Signal to noise ratio (image processing)

The Signal to Noise Ratio (SNR) is used in imaging as a physical measure of the sensitivity

Sensitometry

Sensitometry is the scientific study of light-sensitive materials, especially photographic film. The study has its origins in the work by Ferdinand Hurter and Vero Charles Driffield with early black-and-white emulsions...

of a (digital or film) imaging system

Photography

Photography is the art, science and practice of creating durable images by recording light or other electromagnetic radiation, either electronically by means of an image sensor or chemically by means of a light-sensitive material such as photographic film...

. Industry standards measure SNR in decibels (dB) of power

Power (physics)

In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

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 ISO film speed

Film speed

Film speed is the measure of a photographic film's sensitivity to light, determined by sensitometry and measured on various numerical scales, the most recent being the ISO system....

equivalent; SNR:32.04 dB = excellent image quality and SNR:20 dB = acceptable image quality.

Definition of SNR

Traditionally, SNR has been defined as the ratio of the average signal value

to the standard deviation

Standard deviation

Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

of the background:

However, when presented with a high-contrast scene, many imaging systems clamp

Clamping (graphics)

In computer graphics, clamping is the process of limiting a position to an area. Unlike wrapping, clamping merely moves the point to the nearest available value.To put clamping into perspective, pseudocode for clamping is:...

the background to uniform black, forcing

to zero, artificially making the SNR infinite.
In this case a better definition of SNR is the ratio of the average signal value

to the standard deviation of the signal

which gives a meaningful result in the presence of clamping.

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

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

is fitted

Curve fitting

Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function...

to the array of line data and subtracted from the original array line data. This is done to remove any trends

Trend estimation

Trend estimation is a statistical technique to aid interpretation of data. When a series of measurements of a process are treated as a time series, trend estimation can be used to make and justify statements about tendencies in the data...

. 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

Root mean square

In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...

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.

- = output sequence
  Function (mathematics)
  In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
- = the polynomial order
- = the input sequence
  Function (mathematics)
  In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
  
  (array/line values) from the signal region or background region, respectively.
- = the number of lines
- = the polynomial fit coefficients

The polynomial fit coefficients can thus be calculated by a system of equations.

Which can be written...

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:

where

= average signal value
= average background value
= number of lines in background or signal region
= value of the i^th line in the signal region or background region, respectively.
= value of the i^th output of the second order polynomial.

Hence, the net signal value is determined by :

RMS noise and SNR

The RMS Noise is defined as the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

of the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

of the sum of variances
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

from the signal region.

The SNR is thus given by

Using the industry standard 20 log rule...

Definition of SNR

Explanation

Polynomial and coefficients

Net signal, signal, and background

RMS noise and SNR

See also

Further reading