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Dark and flat calibration

And an illustration and explanation why it's important to do dark and flat calibration


  1. The digital DSLR camera features
  2. Linear and nonlinear
  3. The ideal image
  4. The real image and its components
  5. Vignetting
  6. Making of the calibration images
  7. What happens in the stages when we calibrate?
  8. Did we came back to the ideal picture?
  9. The math behind it

9: The math behind this calibration

Astronomy picture AI1, AI2, AI3, ...

To calibrate the image, we needed to take a large number of extra photos calibration images.

  • 8x dark frames = dai1 ... (which contains bias), need a set for each ISO, exposure time and temperature.
  • 8x dark frames flat = df1 ... (which contains bias) one set for each ISO, flat exposure.
  • 8x bias frames = BF1 ... needed one set for every ISO setting
  • 8x flat frames = FF1 ... needed a set for each telescope/lens and aperture setting and focus setting

In astronomy we usually focus only on the infinite, however is not always so easy!

Astronomical objects have an enormous dynamic range and to manage this object can be photographed with multiple exposures, known as HDR (High Dynamic Range) technology. To avoid having to take a set of dark frames for each temperature, one can assume a set and calculate the other. Will not be as good but saves a lot of time.

It looks like this mathematically for some different approaches:

A is a constant to optimize the subtraction of the dark frame, D is a constant to normalize the calibrated image.

One image and no bias:

  • Calibrated image no 1 =

  • (ai1-(dai1+dai2+dai3+dai4+ dai5+dai6+dai7+dai8)/8) / ((ff1+ff2+ff3+ff4+ff5+ff6+ff7+ ff8-df1-df2-df3-df4-df5-df6-df7-df8)/8)*D

One image and with bias separated for optimizing of dark frames:

  • Calibrated image no 1 =

  • (ai1- (bf1-bf2-bf3-bf4-bf5-bf6-bf7-bf8) / 8 - (dai1+dai2+dai3+dai4+dai5+dai6+dai7+ dai8 -bf1-bf2-bf3-bf4-bf5-bf6-bf7-bf8)*A/8) / ((ff1+ff2+ff3+ff4+ff5+ff6+ff7+ff8 -df1-df2-df3-df4-df5-df6-df7-df8)/8)*D

Usually taken several images of the same object in order to keep down the noise, for each of these images must have above process repeated.

In a modern low noise camera you have alternatives, look under my tutorials about dithering:

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