It is sometimes useful, as in image restoration where we want
to evaluate the quality of the restoration, to compare images
with an objective criterion.
Very few quantitative parameters can be extracted for that.
The correlation between the original image 
and
the restored one 
gives a
classical criterion. The correlation coefficient is:
The correlation is 1 if the images are identical, and less if some differences exist. Another way to compare two pictures is to determine the mean-square error:
can be normalized by:
The Signal-to-Noise Ratio (SNR) corresponding to the above error is:
These criteria are not sufficient, they give no information on the resulting resolution. A complete criterion must take into account the resolution. For each dyadic scale, we can compute the correlation coefficient and the quadratic error between the wavelet transforms of the original and the restored images. Hence, we can compare, the quality of the restoration for each resolution.
Figures 
and show the comparison of
three images with a reference image. Data20 is a simulated noisy image,
median and wave are the output images after respectively applying
a median filter, and a thresholding in the wavelet space. These curves
show that the thresholding in the wavelet space is better than the median
at all the scales.
Figure: Signal to noise ratio.
