The Quadratic Programming Method for Extracting Emission Line Maps from Line-blended Narrowband Images

We present a method to separate blended line emission from a set of narrowband (NB) images whose distinct throughput profiles overlap with each other. This method, based on the principle of quadratic programming (QP), treats line blending as a linearly constrained quadratic optimization problem and finds the solution by imposing a condition that the desired individual line intensities are always positive while allowing measurement uncertainties. We demonstrate the method for the quintessential example of line blending in the Hα complex by the Hα and [N ii] lines using the archived Hubble Space Telescope WFPC2 and WFC3 NB images of the planetary nebula NGC 6720, in comparison with its genuine line maps extracted from a spectral cube obtained by our own long-slit scan mapping observations with the Gemini Multi-Object Spectrograph at Gemini-North. Such line-blended NB images typically permit only qualitative line diagnostics, which can lead to a factor of a few difference in the line flux ratios. This novel QP method, however, can extract individual line emission maps as long as the number of line-blended but well-calibrated NB images of differing throughputs is equal to or grater than the number of emission line maps to be extracted. The proposed QP method provides a viable alternative to the existing line emission calibration methods: a simple but innovative way to enable quantitative fully 2D plasma diagnostics at the pixel scale of input NB images available from various instruments, telescopes, and archives.