original landmarks are mapped to $U$ via semantic correspondences, so that semantically identical landmarks share the same index, while dataset-specific landmarks are preserved. During training, the model predicts the complete unified set $\hat{Y}$, and dataset-specific predictions $\hat{Y}^{(d)}$ are obtained via index selection (Fig.~\ref{map}):
\begin{equation}
\hat{Y}^{(d)}_i = \hat{Y}_{\pi_d(l^{(d)}_i)}, \quad i = 1, \dots, N_d
\end{equation}
where $\hat{Y} \in \mathbb{R}^{124 \times 2}$ is the predicted unified coordinates, $l^{(d)}_i$ is the $i$-th landmark of dataset $d$, and $\pi_d$ maps dataset-specific indices to $U$. This ensures consistent semantic alignment while enabling the network to leverage diverse dataset annotations for unified FLD.



\section{ Self Evaluation}


\begin{table}[!h]
\centering
\fontsize{10}{8}\selectfont
\setlength{\tabcolsep}{8pt}
\renewcommand\arraystretch{1}
\caption{Cross-dataset performance when different datasets are used as the anchor. Values are NME (\%).}
\begin{tabular}{lcccc}
\toprule
Anchor Dataset & 300W-C & WFLW-F & COFW & AFLW \\
\midrule
300W        & 4.64 & 4.69 & 5.07 & 1.70 \\
COFW        & 4.73 & 4.70 & 4.96 & 1.71 \\
AFLW        & -- & 4.65 & 5.02 & 1.69 \\
\midrule
\textbf{WFLW (ours)} & \textbf{4.56} & \textbf{4.52} & \textbf{4.82} & \textbf{1.69} \\
\bottomrule
\end{tabular}
\label{tab:all_anchor_comparison}
\end{table}

\textbf{Evaluation of different anchor datasets in AGGC.}  
To identify the most reliable anchor for resolving multi-task gradient conflicts, we evaluate AGGC with four choices: 300W, AFLW, COFW, and WFLW. As shown in Table~\ref{tab:all_anchor_comparison}, 300W and AFLW yield unstable optimization---improving their own domains but degrading performance on challenging datasets like WFLW and COFW due to limited coverage or biased distributions. COFW also proves suboptimal, as its severe occlusions and small-scale faces produce unrepresentative gradient directions. In contrast, WFLW provides rich and balanced facial variations---pose, illumination, occlusion, blur, makeup, and expression---generating stable and informative gradients. Anchoring AGGC to WFLW consistently improves performance across all benchmarks, achieving the best results on 300W-Challenge, WFLW, and AFLW-full. We thus adopt WFLW as the anchor task for AGGC.







\begin{figure}[H]
\begin{center}
	\includegraphics[width=1\linewidth]{model_visio.png}
	\end{center}
    \vspace{-1em}
	\caption{(a) Original input images, (b) prediction results of the FGC-UFLD model without AGGC, and (c) prediction results of the FGC-UFLD model. The FGC-UFLD model can learn more effective facial structural information, thereby improving the accuracy of facial landmark detection.}
	\label{fig:hf_visualization} 
    \vspace{-1em}
\end{figure}

\textbf{Analysis of the FGC-UFLD model.}
To further illustrate the effectiveness of FGC-UFLD, Fig.~\ref{fig:hf_visualization} provides step-by-step visualizations under challenging conditions (extreme pose, blur, and illumination degradation). Ground-truth and predicted landmarks are shown in red and green, respectively. The first row shows input images. The second row (without AGGC) shows progressive high-frequency refinement enhancing frequency--spatial representations and improving fine-grained localization under severe deformations. The third row (full FGC-UFLD with AGGC) regulates gradient interactions across heterogeneous datasets, yielding more stable and consistent predictions. Overall, the synergy between frequency-aware refinement and anchor-guided correction effectively reduces prediction drift and improves robustness in challenging scenarios.



\begin{table}[!h]
\centering
\fontsize{10}{8}\selectfont
\setlength{\tabcolsep}{8pt}
\renewcommand\arraystretch{1.4}
\caption{Comparison of different multi-task balancing strategies across four benchmark datasets (NME, \%).}
\vspace{1em}
\begin{tabular}{lcccc}
\toprule
Method & 300W-C & WFLW-F & COFW & AFLW \\
\midrule
FGC-UFLD (w/o AGGC) & 4.60 & 4.65 & 4.92 & 1.70 \\
UW~\cite{kendall2018uncertainty} & 4.59 & 4.63 & 4.92 & 1.69 \\
DWA~\cite{liu2019attentionmtl} & 4.63 & 4.71 & 4.91 & 1.70 \\
GradNorm~\cite{chen2018gradnorm} & 4.68 & 4.65 & 4.94 & 1.71 \\
PCGrad~\cite{yu2020pcgrad} & 4.68 & 4.62 &4.99 & 1.71 \\
\midrule
\textbf{AGGC (ours)} & \textbf{4.56} & \textbf{4.52} & \textbf{4.82} & \textbf{1.69} \\
\bottomrule
\end{tabular}
\vspace{1em}
\label{tab:multi_task_comparison}
\end{table}


\textbf{Analysis of AGGC in comparison with other multi-task learning balancing methods. }Table~\ref{tab:multi_task_comparison} compares multi-task balancing strategies across four benchmarks. The baseline FGC-UFLD (w/o AGGC) achieves reasonable performance, while UW~\cite{kendall2018uncertainty} and DWA~\cite{liu2019attentionmtl} offer limited gains, and GradNorm~\cite{chen2018gradnorm} causes overall degradation, indicating simple loss balancing is insufficient for heterogeneous FLD tasks. PCGrad~\cite{yu2020pcgrad} alleviates conflicts on WFLW-full but degrades 300W-Challenge and COFW, suggesting uniform pairwise projection may introduce unnecessary suppression. In contrast, AGGC consistently delivers the best overall performance: 4.56 on 300W-Challenge, 4.52 on WFLW-full, 4.82 on COFW, while maintaining competitive accuracy on AFLW. By coordinating gradients relative to an anchor task rather than treating all pairs equally, AGGC more effectively resolves conflicts and preserves task-relevant directions, leading to more stable convergence and improved generalization for unified FLD.
\begin{table}[!h]
\centering
\caption{Comparative evaluation of computational complexity and inference efficiency among different module setups.}
\setlength{\tabcolsep}{8pt}
\fontsize{12}{10}\selectfont
\begin{tabular}{c|cc}
\toprule
Method & Params (M) & FLOPs(G) \\
\midrule
Trans (baseline) & 24.90 & 18.13 \\
Trans + DiC-Wave & 34.24 & 18.12 \\
\rowcolor{gray!10} Trans + DiC-Wave + PFAM & 34.40 & 17.17 \\
\bottomrule
\end{tabular}
\label{tab:module_efficiency}
\end{table}

\textbf{Sensitivity analysis of parameters.}As reported in~\ref{tab:module_efficiency}, our proposed model introduces additional parameters and computational overhead compared to the baseline. The baseline Transformer model contains 24.90M parameters and requires 18.13 GFLOPs. With the addition of the DiC-Wave module, the number of parameters increases to 34.24M, while the computational cost remains similar at 18.12 GFLOPs. The integration of both DiC-Wave and PFAM modules further increases the parameter count to 34.40M, with a slight reduction in FLOPs to 17.17. Although these extensions introduce extra parameters, the associated computational cost does not increase substantially, and the performance improvements justify the additional complexity.
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