## 3 Methods
### 3D Separability of Convolutional Kernels  
Given a volumetric image, when we employ a 3D convolution kernel to generate a 3D feature volume, the input to the network is the entire volumetric data. By leveraging the kernel sharing across all three dimensions, the network can take full advantage of the volumetric contextual information. Generally, the following equation formulates the exploited 3D convolution operation with stride one in an element-wise fashion:  
\[V^{l}_{jk}(x,y,z)=\sum_{x^{\prime}=1}^{X}\sum_{y^{\prime}=1}^{Y}\sum_{z^{ \prime}=1}^{Z}F^{l-1}_{k}(x-x^{\prime},y-y^{\prime},z-z^{\prime})\times W^{l} _{jk}(x^{\prime},y^{\prime},z^{\prime}) \tag{1}\]  
where \(W^{l}_{jk}\) is the 3D kernel of size \(X\times Y\times Z\) in the \(l\)th layer, which is connected to the \(k\)th input feature volume \(F^{l-1}_{k}\) in the previous layer, and the \(j\)th output feature volume \(F^{l}_{j}\), \(W^{l}_{jk}(x^{\prime},y^{\prime},z^{\prime})\) is the element-wise value of the 3D convolution kernel. Assume the \(l\)th layer has \(K\) input feature volumes, and let \(\sigma(\cdot)\) denote the element-wise nonlinear activation function and \(b^{l}_{j}\) the corresponding bias term; the output feature volume \(F^{l}_{j}\) is obtained as:  
\[F^{l}_{j}=\sigma(\sum_{k=1}^{K}V^{l}_{jk}+b^{l}_{j}) \tag{2}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Method** & **Structure** & **Domain** \\ \hline
[17] & 2D to 1D & general image \\
[18] & 2D to 1D, depthwise separation & general image \\
[20; 21; 22] & depthwise separation & general image \\
[13; 14; 15; 16] & fine tune the trained model (LRA) & general image \\
[19; 23] & 3D to 2D + 1D & medical image \\
3D-DSC & 3D to 1D & medical image \\ \hline \hline \end{tabular}
\end{table} 
Table 1: General comparison with other related methods. LRA, Low Rank Approximation. Mathematically, the 3D kernel tensor \(W_{jk}^{l}\) can be factorized into a linear combination of rank 1 tensors according to the CP decomposition:  
\[W_{jk}^{l}=\sum_{r=1}^{R}a_{jkr}^{l}\otimes b_{jkr}^{l}\otimes c_{jkr}^{l} \tag{3}\]  
where \(R\) is the rank of \(W_{jk}^{l},\otimes\) denotes the outer product operation and \(a_{jkr}^{l}\in\mathbb{R}^{X}\), \(b_{jkr}^{l}\in\mathbb{R}^{Y}\), \(c_{jkr}^{l}\in\mathbb{R}^{z}\) are 1D vectors. Element-wise, the above equation can be rewritten as:  
\[W_{jk}^{l}(x,y,z)=\sum_{r=1}^{R}a_{jkr}^{l}(x)b_{jkr}^{l}(y)c_{jkr}^{l}(z) \tag{4}\]  
Substituting (4) into (1) gives the following equivalent expression for the evaluation of the 3D convolution:  
\[V_{jk}^{l}(x,y,z)=\sum_{r=1}^{R}\big{(}\sum_{z^{\prime}=1}^{Z}\big{(}\sum_{y^{ \prime}=1}^{Y}\big{(}\sum_{x^{\prime}=1}^{X}F_{k}^{l-1}(x-x^{\prime},y-y^{ \prime},z-z^{\prime})a_{jkr}^{l}(x^{\prime})\big{)}\;b_{jkr}^{l}(y^{\prime}) \big{)}\;c_{jkr}^{l}(z^{\prime})\;\big{)} \tag{5}\]  
With this formulation, the 3D convolution can be recast as a sequence of 1D convolutions. From inside out, the calculation within the parentheses can be viewed as: first convolve the feature volume with a 1D filter \(a_{jkr}^{l}\) along the X dimension, then followed by the 1D convolution with \(b_{jkr}^{l}\) and \(c_{jkr}^{l}\) along the Y and Z dimension successively. Vectors \(a_{jkr}^{l}\), \(b_{jkr}^{l}\) and \(c_{jkr}^{l}\) can be viewed as the corresponding horizontal (H), vertical (V) and lateral (L) 1D filters, respectively. Assuming that the rank of kernel tensor \(W_{jk}^{l}\) is equal to one (i.e., \(R=1\)), the 3D convolution can be decomposed into a sequence of three 1D convolutions as shown in Figure 2 (\(R=1\)). Note that convolution is a linear operator, and the 1D filters as shown in Figure 1 can be arranged in any order. Rank 1 is a strong assumption, and the intrinsic rank of \(W_{jk}^{l}\) is generally higher than one in practice, however, the generalization from the rank 1 topology to the rank R case is straightforward.