# Peak signal-to-noise ratio

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.

## Definition

PSNR is most easily defined via the mean squared error (MSE). Given a noise-free m×n monochrome image I and its noisy approximation K, MSE is defined as:

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

The PSNR (in dB) is defined as:

{\displaystyle {\begin{aligned}{\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)\\&=20\cdot \log _{10}\left({{\mathit {MAX}}_{I}}\right)-10\cdot \log _{10}\left({\mathit {MSE}}\right)\end{aligned}}}

Here, MAXI 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, MAXI is 2B−1.

### Application in color images

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 (now for each color, i.e. three as much differences as in a monochrome image) divided by image size and by three. Alternately, for color images the image is converted to a different color space and PSNR is reported against each channel of that color space, e.g., YCbCr or HSL.[1][2]

## Quality estimation with PSNR

PSNR is most commonly used to measure the quality of reconstruction of lossy compression codecs (e.g., for image compression). The signal in this case is the original data, and the noise is the error introduced by compression. When comparing compression codecs, PSNR is an approximation to human perception of reconstruction quality.

Typical values for the PSNR in lossy image and video compression are between 30 and 50 dB, provided the bit depth is 8 bits, where higher is better. For 16-bit data typical values for the PSNR are between 60 and 80 dB.[3][4] Acceptable values for wireless transmission quality loss are considered to be about 20 dB to 25 dB.[5][6]

In the absence of noise, the two images I and K are identical, and thus the MSE is zero. In this case the PSNR is infinite (or undefined, see Division by zero).[7]

Original uncompressed image
Q=90, PSNR 45.53dB
Q=30, PSNR 36.81dB
Q=10, PSNR 31.45dB
Example luma PSNR values for a cjpeg compressed image at various quality levels.

## Performance comparison

Although a higher PSNR generally indicates that the reconstruction is of higher quality, in some cases it may not. One has to be extremely careful with the range of validity of this metric; it is only conclusively valid when it is used to compare results from the same codec (or codec type) and same content.[8][9]

Generally, PSNR has been shown to perform poorly compared to other quality metrics when it comes to estimating the quality of images and particularly videos as perceived by humans.[8][10]

## Variants

PSNR-HVS[11] is an extension of PSNR that incorporates properties of the human visual system such as contrast perception.

PSNR-HVS-M improves on PSNR-HVS by additionally taking into account visual masking.[12] In a 2007 study, it delivered better approximations of human visual quality judgements than PSNR and SSIM by large margin. It was also shown to have a distinct advantage over DCTune and PSNR-HVS.[13]

## References

1. ^ Oriani, Emanuele. "qpsnr: A quick PSNR/SSIM analyzer for Linux". Retrieved 6 April 2011.
2. ^ "pnmpsnr User Manual". Retrieved 6 April 2011.
3. ^ Welstead, Stephen T. (1999). Fractal and wavelet image compression techniques. SPIE Publication. pp. 155–156. ISBN 978-0-8194-3503-3.
4. ^ Raouf Hamzaoui, Dietmar Saupe (May 2006). Barni, Mauro (ed.). Fractal Image Compression. Document and Image Compression. 968. CRC Press. pp. 168–169. ISBN 9780849335563. Retrieved 5 April 2011.
5. ^ Thomos, N., Boulgouris, N. V., & Strintzis, M. G. (2006, January). Optimized Transmission of JPEG2000 Streams Over Wireless Channels. IEEE Transactions on Image Processing , 15 (1).
6. ^ 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.
7. ^ Salomon, David (2007). Data Compression: The Complete Reference (4 ed.). Springer. p. 281. ISBN 978-1846286025. Retrieved 26 July 2012.
8. ^ a b Huynh-Thu, Q.; Ghanbari, M. (2008). "Scope of validity of PSNR in image/video quality assessment". Electronics Letters. 44 (13): 800. doi:10.1049/el:20080522.
9. ^ MIT.edu
10. ^ Huynh-Thu, Quan; Ghanbari, Mohammed (2012-01-01). "The accuracy of PSNR in predicting video quality for different video scenes and frame rates". Telecommunication Systems. 49 (1): 35–48. doi:10.1007/s11235-010-9351-x. ISSN 1018-4864.
11. ^ Egiazarian, Karen, Jaakko Astola, Nikolay Ponomarenko, Vladimir Lukin, Federica Battisti, and Marco Carli (2006). "New full-reference quality metrics based on HVS." In Proceedings of the Second International Workshop on Video Processing and Quality Metrics, vol. 4.
12. ^ Ponomarenko, N.; Ieremeiev, O.; Lukin, V.; Egiazarian, K.; Carli, M. (February 2011). "Modified image visual quality metrics for contrast change and mean shift accounting". 2011 11th International Conference the Experience of Designing and Application of CAD Systems in Microelectronics (CADSM): 305–311.
13. ^ Nikolay Ponomarenko; Flavia Silvestri; Karen Egiazarian; Marco Carli; Jaakko Astola; Vladimir Lukin, "On between-coefficient contrast masking of DCT basis functions" (PDF), CD-ROM Proceedings of the Third International Workshop on Video Processing and Quality Metrics for Consumer Electronics VPQM-07, 25.–26. Januar 2007 (in German), Scottsdale AZ