bcm

🧩 GFSS Problem Settings (Section 3.1)

• 입력 이미지: $X \in \mathbb{R}^{H \times W \times 3} \quad\text{(RGB image of height H and width W)}$

• 세그멘테이션 정답 맵 (GT): $Y \in \mathcal{Y}^{H \times W} \quad\text{(semantic label per pixel)}$

• 클래스 셋 정의:

  $\mathcal{Y}_b: \text{ base classes (e.g., } \{0, 1, 2, 3\} \text{)}$
  $\mathcal{Y}_n: \text{ novel classes (e.g., } \{4, 5\} \text{)}$
  $\mathcal{Y}_{all} = \mathcal{Y}_b \cup \mathcal{Y}_n \quad\text{with } \mathcal{Y}_b \cap \mathcal{Y}_n = \emptyset$

• 픽셀 단위 레이블 정의:

  $[Y]_j = y \Rightarrow \text{class } y \text{ exists at pixel } j \in \{1, \dots, HW\}$

• base-class segmentation model: $g_{b}(X) = \hat{Y}_b \in \mathcal{Y}_b^{H \times W}$

• K-shot novel class support set:

  $\text{K support images per each } y \in \mathcal{Y}_n \quad\Rightarrow\quad K \times |\mathcal{Y}_n| \text{ total annotated images}$

• 최종 목표 (GFSS prediction): $g^{BCM}_b(X) = \hat{Y}^{BCM} \in \mathcal{Y}_{all}^{H \times W}$

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$\forall \beta \in \mathcal{B}, \quad \text{train } g_\beta \text{ using modified } K\text{-shot set where } y \in \mathcal{Y}_n \cup \{0\}$

$\setminus \{\beta\} \rightarrow \beta$

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🔁 BCM Flow (Symbolic Steps)

• 1️⃣: $\hat{Y}_b = g_b(X)$

• 2️⃣: $\forall \beta \in \mathcal{B}, \quad \mathcal{I}_\beta = \{j \mid [\hat{Y}_b]_j = \beta\}$

• 3️⃣: $\hat{Y}_\beta = g_\beta(\text{Feature}(X))$

• 4️⃣: $[\hat{Y}_b]_j \leftarrow [\hat{Y}_\beta]_j, \quad \forall j \in \mathcal{I}_\beta$

• 5️⃣: $\hat{Y}_{BCM} = \hat{Y}_b$

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