October 10, 2023
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FBP(y)\text{FBP}(y)FBP(y)
Poisson model
Gaussian approximation
λ=N0 e−Ax\lambda = N_0~e^{-Ax}λ=N0 e−Ax
y∼Poisson(λ)y \sim \text{Poisson}(\lambda)y∼Poisson(λ)
y∼Normal(λ,λ)y \sim \text{Normal}(\lambda, \lambda)y∼Normal(λ,λ)
DDPM Schedule
p(xt∣x0)=N(αtx0,(1−αt)I)p(x_t \mid x_0) = \mathcal{N}(\sqrt{\alpha_t}x_0, (1 - \alpha_t)\mathbb{I}) p(xt∣x0)=N(αtx0,(1−αt)I)
Tweedie's formula
If xi∼N(ai x0,bi2I)x_i \sim \mathcal{N}(a_i~x_0, b_i^2 \mathbb{I})xi∼N(ai x0,bi2I), then
x^0(xt)=E[x0∣xt]=xt+bi2 ∇xtlogpt(xt)ai\hat{x}_0(x_t) = \mathbb{E}[x_0 \mid x_t] = \frac{x_t + b_i^2~\nabla_{x_t} \log p_t(x_t)}{a_i}x^0(xt)=E[x0∣xt]=aixt+bi2 ∇xtlogpt(xt)
x0x_0x0
x^0\hat{x}_0x^0
t=0t=0t=0
t=1t=1t=1
DPS:
∇xtlogp(y∣xt)≈∇xtlogp(y∣x^0(xt))\nabla_{x_t} \log p(y \mid x_t) \approx \nabla_{x_t} \log p(y \mid \hat{x}_0(x_t))∇xtlogp(y∣xt)≈∇xtlogp(y∣x^0(xt))
Poisson
Gaussian
l^α\hat{l}_{\alpha}l^α
u^α\hat{u}_{\alpha}u^α
u^α−l^α\hat{u}_{\alpha} - \hat{l}_{\alpha}u^α−l^α
By Jacopo Teneggi
