Draft:Continuum subtraction

Narrowband filters (or line filters) are commonly used during imaging to improve the signal-to-noise ratio of narrowband emission image data. However, they cannot produce an image free of broadband sources, since a fraction of the emission of broadband sources will pass through the narrowband filter. Continuum subtraction removes the remaining broadband contribution from these images.

To perform continuum subtraction, two image datasets are needed: a narrowband dataset, and a broadband dataset taken with a filter whose bandpass includes the bandpass of the narrowband filter. In astrophotography, a common choice of filters is a narrowband hydrogen-alpha filter and a broadband red filter. Given narrowband image data D_n and broadband image data D_b, the continuum subtracted data D_cs is calculated by subtracting scaled broadband data from the narrowband data: $${\displaystyle D_{\text{cs}}=D_{\text{n}}-D_{\text{b}}Q}$$ where Q is a proportionality constant to be determined. Given the bandpasses of the narrowband filter W_n and broadband filter W_b,^[a], the contribution of broadband emission to the narrowband data is proportional to the ratio $${\displaystyle Q={\frac {W_{\text{n}}}{W_{\text{b}}-W_{\text{n}}}}}$$ assuming the camera settings and subframe exposure times are identical. The difference W_b - W_n is used instead of W_b because the broadband filter also passes through the narrowband emission.

In practice, narrowband data may be taken with different camera settings; often longer exposures and a higher gain are used with narrowband filters due to the lower total signal. With narrowband exposure time T_n and broadband exposure time T_b, and photon conversion ratios G_n and G_b for the gain settings used, the proportionality constant becomes $${\displaystyle Q={\frac {T_{\text{n}}}{T_{\text{b}}}}{\frac {G_{\text{n}}}{G_{\text{b}}}}{\frac {W_{\text{n}}}{W_{\text{b}}-W_{\text{n}}}}}$$ and the overall formula for the continuum subtracted narrowband data is: $${\displaystyle D_{\text{cs}}=D_{\text{n}}-D_{\text{b}}{\frac {T_{\text{n}}}{T_{\text{b}}}}{\frac {G_{\text{n}}}{G_{\text{b}}}}{\frac {W_{\text{n}}}{W_{\text{b}}-W_{\text{n}}}}}$$

This formula may be rearranged so that the narrowband data is scaled by the exposure time and gain setting factors to match the broadband data: $${\displaystyle D_{\text{cs}}=D_{\text{n}}{\frac {T_{\text{b}}}{T_{\text{n}}}}{\frac {G_{\text{b}}}{G_{\text{n}}}}-D_{\text{b}}{\frac {W_{\text{n}}}{W_{\text{b}}-W_{\text{n}}}}}$$

Astrophotographers often add continuum subtracted data back to broadband data to enhance the visibility of faint narrowband emission. The continuum subtracted data is subtracted from its median, then the difference is scaled by some factor S, chosen to maximize the visibility of faint features without contributing too much noise. $${\displaystyle D_{\text{image}}=D_{\text{b}}+S\left(D_{\text{cs}}-\operatorname {median} \left(D_{\text{cs}}\right)\right)}$$ While the continuum subtracted data is usually added to the broadband data that was used to perform continuum subtraction, astrophotographers may also choose to add this data to other channels. This is often done to replicate the observed color of an emission source. When imaging hydrogen emission, most astrophotographers only use filters that pass its brightest emission line in the visible spectrum, H-alpha, which appears red in isolation. However, ionized hydrogen appears pink due to the contribution from other lines of the Balmer series. To reproduce this color, a small fraction of the continuum subtracted data may be added to the blue channel.