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\title{A Parallel Network for LRCT Segmentation and Uncertainty Mitigation with Fuzzy Sets}

% The standard author block has changed for UAI 2025 to provide
% more space for long author lists and allow for complex affiliations
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% All author information is authomatically removed by the class for the
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\author[1]{\href{mailto:<s.wang22@imperial.ac.uk>?Subject=A Parallel Network for LRCT Segmentation and Uncertainty Mitigation with Fuzzy Sets}{Shiyi Wang}{}}
\author[1]{Yang Nan}
\author[1]{Xiaodan Xing}
\author[1]{Yingying Fang}
\author[2,3]{Simon LF Walsh}
\author[1,2,3,4]{Guang Yang}
% Add affiliations after the authors
\affil[1]{%
    Bioengineering Department and Imperial-
X.\\
    Imperial College London.\\
    UK
}
\affil[2]{%
    National Heart and Lung In-
stitute.\\
    Imperial College London.\\ 
    UK
}
\affil[3]{%
    Cardiovascular Research
Centre.\\ 
    Royal Brompton Hospital.\\
    UK
  }
\affil[4]{
    Imaging Sciences Dept.\\
    King's College London.\\
    UK
  }
  
  \begin{document}
\maketitle

\begin{abstract}
Accurate segmentation of airways in Low-Resolution CT (LRCT) scans is vital for diagnostics in scenarios such as reduced radiation exposure, emergency response, or limited resources. Yet manual annotation is labor-intensive and prone to variability, while existing automated methods often fail to capture small airway branches in lower-resolution 3D data. To address this, we introduce \textbf{FuzzySR}, a parallel framework that merges super-resolution (SR) and segmentation. By concurrently producing high-resolution reconstructions and precise airway masks, it enhances anatomic fidelity and captures delicate bronchi. FuzzySR employs a deep fuzzy set mechanism, leveraging learnable $t$-distribution and triangular membership functions via cross-attention. Through parameters $\mu$, $\sigma$, and $d_f$, it preserves uncertain features and mitigates boundary noise. Extensive evaluations on lung cancer, COVID-19, and pulmonary fibrosis datasets confirm FuzzySR's superior segmentation accuracy on LRCT, surpassing even high-resolution baselines. By uniting fuzzy-logic-driven uncertainty handling with SR-based resolution enhancement, FuzzySR effectively bridges the gap for robust airway delineation from LRCT data.
\end{abstract}

\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{intro.png}
\caption{Figure 1: Introduction. (a) Motivation: need a new biomarker. (b) LRCT can result in false negatives, leakage, and discontinuities. (c) Our solution proposes a parallel network for super-resolution and segmentation. $\mu$, $\sigma$ is learnable for uncertainty.}
\label{fig1}
\end{figure}

\section{Introduction}\label{sec:intro}
Ensuring CT images are sufficiently accurate is crucial for reliable diagnoses, and medical segmentation inevitably suffers from low-resolution scans \cite{r1}. In settings such as coronary CT angiography (CCTA) or vascular and airway imaging, segmentation quality depends heavily on image resolution \cite{r2,r3}. Below a certain threshold, small but clinically significant structures become difficult to delineate, highlighting the need for methods that enhance resolution \emph{and} improve segmentation performance. Airway dilation serves as a biomarker for diagnosing and prognosticating pulmonary fibrosis \cite{r4,r5} (Figure~\ref{fig1}(a)), calling for an automated approach under noisy, low-resolution conditions. Fuzzy set theory helps address intraclass heterogeneity and interclass homogeneity in LRCT, aiding radiologists' decision-making. Yet many LRCT scans are complicated by noise, making sequential super-resolution (SR) and segmentation both time-consuming and suboptimal \cite{r6,r7,r8,r9,r10}. Airway segmentation, especially near terminal bronchi, also remains challenging \cite{r11}, and while fuzzy sets are adept at managing uncertain boundaries \cite{r4,r12,r13,r14}, combining fuzzy logic with SR for airway segmentation remains underexplored (See Figure~\ref{fig1}(b)). For example, Sale’s and Hülleremeier’s \cite{r43,r44} statistical-uncertainty models provide confidence intervals for the final mask and can estimate the mis-segmentation probability of individual branches. However, airway segmentation usually suffers from blurred boundaries and very small, partly invisible branches—limitations rooted in morphology and subjective ambiguity. In addition, the “black-box” nature of deep learning architectures remains an ongoing concern.

To address these challenges, we propose a parallel network called \textbf{FuzzySR}, which merges fuzzy logic with super-resolution and segmentation to enhance interpretability and reduce uncertainty. Rather than proceeding sequentially, FuzzySR performs super-resolution and airway segmentation concurrently, allowing outputs from the super-resolution decoder to directly benefit the segmentation process. We incorporate fuzzy attention to tackle borderline or noisy regions, where binary logic often omits fine details \cite{r15}. Specifically, \emph{triangular} and \emph{t-distribution} membership functions enable channel-specific fuzzy attention, mitigating feature heterogeneity. By learning parameters ($\mu$, $\sigma$), the network adapts to sub-branches that significantly vary yet still belong to the airway, while heavier tails in the $t$-distribution or localized windows in the triangular function refine airway versus vascular distinctions. Cross-attention further compares membership values adaptively, avoiding pointwise minima and thus more effectively capturing subtle intensity variations in LRCT (See Figure~\ref{fig1}(c)).

Our comparative experiments yielded four critical findings:
\newcounter{mypara}
\renewcommand{\themypara}{\alph{mypara})}
\newcommand{\mypara}[1]{
  \refstepcounter{mypara}
  \par
  \noindent
  \hspace*{1em}%
  \themypara\ #1
}

\mypara {Performing super-resolution and segmentation synchronously is more efficient and yields better outcomes than separate steps.}
\mypara {Including fuzzy attention led to significantly improved results compared to networks without fuzzy logic.}
\mypara {Our proposed FuzzySR achieved performance on LRCT comparable to HRCT-based UNet and FuzzyAttentionUnet, with no significant differences (p=0.89 and p=0.33).}
\mypara {Explainability: Visualizing $\mu,\sigma$ over training epochs and attention maps shows some channels becoming more sensitive to airway features.}

\section{Methodology}
Joint multi-task learning in medical imaging typically outperforms sequential pipelines \cite{r45,r46,r47}. Accordingly, we adopt a Y-shaped network with a shared encoder and two parallel decoders—one for super-resolution (SR), one for segmentation—linked by a fuzzy-gating layer that forwards only SR features useful for segmentation. This design (i) provides mutual regularisation, with SR safeguarding pixel-level detail while segmentation enforces structural semantics; (ii) reduced error propagation - Unlike a sequential pipeline, the segmentation branch in our parallel framework is not solely conditioned on the SR output, thereby preventing the amplification of reconstruction errors. (iii) enables true end-to-end optimisation, letting the segmentation loss directly shape encoder filters; (iv) permits real-time cross-reinforcement between fine-detail and semantic cues that a serial SR→Seg cascade cannot exploit; and (v) handles boundary ambiguity through fuzzy weighting. These considerations furnish both the theoretical and empirical basis for expecting the parallel scheme to surpass sequential SR + Seg approaches.
\subsection{Super Resolution}
Inspired by Luo's work \cite{r16}, our FuzzySR employs a super-resolution network resembling a plain CNN, which outperforms variants like Auto Encoder and UNet that utilize up/down sampling. Although UNet's skip connections slightly improve detail reuse, they still underperform compared to the plain CNN. In our design, eight convolutional layers are followed by LeakyReLU activation, and two fuzzy attention layers compute feature-weight maps for the 3rd and 4th encoder blocks, as well as the 1st and 2nd decoder blocks. Specifically, \textit{att4} is used within the super-resolution model, while \textit{att5} feeds information to the segmentation decoder. Figure~\ref{fig3} (left) outlines this architecture, and Table~\ref{tab:table3} illustrates how fuzzy approaches further enhance super-resolution performance.

\subsection{Proposed FuzzySR}
\begin{figure*}
\centerline{\includegraphics[width=\textwidth]{main_structure.png}}
\caption{Network Structure of FuzzySR. *LR is low-resolution CT. HR is high-resolution CT. IN is InstanceNorm3d. *$E^i$ means the output of encoder. *$D^i$means the output of decoder. "512" denotes the resolution in the x and y axes, whereas "z" specifies the resolution along the z-axis in CT. FAL is Fuzzy Attention Layer.}
\label{fig3}
\end{figure*}

\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{Fuzzy_attention_layer.png}
\caption{Fuzzy Attention Layer. *$E^i$ means the output of encoder. *$D^i$means the output of decoder. The parameters used for the membership functions, such as $\mu$, $\sigma$ and df are trainable. T-distribute,Triangular and Gaussian are fuzzy membership functions. CAL is Cross Attention Layer.}
\label{fig4}
\end{figure}

\begin{figure*}[t]
\centerline{\includegraphics[width=\textwidth]{CAL_CAG.png}}
\caption{Cross Attention Layer \& Cross Attention Gate (CAG).  \( Q_i\) and \( K_i\) are multiple chunk size of flatten query and key arries. \( WQ_i\) and \( WK_i\) are weighted \( Q_i\) and \( K_i\).}
\label{fig5}
\end{figure*}

The fuzzy attention layers form the core of FuzzySR, promoting interaction between super-resolution (SR) and segmentation. The SR decoder’s high-resolution output flows to the segmentation decoder, refining segmentation accuracy, while the segmentation model’s attention map influences the MSE loss in SR. This bidirectional exchange fosters improved predictions of both SR images and airways. An attention gate suppresses irrelevant activations, and because different channels capture different features \cite{r18}, the map must be channel-specific. Unlike conventional attention gates reliant on sigmoid thresholds \cite{r4}, the fuzzy layer leverages T-distribution, Triangular, and Gaussian membership functions, endowing mean and variance with learnable parameters to address LRCT noise, heterogeneous features, and uncertainty.

To achieve parallel SR and airway segmentation within a single large network (FuzzySR), the third and fourth encoder layers for SR are processed by fuzzy attention layers and feed into the SR decoder. In turn, the first and second SR decoder layers pass through fuzzy attention for use as initial features and decoder inputs in the airway segmentation network. These inputs undergo typical UNet-like operations (convolution, up/downsampling). LRCT is provided to FuzzySR, which outputs a super-resolved CT (SRCT) and a predicted airway (Figure~\ref{fig3}). In FuzzySR, the segmentation network fuses outputs from four decoding layers, iteratively computing each layer’s loss against the ground truth to extract richer details from diverse convolutional kernels.

{\bfseries Fuzzy Attention Layer.\bfseries} In the segmentation network, the fuzzy attention layer is applied along the pathway of the skip connections, taking as input the spatial output features from both the encoder and decoder. These two input feature vectors are first passed through a 3-D convolution layer of size 1×1×1, instance normalization, and LeakyReLU for feature reconstruction. This is followed by voxel-level addition operations for information fusion, then another LeakyReLU. Subsequently, the feature representations are fed into the cross attention layer(CAL) combined with fuzzy logic for comparison, generating voxel-level attention maps (Figure~\ref{fig4}).\par 
{\bfseries Cross Attention Layer. \bfseries} After the final LeakyReLU in the Fuzzy Attention Layer, a 3D $1\times1\times1$ convolution and instance normalization produce a multi-channel attention map, which serves as input to the Cross Attention Layer (CAL) \cite{r5} (Figure~\ref{fig5}). This map is fed through membership functions with trainable parameters ($df$, $\sigma$, $\mu$) to determine channel-specific feature importance. Query and Key feature representations are flattened (height, width, depth) and segmented into chunks (defined by chunk size) for the Cross Attention Gate (CAG) (Figure~\ref{fig5}). After cosine similarity and softmax, the weighted Query and Key chunks are reconstructed into a weighted feature representation. Fuzzy logic then selects critical features by combining these weighted representations with encoder feature maps, producing an uncertainty map that reflects overall ambiguity.\par
{\bfseries Fuzzy Membership Functions\bfseries}. In fuzzy systems, Membership Functions are utilized to delineate the fuzziness of pixels, assigning a degree of membership (ranging between 0 and 1) to each element, thereby facilitating fuzzy inference and managing the complexity and uncertainty inherent in real-world scenarios\cite{r19}. Membership Functions can be implemented through various methodologies. For instance, the triangular membership function\cite{r20} is defined by three parameters (a, b, c), forming a triangle where b represents the peak's position, and a and c are the base points. 

This function is straightforward and computationally efficient, making it suitable for applications demanding high computational performance:
\begin{equation}
\text{Triangular}(x) = \max\left(\min\left(\frac{x - a}{b - a}, \frac{c - x}{c - b}\right), 0\right).
\label{eq:triangular}
\end{equation}
where, x represents the fusion generated by the outputs of encoder and decoder. $a$ is the left endpoint, $b$ is the peak point, $c$ is the right endpoint. Where a = $\mu$ - $\sigma$,\quad b = $\mu$,\quad c = $\mu$ + $\sigma$.  $\mu$ is the mean, \( \sigma \) is the standard deviation. When $\sigma$ is small and $\mu$ remains unchanged, the triangular shape becomes more ``pointed,'' assigning higher membership values only near $\mu$. Conversely, if $\sigma$ is larger, 
the function grants higher membership across a broader range.

The t-distribution membership function\cite{r21}, a type of probability distribution function, resembles the standard normal distribution but with ``heavier'' tails. It is typically defined by the degrees of freedom parameter. The t-distribution is suitable for situations with smaller sample sizes, especially when estimating standard deviations. In fuzzy logic, it can be employed to better handle data uncertainty and skewed distributions. In Equation (2), \( \nu \) is the degrees of freedom:
\begin{equation}
\text{T-distribution}(x) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu \pi} \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{(x - \mu)^2}{\nu \sigma^2}\right)^{-\frac{\nu + 1}{2}}.
\label{eq:t-dis}
\end{equation}
Concurrently, fuzzy logic operations (such as AND, OR) define the methodology for amalgamating the membership degrees of two or more fuzzy sets to derive a new membership degree. In the process of fuzzy logic reasoning, it is imperative to initially delineate the fuzzy sets and the membership degrees of elements via fuzzy membership functions. Subsequently, these membership degrees are combined through the application of fuzzy logic operations (e.g., AND, OR), thereby facilitating fuzzy inference and decision-making\cite{r22}. The outcomes of fuzzy logic operations (i.e., the new membership degrees) are then reflected upon the fuzzy membership functions. The higher the membership value, the greater the probability of occurrence. Ultimately, the precise output is derived through the process of defuzzification\cite{r23}. \par

{\bfseries Cross Attention for membership functions \bfseries}Cross Attention forms a similarity matrix (score\_chunk) between query\_chunk and key\_chunk, leveraging global contextual information for feature fusion instead of purely local operations like element-wise minima \cite{r24}. Chunk\_size controls feature partitioning granularity in 3D or high-dimensional data, progressively accumulating local features into global representations. This multi-scale approach captures object boundaries and fine details more effectively than single-scale element-wise minima \cite{r25}, and dynamically balances uneven or noisy inputs via adaptive weight allocation \cite{r26}. In the membership function processing, $Q$ and $K$ are query/key feature tensors, while $Q_{\text{flat}}$ and $K_{\text{flat}}$ denote flattened forms. The parameter (chunk\_size) specifies how many channels $C$ are processed in each chunk. Spatial size is $D \times H \times W$. Query/key chunks undergo cosine similarity $S_i$ and softmax normalization. Weighted results $W_q$ and $W_k$ merge into the global tensor, and their element-wise minimum yields the final attention map, where smaller values indicate greater distance from the fuzzy center.

{\bfseries Attention Loss Functions\bfseries} 
\begin{equation}
\mathcal{L}_{\text{att}} = \lambda \cdot \frac{1}{C} \sum_{i=1}^{C} \frac{1}{N} \sum_{n=1}^{N} \mathbf{A}_i^{(n)} \odot \left( \mathbf{P}_{\text{hr}}^{(n)} - \mathbf{H}^{(n)} \right)^2.\\
\label{eq:attention_loss}
\end{equation}
where The values \( \mathbf{P}_j \) and \( \mathbf{T}_j \) represent the predicted and target values at pixel \( j \), where the intersection is calculated as \( \sum_{j} \mathbf{P}_j^{(n)} \cdot \mathbf{T}_j^{(n)} \), the denominator is obtained by summing the target values \( \sum_{j} \mathbf{T}_j^{(n)} \), and a smoothing term \( \epsilon \) is added to avoid division by zero and improve numerical stability. The weighted sum of the attention map channels \( i \) (total \( C \) channels) is calculated, where each channel's weight map \( \mathbf{A}_i \) is element-wise multiplied by the mean squared error \( \left( \mathbf{P}_{\text{hr}} - \mathbf{H} \right)^2 \) using \(\odot\), followed by batch averaging over \( N \) samples, and scaled by a fixed weight coefficient \( \lambda \).

\begin{table*}
\caption{Comparative Analysis of Model Performance.}
\label{tab:table1}
\resizebox{\textwidth}{!}{
\begin{tabular}{llllll}  % Four columns, alignment: all left
\toprule % Top horizontal line
\multirow{2}{*}{Our Models} & \multicolumn{5}{l}{Evaluation Metrics} \\ \cline{2-6}\\
                         & IoU & DSC & TD & BD & Precision\\

\midrule % In-table horizontal line
UNet (Use LRCT)& $0.7815 \pm 0.067^{*}$
 & $0.8607 \pm 0.042^{*}$
 & $0.8436 \pm 0.076^{*}$
 & $0.8036 \pm 0.100^{*}$
 & $0.8587 \pm 0.048^{*}$\\
UNet (Use HRCT) & $0.8162 \pm 0.056$
& $0.8978 \pm 0.034$	& $0.8709 \pm 0.067^{*}$	& $0.8269 \pm 0.095^{*}$	& $0.8751 \pm 0.050^{*}$\\ % Content row 1
AttentionUnet (Use LRCT)  & $0.7914 \pm 0.050^{*}$
	& $0.8827 \pm 0.031^{*}$	& $0.8744 \pm 0.068^{*}$	& $0.8259 \pm 0.090^{*}$	& $0.8697 \pm 0.042^{*}$ \\% Content row 2
AttentionUnet (Use HRCT)\cite{r33}	& $0.7746 \pm 0.051^{*}$
	& $0.8720 \pm 0.033^{*}$	& $0.9071 \pm 0.054^{*}$	& $0.8761 \pm 0.074^{*}$	& $0.8461 \pm 0.057^{*}$ \\ % Content row 3
FuzzyAttentionUnet (Use LRCT)	&$0.7972 \pm 0.069^{*}$
	&$0.8787 \pm 0.069*$ &$0.8329 \pm 0.095^{*}$	&$0.7828 \pm 0.123^{*}$	&$0.8942 \pm 0.063^{*}$\\

FuzzyAttentionUnet (Use HRCT)\cite{r4}	& \textcolor{red}{$0.8243 \pm 0.062$}	& \textcolor{red}{$0.9032 \pm 0.039$}
	& $0.8566 \pm 0.086^{*}$	& $0.8392 \pm 0.111^{*}$	& $0.9027 \pm 0.049$\\

AttentionUNet + SR (sequence) (Use LRCT)	&$0.7876 \pm 0.052*$ &$0.8802 \pm 0.033*$
	&$0.8963 \pm 0.055*$	& $0.8587 \pm 0.080*$
	&$0.8643 \pm 0.049*$
\\

FuzzyAttentionUnet + SR (sequence) (Use LRCT)	& $0.7934 \pm 0.064*$	& $0.8834 \pm 0.040*$	& $0.8701 \pm 0.066*$	& $0.8457 \pm 0.090*$	& $0.8745 \pm 0.062*$\\

AttentionUNet + SR (parallel) (Use LRCT) &$0.7866 \pm 0.045*$ &$0.8799 \pm 0.028*$ &$0.9136 \pm 0.046*$ &$0.8808 \pm 0.067*$ &$0.8582 \pm 0.036*$\\
\bfseries FuzzySR (parallel) t-dis + triangular (Use LRCT) &\bfseries $0.8174 \pm 0.043$	&\bfseries $0.8989 \pm 0.026$
 &\bfseries $0.9014 \pm 0.056$
	&\bfseries $0.8663 \pm 0.080$
 &\bfseries\textcolor{red}{ $0.9155 \pm 0.032$} \\
\bottomrule % Bottom horizontal line
\multicolumn{6}{p{600pt}}{*All values are denoted by the mean value ± standard deviation. The red font indicates the best results and the bold font indicates our best proposed FuzzySR with best combination of membership functions. TD/BD: Tree/Branch detected ratio. “*” represents statistical significance (with Wilcoxon Signed-Rank Test \( p-value < 0.05 \)) compared with our proposed best FuzzySR t-dis + triangular.}
\end{tabular}
}
\end{table*}

\begin{table*}
\caption{ Comparative Analysis of Model Performance With Different Fuzzy Logic (AND/OR).}
\label{tab:table2}
\begin{tabular}{llllll} % Four columns, alignment: all left
\toprule % Top horizontal line
\multirow{2}{*}{Our Models (Use HRCT)} & \multicolumn{5}{l}{Evaluation Metrics} \\ \cline{2-6}\\
                         & IoU & DSC & TD & BD & Precision\\

\midrule % In-table horizontal line
 % Content row 1
Fuzzy\_Unet (AND) & $0.798 \pm 0.072$ & $0.886 \pm 0.047$ & $0.794 \pm 0.141$ & $0.735 \pm 0.179$ & $0.880 \pm 0.041$ \\ 
\bfseries Fuzzy\_Unet (OR)	& \bfseries $0.824 \pm 0.047$	& \bfseries $0.903 \pm 0.035$	& \bfseries $0.920 \pm 0.067$	& \bfseries $0.879 \pm 0.095$	& \bfseries $0.903 \pm 0.049$\\
\bottomrule % Bottom horizontal line
\end{tabular}
\end{table*}

\begin{table}
\caption{Super-Resolution Network Comparison.}\label{tab:table3}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{llllll}
\toprule % Top horizontal line
Our Models (Use LRCT) & \multicolumn{3}{l}{Evaluation Metrics} \\ \cline{2-4}\\
                         & SSIM & PSNR & RMSE\\
\midrule % In-table horizontal line
UNet & $0.967 \pm 0.026$ & $74.97 \pm 2.52$	& $0.05 \pm 0.02$\\
AutoEncoder & $0.956 \pm 0.030$
 & $75.39 \pm 3.03$
 & $0.05 \pm 0.02$\\
Plain CNN (no fuzzy)& $0.953 \pm 0.030$
 & $73.51 \pm 2.25$
 & $0.06 \pm 0.02$\\
Plain CNN (with fuzzy)& \textcolor{red}{$0.971 \pm 0.024$}
 & \textcolor{red}{$78.55 \pm 3.84$}
 & \textcolor{red}{$0.002 \pm 0.02$}\\
 % Content row 1
SRCNN \cite{r34}& $0.876 \pm 0.021$ & $32.26 \pm 2.99$ & $6.22 \pm 0.19$\\
ESPCN \cite{r35}& $0.872 \pm 0.046$ & $32.06 \pm 2.93$ & $6.36 \pm 0.55$\\
VDSR \cite{r36}& $0.886 \pm 0.054$ & $33.05 \pm 3.22$ & $5.68 \pm 0.36$\\
IMDN \cite{r37}& $0.884 \pm 0.030$ & $34.04 \pm 1.98$ & $5.06 \pm 1.03$\\
PAN \cite{r38}& $0.885 \pm 0.048$ & $33.93 \pm 2.76$ & $5.13 \pm 0.60$\\
DRIDSR \cite{r39}& $0.887 \pm 0.037$ & $34.47 \pm 3.11$ & $4.82 \pm 0.45$\\
\bfseries Proposed FuzzySR(with cl, MF) & \bfseries $0.924 \pm 0.045$ & \bfseries $78.15 \pm 4.83$ & \bfseries $0.04 \pm 0.03$\\
\bottomrule % Bottom horizontal line
\multicolumn{3}{p{251pt}}{*MF is membership function. *cl is central line.}
\end{tabular}%
}
\end{table}

\begin{table*}
\caption{State-Of-The-Art Tansformer Based Models Performed On HRCT Compares To Proposed FuzzySR Performed On LRCT.}
\label{tab:table4}
\resizebox{\textwidth}{!}{
\begin{tabular}{llllll} % Four columns, alignment: all left
\toprule % Top horizontal line
\multirow{2}{*}{Models} & \multicolumn{5}{l}{Evaluation Metrics} \\ \cline{2-6}\\
                         & IoU & DSC & TD & BD & Precision\\

\midrule % In-table horizontal line
FuzzyAttentionUnet (Use HRCT)\cite{r4}	& \textcolor{red}{$0.8243 \pm 0.062$}	& \textcolor{red}{$0.9032 \pm 0.039$}
	& $0.8566 \pm 0.086^{*}$	& $0.8392 \pm 0.111^{*}$	& $0.9027 \pm 0.049$\\

Transformer-based 3D U-Net (HRCT)\cite{r40} &$0.7106 \pm 0.124^{*}$
 &$0.8232 \pm 0.098^{*}$
	&$0.8454 \pm 0.067^{*}$
	& $0.7975 \pm 0.100^{*}$
	&$0.8227 \pm 0.156^{*}$
\\
Segformer (HRCT)\cite{r41}	& $0.7350 \pm 0.066^{*}$ & $0.8455 \pm 0.045^{*}$ & $0.7220 \pm 0.101^{*}$ & $0.6244 \pm 0.124^{*}$ & $0.8337 \pm 0.043^{*}$ \\
SwinUnet (HRCT)\cite{r42} &$0.8134 \pm 0.048$
 &$0.8963 \pm 0.029$
 &$0.7955 \pm 0.097^{*}$
 &$0.7284 \pm 0.119^{*}$
 &$0.9166 \pm 0.030$\\
\bfseries Proposed FuzzySR  (Use LRCT) &\bfseries $0.8174 \pm 0.043$	&\bfseries $0.8989 \pm 0.026$
 &\bfseries $0.9014 \pm 0.056$
	&\bfseries $0.8663 \pm 0.080$
 &\bfseries\textcolor{red}{ $0.9155 \pm 0.032$} \\
\bottomrule % Bottom horizontal line
\multicolumn{6}{p{600pt}}{*All values are denoted by the mean value ± standard deviation. The red font indicates the best results and the bold font indicates our proposed FuzzySR. “*” represents statistical significance (with Wilcoxon Signed-Rank Test \( p-value < 0.05 \)) compared with our proposed best FuzzySR t-dis + triangular.}
\end{tabular}
}
\end{table*}

\section{Results and Discussion}
\subsection{Datasets}
We used 140 instances by combining EXACT'09~\cite{r27}, LIDC-IDRI~\cite{r28}, and an in-house 3D HRCT dataset. We split the entire dataset into training (90), validation (25), and testing (25), each containing original images and validated ground truths. As a preprocessing step, data were cropped into $[128, 96, 64]$ patches centered on pulmonary airways. The final 50 in-house cases were reserved for validation and testing to prevent overlap, ensuring distinct training/validation and test samples. \par
(1) Public binary dataset. The training set included three-dimensional CT scan images depicting typical lung structures, each accompanied by accurate ground truth.\par 
(2) Our in-house dataset. The validation and test sets contained three-dimensional CT scan images of patients diagnosed with COVID-19 and pulmonary fibrosis, specifically for the lung airway segmentation task, effectively reducing the risk of model overfitting. Fibrosis data is now avaliable at MICCAI AIIB23 \cite{r29}.  Meanwhile, this study has ethical approval from the Sydney Local Health District (protocol no. X14–0264) and the local Ethics Committee (code 934/2021/OSS/AOUPR -11.01.2022) at the University Hospital of Parma (UHP).\par 
All models operate on input patches of size [1,1,128,96,144]. The proposed Parallel FuzzySR yields lower MACs (333.46 GMac) than AttentionUNet (364.68 GMac), and its parameter count (15.52 M) remains comparable to other high-performance networks. The inference time is approximately 1.08 seconds, which is still feasible for 3D medical datasets. Although this results in a slight increase in MACs and inference time, it achieves higher accuracy overall.

\begin{figure*}
\includegraphics[width=\textwidth]{all_pred_airway.png}
\caption{Visualization Results of all baseline models and proposed models. *Unet, AteentionUnet and FuzzyAttentionUnet are comparable baseline models. Green areas indicate false positive (FP), blue areas indicate false negative(FN). The blod font model is the proposed best FuzzySR model.}
\label{fig7}
\end{figure*}

\subsection{Quantitative and Qualitative Findings}
{\bfseries Comparison between methods with and without the fuzzy attention mechanism. \bfseries}  In Table~\ref{tab:table1}, FuzzyAttentionUnet using LRCT surpasses UNet in IoU, DSC, and Precision (0.7972, 0.8787, 0.8942). On HRCT, FuzzyAttentionUnet similarly outperforms UNet and AttentionUNet in IoU, DSC, and Precision (0.8243, 0.9032, 0.9027), while AttentionUNet attains higher TD/BD (0.9071/0.8761). For sequential setups, FuzzyAttentionUnet + SR yields higher IoU/DSC/Precision (0.7934, 0.8834, 0.8745) than AttentionUNet + SR but scores lower on TD/BD (0.8701/0.8963 vs.\ 0.8457/0.8587). In the parallel approach, FuzzySR with Gaussian MF achieves better IoU, DSC, Precision, and BD (0.8057, 0.8918, 0.8966, 0.8817) compared with AttentionUNet + SR, albeit slightly lower TD (0.9074/0.9136). Figure~\ref{fig7} shows that among the six baseline models (LRCT and HRCT), FuzzyUnet achieves the best IoU/DSC/Precision, with minimal false negatives and the capacity to predict small unannotated branches, while AttentionUnet achieves higher TD/BD but lower IoU/DSC/Precision. Although this suggests strong terminal-branch prediction, some outputs may reflect tiny branches missed by experts rather than leakage. Comparing both serial and parallel AttentionUnet + SR with FuzzyAttentionUnet + SR confirms that fuzzy-enabled models exhibit fewer false negatives, reduced leakage, and improved detection of small terminal branches.\par
{\bfseries The comparison between parallel and sequential multi-model networks. \bfseries}In comparing sequential vs.\ parallel AttentionUNet + SR, the parallel setup shows only minor differences in IoU, DSC, and Precision (0.001, 0.0003, 0.0061) but achieves higher TD and BD (+0.0173, +0.0221). With comparable leakage, the parallel AttentionUnet + SR yields fewer false negatives and better terminal airway detection (Figure~\ref{fig7}). Meanwhile, the sequential FuzzyAttentionUnet + SR underperforms the proposed FuzzySR, which applies fuzzy logic (OR) and the T-Distribution + Triangular membership functions via cross-attention. This T-Distribution + Triangular setup attains near-zero leakage and minimal false negatives, scoring 0.8174 (IoU), 0.8989 (DSC), 0.9014 (TD), 0.8663 (BD), and 0.9155 (Precision) (Table~\ref{tab:table1}), while preserving robust predictive capabilities for terminal airways.

{\bfseries Comparison between the use of LRCT and HRCT. \bfseries}The UNet using HRCT outperforms its LRCT counterpart on all five metrics (IoU, DSC, TD, BD, Precision) \cite{r1,r2,r3}, while AttentionUNet using LRCT surprisingly surpasses AttentionUNet with HRCT, showing minimal leakage and better terminal airway predictions on HRCT. FuzzyAttentionUnet further improves upon LRCT results by employing HRCT, achieving higher IoU, DSC, TD, BD, and Precision \cite{r4,r5} (Figure~\ref{fig7}). Notably, our FuzzySR with LRCT and T-Distribution + Triangular surpasses the classic UNet with HRCT, scoring 0.8174 (IoU), 0.8989 (DSC), 0.9014 (TD), 0.8663 (BD), and 0.9155 (Precision). Compared with FuzzyAttentionUnet using HRCT, FuzzySR achieves similar IoU, DSC, TD, BD, and even higher Precision, despite slightly lower IoU, DSC, and Precision stemming from more terminal branch predictions (some false positives). Table~\ref{tab:table1} reveals no significant difference (p=0.33 or p=0.89) between FuzzySR and other HRCT-based models, indicating their performance is statistically comparable. Although FuzzyAttentionUnet (HRCT) reports slightly better metrics, it can underperform on certain individual cases, suggesting weaker generalization.

{\bfseries Comparison of Fuzzy Logic. \bfseries}In determining the trachea’s importance across channels and performing fuzzy logic operations, the AND (intersection) operator marks a voxel as airway only if multiple features concur, whereas the OR (union) operator requires at least one feature to indicate airway membership. Although the AND operator enforces stricter segmentation, it may under-segment terminal bronchi, while the OR operator often predicts more terminal branches but admits some false positives. Table~\ref{tab:table2} confirms that AND logic causes a larger decline in TD/BD metrics (by 0.162 and 0.144) compared to OR logic, whereas IoU, DSC, and Precision drop by 0.026, 0.017, and 0.004, respectively. This finding aligns with the theoretical trade-off and justifies using OR logic for all airway prediction models \cite{r10}.\par

\begin{figure}
\includegraphics[width=1.0\linewidth]{SRCT.png}
\caption{ Comparison between original HRCT (Ground Truth) images, LRCT (Low-Resolution CT input images), and SRCT (Super-Resolution CT output images).}
\label{fig8}
\end{figure}

\begin{figure}
\includegraphics[width=1.0\linewidth]{att_map.png}
\caption{Visualization of attention maps for different feature channel indices. (a) Feature channel index = 3, (b) Feature channel index = 10.}
\label{fig9}
\end{figure}

{\bfseries Comparison of the super resolution network. \bfseries} In line with Section I.A, simpler architectures without up/downsampling (e.g., Plain CNN) can deliver better results without extra computational load. Table~\ref{tab:table3} compares PlainCNN with and without the fuzzy algorithm, revealing that PlainCNN with fuzziness yields superior SSIM, PSNR, and RMSE (0.971, 78.55, and 0.002) than UNet and AutoEncoder (0.967/0.956, 74.97/75.39, and 0.05/0.05). Also, PlainCNN (with fuzzy) also superior the metrics than PlainCNN (without Fuzzy) in SSIM, PSNR and RMSE (0.953, 73.51, 0.06). In FuzzySR, the super-resolution CT output alongside airway predictions outperforms most SR methods \cite{r34,r35,r36,r37,r38,r39}, achieving SSIM, PSNR, and RMSE of 0.924 ± 0.045, 78.15 ± 4.83, and 0.038 ± 0.028 (Table~\ref{tab:table3}). As shown in Figure~\ref{fig8}, the SRCT predicted by FuzzySR exhibits finer textures and is visually closer to HRCT than LRCT.

\begin{figure}
\includegraphics[width=1.0\linewidth]{mu_sigma.png}
\caption{Evolution of the learnable $\mu$ (mu) and $\sigma$ (sigma) across different feature channels,  
as training epochs increase and Dice scores vary. fuzzy in this figure means different fuzzy subsets. r,p were calculated by Pearson correlation coefficient.}
\label{fig10}
\end{figure}

\begin{figure*}
\includegraphics[width=0.9\textwidth]{uai2025-template/MF_00.png}
\caption{Comparison between the ground-truth airway annotation and the corresponding fuzzy membership values. 1(a) and 1(b) visualize the global airway fuzzy membership, where the color gradient from blue to red indicates 
membership values ranging from high to low (0–1). Meanwhile, 2(a)/2(b), 3(a)/3(b), and 4(a)/4(b) each display the 
zoomed-in view of local membership regions.}
\label{fig11}
\end{figure*}

{\bfseries SOTA Transformer-based Model Compares With Our Proposed FuzzySR.  \bfseries} Due to the proposed FuzzySR employs a t-distribution and triangular MF for cross-attention fusion, we not only compared it against the baseline AttentionUnet but also against several state-of-the-art methods, including our benchmark model FuzzyAttentionUnet (using HRCT)\cite{r4}, Transformer-based 3D U-Net (HRCT)\cite{r40}, Segformer (HRCT)\cite{r41}, and SwinUnet (HRCT)\cite{r42}. Except for the benchmark model, the other three transformer-based methods—despite utilizing HRCT for training—performed worse than the proposed FuzzySR framework, which was trained solely on LRCT, across all five evaluation metrics (See Table~\ref{tab:table4}). Although SwinUnet’s performance is close, it remains slightly below that of the proposed FuzzySR in IoU (0.8134 vs. 0.8174) and DSC (0.8963 vs. 0.8989). Two metrics are notably lower than those of the proposed FuzzySR—TD (0.7955 vs. 0.9014) and BD (0.7284 vs. 0.8663). SwinUnet achieves a marginally higher Precision value, surpassing the proposed FuzzySR by only 0.0011. 

{\bfseries Explainability of Our Proposed FuzzySR.  \bfseries} In medical image segmentation, certain structures (e.g., pulmonary airways) often occupy a limited grayscale range but may face uncertainty from noise or pathologies. The membership function provides higher membership near $\mu$ and decreases it for more distant intensities. To validate learnable $\mu,\sigma$ in our FuzzySR for mitigating airway-segmentation uncertainty, we examined two feature channels---one (a)~Channel $=3$ targeting brighter intensities and another (b)~Channel $=10$ focusing on darker intensities (Figure~\ref{fig9}), revealing distinct types of airway boundaries or noise regions. As shown in Figure~\ref{fig10}, the network gradually shifts each channel’s $\mu$ toward relevant intensity ranges $(a,c,e)$, indicating stronger resilience to fuzziness and noise. Once $\mu,\sigma$ stabilize, the channel gains “confidence” in that intensity range: a high $\mu$ highlights brighter zones $(a)$, while a low $\mu$ focuses on lower-intensity areas $(c,e)$. A larger $\sigma$ tolerates outliers $(b,d)$, whereas a smaller $\sigma$ narrows the focus, improving detail yet risking noise sensitivity $(f)$. Pearson’s correlation analysis reveals significant positive or negative links between Dice scores and $\mu,\sigma$ in certain channels ($p<0.05$ or $p<0.01$), confirming strong interpretability for both high- and lower-intensity regions. 

In Figure ~\ref{fig10}, certain high–sensitivity channels appear to map to specific airway sub-structures. 
Because the pulmonary tree forms a hierarchical network whose branches vary in calibre and morphology, 
some channels specialise in detecting peripheral bronchioles, segmental bronchi, or the main bronchus. 
Abrupt shifts in a channel’s learnable $\mu$ or $\sigma$ thus mark strong responsiveness and exert a major 
influence on the final segmentation. 
Channels that focus on ultra–thin branches or high–contrast edges rapidly adapt their fuzzy functions during 
training and can markedly raise the Dice score. 
When the channel-wise $(\mu,\sigma)$ values correlate strongly with Dice (e.g., $r\approx0.97\text{--}0.98$), 
the channel is likely performing critical \emph{discrimination} or \emph{selection}, thereby driving the overall improvement.
Clinically, if a channel “locks onto’’ a distinctive morphology—such as tiny peripheral airways—and exhibits such a 
high performance correlation, it indicates refined differentiation of those structures or pathologies. 
Hence, these high–sensitivity channels capture regions that are otherwise difficult to delineate 
(e.g., very fine or distorted bronchi), implying enhanced discriminative power precisely where segmentation 
is most challenging and providing potentially valuable insights for diagnosis and intervention.

Figure ~\ref{fig11} visualises global and local uncertainty maps: high-uncertainty regions (green) cluster along airway boundaries, indicating that the super-resolution branch—by reconstructing fine details—also contributes to boundary awareness instead of relying solely on the segmentation branch. In clinical airway assessment, one must evaluate not only airway location but also subtle deformities, stenoses, and other fine-scale changes. By retaining boundary uncertainty, fuzzy logic supports more precise downstream quantification—for example, detecting incipient luminal narrowing. A hard-threshold mask risks discarding the borderline voxels (e.g., a value of 0.49 forced to 0), thereby masking early pathology. Fuzzy membership maps, in contrast, allow graded updates, prevent abrupt structural jumps, and give clinicians a smoother, more reliable basis for local refinement and intervention planning.

In two separate models (super-resolution and segmentation), feature maps from the super-resolution decoder do not undergo fuzzy attention nor interact with the segmentation decoder. By contrast, our proposed FuzzySR embeds the fuzzy algorithm within the SR network and fuses intermediate four-layer features through a fuzzy attention gate, subsequently feeding them into the segmentation network (Fig.~\ref{fig3}). We also integrate central line assistance, fuzzy logic (OR), central line loss, attention loss, cross-attention fusion, and triangular/t-distribution membership functions. As shown in Table~\ref{tab:table1} and Table~\ref{tab:table4}, this combination achieves the best airway prediction accuracy. FuzzySR significantly outperforms sequential SR+segmentation models across all metrics ($p < 0.01$), while the single SR model (PlainCNN with fuzzy) surpasses baseline UNet and AutoEncoder in super-resolution precision. FuzzySR likewise outperforms many state-of-the-art SR approaches. Furthermore, fuzzy membership functions combined with logical operations (OR/AND/min) enable ``multiple parallel judgments'' across scales and channels, preserving high-likelihood branches and avoiding overly strict thresholds that may exclude relevant structures.

\begin{acknowledgements} 
    Guang Yang was supported in part by the ERC IMI (101005122), the H2020 (952172), the MRC (MC/PC/21013), the Royal Society (IEC\textbackslash{}NSFC\textbackslash{}211235), the NVIDIA Academic Hardware Grant Program, the \textit{SABER} project supported by Boehringer Ingelheim Ltd, the NIHR Imperial Biomedical Research Centre (RDA01), the Wellcome Leap \textit{Dynamic Resilience} programme (co-funded by Temasek Trust), UKRI guarantee funding for Horizon Europe MSCA Postdoctoral Fellowships (EP/Z002206/1), the UKRI MRC Research Grant, TFS Research Grants (MR/U506710/1), and the UKRI Future Leaders Fellowship (MR/V023799/1).
\end{acknowledgements}

% References
\bibliography{uai2025-template}

\newpage

\onecolumn

\title{A Parallel Network for LRCT Segmentation and Uncertainty Mitigation with Fuzzy Sets\\(Supplementary Material)}
\maketitle

\appendix
\section{Introduction}
\textbf{Integration of Central Lines and Attention Loss. }%
(1) Central-Line-Based Loss. We incorporate manually extracted and expert-modified central lines of the airways into the loss function, thus improving the detection around branching areas. This step addresses the difficulty in capturing fine, distal airways and ensures that the parallel network remains sensitive to small structural details. (2) Attention Loss. We additionally propose an Attention Loss, which uses the fuzzy attention maps generated by the segmentation branch to guide the super-resolution branch, ensuring that terminal airway regions receive higher emphasis. This design further facilitates the capture of subtle airway structures in low-resolution scans and reinforces the synergy between super-resolution and segmentation.
\section{Additional  Methodology}
\subsection{Central Line Extraction}
Certain Python packages offer functions frequently used for segmenting tubular structures like airways and vessels. For instance, one function from the Skimage package processes a three-dimensional Boolean array (a pre-processed binary image) and produces another Boolean array of identical dimensions, indicating the skeleton of the original shape. This skeletonization process iteratively strips away outer pixels until only the core structure remains. However, its output can be inaccurate, often retaining redundant looping centrelines that diminish segmentation precision. Consequently, we manually extract centrelines during data preprocessing and then rely on expert corrections to ensure more accurate centreline segmentation.\par
After medical experts annotated the tracheal central line, we incorporated it as a corrective measure in the training process. Comparative analyses demonstrated pronounced improvements in quantitative metrics (e.g., Dice Similarity Coefficient, Intersection over Union, and precision) once the central line was included. Meanwhile, removing falsely segmented regions (false positives) led to minor decreases in the branch score (BD) and tree detected ratio (TD). Nonetheless, these findings underscore the clinical importance of integrating the central line, which enhances the model’s predictive accuracy and delivers better diagnostic support for airway-related pathologies—consistent with the primary objective of our study.
{\bfseries Refinement of Pulmonary Airway Central Lines: \bfseries} The skeletonize\_3d() function in the skimage.morphology module of the SciKit-Image Python library is frequently used to obtain object skeletons in 3D images. However, since it is designed for general 3D image processing rather than tracheal anatomy, it may yield tracheal centerlines lacking the smoothness and branching clarity needed for accurate analyses. In contrast, MIMICS\cite{r17} provides superior centerline extraction compared to skeletonize\_3d(), though further radiologist refinement remains necessary to ensure data reliability. Consequently, expert manual corrections are indispensable for obtaining accurate tracheal centerlines, even after the use of skeletonize\_3d() and MIMICS (see Figure~\ref{fig2}).

\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{cl.png}
\caption{Visualization of Airway Parsing Results. (a) represents the tree parsing results based on the central line extracted by skeletonize\_3d() function. (b) is the inaccurate central line extracted by skeletonize\_3d() function. (c) shows the accurate tree parsing results based on the central line refined by radiologists. (d) represents an accurate, smooth, and nodule-free refined central line.}
\label{fig2}
\end{figure}

Figure~\ref{fig2} highlights the limitations of traditional centerline extraction functions, which, although convenient, are not tailored to the trachea’s spatial configuration and can generate central lines with looping and nodules. Conversely, medical software specifically designed for pulmonary airways removes most looping and nodules to produce smoother, more coherent tracheal centerlines. Nonetheless, small branches still require radiologist intervention. These refined centerlines afford more precise airway parsing, reducing inaccurate branch counts and diminishing erroneous segments from 236 to 145. In Section IV (RESULTS AND DISCUSSION), all airway prediction metrics reported are derived using the refined centerlines described in this section to support model training.

{\bfseries Proposed Loss Functions\bfseries} The proposed FuzzySR incorporates two novel loss functions: the Central Line Loss and the Attention Loss. Each branch is assigned a unique numerical identifier, which is determined through computational processes. The key to this calculation lies in the hierarchical structure defined by the parent-child relationships. This computation employs tree parsing methodologies to delineate the branches pertinent to each segment. The intersection metric is derived by multiplying the forecasted branch for every segment with its respective ground truth (GT) branch and then aggregating these multiplicative values. The aggregate of the branches delineated in the ground truth across all segments constitutes the calculation's denominator. 
Therefore, the formula for branch loss is given by equation (3) and (4). 
The formula for central line loss is given by following equations:
\begin{equation}
\mathcal{L}_{\text{CL}} = \frac{1}{N} \sum_{n=1}^{N} \left( 1 - \frac{\sum_{j} \mathbf{P}^{(n)}_j \cdot \mathbf{T}^{(n)}_j + \epsilon}{\sum_{j} \mathbf{T}^{(n)}_j + \epsilon} \right).\\
\label{eq:CL_loss}
\end{equation}

\begin{equation}
\begin{aligned}
\mathcal{L}_{\text{airway}} = \lambda \cdot \sum_{i=1}^{C} 
\left( \mathcal{L}_{\text{BCE}}(\mathbf{P}_i, \mathbf{T}) + \mathcal{L}_{\text{Dice}}(\mathbf{P}_i, \mathbf{T}) \right) \\
+ \lambda \cdot \sum_{i=1}^{C} \mathcal{L}_{\text{CL}}(\mathbf{P}_i, \mathbf{C}).
\end{aligned}
\label{eq:airway_loss}
\end{equation}

Inaccuracies in airway central lines extracted via a Python library function, including imprecise branching and looping, led our experts to refine these lines. The improved lines were then used as a loss metric in our model's training. The proposed attention loss is designed by weighting the computed mean squared error (MSE) with the weighted attention map extracted from FuzzySR. This mechanism provides feedback to the super-resolution regions based on the varying importance of airway feature distributions. Theoretically, the proposed attention loss allows FuzzySR to allocate more focus to the pixels surrounding the airway features, thereby improving the accuracy of airway predictions.

\section{Experiment Setup}
{\bfseries Coding Environment and Radiologist Annotation tools. \bfseries} The study employed a quad-GPU setup featuring an NVIDIA TITAN X (12GB VRAM). For annotation, we utilized 3D Slicer\cite{r30}, ITK-Snap\cite{r31}, MeshLab\cite{r32}, and MIMICS\cite{r17}. Specifically, experts used ITK-Snap for the annotation of the pulmonary airways and corrections of the central line. MeshLab converted airway ground truths (annotated in ITK-Snap in NIFTI format) into meshes. 3D Slicer transformed 3D CT data into DICOM, a medical imaging standard. These DICOM CT scans and meshes were then processed in MIMICS to generate central lines, followed by expert annotation. We plan to share our findings, including source code, with the scientific community in due course.
{\bfseries Performance Evaluation Metrics: \bfseries} The efficacy of the model was assessed through predictions conducted on the test dataset, subsequently benchmarked against the ground truth annotations provided by radiologists. Four key metrics were employed for this evaluation: the Dice Similarity Coefficient (DSC), Precision, Tree Detected Ratio (TD), and Branch Detected Ratio (BD). These measures facilitated a comprehensive analysis of the model's performance in terms of accuracy, precision, and its ability to accurately detect tree and branch structures within the pulmonary airway. To validate the performance of the proposed FuzzySR in super-resolution, three evaluation metrics were introduced: SSIM(Structural Similarity Index Measure), PSNR(Peak Signal-to-Noise Ratio), and RMSE(Root Mean Square Error). SSIM  is an index that measures the similarity in visual effects between two images. It considers changes in luminance, contrast, and structure of the images, with values ranging from -1 to 1, where 1 indicates identical images. This measure aligns more closely with human visual perception of image quality. PSNR  is one of the most commonly used metrics for assessing the quality of image reconstruction. It evaluates image quality by calculating the relationship between the maximum possible pixel value of the original image and the mean squared error (MSE) with the reconstructed image. A higher PSNR value indicates better image quality. RMSE  directly measures the pixel-level differences between the original and reconstructed images. A lower RMSE value indicates that the image is closer to the original, signifying higher image quality.

\begin{algorithm}[t]
\caption{Cross Attention}\label{alg:alg1}
\begin{algorithmic}
\STATE \textbf{class} \textsc{Cross\_Attention}$(\text{Q,K})$
\STATE \hspace{0.5cm}\textbf{initialize:}
\STATE \hspace{1cm} $\text Q_{\text{flat}}, K_{\text{flat}} \in \mathbb{R}^{B \times C \times (DHW)}$
\STATE \hspace{1cm} $ W_Q, W_K \in \mathbb{R}^{B \times C \times D \times H \times W}$
\STATE \hspace{1cm} $\hat{S}_i \in \mathbb{R}^{B \times \text{chunk\_size} \times \text{chunk\_size}}$

\STATE \textbf{function} \textsc{forward}$(Q,K)$
\STATE \hspace{0.5cm} \textbf{for i = 0 to spatial\_size by chunk\_size:}
\STATE \hspace{1cm} $ Q_i = Q_{\text{flat}}[:, :, i:i+\text{chunk\_size}]$
\STATE \hspace{1cm} $ K_i =  K_{\text{flat}}[:, :, i:i+\text{chunk\_size}]$
\STATE 
\STATE \hspace{1cm} $ S_i = \text{cosine\_similarity}(Q_i, K_i) = \frac{Q_i^\top \cdot K_i}{\|Q_i\| \cdot \|K_i\|}$
\STATE 
\STATE \hspace{1cm} $ \hat{S}_i = \text{softmax}\left(\frac{S_i}{\text{scale}}\right)$
\STATE 
\STATE \hspace{1cm} $ W_{Q_i} = \hat{S}_i \cdot Q_i^\top, \quad W_{K_i} = \hat{S}_i \cdot K_i^\top$
\STATE \hspace{1cm} $ W_Q[:, :, i:i+\text{chunk\_size}] += W_{Q_i}$
\STATE \hspace{1cm} $ W_K[:, :, i:i+\text{chunk\_size}] += W_{K_i}$
\STATE
\STATE \hspace{0.5cm} $ attention\_map = \min(W_Q, W_K)$
\STATE \hspace{0.5cm} \textbf{return} $ attention\_map$
\end{algorithmic}
\end{algorithm}

\end{document}