Signal-to-noise ratio (imaging)

The signal-to-noise ratio (SNR) is used in imaging as a physical measure of the sensitivity of a (digital or film) imaging system. Industry standards measure SNR in decibels (dB) of power and therefore apply the 10 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 ISO film speed equivalent; SNR:32.04 dB = excellent image quality and SNR:20 dB = acceptable image quality.^[1]

Definition of SNR

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 intensity), is used to determine the average signal and background values.

Traditionally, SNR has been defined as the ratio of the average signal value $\mu_\mathrm{sig}$ to the standard deviation $\sigma_\mathrm{bg}$ of the background:

$\mathrm{SNR} = \frac{\mu_\mathrm{sig}}{\sigma_\mathrm{bg}}$

However, when presented with a high-contrast scene, many imaging systems clamp the background to uniform black, forcing $\sigma_\mathrm{bg}$ to zero, artificially making the SNR infinite.^[2] In this case a better definition of SNR is the ratio of the average signal value $\mu_\mathrm{sig}$ to the standard deviation of the signal $\sigma_\mathrm{sig}$:

$\mathrm{SNR} = \frac{\mu_\mathrm{sig}}{\sigma_\mathrm{sig}}$

which gives a meaningful result in the presence of clamping.

Calculations

Explanation

The line data is gathered from the arbitrarily defined signal and background regions and input into an array (refer to image to the right). To calculate the average signal and background values, a second order polynomial is fitted to the array of line data and subtracted from the original array line data. This is done to remove any 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.

$f_i = \sum_{j=0}^m \sum_{i=1}^n a_j x_i^j$

• • $f\,$ = output sequence
  • $m\,$ = the polynomial order
  • $x\,$ = the input sequence (array/line values) from the signal region or background region, respectively.
  • $n\,$ = the number of lines
  • $a_j\,$ = the polynomial fit coefficients
• The polynomial fit coefficients can thus be calculated by a system of equations.^[3]

$\begin{bmatrix} 1 & x_1 & x_1^2 \\ 1 & x_2 & x_2^2 \\ \vdots & \vdots & \vdots \\ 1 & x_n & x_n^2 \end{bmatrix} \begin{bmatrix} a_2 \\ a_1 \\ a_0 \\ \end{bmatrix} = \begin{bmatrix} f_1 \\ f_2 \\ \vdots \\ f_n \end{bmatrix}$

• Which can be written...

$\begin{bmatrix} n & \sum x_i & \sum x_i^2 \\ \sum x_i & \sum x_i^2 & \sum x_i^3 \\ \sum x_i^2 & \sum x_i^3 & \sum x_i^4 \end{bmatrix} \begin{bmatrix} a_2 \\ a_1 \\ a_0 \end{bmatrix} = \begin{bmatrix} \sum f_i \\ \sum f_i x_i \\ \sum f_i x_i^2 \end{bmatrix}$

• 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:

$\mu_\text{sig} = \frac{\sum_{i=1}^n (X_i - f_i)}{n} \qquad \qquad \mu_\text{bkg} = \frac{\sum_{i=1}^n (X_i-f_i)}{n}$

where

• $\mu_\text{sig}$ = average signal value
• $\mu_\text{bkg}$ = average background value
• $n\,$ = number of lines in background or signal region
• $X_i\,$ = value of the i^th line in the signal region or background region, respectively.
• $f_i\,$ = value of the i^th output of the second order polynomial.

Hence, the net signal value is determined by :

$\text{signal} = \mu_\text{sig} - \mu_\text{bkg}$.

RMS noise and SNR

• The RMS Noise is defined as the square root of the mean of variances from the background region.^[2]

$\text{RMS noise} = \sqrt{\frac{\sum_{i=1}^n (X_i-\frac{\sum_{i=1}^n X_i}{n})^2}{n}}$

The SNR is thus given by

$\text{SNR} = \frac{\text{signal}}{\text{RMS noise}}$

Using the industry standard 20 log rule^[4]...

$\text{SNR} = 20 \log_{10} \frac{\text{signal}}{\text{RMS noise}}\,\mbox{dB}$

See also

• Coefficient of variation
• Minimum resolvable contrast
• Minimum resolvable temperature difference
• Optical transfer function
• Signal transfer function

References

1. ISO 12232: 1997 Photography – Electronic Still Picture Cameras – Determining ISO Speed here
2. Mazzetta, J.; Caudle, Dennis; Wageneck, Bob (2005). "Digital Camera Imaging Evaluation" (PDF). Electro Optical Industries. p. 8. Retrieved 28 March 2010.
3. Aboufadel, E.F., Goldberg, J.L., Potter, M.C. (2005).Advanced Engineering Mathematics (3rd ed.).New York, New York: Oxford University Press
4. Test and Measurement World (2008). SNR. In Glossary and Abbreviations.http://www.tmworld.com/info/CA6436814.html?q=SNR