Which is the link between a digital ISO value (e.g. 100) and the sensor sensitivity?

Question

There are many questions about ISO (such as 1 and 2), but none of them answers my precise question. My doubt starts from the exposure formula (for instance let's consider it in linear scale):

My questions are:

1. There is a unit of measurement misunderstanding in my formula, and I do not understand the solution. As written there,

ISO speed ratings of a digital camera are based on the properties of the sensor and the image processing done in the camera, and are expressed in terms of the luminous exposure H (in lux seconds) arriving at the sensor.

it appears to be that ISO is expressed as lux*seconds.

But if it were, the exposure unit of measurement would be, by the previous formula, (lux*s)^2! What I'm missing?

2. My really basic academic studies of CCD and CMOS sensors have defined the single pixel signal power as:

Such a signal power corresponds to a lumen value through the Watt-Lumen relationship. Such a signal is multiplied by the gain of the sensor output amplifier. This gain is tunable and it's the quantity we modify when we change the ISO setting?

So, what is exposure in this formula? Is the signal power converted in lumen? It does not appear to be that. As far as I'm concerned, exposure is measured in lux ( = lumen/m^2), not lumen.

3. To increase the overall confusion, there is the following statement about Base ISO:

The terms "Base ISO" refers to the unamplified sensitivity of the camera.

So, what is the unamplified sensitivity? Let's consider the exposure equation, and let's imagine the Base ISO being 100 (the same value of the reference value). It means ISO/100 = 1 in such an equation. Exposure would equal the subject luminance times the shutter time divided by the f number squared.

The overall sensor sensitivity would appear to be 1, independently on the quantum efficiency and pixel size. It looks like so strange.

Answer 1

The basic exposure formula is fine/simple... exposure = sensor illuminance x sensitivity ÷ 100.

But digital exposure is a ratiometric value.

First the sensor is given an electrical charge (like a static charge). Then when a photon of visible light hits the sensor it frees an electron from that charge. That electron is then stored in the photodiode (acting as a capacitor). The stored electrons are the conversion gain, and generally described in terms of photo electrons (photons converted into freed electrons). And the photosite's exposure is the conversion gain (voltage) achieved as a ratio of what the photodiode (capacitor) could store at maximum exposure (full well capacity). Strictly in terms of size/area (m²) a larger photosite requires an equally larger conversion gain in order to be considered equally exposed.

So base ISO is sensor fill efficiency (microlenses, color filters, circuitry, etc), combined with the photosite conversion gain requirement, and limited by photosite read (packet) error and noise (switching).

At its' base, 1 photon hitting the sensor results in one freed electron of conversion gain. And since a photosite of 4x area (2x LxW) would theoretically receive 4x as many photons in a given period of time, the larger photosite remains equally charged/exposed; base ISO is effectively 1 irrespective of photosite/sensor area... one would think it should be the same for all sensors.

But it doesn't always work that way. e.g. a smaller photosite's lower conversion gain requirement can result in a lower read error/noise making the smaller photosite effectively more sensitive... that's why many modern sensors are using smaller photosites of dual gain design (secondary capacitors/dual ISO); and with better fill efficiencies (e.g. back side illuminated to avoid the circuitry limitation). And if you can combine lower conversion gain requirements with lower read noise you can generate fewer ADC errors (write noise) and a much higher overall QE (results in useful ISO invariance).

However ISO/exposure is not exactly standardized... especially the Recommended Exposure Index (REI) method used by most DSLR's and larger format cameras. But the recommended exposure (rated ISO) isn't usually more than 1/2 stop offset (reduced) from a more rigid ISO rating method (such as SOS)... the CIPA standard does say that the camera's ISO should result in an exposure similar to the result given by using a handheld/external exposure meter (and those sensitivities/exposures aren't exactly standardized either).

Any analog voltage gain added to the conversion gain voltage prior to ADC conversion is analog ISO amplification above base ISO (or base ISO's in dual gain designs). Digital ISO (after ADC) is also possible, but that could be just about any type of exposure manipulation.

Answer 2

If the base ISO is 100 the digital values are directly converted from analogue signal from sensor. think about schematic like this:

sensor->amplifier->ADC

So in case of base ISO you have no amplification, amplification=1

P.S. Some amplification (multiplication) may be done after the DAC, but this will require DAC with bigger resolution than RAW files of the camera.