PSNR SSIM LPIPS Image Quality

Why Does This Exist?

Restored images need numbers, but pixel error alone rewards blur. Shift a sharp edge by one pixel and the squared error screams, yet any viewer prefers it over a blurry smear with a smaller error. So restoration papers report three lenses at once: PSNR prices mean squared error, SSIM prices local structure, and LPIPS prices distance in deep features. This page covers all three formulas, one fully recomputed toy patch, and the traps of reading any single number alone. Denoising pipelines that consume these scores live in image denoising.

Think of It Like This

How It Actually Works

For maximum pixel value $M$ and mean squared error $\text{MSE}$, the peak signal-to-noise ratio is $\text{PSNR} = 10 \log_{10}(M^2 / \text{MSE})$, in decibels, where higher is better. SSIM compares two windows through luminance, contrast, and structure terms and returns a value up to 1, where 1 means identical and higher is better. LPIPS feeds both patches through a fixed network (AlexNet or VGG in the original), normalizes the feature maps, and averages squared differences, so it is a distance where lower is better and 0 means identical.

Worked toy patch

Take $M = 1$ with $\text{MSE} = 0.01$: $\text{PSNR} = 10 \log_{10}(1 / 0.01) = 10 \times 2 = 20$ dB. Quarter the error to $0.0025$ and the score rises to $10 \log_{10}(400) \approx 26.02$ dB, so a 6.02 dB lift always means the error fell by a factor of four.

For SSIM, compare $x = [100, 100, 100, 104]$ against $y = [100, 100, 100, 100]$ on a 0 to 255 scale. The means are $\mu_x = 101$ and $\mu_y = 100$, where $\mu$ denotes the window average. The population variances are $\sigma_x^2 = 3.0$ and $\sigma_y^2 = 0.0$, where $\sigma^2$ denotes spread around the mean, and the covariance $\sigma_{xy} = 0.0$, because $y$ never moves from its mean. With the standard constants $C_1 = (0.01 \times 255)^2 = 6.5025$ and $C_2 = (0.03 \times 255)^2 = 58.5225$, which stabilize the division:

One deviant pixel in four costs about five SSIM points, while the same patch pair would show a modest MSE of $4.0$. LPIPS needs a real network, so treat it qualitatively here: identical patches score exactly 0, and a blurred patch scores farther from the original than a sharp shifted one, which is the opposite of the PSNR verdict.

Code

1import math2
3
4def psnr(mse: float, peak: float = 1.0) -> float:5    """Peak signal-to-noise ratio in dB from mean squared error."""6    return 10 * math.log10(peak ** 2 / mse)7
8
9print(round(psnr(0.01), 2), round(psnr(0.0025), 2))10# -> 20.0 26.02

MSE 0.01 gives 20 dB and quarter error gives about 26.02 dB, replaying the worked patch. SSIM and LPIPS need real windows and networks, so they stay qualitative on this page.

Watch Out For

The Quick Version

• PSNR equals $10 \log_{10}(M^2 / \text{MSE})$ in decibels, and higher is better.
• SSIM prices local structure with 1 as perfect, and one pixel in four off costs about five points here.
• LPIPS is deep-feature distance with 0 as identical, and lower is better.
• MSE 0.01 gives 20 dB while 0.0025 gives about 26.02 dB at peak 1.
• Report all three, because a pixel win can look worse to a viewer.

What to Read Next

• Super-Resolution and Image Restoration
• Image Enhancement
• Image Generation