Which is the physical cause of the increasing noise at high iso?
I know that Iso represents the sensor/film sensitivity to light: at lower iso we need an higher exposure time to get the image of the desired brightness, while at higher iso we need a lower exposure time. Which is the connection of this with noise?
ISO for Analog vs Digital
ISO is confusing in digital photography in part because it was actually meant for film photography. In film photography, ISO 400 really is more sensitive than ISO 100 film.
This isn't actually true of digital photography. The sensitivity of your camera sensor is whatever it is and does not change. The sensor works a bit like an array of very tiny solar panels. Photons of light are particles that carry energy. That energy is absorbed by the photo-site on your camera's sensor array and creates a tiny charge (voltage). This is analog information -- it is not yet in digital form.
When the exposure is completed and the shutter closes, the camera will perform a read-out of the information from the sensor.
Upstream and Downstream Gain
Two things can happen and how this works will depend on the camera.
The camera can apply analog amplification. Cameras typically only do this for a handful of "stops" of gain. Since the analog information has not yet been converted to digital form, this type of gain is sometimes referred to as "upstream gain".
The analog information is converted to digital by converting voltage into digital units. This is the ADC or Analog to Digital Conversion. The output values are sometimes referred to as ADUs - short for Analog Digital Units. This is the digital output.
This digital information can also be increased by simple multiplication of the numeric values. Since this is occurring after the digital to analog conversion this is sometimes referred to as "downstream gain".
Some cameras do exclusively or mostly digital gain (downstream gain), some use a combination of both upstream and downstream gain. Since this varies by camera model there's no one right answer for how it is done.
At this point, much of the information in the digital format represents actual "signal" -- meaning this is information representing the light collected during the exposure.
Bias
But there are many interesting nuances. For example, if you power up a sensor, keep the lens covered, capture the shortest possible image, and then perform a read-out, you might think that all the pixel values would read zeros. But that's not what happens... you'll find they all read some very small values which are close to zero... but not quite zero. This represents the bias value of the sensor. BTW, modern digital cameras internally offset the bias before writing the image data.
Thermal noise
Also if you continue to keep the lens covered but take a longer exposure, you'll find many of the pixels increase their accumulated ADUs -- even though no photons of visible light are entering the camera. There are several reasons for this and one is thermal noise (and there are camera sensors that are chilled to reduce to reduce this noise.)
There is also noise due to quantum effects. If a camera is slow to perform a read-out you can get noise caused by something called amp-glow. And while rare... the occasional high-energy photon can come wandering by and penetrate the camera.
The take-away here is that there are many causes of 'noise'.
High ISO doesn't cause noise
ABSENT from this list is ISO. ISO doesn't cause 'noise' per se. The noise captured when the exposure is complete and the camera completes the read-out is whatever it is and does not change. ISO is a gain applied after the exposure is completed (it is technically not part of the exposure - even though it's much easier to think of it as if it is part of exposure.)
Imagine a very poor quality audio recording of someone speaking ... but the microphone is too far away from the person speaking. You turn up the volume while playing the audio back and you hear lots of background noise, hiss, hum, and other nuances (maybe even including other people in the room). But because you turn up the volume, you hear all these background sounds which are much more apparent. If the microphone had been closer to the person speaking, you would have stronger "signal" and you would not have needed to increase the play-back volume.
This is a good analogy for the signal-to-noise ratio. The noise is relatively constant. It's the signal that was stronger or weaker. But this dynamic changes the "signal-to-noise ratio" (SNR).
If you have poor signal, you'll be tempted to "turn up the volume". In photography you "turn up the volume" by increasing the ISO (really the gain). But this increases ALL the information... both the signal and the noise.
Noise becomes apparent as the result of insufficient exposure
The noise simply becomes more apparent because the photograph had insufficient signal. Ultimately it is the Signal-to-Noise Ratio (SNR) that dictates how much noticeable noise you see in an image. The noise is always there ... but if the signal values are very high then the information doesn't need to be amplified so the noise isn't noticeable. Basically the signal overwhelms the noise to the point that our eyes don't notice it. If the signal is poor then we have to amplify the information. This results in a poor SNR where the noise is a decent percentage of the overall information and now we do notice the noise.
The take-away here is that the "noise" level of your camera doesn't actually change so much (it would in very long exposures where you get thermal buildup resulting in more noise). If you see "noise" in your images, it means you had insufficient signal. And since ISO isn't technically part of exposure... what it really means is you had an insufficient exposure.
What I just said is probably very contrary to what many photographers have learned. But we learn and teach photography based on the analog concepts of analog film cameras ... and we usually don't go into the depths of how digital sensors and cameras actually work. This mostly serves our needs well, but in the case of ISO and noise it creates confusion.
Think of ISO like an amplifier - it increases the gain on the incoming signal.
If you have an audio amplifier & an old AM radio tuned to a distant signal, you have to turn up the amp in order to hear the station properly. Unfortunately that brings up a whole lot of other noise at the same time - so you can still barely hear the music over the crackling & hiss.
ISO works in just the same way. It amplifies the 'light', but at a cost of also increasing the unwanted random signals already present in the incoming data. The noise is always there, it's just not amplified if you're at ISO 100.
Cameras that are more effective at higher ISO values basically have better signal/noise ratio right from the beginning. A 'quieter' sensor & the capability to actually take in more light, which is why larger sensors are usually better at this - they have a larger area to capture more light.
ISO on a digital camera isn't really ISO like it was on film, but it's an easy way to preserve the familiarity & exposure relationship.
High ISO does not cause more noise. If you take some photos with equal shutter and aperture at different ISO settings, and adjust them to the same exposure in post-processing, you will not find an increase in noise. In fact, you may even find a decrease in noise.
High ISO is a symptom of insufficient light, which is the real cause of noise. The name for this phenomenon is shot noise, and it is the primary source of noise in most digital cameras.
Shot noise is a direct product of the quantum nature of light, which is just a fancy way of saying that light consists of individual units called photons rather than being a continuous quantity. Shot noise is an inherent property of physics, meaning that even if you had a perfect camera sensor that counted exactly how many photons landed on it, there would still be just as much shot noise.
The most popular analogy for shot noise seems to be raindrops falling into a bucket, so let's go with that. The raindrops are photons and the bucket is your sensor. If I record the number of raindrops falling into the bucket over an interval of 10 seconds, intuitively, I'm going to get different numbers basically every time. I might see six drops fall into the bucket in one measurement, and seven in the next.
Let's say I extend the measurement to an entire minute instead of 10 seconds. There's still going to be a lot of variance in our measurement, but now, the magnitude of our measurements is greater too. The amount of error in our measurement has gone down. In technical terms, the signal-to-noise ratio has improved. If you do the math, it turns out that SNR increases proportional to a square-root curve, meaning that quadrupling your exposure time will double the SNR.
These details aren't really important to know; the main takeaway is that insufficient light (caused by low-light conditions or too fast shutter speed) will lead to more noise. Notice that these are also the exact conditions that lead photographers to bump up their ISO. But increasing the ISO can't magically summon more photons; it merely amplifies the signal, leaving SNR the same. Thus, we have come to falsely associate high ISO with more noise, when the amplification process itself is not responsible for the graininess most people think of when they hear "high ISO".
Increasing the ISO on a modern digital camera has two effects:
Both of these combine to increase the noise.
Most of the noise you see in a digital photo is photon shot noise. By lowering the exposure the noise makes up a greater percentage of the final result. The increased amplification makes that noise more visible.
Tetsuijn gives already a very good explanation. I try to give a similar explanation but slightly more theoretical.
Assume the sensor retrieves the light an all values are between 0 (dark) to 100 (maximum light detectable by the sensor). The sensor which has a width X (with columns 0 to x) and height Y (with rows 0 to y) has pixels X * Y (e.g. a sensor of 100x100 has 10,000 pixels). Every pixel can have a light intensity between 0 and 100. Also, because there is always 'noise' (no sensor is perfect), assume the noise is max. 2. So that means 2% (2 out of 100) is the noise ratio, 2% of the (maximum) signal is noise.
Now assume that it is very dark, so the sensor reads values from 0 to 5 only. So for all pixels, all values are between 0 and 5. Now by changing the ISO the amplification factor is changed, thus the ISO is set to a value that the amplification factor is increased with 20. This means the maximum range can be sampled, from 0 * 20 to 5 * 20, thus 0 to 100. However, the noise (2) is also increased with the same value, thus 2 * 20 = 40. And 40/100 = 40%, thus now we have a noise ratio of 40%.
The bigger the sensor the more light it can sample, so the maximum value is instead of 100, maybe 200 or 500. With the same noise level of 2, the noise ratio is 2/500 = 0.4%, or when the ISO is increased to an amplification level of 5, it will be 2% (instead of 10% with the sensor of the first example).
If you increase the shutter time (i.e. a longer exposure), then the sample time is increased, making the value of 100 in the first example for the maximum amount of light longer. Let's say 10 times more, thus 1,000. However, the noise will still be 2, thus the noise ratio will be 2 / 1,000 = 0.2%.
So that is why increasing ISO will make noise worse, but increasing the shutter time will not. But you can only use this when the image you are shooting does not move (no fast moving items), and if the camera/lens is stable.
The primary cause is Photon Shot Noise which is the randomness of light photons.
The common analogy is of pixels being cups (wells) collecting rain (photons). And in low light it is like walking to your car in a light sprinkle... large areas of your body may remain completely dry while other areas get wet. Whereas in bright light it is like trying to walk to your car in a heavy downpour... you're going to be completely soaked. It is the lack of sufficient light/data from the "dry pixels" that causes the noise. And many cameras also add their own noise to the signal chain, so they require a stronger collected signal (more light collected) to overwhelm it.
The digital image sensor is covered with photo-sites. The camera lens focuses an image of the outside world on the surface of this sensor. The shutter opens and light plays on the sensor. Each photo-site is bombarded by photon hits. The number of hits at any given site is proportional to scene brightness. Each photon hit generates a charge within the photo-site. This charge is incredibly weak. At the end of the exposure, the charges are measured and converted to a voltage. The level of the voltage is proportional to scene brightness. Depending on the ISO (International Organization of Standards) who sets the rules, amplification is applied. The now amplified signal is assigned a numeric value in proportion to scene brightness. When amplification is applied, some static always creeps in. This is equivalent to the static we hear when we turn the volume up too high on a radio or TV. In the jargon of digital imaging, we change the name “static” to noise. This “noise” shows up as unwanted artifacts imbedded in the finished image. These artifacts generally give the image “granularity”. This is the digital counterpart of a grainy pattern seen in chemical based photographic film photography. The higher ISO settings allow photography under feeble light conditions. The downside is, the amplification is turned up so the signal has a higher amount of noise. The countermeasure is noise reduction software and digital imaging chips with inherently more sensitivity to light.
