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[[Image:Exsecant and exosecant plot.png|frame|right|exsecant (blue) and excosecant (green)]]
'''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.
[[Image:Circle-trig6.svg|right|thumb|320px|The trigonometric functions, including the exsecant, can be constructed geometrically in terms of a unit circle centered at ''O''.  The exsecant is the portion ''DE'' of the secant ''exterior'' to (''ex'') the circle.]]
The '''exsecant''', also abbreviated '''exsec''', is a [[trigonometric function]] defined in terms of the secant function sec(θ):


:<math>\operatorname{exsec}(\theta) = \sec(\theta) - 1. \,</math>
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. 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>{{cite doi|10.1049/el:20080522}}</ref><ref>[http://web.mit.edu/xiphmont/Public/theora/demo7.html MIT.edu]</ref>


Once important in fields such as [[surveying]], [[astronomy]], and [[spherical trigonometry]], the exsecant function is now little-used.  Mainly, this is because the availability of [[calculator]]s and [[computer]]s has removed the need for trigonometric tables of specialized functions such as this one.
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>


A related function is the '''excosecant''' ('''excsc'''), the exsecant of the complementary angle:
The PSNR (in 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>


:<math>\operatorname{excsc}(\theta) = \operatorname{exsec}(\pi/2 - \theta) = \csc(\theta) - 1. \!</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. 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 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>


The reason to define a special function for the exsecant is similar to the rationale for the [[versine]]: for small [[angle]]s <var>θ</var>, the sec(<var>θ</var>) function approaches [[1 (number)|one]], and so using the above formula for the exsecant will involve the [[subtraction]] of two nearly equal quantities and exacerbate roundoff errors. Thus, a table of the secant function would need a very high accuracy to be used for the exsecant, making a specialized exsecant table useful. Even with a computer, [[floating point]] errors can be problematic for exsecants of small angles. A more accurate formula in this limit would be to use the identity:
Typical values for the PSNR in [[lossy compression|lossy]] image and video compression are between 30 and 50&nbsp;[[decibel|dB]], provided the bit depth is 8&nbsp;[[bit|bits]], where higher is better. For 16-bit data typical values for the PSNR are between 60 and 80&nbsp;[[decibel|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=http://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 journal|author=Raouf Hamzaoui, Dietmar Saupe|title=Fractal Image Compression|journal=Document and image compression|date=May 2006|volume=968|pages=168–169|url=http://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>


:<math>\operatorname{exsec}(\theta) = \frac{1-\cos(\theta)}{\cos(\theta)}
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=http://books.google.com/books?id=ujnQogzx_2EC&lpg=PA281&ots=FolwqB8qsN&dq=PSNR%20infinite&pg=PA281#v=onepage&q=PSNR%20infinite&f=false|edition=4|accessdate=26 July 2012|page=281}}</ref>
= \frac{\operatorname{versin}(\theta)}{\cos(\theta)}
= 2 \sin^2(\theta/2) \sec(\theta).\ </math>


Prior to the availability of computers, this would require time-consuming multiplications.
{{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


The name ''exsecant'' can be understood from a graphical construction, at right, of the various trigonometric functions from a [[unit circle]], such as was used historically. sec(<var>θ</var>) is the [[secant line|secant]] <math>\overline{OE}</math>, and the exsecant is the portion <math>\overline{DE}</math> of this secant that lies ''exterior'' to the circle (''ex'' is [[Latin]] for ''out of'').
(full resolution is [http://upload.wikimedia.org/wikipedia/commons/d/d3/PSNR-example-base.png here].)
  }}


==See also==
==See also==
* [[List of trigonometric identities#Historic shorthands|Trigonometric identities]]
* [[Data compression ratio]]
* [[Versine|Versine and haversine]]
* [[PEVQ|Perceptual Evaluation of Video Quality (PEVQ)]]
* [[Signal-to-noise ratio]]
* [[Structural similarity|Structural similarity (SSIM) index]]
* [[Subjective video quality]]
* [[Video quality]]


==References==
==References==
* M. Abramowitz and I. A. Stegun, eds., ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables'' (Dover: New York, 1972), p.&nbsp;78.  (See [[Abramowitz and Stegun]].)
{{Reflist}}
* James B. Calvert, [http://www.du.edu/~jcalvert/math/trig.htm Trigonometry] (2004).  Retrieved 25 December 2004.


[[Category:Trigonometry]]
{{Noise}}
[[Category:Elementary special functions]]
 
[[Category:Image compression]]
[[Category:Noise]]
[[Category:Film and video technology]]
[[Category:Digital television]]
[[Category:Engineering ratios]]
 
[[vi:PSNR]]

Revision as of 11:38, 13 August 2014

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.

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. 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.[1][2]

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:

MSE=1mn∑i=0m−1∑j=0n−1[I(i,j)−K(i,j)]2

The PSNR (in dB) is defined as:

PSNR=10⋅log10(MAXI2MSE)=20⋅log10(MAXIMSE)=20⋅log10(MAXI)−10⋅log10(MSE)

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. 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. 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.[3][4]

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.[5][6] Acceptable values for wireless transmission quality loss are considered to be about 20 dB to 25 dB.[7][8]

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).[9]

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See also

References

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Template:Noise

vi:PSNR

  1. ↑ Template:Cite doi
  2. ↑ MIT.edu
  3. ↑ Template:Cite web
  4. ↑ Template:Cite web
  5. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  6. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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  7. ↑ 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).
  8. ↑ 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.
  9. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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