View source for 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 (information theory)|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 [[logarithm]]ic [[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:
:<math>\mathit{MSE} = \frac{1}{m\,n}\sum_{i=0}^{m-1}\sum_{j=0}^{n-1} [I(i,j) - K(i,j)]^2</math>

The PSNR (in [[decibel|dB]]) is defined as:
:<math>\begin{align}\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{align}</math>

Here, ''MAX<sub>I</sub>'' 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 [[Pulse-code modulation|PCM]] with ''B'' bits per sample, ''MAX<sub>I</sub>'' is 2<sup><var>B</var></sup>−1.

=== Application in color images ===
For [[color image]]s 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 times as many 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 and HSV|HSL]].<ref>{{cite web|last=Oriani|first=Emanuele|title=qpsnr: A quick PSNR/SSIM analyzer for Linux|url=http://qpsnr.youlink.org/|accessdate=6 April 2011}}</ref><ref>{{cite web|title=pnmpsnr User Manual|url=http://netpbm.sourceforge.net/doc/pnmpsnr.html|accessdate=6 April 2011}}</ref>

== Quality estimation with PSNR ==
PSNR is most commonly used to measure the quality of reconstruction of lossy compression [[codec]]s (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 compression|lossy]] image and video compression are between 30 and 50&nbsp;dB, provided the bit depth is 8&nbsp;[[bit]]s, where higher is better. For 16-bit data typical values for the PSNR are between 60 and 80&nbsp;dB.<ref>{{cite book|last=Welstead|first=Stephen T.|title=Fractal and wavelet image compression techniques|year=1999|publisher=SPIE Publication|isbn=978-0-8194-3503-3|pages=155–156|url=https://books.google.com/books?id=evGyv-mogukC&lpg=PA155&dq=image%20compression%20acceptable%20PSNR&pg=PA156#v=onepage&q=image%20compression%20acceptable%20PSNR&f=false}}</ref><ref>{{cite book|author=Raouf Hamzaoui, Dietmar Saupe|title=Fractal Image Compression|journal=Document and Image Compression|date=May 2006|volume=968|pages=168–169|url=https://books.google.com/books?id=FmqPOsEYRsEC&lpg=PA229&dq=image%20compression%20acceptable%20PSNR&pg=PA168#v=onepage&q=PSNR&f=false|accessdate=5 April 2011|editor1-first=Mauro|editor1-last=Barni|publisher=CRC Press|isbn=9780849335563}}</ref> Acceptable values for wireless transmission quality loss are considered to be about 20&nbsp;dB to 25&nbsp;dB.<ref>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).</ref><ref>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]].</ref>

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]]).<ref>{{cite book|last=Salomon|first=David|title=Data Compression: The Complete Reference|year=2007|publisher=Springer|isbn=978-1846286025|url=https://books.google.com/?id=ujnQogzx_2EC&lpg=PA281&dq=PSNR%20infinite&pg=PA281#v=onepage&q=PSNR%20infinite&f=false|edition=4|accessdate=26 July 2012|page=281}}</ref>

{{multiple image
   | width     = 200
   | align = center
   | footer    = Example [[Luma (video)|luma]] PSNR values for a [[libjpeg|cjpeg]] compressed image at various quality levels.
   | image1    = PSNR-example-base.png
   | caption1  = Original uncompressed image 
   | image2    = PSNR-example-comp-90.jpg
   | caption2  = Q=90, PSNR 45.53dB
   | image3    = PSNR-example-comp-30.jpg
   | caption3  = Q=30, PSNR 36.81dB
   | image4    = PSNR-example-comp-10.jpg
   | caption4  = Q=10, PSNR 31.45dB
  }}

== 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.<ref name=":0">{{Cite journal|last1=Huynh-Thu|first1=Q.|last2=Ghanbari|first2=M.|year=2008|title=Scope of validity of PSNR in image/video quality assessment|journal=Electronics Letters|volume=44|issue=13|pages=800|doi=10.1049/el:20080522|pmc=|pmid=}}</ref><ref>[http://web.mit.edu/xiphmont/Public/theora/demo7.html MIT.edu]</ref>

Generally, PSNR has been shown to perform poorly compared to other [[Video quality|quality metrics]] when it comes to estimating the [[Image quality|quality of images]] and particularly videos as perceived by humans.<ref name=":0" /><ref>{{Cite journal|last=Huynh-Thu|first=Quan|last2=Ghanbari|first2=Mohammed|date=2012-01-01|title=The accuracy of PSNR in predicting video quality for different video scenes and frame rates|journal=Telecommunication Systems|language=en|volume=49|issue=1|pages=35–48|doi=10.1007/s11235-010-9351-x|issn=1018-4864}}</ref>

== Variants ==
PSNR-HVS<ref>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.</ref> is an extension of PSNR that incorporates properties of the human visual system such as [[Contrast (vision)|contrast perception]].

PSNR-HVS-M improves on PSNR-HVS by additionally taking into account [[visual masking]].<ref>{{Cite journal|last=Ponomarenko|first=N.|last2=Ieremeiev|first2=O.|last3=Lukin|first3=V.|last4=Egiazarian|first4=K.|last5=Carli|first5=M.|date=February 2011|title=Modified image visual quality metrics for contrast change and mean shift accounting|url=https://ieeexplore.ieee.org/abstract/document/5744476/|journal=2011 11th International Conference the Experience of Designing and Application of CAD Systems in Microelectronics (CADSM)|pages=305–311}}</ref> In a 2007 study, it delivered better approximations of human visual quality judgements than PSNR and [[structural similarity|SSIM]] by large margin. It was also shown to have a distinct advantage over [[DCTune]] and PSNR-HVS.<ref name="PSNR-HVS-M">{{citation|surname1=Nikolay Ponomarenko|surname2= Flavia Silvestri|surname3= Karen Egiazarian|surname4= Marco Carli|surname5= Jaakko Astola|surname6= Vladimir Lukin|periodical=CD-ROM Proceedings of the Third International Workshop on Video Processing and Quality Metrics for Consumer Electronics VPQM-07, 25.–26. Januar 2007|title=On between-coefficient contrast masking of DCT basis functions |location=Scottsdale AZ|language=German|url=http://ponomarenko.info/vpqm07_p.pdf}}</ref>

==See also==
* [[Data compression ratio]]
* [[PEVQ|Perceptual Evaluation of Video Quality]] (PEVQ)
* [[Signal-to-noise ratio]]
* [[Structural similarity]] (SSIM) index
* [[Subjective video quality]]
* [[Video quality]]

==References==
{{Reflist}}

{{Noise}}

[[Category:Image compression]]
[[Category:Noise (graphics)]]
[[Category:Film and video technology]]
[[Category:Digital television]]
[[Category:Engineering ratios]]

[[vi:PSNR]]