What is Photon transfer Curve?

What is Photon transfer Curve?
Linear Signal Model

Figure 1-1: a. Physical model of the camera and b. Mathematical model of a single pixel. Figures separated by comma represent the mean and variance of a quantity; unknown model parameters are marked in red.

As illustrated in Fig. 1-1, a digital image sensor essentially converts photons hitting the pixel area during the exposure time texp by a sequence of steps finally into a digital number. During the exposure time on average () photons hit the whole area A of a single pixel. A fraction of them, the total quantum efficiency, is absorbed and accumulates () charge units. The total quantum efficiency as defined here refers to the total area occupied by a single sensor element (pixel) not only the light sensitive area.

Consequently, this definition includes the effects of fill factor and microlenses. As expressed in Eq. (1), the quantum efficiency depends on the wavelength of the photons irradiating the pixel.
The mean number of photons that hit a pixel with the area A during the exposure time can be computed from the irradiance E on the sensor surface in W/m2 by

using the well-known quantization of the energy of electromagnetic radiation in units of  h.These equations are used to convert the irradiance calibrated by radiometers in units of W/cm2 into photon fluxes required to characterize imaging sensors.In the camera electronics, the charge units accumulated by the photo irradiance is converted into a voltage, amplified, and finally converted into a digital signal y by an analog digital converter (ADC). The whole process is assumed to be linear and can be described by a single quantity, the overall system gain K with units DN/e , i. e., digits per electrons. Then the mean digital signal ( )results in

Where()is the mean number electrons present without light, which result in the mean dark signal(  )in units DN with zero irradiation. Note that the dark signal will generally depend on other parameters, especially the exposure time and the ambient temperature.With Eqs. (1) and (2), Eq. (3) results in a linear relation between the mean gray value()and the number of photons irradiated during the exposure time onto the pixel:

This equation can be used to verify the linearity of the sensor by measuring the mean gray value in relation to the mean number of photons incident on the pixel and to measure the responsivity Kn from the slope of the relation. Once the overall system gain K is determined from Eq. (7), it is also possible to estimate the quantum efficiency from the responsivity Kn.

Noise Model

The number of charge units (electrons) fluctuates statistically. According to the laws of quantum mechanics, the probability is Poisson distributed. Therefore the variance of the fluctuations is equal to the mean number of accumulated electrons:

This noise, often referred to as shot noise is given by the basic laws of physics and equal for all types of cameras.
All other noise sources depend on the specific construction of the sensor and the camera electronics. Due to the linear signal model, all noise sources add up. For the purpose of a camera model treating the whole camera electronics as a black box it is sufficient to consider only two additional noise sources. All noise sources related to the sensor read out and amplifier circuits can be described by a signal independent normal distributed noise source with the variance (). The final analog digital conversion (Fig. 1-1-b) adds another noise source that is uniform-distributed between the quantization intervals and has a variance (  ). Because the variances of all noise sources add up linearly, the total temporal variance of the digital signal  (   )  is given according to the laws of error propagation by

Using Eqs. (3) and (5), the noise can be related to the measured mean digital signal:

This equation is central to the characterization of the sensor. From the linear relation between the variance of the noise and the mean photo-induced gray value () it is possible to determine the overall system gain K from the slope and the dark noise variance () from the offset. This method is known as the photon transfer method.

Evaluation of the Measurements according to the Photon Transfer Method

The application of the photon transfer method and the computation of the quantum efficiency requires the measurement of the mean gray values and the temporal variance of the gray together with the irradiance per pixel in units photons/pixel. The mean and variance are computed in the following way:
Mean gray value. The mean of the gray values () over all N pixels in the active area at each irradiation level is computed from the two captured M x N images ya and yb as
averaging over all rows i and columns j . In the same way, the mean gray value of dark images,(), is computed.
Temporal variance of gray value. Normally, the computation of the temporal variance would require the capture of many images. However, the noise is stationary and homogenous, so that it is sufficient to take the mean of the squared difference of the two images mean of the squared difference of the two images

Because the variance of the difference of two values is the sum of the variances of the two values, the variance computed in this way must be 
divided by two as indicated in Eq. (9).
Saturation. The saturation gray value is given as the mean gray value where the variance () has a maximum value (see green square in Fig. 1-1). To find this value the following procedure is recommended: The saturation point is given by scanning the photon transfer curve from the right and given by the first point where the next two points are lower. For a smooth photon transfer curve this is equivalent to taking the absolute maximum.
Responsivity R. According to Eq. (4), the slope of the relation

(with zero offset) gives the responsivity R=Kn. For this regression all data points must be used in the range between the minimum value and 70% saturation 

Overall system gain K. According to Eq. (7), the slope of the relation

(with zero offset) gives the absolute gain factor K. Select the same range of data points as for the estimation of the responsivity (see above, and Fig. 1-3).

Compute a least-squares linear regression of () versus () over the selected range and specify the gain factor K.

Figure 1-2: Example of a measuring curve to determine the responsivity () of a camera. The graph draws the measured mean photo-induced gray values () versus the irradiation H in units photons/pixel and the linear regression line used to determine (). The red dots marks the 0 - 70% range of saturation that is used for the linear regression. For color cameras, the graph must contain these items for each color channel. If the irradiation is changed by changing the exposure time, a second graph must be provided which shows () as a function of the exposure time texp.

Figure 1-3: Example of a measuring curve to determine the overall system gain K of a camera (photo transfer curve). The graph draws the measured photo-induced variance () versus the mean photo-induced gray values () and the linear regression line used to determine the overall system gain K. The green dots mark the 0 - 70% range of saturation that is used for the linear regression. The system gain K is given with its one-sigma statistical uncertainty in percent, computed from the linear regression.

Quantum Efficiency

The quantum efficiency is given as the ratio of the responsivity () and the overall system gain K.

For monochrome cameras, the quantum efficiency is thus obtained only for a single wavelength band with a bandwidth no wider than 50 nm. Because all measurements for color cameras are performed for all color channels, quantum efficiencies for all these wavelengths bands are obtained and to be reported. For color camera systems that use a color filter pattern any pixel position in the repeated pattern should be analyzed separately. For a Bayer pattern, for example, there are four color channels in total, mostly two separate green channels, a blue channel, and a red channel.