Nidelva river through Trondheim Norway

Nidelva river through Trondheim Norway
by Saaru Lindestokke                

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How does the value of a pixel at any point correspond to the intensity of light photons striking at that point? Is there any mathematical relation between these two? 1 pixel value = ?? lumen.

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The structure of a three-FET sensor cell is (from Wikipedia): Three-FET CMOS sensor cell

Now, how it works is very simple. The Mrst allows some current to Msf's gate, so there will be a fixed amount of charge there, then shuts off. Then the photodiode will draw some charge, the higher the light intensity, the more charge will be discharged. Msf will just amplify that final charge value and convert it to current, and Msel just select this cell when there is need to read this value.

Now, you will have to know how a photodiode works.

First of all, it should respond to incoming light intensity linearly:

Photodiode I-U chart

(check out the left bottom quadrant).

However, photodiodes have wavelength-sensitivity:

Photodiode wavelength sensitivity

so there is no clear one-to-one mapping between light intensity and pixel value.

Also, there are additional factors here, the most important is the Bayer filter. As you can see, R, G and B pixels have different intensity response areas:

enter image description here

This basically shows the diode sensitivity + the color filter effect.

So, to answer your question: is there a mathematical relation between these two? No, because

  • sensitivities are frequency dependent,
  • incoming light is not pure spectrally,
  • sensors do not have a mathematical model (there are non-linearities, etc.)
  • and let's not even go into the definition of "lumen", which involves frequency dependent intensity response of the human eye...

However, there are tools that can simulate photodiode physics and calculate the electrical response (latter is HSPICE or similar tool), and those can also simulate the entire circuit response to different incoming light intensity and spectra. (Randomly picking one: see this link).

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There is no correspondence. The actual photosite is an analog device which accumulates charge as it is struck by photons. Then, that charge is converted to a numerical scale by an analog-digital converter. The "meaning" of the number depends on the size of the well, the time of the exposure, the level chosen as zero, the amount of gain applied, and so on. All of this is meant to produce photographs, not scientific measurements — if the latter is what you want, you need a different device.

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Technically, there is a correspondence; it is just not consistent between camera models or exposure settings. For a given setting on a particular model though, the same intensity will result in the same pixel value. – AJ Henderson Jun 10 '14 at 13:56

While there is a correspondence, it is not uniform from one camera to another or from one setting to another. Camera sensors convert photos striking the sensor in to an electrical charge that accumulates in each pixel. The raw measure of intensity is the count of photons that struck which is measured by the charge based on the efficiency of the sensor building a charge from the photon strike.

Then, when the image is read from the sensor, the size of the charge on each photosite is measured and a raw digital value is formed based on how full the photosite was. Depending on the response characteristic of the sensor, this may or may not be a linear relationship.

Then we move in to image processing which converts the raw sensor data to an image. The selected white and black points as well as the selected gamma of the image will impact the final pixel values that are derived from the RAW sensor data.

So in short, yes, there is a relationship, but it is long and complicated and dependent on a large number of complex factors, so practically, there isn't a direct relationship that you can rely on to map pixel data back to intensity values, at least not without knowing the response curves of all the processing steps and having reference levels to work from.

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