Understanding of Eq.2 in paper.

[yanqswhu]

Eq. 2 in SEA-RAFT is

$$ -log[MixLap(x,\alpha,\beta_1,\beta_2,\mu)] $$

where

$$ MixLap(x,\alpha,\beta_1,\beta_2,\mu)=\alpha\frac{e^{-\frac{|x-\mu|}{\beta_1}}}{2\beta_1} + (1-\alpha)\frac{e^{-\frac{|x-\mu|}{\beta_2}}}{2\beta_2} $$

replace $\alpha=\frac{exp(\alpha_1)}{exp(\alpha_1)+exp(\alpha_2)}$

$$ MixLap(x,\alpha,\beta_1,\beta_2,\mu)=\frac{exp(\alpha_1)}{exp(\alpha_1)+exp(\alpha_2)}\frac{e^{-\frac{|x-\mu|}{\beta_1}}}{2\beta_1}+\frac{exp(\alpha_2)}{exp(\alpha_1)+exp(\alpha_2)}\frac{e^{-\frac{|x-\mu|}{\beta_2}}}{2\beta_2} $$

rewrite as follows

$$ MixLap(x,\alpha,\beta_1,\beta_2,\mu)=\frac{exp(\alpha_1)\frac{e^{-\frac{|x-\mu|}{\beta_1}}}{2\beta_1}+exp(\alpha_2)\frac{e^{-\frac{|x-\mu|}{\beta_2}}}{2\beta_2}}{exp(\alpha_1)+exp(\alpha_2)} $$

therefore,

$$ -log[MixLap(x,\alpha,\beta_1,\beta_2,\mu)]=-(log[{exp(\alpha_1)\frac{e^{-\frac{|x-\mu|}{\beta_1}}}{2\beta_1}+exp(\alpha_2)\frac{e^{-\frac{|x-\mu|}{\beta_2}}}{2\beta_2}}]-log[{exp(\alpha_1)+exp(\alpha_2)}]) $$

remove the negtive

$$ -log[MixLap(x,\alpha,\beta_1,\beta_2,\mu)]=log[{exp(\alpha_1)+exp(\alpha_2)}]-log[{exp(\alpha_1)\frac{e^{-\frac{|x-\mu|}{\beta_1}}}{2\beta_1}+exp(\alpha_2)\frac{e^{-\frac{|x-\mu|}{\beta_2}}}{2\beta_2}}] $$

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Notes in code raft.py Line145-Line156 , where $\alpha,\beta,\mu$ is replace by a, b, u：

obtain $log(\beta)$ with shape $(N,2,H,W)$

raw_b = info_predictions[i][:, 2:]
log_b = torch.zeros_like(raw_b)
# Large b Component                
log_b[:, 0] = torch.clamp(raw_b[:, 0], min=0, max=var_max)
# Small b Component
log_b[:, 1] = torch.clamp(raw_b[:, 1], min=var_min, max=0)

obtain $\alpha$ with shape $(N,2,H,W)$, which can be splited into $\alpha_1,\alpha_2$ on dim=1

weight = info_predictions[i][:, :2]

obtain $|x-\mu|/\beta$ with shape $(N,2,H,W)$

# |x-u|*exp[-log(b)]
# |x-u|/b
term2 = ((flow_gt - flow_predictions[i]).abs().unsqueeze(2)) * (torch.exp(-log_b).unsqueeze(1))

obtain $\alpha-log(2)-log(\beta)$ with shape $(N,2,H,W)$

# term1: [N, m, H, W]
# a-log(2)-log(b)
term1 = weight - math.log(2) - log_b

obtain nf_loss= $-log[MixLap(x,\alpha,\beta_1,\beta_2,\mu)]$ with shape $(N,1,H,W)$

nf_loss = torch.logsumexp(weight, dim=1, keepdim=True) - torch.logsumexp(term1.unsqueeze(1) - term2, dim=2)

where torch.logsumexp(weight, dim=1, keepdim=True) = $log[exp(\alpha_1)+exp(\alpha_2)]$,
torch.logsumexp(term1.unsqueeze(1) - term2, dim=2) = $log[exp(\alpha_1-log(2)-log(\beta_1)-|x-\mu|/\beta_1)+exp(\alpha_2-log(2)-log(\beta_2)-|x-\mu|/\beta_2)]$
$=log[exp (\alpha_1) \frac{1}{2\beta_1} e^{-\frac{|x-\mu|}{\beta_1}}+exp (\alpha_2) \frac{1}{2\beta_2} exp^{-\frac{|x-\mu|}{\beta_2}}$

Hope this helps guys read the code！

[MemorySlices]

Thanks for the explanation!

[YangHai-1218]

Very good explanation. Thanks!

[infinity1096]

This greatly helps to understand the code. Thanks!