\section{Introduction}
Decreasing spatial resolution in CNN's forward propagation engenders difficulty in learning high-resolution global features which affects pixel-wise image segmentation quality. Preserving spatial resolution therefore, has been a consistent design choice among a number of high-precision CNN models \cite{wang2020deep} proposed in recent times. However, such design choices \cite{han2022seethroughnet} lead to computational overhead due to having multiple ResNet and transformer blocks running in parallel at high-resolution (spatial). Despite their high precision, training such models are often challenging in the presence of constraints in computational budget and n-dimensional imagery where $n \in \{\mathcal{Z}, n>2\}$. To mitigate this bottleneck, we propose \textbf{R}eceptive \textbf{F}ield \textbf{C}hain \textbf{Net}work \textbf{(RFC-Net)} by means of devising the following:
\begin{equation}
  \|\Theta_{SDCS}\|_{l}^{l+1} = d_{l+1} (\mathbf{[k^{2}]} d_{l} + \mathbf{d_{l+1}})
  \label{eqn:param_SDCS}
\end{equation}
\begin{itemize}
    \item Introduce \textbf{L}oose \textbf{D}ense \textbf{C}onnection \textbf{S}trategy \textbf{(LDCS)} for reducing the number of parameters.
    \item Design a \textbf{m-way Tree structure} to learn features through a chain constituting all possible combination of different receptive fields from a selected set of kernels.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[height=3.95cm, width=\columnwidth]{images/RFC.png} 
\caption{Enhanced view recommended. \textbf{(a):} Conventional Strong Dense Connection Strategy (SDCS). \textbf{(b):} Proposed Loose Dense Connection Strategy (LDCS). \textbf{(c):} Proposed RFC-Net Architecture: Expanding with m-way tree on uncompressed resolution with LDCS (DN: Downsample). \textbf{(d):} Comparison between the quality of RFC-Net and UNet's prediction $(\hat{y})$ mask on Kvasir dataset (GT: Ground Truth).}
\label{fig:illustration}
\end{figure}
\section{Methodology}
As constituent of our proposed RFC-Net we first introduce Loose Dense Connection Strategy (LDCS) which in comparison with conventional Strong Dense Connection Strategy (SDCS) aids in reducing the number of parameters as illustrated in Figure ~\ref{fig:illustration}(a-b). We define strong connection between two neurons as being filtered through a $k \times k$ convolutional kernel $(\forall k>1)$ since such connections enable learning spatial correlations whereas, we define loose connection as using a $k \times k$ kernel $(\forall k=1)$ and this operation is equivalent to that of a linear layer with no propagation of spatial information about neighbouring pixels. Using $k \times k$ kernels, conventional SDCS \cite{ronneberger2015u} strongly $(k>1)$ connects every neurons from $l^{th}$ layer to accrue each neuron in the $(l+1)^{th}$ layer and Equation ~\ref{eqn:param_SDCS} denotes the number of parameters $(\|\Theta\|)$ to constitute the entire $(l+1)^{th}$ layer where $d_{l}$ is the total number of neurons in layer $l$, $l=\{1,2,..,L\}$ and L is the total number of layers. However, in LDCS we split the number of neurons in $l^{th}$ layer to $n_{l}$ groups. Using our proposed LDCS, we first of all strongly $(k>1)$ connect one of those $n_{l}$ groups in $l^{th}$ layer to one of those $n_{l+1}$ groups in $(l+1)^{th}$ layer. As illustrated in Figure ~\ref{fig:illustration}(b), we then concatenate the other ($d_{l}-n_{l}$) groups in $l^{th}$ layer and loosely $(k=1)$ connect them to the previously chosen $n_{l+1}^{th}$ group in $(l+1)^{th}$ layer where we performed the strong connection with $n_{l}^{th}$ group. Finally, we concatenate the strongly and loosely connected neurons to the aforementioned $n_{l+1}^{th}$ group and pass through one more $(1\times1)$ kernel to construct the final $n_{l+1}^{th}$ group in $(l+1)^{th}$ layer. Equation ~\ref{eqn:param_LDCS} depicts the number of parameters $(\|\Theta\|)$ required to construct the entire $(l+1)^{th}$ layer using LDCS. The computation reduction factors driven by LDCS are \textbf{highlighted} in Equation ~\ref{eqn:param_SDCS} and ~\ref{eqn:param_LDCS}. 
\begin{equation}
  \|\Theta_{LDCS}\|_{l}^{l+1} = d_{l+1} (\left[\mathbf{\frac{k^{2}}{n_{l}}}+\frac{n_{l}-1}{n_{l}}\right] d_{l} + \mathbf{\frac{d_{l+1}}{n_{l+1}}})
  \label{eqn:param_LDCS}
\end{equation}
\begin{equation}
  x_{i}^{l+1} = \mathcal{F}_{1\times1}\mathbf{(}\mathcal{F}_{k\times k}(x_{\lceil i/m \rceil}^{l}) \oplus \mathcal{F}_{1\times1}(\oplus_{j \in n_{l}}x_{j\neq \lceil i/m \rceil}^{l})\mathbf{)}
  \label{eqn:propagation_forward}
\end{equation}

Secondly, we introduce a m-way Tree structure into the network and the number of groups at layer $l$, $n_{l}=m^{l}$. We repeatedly deploy m kernels $(k>1)$ of different receptive fields throughout each layers across the network as shown in Figure ~\ref{fig:illustration}(c). Equation ~\ref{eqn:propagation_forward} shows how connecting these groups across layers with LDCS using a m-way Tree structure thus facilitates the propagation of features through a chain constituting all possible combination of different receptive fields with different values of $k$, where $\mathcal{F}_{k\times k}$ is a convolution kernel of size $k\times k$ and $x_{i}^{l}$ denotes $i^{th}$ intermediate feature representation at layer $l$. However, connecting only one group from a layer to that of the next one (considering only the $\mathcal{F}_{k\times k}$ portion in Equation ~\ref{eqn:propagation_forward}) leads to amassing $m^{L}$ number of week-performing segregated networks due to lack of information exchange. On the contrary, connecting all of the groups using the conventional SDCS strategy will result in a combinatorial explosion in computation. We observe that, replacing $n_{l}$ in Equation ~\ref{eqn:param_LDCS} with $m^{l}$ exponentially reduces the computation constraints. Therefore, juxtaposing m-way Tree structure with different receptive fields and LDCS enables Receptive Field Chain Network termed as RFC-Net (Figure ~\ref{fig:illustration}(c)) to learn high-resolution aware features within computational budget. Finally, learning features on a hierarchical space of cascaded receptive fields without severe spatial downsampling (we only downsample twice before passing through the m-way tree, as illustrated in Figure ~\ref{fig:illustration}(c)) helps RFC-Net learn more robust global features.
\section{Experiments, Results And Observations}
We evaluate RFC-Net on three binary segmentation benchmarks- (1) Kvasir SEG Polyp Dataset (Gastrointestinal Disease Detection, train:test=850:150), (2) GlaS Dataset (Gland Segmentation in Colon Histology, train:test=132:33), (3) CVC-ClinicDB (Polyp segmentation from colonoscopy video frames, train:test=521:91). We use SGD optimizer with momentum=0.9, weight decay=0.0005. Furthermore, we train (without data augmentation) all our models on the benchmarks with Online Hard Example Mining Cross Entropy (OHEM CE) loss (threshold=0.7). Firstly, we train Kvasir SEG Polyp Dataset (resized to 200$\times$200, batch size of 4) for 160 epochs (base learning rate=0.01, step size=45, converges in 77 epochs). Secondly, we train GlaS Dataset (resized to 300$\times$300, batch size of 2) for 500 epochs (base learning rate=0.01, step size=60, converges in 259 epochs). Thirdly, we train CVC-ClinicDB Dataset (resized to 200$\times$300, batch size of 3) for 160 epochs (base learning rate=0.01, step size=45, converges in 85 epochs). We ran all of our experiments on PyTorch using one Nvidia RTX 3090 and one Nvidia RTX 3070 GPU.

In Table ~\ref{tab:results_and_ablation}, we narrate how RFC-Net performs in comparison to existing heavier and lighter models. We observe that while reducing parameters RFC-Net$^{a}$ outperforms the existing models in Kvasir and CVC-ClinicDB benchmarks and exhibits comparable performance on GlaS dataset. We further perform an ablation study (Table ~\ref{tab:results_and_ablation}), to inspect the efficacy of our model. We notice that RFC-Net$^{b}$ with even lesser parameters outperforms the existing models on kvasir and CVC-Clinic DB benchmark. We further observe that, it is the exploitation of the m-way tree structure by RFC-Net$^{a}$ for using enhanced receptive field that leads to a sharp performance gain (RFC-Net$^{b}\mapsto$RFC-Net$^{a}$, RFC-Net$^{d}\mapsto$RFC-Net$^{c}$). Additionally, the performance of RFC-Net's lightweight versions (RFC-Net$^{c}$, RFC-Net$^{d}$) is substantially higher than that of ESPNet-C which is another existing lightweight CNN for segmentation. Figure ~\ref{fig:illustration}(d) further depicts the higher quality (in comparison to UNet) of prediction masks produced by RFC-Net. Being computationally inexpensive, RFC-Net can be further applied to 3D and 4D image segmentation for \textit{Computer Aided Diagnosis}.
\begin{table}[t]
\centering
\resizebox{.99\columnwidth}{!}{
\begin{tabular}{l|cc|ccc}
    \hline
     & \multicolumn{2}{c|}{\textit{Computation}} & \multicolumn{3}{c}{\textit{Performance (mIOU \%)}} \\ \hline
    \textit{Model} & \textit{Params} & \textit{GFLOPs} & \textit{Kvasir} & \textit{GlaS} & \textit{CVC-DB} \\ \hline
    U-Net & 07.76M & 30.75B & 74.39 & 83.51 & 75.91 \\ \hline
    U-Net++ & 09.04M & 34.91B & 75.95 & 84.02 & 79.27 \\ \hline
    ResUNet & 13.04M & 43.56B & 77.77 & \textbf{85.67} & 81.33 \\ \hline 
    ESPNet-C & 00.41M & 02.49B & 75.95 & 65.43 & 72.53 \\ \hline
    \textbf{RFC-Net$^{a}$} & \textbf{05.76M} & \textbf{18.13B} & \textbf{81.31} & 77.88 & \textbf{85.90} \\ \hline
    RFC-Net$^{b}$ & 04.49M & 14.03B & 79.15 & 75.34 & 83.34 \\ \hline
    RFC-Net$^{c}$ & 00.39M & 01.27B & 76.41 & 75.49 & 79.51 \\ \hline
    RFC-Net$^{d}$ & 00.28M & 00.91B & 73.24 & 66.17 & 77.68 \\ \hline
\end{tabular}
}
\caption{Results on Kvasir, GlaS, CVC-ClinicDB Datasets. GFLOPs are calculated on $224\times 224$ RGB image. RFC-Net$^{a}$:[m=3,k=3,5,7], RFC-Net$^{b}$:[m=3,k=3,3,3], RFC-Net$^{c}$:[m=2,k=3,5], RFC-Net$^{d}$:[m=2,k=3,3].}
\label{tab:results_and_ablation}
\end{table}
