SNR Curves and IQ in Digital Cameras

In photography the higher the ratio of Signal to Noise, the better the final image normally looks.  The signal-to-noise-ratio SNR is therefore a key component of IQ.  Let’s take a closer look at it.

We know that for a given number of impinging photons during exposure our cameras’ digital sensors produce a proportional output signal S in units of photoelectrons according to the simple relationship:

(1)   \begin{equation*} S=EQE \cdot N_{photons} \: \: \: e^- \end{equation*}

where EQE is Effective Quantum Efficiency.  We typically model three macro ‘random’ noise components (\sigma) present at the output of the sensor with signal S.  Shot Noise inherently present in the incoming light and photoelectric conversion process

(2)   \begin{equation*} \sigma_{shot}=\sqrt{S} \:, \end{equation*}

noise introduced by the sensor and related electronics (\sigma_{read});  and unpredictable sensor non-uniformities (\sigma_{prnu}) which are proportional to signal S – all in units of e^-.   Ignoring Pattern and other types of noise, Total Random Noise is then obtained by summing the three components in quadrature

(3)   \begin{equation*} \sigma_{total}=\sqrt{S+\sigma_{read}^2+(S\cdot\sigma_{prnu})^2} \end{equation*}

You can learn more about modeling noise in digital cameras from these excellent pages by Emil Martinec and DxOLabs.  If we plot the signal, the three noise components and their total in units of photoelectrons this is the graph that results when using average values for the parameters above from a Nikon D610 at base ISO:

Photon Transfer Model D610

The horizontal axis is interchangeable with the number of arriving photons which are directly related to the output signal S by EQE per equation (1).  Note that the Total Noise curve (solid black line) has three very distinct areas, depending on the relative values of the three components of noise: at very low signals it is mainly made up of  shot and read noise; at 5 or 6 stops below saturation it is mainly shot noise; and approaching saturation it is mainly shot noise and PRNU.  If we calculate and plot the resulting signal S to total noise \sigma_{total} ratio (SNR) this is what it looks like

SNR-Photon-Transfer-Model-D610-4

The red curve is the measured SNR curve, what DxOMark.com calls a Full SNR Curve. They present their x-axis as a relative percentage of signal full scale while the graph above has an absolute scale in units of lx-s, which will come in handy in future posts.

SNR curves are a good way to characterize a sensor’s IQ by showing at a glance its noise performance  throughout the operating range – from deep shadows, to mid-tones to highlights.   Note how the measured SNR curve dips from the ideal at low signal levels because of the increasing contribution of the read noise ‘floor’; and near saturation, although less dramatically, because of PRNU.  The ‘engineering’ Dynamic Range of the sensor can be read off the x-axis by simply counting the number of stops between saturation Exposure and when the SNR is equal to 1 (zero as a logarithm above), in this case from 0.06 to -13.5, for a total of 13.6 stops of pixel level DR.

Also note that SNR is highest near saturation and it continues to decreases monotonically until it hits eDR’s lower limit .  Therefore in order to maximize SNR (hence IQ) if a photographer can they should always use the full dynamic range at their disposal by exposing the image so that the brightest desirable highlights are recorded just below saturation, a strategy sometimes descibed as Exposing To The Right (ETTR).

In clean sensors at low ISOs (for instance Exmor class sensors) there is usually an area where the SNR curve is effectively shot noise dominated, in this case 5 or 6 stops below saturation.  This area is quite significant because it allows us to estimate the signal in e^- by simply squaring the ratio of the mean and standard deviation of a uniform patch in the raw file, measured in ADU with a program like RawDigger.    This bit of magic allows us to jump straight from output raw file in ADU to input signal in e^- (photons if we know EQE) bypassing all the complicated steps and parameters that we would need to know otherwise.  It is called Photon Transfer and it works because shot noise follows Poisson statistics where

(4)   \begin{equation*} SNR_{shot} = \sigma_{shot} = \sqrt{S} \end{equation*}

so in a shot noise dominated area of the picture signal S={SNR}^2 in units of e^-.

Next, how to use SNR curves to compare cameras.