1. Peak signal-to-noise ratio (PSNR)

• Commonly used to quantify reconstruction quality for images and video subject to lossy compression.

$$ PSNR = 10 \cdot\log_{10}\begin{pmatrix} \frac{MAX^{2}_I} {MSE} \end{pmatrix} $$

• A ratio between
  • $MAX$: maximum possible power of signal
  • Power of corrupting noise (easily defined via MSE) that effects the fidelity of the image’s representation

$$ MSE = \frac{1}{mn}\sum^{m-1}{i=0}\sum^{n-1}{j=0}[I(i,j)-K(i,j)]^2 $$

$I$ : image

$K$: noisy approximation

2. Structural similarity index (SSIM)

• SSIM papers
• Used to compute the perceptual distance between translated image and its gt
• $\uparrow$ similarity of luminance, contrast and structure of images → $\uparrow$SSIM

$$ SSIM(x,y) = \Bigl( l(x,y)\Bigr)^{\alpha} \Bigl( c(x,y)\Bigr)^{\beta} \Bigl( s(x,y)\Bigr)^{\gamma} $$

3. Inception Score(IS)

• IS papers
• Used to assess the quality of images created by a generative model
• $\uparrow$IS → $p_{gen}$ is a “sharp and distinct” collection of images → $\uparrow$diversity, fidelity

$$ \text{IS} = \exp(\mathbb{E}{x \sim p_g} D{KL}(p(y|x)||p(y)) ) $$

$$ D_{KL}(p(y|x)||p(y))=p(y|x)\log \begin{pmatrix} \frac{p(y|x)}{p(y)} \end{pmatrix} $$

• Inception Model Classification

• Fidelity
  • High probability for a few select classes, low probability for remaining classes
  • $p(y|x)$ → conditional distribution
  • Low entropy