Editing Signal-to-noise ratio (section)

== Alternative definition ==
An alternative definition of SNR is as the reciprocal of the [[coefficient of variation]], i.e., the ratio of [[mean]] to [[standard deviation]] of a signal or measurement:<ref>
{{cite book | title = Astronomical optics | author = D. J. Schroeder | edition = 2nd | publisher = Academic Press | year = 1999 | isbn = 978-0-12-629810-9 | page = 278 | url = https://books.google.com/books?id=v7E25646wz0C&pg=PA433}}, [https://books.google.com/books?id=VZvqqaQ5DvoC&q=signal%20to%20noise&pg=PA278 p.278]</ref><ref name=b1>Bushberg, J. T., et al., ''[https://books.google.com/books?id=VZvqqaQ5DvoC&pg=PA280 The Essential Physics of Medical Imaging],'' (2e). Philadelphia: Lippincott Williams & Wilkins, 2006, p. 280.</ref>
:<math>
 \mathrm{SNR} = \frac{\mu}{\sigma}
</math>
where <math>\mu</math> is the signal mean or [[expected value]] and <math>\sigma</math> is the standard deviation of the noise, or an estimate thereof.<ref group="note">The exact methods may vary between fields. For example, if the signal data are known to be constant, then <math>\sigma</math> can be calculated using the standard deviation of the signal. If the signal data are not constant, then <math>\sigma</math> can be calculated from data where the signal is zero or relatively constant.</ref> Notice that such an alternative definition is only useful for variables that are always non-negative (such as photon counts and [[luminance]]), and it is only an approximation since <math>\operatorname{E}\left[X^2 \right] = \sigma^2 + \mu^2 </math>. It is commonly used in [[image processing]],<ref>{{cite book|url=https://books.google.com/books?id=8uGOnjRGEzoC&pg=PA354|title=Digital image processing|page=354|author=Rafael C. González, Richard Eugene Woods|publisher=Prentice Hall|year=2008|isbn=978-0-13-168728-8}}</ref><ref>{{cite book|url=https://books.google.com/books?id=VmvY4MTMFTwC&pg=PA471|title=Image fusion: algorithms and applications|page=471|author=Tania Stathaki|publisher=Academic Press|year=2008|isbn=978-0-12-372529-5}}</ref><ref>{{cite book|url=https://books.google.com/books?id=7s8xpR-5rOUC&pg=PA471 |title=Multi-Sensor Data Fusion: Theory and Practice|author=Jitendra R. Raol|publisher=CRC Press|year=2009|isbn=978-1-4398-0003-4}}</ref><ref>{{cite book|url=https://books.google.com/books?id=Vs2AM2cWl1AC&pg=PA26 |title=The image processing handbook|author=John C. Russ|publisher=CRC Press|year=2007|isbn=978-0-8493-7254-4}}</ref> where the SNR of an [[image]] is usually calculated as the ratio of the [[mean]] pixel value to the [[standard deviation]] of the pixel values over a given neighborhood.

Sometimes{{explain|date=March 2021}} SNR is defined as the square of the alternative definition above, in which case it is equivalent to the [[#Definition|more common definition]]:
:<math>
 \mathrm{SNR} = \frac{\mu^2}{\sigma^2}
</math>

This definition is closely related to the [[sensitivity index]] or ''d{{'}}'', when assuming that the signal has two states separated by signal amplitude <math>\mu</math>, and the noise standard deviation <math>\sigma</math> does not change between the two states.

The ''Rose criterion'' (named after [[Albert Rose (physicist)|Albert Rose]]) states that an SNR of at least 5 is needed to be able to distinguish image features with certainty. An SNR less than 5 means less than 100% certainty in identifying image details.<ref name=b1/><ref>
{{cite book |last= Rose |first= Albert |title= Vision – Human and Electronic |publisher=  Plenum Press  |isbn= 9780306307324| year = 1973 | page=[https://archive.org/details/visionhumanelect01rose/page/10 10] | url=https://archive.org/details/visionhumanelect01rose |url-access= registration |quote= [...] to reduce the number of false alarms to below unity, we will need [...] a signal whose amplitude is 4–5 times larger than the rms noise.}}</ref>

Yet another alternative, very specific, and distinct definition of SNR is employed to characterize [[film speed|sensitivity]] of imaging systems; see [[Signal-to-noise ratio (imaging)]].

Related measures are the "[[contrast ratio]]" and the "[[contrast-to-noise ratio]]".