## 2 Methods
### Algorithm Workflow
Following the previous work [3; 23; 26], we adopted an upper bound optimization technique to simplify the optimization of Equation (1) by deriving and optimizing a series of upper bound functions of Equation (1), assuming that the upper bound function is easier to minimize than the original formulations. Briefly, for any given segmentation \(\tilde{u_{l}}\), \(l\in L\), \(x\in\Omega\), previous studies [26] showed that the following is an upper bound function of Equation (1):\[\sum_{l\in L}\left\langle\frac{X\mathbf{1}\hat{u}_{l}^{T}X\hat{u}_{l}}{(1X\hat{u}_ {l})^{2}}-\frac{2X\hat{u}_{l}}{\mathbf{1}X\hat{u}_{l}},u_{l}\right\rangle\,+\, \int_{\Omega}g(x)|\
abla u_{l}(x)|dx\,+\,\int_{\Omega}\omega(x)\cdot|u_{l}(x)- \hat{s}_{l}(x)|dx\,,\quad u_{l}\in\{0,1\},\] (2) where \(\langle\,,\rangle\) and \({}^{T}\) denote inner product and transpose, respectively; the first term in Equation (2) is linear with respect to \(u_{l}\) that we aim to solve. Through convex relaxation, i.e., by relaxing \(u_{l}\in\{0,1\}\) to \(u_{l}\in[0,1]\), we can derive a convex relaxed formulation of Equation (2), which can be efficiently and globally optimized using a continuous min-cut/max-flow algorithm on a graphics card [27; 28]. We refer readers to the previous studies [27; 28] for the details of the continuous min-cut/max-flow algorithm. Please note that Equation (1) was optimized iteratively; for each iteration, we derived an upper bound using Equation (2) and minimized the upper bound to generate a solution \(\hat{u}_{l}\), which was used to update the upper bound for the next iteration. In particular, for the first iteration we used the derived CNN \"mean\" segmentation \(\hat{s}_{l}\)l as the given solution \(\hat{u}_{l}\). This process was iterated several times until convergence (we observed convergence typically within five iterations in this study) to derive the final solution to Equation (1). We refer readers to previous studies [3; 24] for the details of minimizing the CKC segmentation model in Equation (1). Upon CKC algorithm convergence, the final segmentation of the fine-tuning dataset (without manual labels) was saved and used to update the trained U-net model for another 20 epochs in \(\sim\)10 min. This procedure was iterated until convergence and the final U-net model was tested on the remaining 90 ACDC subjects for LV indices quantification. We also implemented several commonly used methods for comparison, including:  
1. A naive method (Naive): The trained U-net was used to segment the 90 ACDC test subjects directly. 2. A combined method (Combined) that integrated MCD, spatial augmentation, and style-intensity augmentation method. We explored the effects of MCD, spatial augmentation, and advanced style-intensity augmentation for U-net training; the optimal combination of the three components constitutes the combined method. A recent study [17] proposed style-intensity augmentation during network training to tackle the domain shift issue and demonstrated state-of-the-art performance in breast segmentation in MRI datasets from a different domain. Style-intensity augmentation comprises style transfer and intensity remapping, which produce non-realistic looking MR scans while preserving the image shapes. The style transfer procedure uses features extracted from style images to augment the training images, randomizing the color, texture and contrast but preserving the geometry [29]. The intensity remapping technique generates a random mapping function to map the original image signal intensities to new values. This method is based on the assumption that by considerably changing the appearance of training images, the network will focus on non-domain specific features, e.g., the geometric shape of breast that is preserved in different breast MR datasets [17]. The optimized combined method was applied to the ACDC test dataset. 3. DeepLab: DeepLabV3+ [30], a top performing neural network in several medical image segmentation challenges, was trained on the LVSC dataset and tested on the ACDC test dataset.