Diffusion Inversion
: Reverse diffusion process의 initial state $x_T$를 $x_0$로 부터 sampling하는 것.
(Reverse process와 Forward process가 서로 adjoint 되어 있음이 보장되면 가능.)
Learned reverse transition probabilistic model $\epsilon_\theta(x_t)$을 이용하여 opposite direction으로 이동할 수 있음.
$\hat x_0=D_\theta(x_T,0)\approx x_0$
Here, $D_\theta$ is reverse diffusion process.
This mechanism depends on the assumption that $x_{t+1}-x_t\approx x_{t}-x_{t-1}$ (논문에 표기된 수식은 오타인 듯)
\[x_{t+1} = \sqrt{\bar \alpha_{t+1}}\hat x_0^t + \sqrt{1-\bar \alpha_{t+1}}\epsilon_\theta(x_t).\]$D^\dagger_\theta$ denotes inversion process by above equation.
정리하자면, $\hat x_T=D^\dagger_\theta(x_0)\approx x_T$
Tree-Ring Watermarking
: Choosing the initial noise state so that its Fourier transform contains a carefully constructed pattern near its center. (Center = Low frequency region) This pattern is called as ‘key’.
바로 $x_T$에 key를 삽입하면 결과 이미지에 알아챌만한 패턴이 나타날 수 있음.
Fourier space를 이용하는 이유는 invariant property들이 잘 알려져 있음.