## 2 Methods
### Algorithm Workflow  
We used a 2D U-net [21] that comprised a symmetric contracting and expanding path with five levels. Each level consists of two blocks of \(3\times 3\) convolution and a rectified linear unit, followed by max-pooling in the contracting path or up-sampling in the expanding path; the number of feature maps was 16 in the top level and increased to 256 in the bottom level. The network was pre-trained on 45 images from the LVSC dataset for 200 epochs by minimizing the cross-entropy between model prediction and manual reference segmentation using an ADAM optimizer (learning rate = \(10^{-4}\)). Spatial data augmentation, including translation (\(-\)50-50 pixels), random rotation (\(-\)50-50\({}^{\circ}\)), voxel size and intensity scaling (0.75-1.25 times), and elastic deformation, was performed in parallel. To further minimize overfitting and improve CNN segmentation generalizability, Monte-Carlo dropout [22] (MCD, dropout rate = 0.5) was applied to each block in the bottom three levels of the 2D U-net. These settings were adopted for the following fine-tuning procedure. Figure 1 provides the schematic of our proposed algorithm. Briefly, the trained U-net was applied to the 10 ACDC fine-tuning subjects. For each subject, test-time MCD was applied to generate 50 segmentation samples \((s_{1}(x),s_{2}(x),\ldots,s_{50}(x),\,x\in\Omega)\); the mean of the associated probability maps were calculated to derive the \"mean\" segmentation \(\tilde{s}(x)\). In addition, the standard deviation of the 50 segmentation samples was calculated for each pixel and used as pixel-wise U-net segmentation uncertainty \(\omega(x)\), i.e., \(\omega(x)\propto\frac{1}{std(\{s1(x),s2(x),\ldots,s50(x)\})}\), \(x\in\Omega\). The derived \"mean\" segmentation \(\tilde{s}(x)\) was post-processed using a recently developed continuous kernel-cut (CKC) segmentation method, which demonstrated effectiveness in post-processing cardiac MRI CNN segmentation outputs [3; 23; 24]. The CKC segmentation algorithm employs normalized cut for balanced pair-wise feature clustering and continuous regularization on image grids to generate spatially smooth contours. In addition, we proposed to use the derived CNN \"mean\" segmentation as descent initialization of the CKC algorithm such that in regions with high U-net segmentation uncertainty (i.e., \(\omega(x)\) is relatively low), the final segmentation \(u(x)\) can be more different from the \"mean\" segmentation \(\tilde{s}(x)\) and vice versa. To this end, we derived the deep-learning uncertainty-guided CKC segmentation algorithm by minimizing the following function:  
\[\sum_{l\in L}-\frac{u_{l}^{T}Xu_{l}}{\mathbf{1}Xu_{l}}\,+\,\int_{\Omega}g(x)| \
abla u_{l}(x)|dx\,+\,\int_{\Omega}\omega(x)\cdot|u_{l}(x)-\tilde{s_{l}}(x) |dx\,,\quad u_{l}\in\{0,1\}, \tag{1}\]  
subject to \(\sum_{l\in L}u_{l}(x)=1\), \(\forall x\in\Omega\). In Equation (1), \(u_{l}(x)\in\{0,1\}\) is decomposed from the final segmentation \(u(x)\) and indicates if voxel \(x\) is in region \(l\in L=\{LVM,LVC,background\}\), \(X\) is a matrix where each element \(X(i,j)\) indicates if voxel \(j\) is within the K-nearest neighbor of voxel \(i\), \(\mathbf{1}\) is an all-ones matrix, \(g(x)\) is a boundary weight function based on image contrast edges, and \(\omega(x)\) enforces the similarity of CNN initial segmentation \(\tilde{s_{l}}(x)\) and CKC final segmentation \(u_{l}(x)\) for each region \(l\). Of note, \(\tilde{s_{l}}\) was decomposed from \(\tilde{s_{l}}\) similar to \(u_{l}(x)\). The CKC algorithm in Equation (1) integrates the advantages of balanced portioning of image features in high-dimensional space and spatially smooth segmentation that mimics the behavior of manual delineation [3; 24; 25]. Direct optimization of the high-order and non-smooth function in Equation (1) is particularly challenging.