Peak signal to noise ratio

The peak signal-to-noise ratio ( PSNR ) is an abbreviation for PSNR and is an engineering term for the relationship between the maximum possible signal value and the noise power that distorts the signal value. Since many signals have a wide dynamic range, PSNR is usually measured on a logarithmic scale in decibels .

PSNR is most often used to measure distortion levels in image compression . The easiest way to determine it is through the standard error ( MSE ) or MSE ( English mean square error ).

1. In the case of using MSE, this indicator for two monochrome images I and K of size m × n , one of which is considered a noisy approximation of the other, is calculated as follows:

{\mathit {MSE}}={\frac {1}{mn}}\sum _{i=0}^{m-1}\sum _{j=0}^{n-1}|I(i,j)-K(i,j)|^{2}

{\ displaystyle {\ mathit {MSE}} = {\ frac {1} {mn}} \ sum _ {i = 0} ^ {m-1} \ sum _ {j = 0} ^ {n-1} | I (i, j) -K (i, j) | ^ {2}}

{\ mathit {MSE}} = {\ frac {1} {mn}} \ sum _ {{i = 0}} ^ {{m-1}} \ sum _ {{j = 0}} ^ {{n -1}} | I (i, j) -K (i, j) | ^ {2}

PSNR is defined as follows:

{\mathit {PSNR}}=10\log _{10}\left({\frac {{\mathit {MAX}}_{I}^{2}}{\mathit {MSE}}}\right)=20\log _{10}\left({\frac {{\mathit {MAX}}_{I}}{\sqrt {\mathit {MSE}}}}\right)

{\ displaystyle {\ mathit {PSNR}} = 10 \ log _ {10} \ left ({\ frac {{\ mathit {MAX}} _ {I} ^ {2}} {\ mathit {MSE}}} \ right) = 20 \ log _ {10} \ left ({\ frac {{\ mathit {MAX}} _ {I}} {\ sqrt {\ mathit {MSE}}}} right)}

{\ mathit {PSNR}} = 10 \ log _ {{10}} \ left ({\ frac {{\ mathit {MAX}} _ {I} ^ {2}} {{\ mathit {MSE}}}} \ right) = 20 \ log _ {{10}} \ left ({\ frac {{\ mathit {MAX}} _ {I}} {{\ sqrt {{\ mathit {MSE}}}}}} right )

where MAX _I is the maximum value accepted by the image pixel. When the pixels have a resolution of 8 bits, MAX _I = 255 . Generally speaking, when signal values are presented linearly ( PCM ) with B bits per value, the maximum possible value of MAX _I will be 2 ^B -1.

2. In the case of using standard deviation ( English root-mean-square error ), this indicator for two monochrome images I and K of size m × n , one of which is considered a noisy approximation of the other, is calculated as follows:

{\mathit {RMSE}}={\sqrt {{\frac {1}{mn}}\sum _{i=0}^{m-1}\sum _{j=0}^{n-1}|I(i,j)-K(i,j)|^{2}}}

{\ displaystyle {\ mathit {RMSE}} = {\ sqrt {{\ frac {1} {mn}} \ sum _ {i = 0} ^ {m-1} \ sum _ {j = 0} ^ {n -1} | I (i, j) -K (i, j) | ^ {2}}}}

{\ displaystyle {\ mathit {RMSE}} = {\ sqrt {{\ frac {1} {mn}} \ sum _ {i = 0} ^ {m-1} \ sum _ {j = 0} ^ {n -1} | I (i, j) -K (i, j) | ^ {2}}}}

PSNR is defined as follows:

{\mathit {PSNR}}=10\log _{10}\left({\frac {{\mathit {MAX}}_{I}^{2}}{{\mathit {RMSE}}^{2}}}\right)=20\log _{10}\left({\frac {{\mathit {MAX}}_{I}}{\mathit {RMSE}}}\right)

{\ displaystyle {\ mathit {PSNR}} = 10 \ log _ {10} \ left ({\ frac {{\ mathit {MAX}} _ {I} ^ {2}} {{\ mathit {RMSE}} ^ {2}}} \ right) = 20 \ log _ {10} \ left ({\ frac {{\ mathit {MAX}} _ {I}} {\ mathit {RMSE}}} \ right)}

{\ displaystyle {\ mathit {PSNR}} = 10 \ log _ {10} \ left ({\ frac {{\ mathit {MAX}} _ {I} ^ {2}} {{\ mathit {RMSE}} ^ {2}}} \ right) = 20 \ log _ {10} \ left ({\ frac {{\ mathit {MAX}} _ {I}} {\ mathit {RMSE}}} \ right)}

where MAX _I is the maximum value accepted by the image pixel. When the pixels have a resolution of 8 bits, MAX _I = 255 . Generally speaking, when signal values are presented linearly ( PCM ) with B bits per value, the maximum possible value of MAX _I will be 2 ^B -1.

It should be noted that the term “peak signal-to-noise ratio” is a frequently used, but not entirely correct, literal translation of the English term “peak signal-to-noise ratio”. The correct translation will be “the ratio of the peak signal level to noise”. This takes into account the fact that when calculating the PSNR, it is precisely the ratio of the maximum possible ("peak") signal to the noise level that is calculated, and not the maximum ("peak") ratio of the calculated signal-to-noise value is searched, as one could understand from an incorrect verbatim translation.

For color images with three RGB components per pixel, the same PSNR definition is applied, but MSE is counted across all three components (and divided by three times the image size).

Typical PSNR values for image compression range from 30 to 40 dB. [one]

Links

Program for calculating PSNR for BMP files and videos

Peak signal to noise ratio

See also

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