Peak signal-to-noise ratio

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The phrase peak signal-to-noise ratio, often abbreviated PSNR, is an engineering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation. Because many signals have a very wide dynamic range, PSNR is usually expressed in terms of the logarithmic decibel scale.

The PSNR is most commonly used as a measure of quality of reconstruction in image compression etc. It is most easily defined via the mean squared error (MSE) which for two m×n monochrome images I and K where one of the images is considered a noisy approximation of the other is defined as:

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

The PSNR is defined as:

$\mathit{PSNR} = 10 \cdot \log_{10} \left( \frac{\mathit{MAX}_I^2}{\mathit{MSE}} \right) = 20 \cdot \log_{10} \left( \frac{\mathit{MAX}_I}{\sqrt{\mathit{MSE}}} \right)$

Here, MAX_i is the maximum possible pixel value of the image. When the pixels are represented using 8 bits per sample, this is 255. More generally, when samples are represented using linear PCM with B bits per sample, MAX_I is 2^B-1.

For color images with three RGB values per pixel, the definition of PSNR is the same except the MSE is the sum over all squared value differences divided by image size and by three.

Typical values for the PSNR in lossy image and video compression are between 30 and 50 dB, where higher is better. Acceptable values for wireless transmission quality loss are considered to be about 20 dB to 25 dB ^[1]^[2].

An identical image to the original will yield an undefined PSNR as the MSE will become equal to zero due to no error. In this case the PSNR value can be thought of as approaching infinity as the MSE approaches zero; this shows that a higher PSNR value provides a higher image quality. At the other end of the scale an image that comes out with all zero value pixels (black) compared to an original does not provide a PSNR of zero ^[3]. This can be seen by observing the form, once again, of the MSE equation. Not all the original values will be a long distance from the zero value thus the PSNR of the image with all pixels at a value of zero is not the worst possible case.

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[edit] References

^ Boulgouris, N. V., Thomos, N., & Strintzis, M. G. (2006, January). Optimized Transmission of JPEG2000 Streams Over Wireless Channels. IEEE Transactions on Image Processing , 15 (1).
^ Xiangjun, L., & Jianfei, C. ROBUST TRANSMISSION OF JPEG2000 ENCODED IMAGES OVER PACKET LOSS CHANNELS. ICME 2007 (pp. 947-950). School of Computer Engineering, Nanyang Technological University.
^ Munadi, K., Kurosaki, M., Nishikawa, K., & Kiya, H. (2003). A Robust Error Protection Technique for JPEG2000 Codestream and Its Evaluation in CDMA Environment. (pp. 654-658). Department of Electrical Engineering, Tokyo Metropolitan University.