Computer Vision

O = \lfloor (W - F + 2P)/S \rfloor + 1

\text{IoU}(A,B) = \frac{|A \cap B|}{|A \cup B|}

\text{GIoU} = \text{IoU} - \frac{|C \setminus (A \cup B)|}{|C|}

\text{Dice} = \frac{2|A \cap B|}{|A| + |B|} = \frac{2 \,\text{IoU}}{1 + \text{IoU}}

FL(p_t) = -(1-p_t)^{\gamma} \, \log p_t

\text{Attn}(Q,K,V) = \text{softmax}\!\left(\tfrac{QK^\top}{\sqrt{d_k}}\right) V

\text{MHA}(Q,K,V) = \text{Concat}(h_1,\dots,h_H)\,W^O,\ h_i = \text{Attn}(QW_i^Q, KW_i^K, VW_i^V)

L_i = -\log \frac{\exp(\text{sim}(z_i,z_i^+)/\tau)}{\sum_j \exp(\text{sim}(z_i,z_j)/\tau)}

L = -\tfrac{1}{n^2}\sum_{i,j}\Big[y_{ij}\log \sigma(z_{ij}) + (1-y_{ij})\log(1-\sigma(z_{ij}))\Big]

\text{PSNR} = 10 \log_{10}\!\left(\tfrac{R^2}{\text{MSE}}\right)

\begin{bmatrix} q'_{2i}\\ q'_{2i+1}\end{bmatrix} = \begin{bmatrix}\cos m\theta_i & -\sin m\theta_i\\ \sin m\theta_i & \cos m\theta_i\end{bmatrix}\begin{bmatrix} q_{2i}\\ q_{2i+1}\end{bmatrix}

y = \gamma \cdot \frac{x}{\sqrt{\tfrac{1}{d}\sum_i x_i^2}}

y = \gamma \cdot \frac{x - \mu}{\sigma} + \beta

C = \sum_i c_i\,\alpha_i \prod_{j<i}(1 - \alpha_j)

\Sigma = R\,S\,S^\top R^\top

M(\beta,\theta) = W(T_P(\beta,\theta),\, J(\beta),\, \theta,\, \mathcal{W})

\text{PCK@}\alpha = \tfrac{1}{N}\sum_i \mathbb{1}\!\left[\|\hat p_i - p_i\| \le \alpha \cdot d_{\text{ref}}\right]

L = \lambda_{\text{coord}}\sum_{ij}^{\text{obj}}[(x-\hat x)^2 + (y-\hat y)^2] + \lambda_{\text{coord}}\sum_{ij}^{\text{obj}}[(\sqrt w - \sqrt{\hat w})^2 + (\sqrt h - \sqrt{\hat h})^2] + \sum_{ij}^{\text{obj}}(C-\hat C)^2 + \lambda_{\text{noobj}}\sum_{ij}^{\text{noobj}}(C-\hat C)^2 + \sum_i^{\text{obj}}\sum_c (p(c)-\hat p(c))^2

L = \max(0,\ \|A-P\|^2 - \|A-N\|^2 + \alpha)