What is the difference between MSE and PSNR of images?

Question

I have some doubts.

1. What is the difference between quality parameters MSE and PSNR?
2. Which is used when?
3. Is there any advantage of PSNR over MSE or vice versa?

Also I have heard people saying that MSE and PSNR should never be used to check image quality. Instead one should use UQI and SSIM. I read about UQI and SSIM and came to know they are close to human visual system but I read a technical report which says that SSIM is sometimes better than MSE and sometimes MSE is better. Here is the link http://www.cs.uregina.ca/Research/Techreports/2008-02.pdf

Now I am confused which one to use(among MSE and SSIM) and why?

Also it is said that SSIM is upgraded version of UQI. Then what's the difference between these two? Why do people still use UQI irrespective of the fact that SSIM is there?

Answer 1

What is the difference between MSE and PSNR of images?

Very little. Almost nothing, in a relative way. Most error measures are relative to certain quantities or references. If you have $K$ samples $d_k$ from "original" data, and an estimated, compressed or denoised version $\hat{d}_k$, then:

• MSE, the mean squared error, is realtive to the zero (the lower the better) in the Euclidean metric:

$$\textrm{MSE} = \frac{1}{K}\sum_{k=1}^{K} (d_k - \hat{d}_k)^2 $$

Signal-to-noise ratio (SNR) or peak signal-to-noise ratio (PSNR) are directly related quantities, in an inverse logarithmic scale (the higher the better), with respect to the data energy (SNR):

$$\textrm{SNR} = 10\log_{10} \left( \frac{\frac{1}{K}\sum_{k=1}^{K} (d_k )^2}{\textrm{MSE}} \right)$$

or the maximum possible data power (PSNR):

$$\textrm{PSNR} =10\log_{10} \left( \frac{ (\max{d_k} )^2}{\textrm{MSE}}\right) $$

where $\max{d_k}$ is sometimes set to $2^b-1$, where $b$ is the number of bits onto which image channel values are coded. Classically, it is 255. So, all in all, there is a one-to-one monotonous mapping between them, so they are closely related.

All them three are mere scalar $L_2$ quantities with a twist. So basically unrelated to vision. SSIM was thought to be more realistic with respect to perceptual comment. With mean, scale and covariance reweighting, it can correct some standard display issues.

But it remains debated, see for instance:

• A Formal Assessment of the Structural Similarity Index

Answer 2

These metrics are used to compute certain signal properties. They are roughly divided into 2 categories as 1-objective, 2-subjective. Objective criteria are too mechanical and do not take any human perception weighting. Subjective criteria are either model based computations or directly opinion based measurements. Which is to use when is a matter of application.

Answer 3

I think there are two differences:

• The MSE is meaningless without knowing the peak intensity or the number of encoding bits. According to the encoding, the same MSE can represent a "strong" or "weak" error (e.g., MSE=240 has not the same signification for 8-bit and 24-bit images). By taking into account the peak intensity, the PSNR does not have this problem.

• The logarithm enables to get scores with a few number of digits (e.g., it can be more convenient to use 90 $dB$ instead of 1,000,000,000).

Thus, the advantages of the PSNR over the MSE are: (1) it enables to compare results on images encoded with a different number of bits, (2) concision. However, by definition, PSNR is nothing more than a normalized version of the MSE.