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{{See also|Film speed#Measurements and calculations}}
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The '''signal-to-noise ratio''' ('''SNR''') is used in imaging as a physical measure of the [[sensitometry|sensitivity]] of a [[photography|(digital or film) imaging system]]. Industry standards measure '''SNR''' in [[decibels]] (dB) of [[power (physics)|power]] and therefore apply the [[20 log rule]] to the "pure" '''SNR''' ''ratio'' (a ratio of 1:1 yields 0 decibels, for instance). In turn, yielding the "sensitivity." Industry standards measure and define sensitivity in terms of the [[Film speed|ISO film speed]] equivalent; SNR:32.04 dB = excellent image quality and SNR:20 dB = acceptable image quality.<ref>ISO 12232: 1997  Photography – Electronic Still Picture Cameras – Determining ISO Speed here</ref>
 
==Definition of SNR==
[[Image:SiTF Half Moon Target.jpg|thumb|right|250px|An operator arbitrarily defines a box area in the signal and background regions of a back-illuminated '''half moon''' or '''knife-edge''' test target. The data, (such as [[pixel|pixel intensity]]), is used to determine the average signal and background values.]]
 
Traditionally, '''SNR''' has been defined as the ratio of the average signal value <math>\mu_\mathrm{sig}</math> to the [[standard deviation]] <math>\sigma_\mathrm{bg}</math> of the background:
 
:<math> \mathrm{SNR} = \frac{\mu_\mathrm{sig}}{\sigma_\mathrm{bg}}</math>
 
However, when presented with a high-contrast scene, many imaging systems [[clamping (graphics)|clamp]] the background to uniform black, forcing <math>\sigma_\mathrm{bg}</math> to zero, artificially making the SNR infinite.<ref name="EO">{{Cite web
  | last1 = Mazzetta
  | first1 = J.
  | last2 = Caudle
  | first2 = Dennis
  | last3 = Wageneck
  | first3 = Bob
  | title = Digital Camera Imaging Evaluation
  | publisher = Electro Optical Industries
  | url = http://www.electro-optical.com/pdf/EOI%20OSG%202005%20Paper%20final.pdf
  | page = 8
  | year = 2005
  | format = PDF
  | accessdate = 28 March 2010}}
</ref>
In this case a better definition of SNR is the ratio of the average signal value <math>\mu_\mathrm{sig}</math> to the standard deviation of the ''signal'' <math>\sigma_\mathrm{sig}</math>:
 
:<math> \mathrm{SNR} = \frac{\mu_\mathrm{sig}}{\sigma_\mathrm{sig}}</math>
 
which gives a meaningful result in the presence of clamping.
 
==Calculations==
 
=== Explanation ===
The [[line (video)|line data]] is gathered from the arbitrarily defined signal and background regions and input into an [[Array data structure|array]] (refer to image to the right). To calculate the average signal and background values, a [[polynomial|second order polynomial]] is [[curve fitting|fitted]] to the array of line data and subtracted from the original array line data. This is done to remove any [[trend estimation|trends]]. Finding the mean of this data yields the average signal and background values. The net signal is calculated from the difference of the average signal and background values. The '''RMS''' or [[root mean square]] noise is defined from the signal region. Finally, '''SNR''' is determined as the ratio of the net signal to the '''RMS noise.'''
 
===Polynomial and coefficients===
*The second order polynomial is calculated by the following double summation.
<br />
<math>f_i = \sum_{j=0}^m \sum_{i=1}^n a_j x_i^j</math>
 
**<math>f\,</math> = [[function (mathematics)|output sequence]]
**<math>m\,</math> = the polynomial order
**<math>x\,</math> = the [[function (mathematics)|input sequence]] (array/line values) from the signal region or background region, respectively.
**<math>n\,</math> = the number of [[line (video)|lines]]
**<math>a_j\,</math> = the polynomial fit [[coefficients]]
 
*The polynomial fit coefficients can thus be calculated by a [[system of equations]].<ref>Aboufadel, E.F., Goldberg, J.L., Potter, M.C. (2005).''Advanced Engineering Mathematics (3rd ed.).''New York, New York: Oxford University Press</ref>
<br />
<math>
\begin{bmatrix}
    1      &    x_1    &    x_1^2  \\
    1      &    x_2    &    x_2^2  \\
    \vdots &    \vdots  &    \vdots \\
    1      &    x_n    &      x_n^2
\end{bmatrix}
 
\begin{bmatrix}
    a_2      \\
    a_1      \\
    a_0      \\
\end{bmatrix}
=
\begin{bmatrix}
    f_1      \\
    f_2      \\
    \vdots    \\
    f_n     
\end{bmatrix}
</math>
 
*Which can be written...
<br />
<math>
\begin{bmatrix}
    n            &    \sum x_i      &    \sum x_i^2  \\
    \sum x_i      &    \sum x_i^2    &    \sum x_i^3  \\
    \sum x_i^2    &    \sum x_i^3    &    \sum x_i^4 
\end{bmatrix}
 
\begin{bmatrix}
    a_2      \\
    a_1      \\
    a_0 
\end{bmatrix}
=
\begin{bmatrix}
    \sum f_i      \\
    \sum f_i x_i  \\
    \sum f_i x_i^2
\end{bmatrix}
</math>
 
*Computer software or rigorous [[row operations]] will solve for the coefficients.
 
===Net signal, signal, and background===
The second-order polynomial is subtracted from the original data to remove any trends and then averaged. This yields the signal and background values{{Clarify|date=March 2010}}:
:<math>\mu_\text{sig} = \frac{\sum_{i=1}^n (X_i - f_i)}{n} \qquad \qquad \mu_\text{bkg} = \frac{\sum_{i=1}^n (X_i-f_i)}{n}</math>
where
*<math>\mu_\text{sig}</math> = average signal value
*<math>\mu_\text{bkg}</math> = average background value
*<math>n\,</math> = number of lines in background or signal region
*<math>X_i\,</math> = value of the i<sup>th</sup> line in the signal region or background region, respectively.
*<math>f_i\,</math> = value of the i<sup>th</sup> output of the second order polynomial.
Hence, the net signal value is determined by {{Citation needed|date=March 2010}}:
:<math>\text{signal} = \mu_\text{sig} - \mu_\text{bkg}</math>.
 
===RMS noise and SNR===
*The '''RMS Noise''' is defined as the [[square root]] of the [[variance|sum of variances]] from the signal region.<ref name="EO" />
:<math>\text{RMS noise} = \sqrt{\frac{\sum_{i=1}^n (X_i-\frac{\sum_{i=1}^n X_i}{n})^2}{n}}</math>
The SNR is thus given by
: <math> \text{SNR} = \frac{\text{signal}}{\text{RMS noise}}</math>
 
Using the industry standard [[20 log rule]]<ref>Test and Measurement World (2008). SNR. In ''Glossary and Abbreviations''.http://www.tmworld.com/info/CA6436814.html?q=SNR</ref>...
:<math> \text{SNR} = 20 \log_{10} \frac{\text{signal}}{\text{RMS noise}}\,\mbox{dB}</math>
 
==See also==
* [[Coefficient of variation]]
* [[Minimum resolvable contrast]]
* [[Minimum resolvable temperature difference]]
* [[Optical transfer function]]
* [[Signal transfer function]]
 
==References==
<references/>
 
==Further reading==
* [http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=28835 ISO 15739:2003, Photography -- Electronic still-picture imaging -- Noise measurements]: specifies methods for measuring and reporting the noise versus signal level and dynamic range of electronic still-picture cameras. It applies to both monochrome and colour electronic still-picture cameras.
* [http://www.iso.org/iso/iso_catalogue/catalogue_ics/catalogue_detail_ics.htm?ics1=37&ics2=040&ics3=10&csnumber=37777 ISO 12232:2006, Photography -- Digital still cameras -- Determination of exposure index, ISO speed ratings, standard output sensitivity, and recommended exposure index]: specifies the method for assigning and reporting ISO speed ratings, ISO speed latitude ratings, standard output sensitivity values, and recommended exposure index values, for digital still cameras. ISO 12232:2006 is applicable to both monochrome and colour digital still cameras. It revises ISO 12232:1998.
 
{{Noise}}
{{photography subject}}
{{Use dmy dates|date=September 2010}}
 
{{DEFAULTSORT:Signal To Noise Ratio (Imaging)}}
[[Category:Image processing]]
[[Category:Optics]]
[[Category:Science of photography]]
[[Category:Units of measurement]]

Latest revision as of 11:34, 9 August 2014

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