# ConceptVAE: Self-Supervised Fine-Grained Concept Disentanglement from 2D Echocardiographies
## 5 Quantitative Model Analysis
### Semantic Segmentation  
The second task we employ is semantic segmentation, where features from the pre-trained models are projected to match a down-sampled ground-truth mask. For this task, we use five labels corresponding to heart chambers: left and right ventricles and atria in apical views (A2C, A3C, and A4C views) and the left ventricle in the short-axis (SAX) view. Starting with frozen model latent codes, a linear 2D convolutional kernel is fitted to predict low-resolution (stride 16\(\times\)) segmentation maps. Channel-wise softmax activation is applied on top of the predicted linear logits, as shown in Equation (4). Here, \(p_{ij}(s)\) represents the probability that location \(i,j\) to contain chamber \(s\), \(x_{input}\) is the frozen latent feature map, and \(W_{k}\) and \(w_{b}\) are the kernel weight matrix and bias vector, respectively, and containing 6 rows for the 5 prediction targets and one background channel. \[p_{ij}(s)=Softmax(W_{k}\cdot x_{input}^{ij}+w_{b}) \tag{4}\]  
The ground-truth was obtained by down-sampling the full scale chamber masks using the area interpolation method. We perform training on an independent set consisting of 5000 training examples, and test the outcomes using an independent test set of 500 samples. The Dice loss was employed as in Equation (5), where \(p_{ij}\) and \(t_{ij}\) are the predicted and target chamber presence probabilities at location \(i,j\), respectively. \[\mathcal{L}_{Dice}=1-\frac{2\sum_{i,j}p_{ij}t_{ij}}{\sum_{i,j}p_{ij}^{2}+t_{ij} ^{2}} \tag{5}\]  
We explore three scenarios: (i) using only the concepts \(x_{concept}\) as input, (ii) using the full latent space (\(x_{latent}=[x_{concept},x_{style}]\)) as input, and (iii) using only the style map \(x_{style}\) as input. We also investigate the influence of the linear kernel spatial size \(k\) for the _Feature decoder_ block on the evaluation scores, with different ranges, \(k\in\{1,3,5,7,9\}\). To investigate the effect of the proposed training procedure, we first compare with a randomly initialized frozen model. The same random seed, dataset and number of linear-classifier optimization iterations were used throughout all scenarios. Table 2 presents the linear evaluation results in terms of Dice Loss, which is equivalent to subtracting the Dice Score from 1. For both types of models (trained and randomly initialized) and across all \(x_{input}\) setups, larger values of \(k\) result in lower test set losses. This is expected, as larger kernels capture more local information, and concepts cooperate locally to form larger anatomical structures. When \(x_{input}:=x_{latent}\) and the model is trained, the loss decreases only marginally when \(k\) exceeds 5 (i.e., the receptive field size used in the _Feature Decoder_ block). In all scenarios, ConceptVAE achieves lower test losses. For both models, the lowest losses occur when \(x_{input}:=x_{latent}\) (i.e., both concepts and styles are used for segmentation). When using only the concepts from the trained model, the losses are slightly higher but still significantly lower than when using only styles. Additionally, when \(x_{input}:=x_{style}\) \begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{2}{c}{**Model**} \\ \cline{2-3}
**Landmark** & **ConceptVAE** & **Baseline** \\ \hline left annulus & 0.418 & 0.148 \\ mid-septum & 0.281 & 0.098 \\ apex & 0.518 & 0.345 \\ mid-free-wall & 0.263 & 0.094 \\ right annulus & 0.371 & 0.128 \\ _average_ & 0.370 & 0.163 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Region-based instance retrieval mAP metric values. the differences between the ConceptVAE and the random-init model are the smallest among all three input scenarios. This result brings further evidence that \(x_{concept}\) contains semantic information useful for downstream tasks like segmentation, while \(x_{style}\) focuses on local stylistic features.