Using PixInsight PixelMath to Correct a Flat Frame Issue

In my last post I talked about a problem I had with flat frames collected at f/2. To correct the issue I used the processes AutomaticBackgroundExtraction and GradientCorrection. This worked very well for two of my images, but I recognized it might not with the huge nebula Sh2-264, (AKA the Lambda Orionis / Angelfish Nebula).

I tried the ABE+GC method on this and it failed spectacularly by obliterating the eastern half of the nebula. So my approach will instead be to use a method that employs PixelMath and an ad hoc model of the donut hole.

Here is the hole as seen in the green master frame from one the other images collected the same night:

Green Master Frame with the donut at center in all its glory

This donut appears in the master green frame of Sh2-264, if not as prominently. So I'll use the above photo as the basis for the correction's structure. It's worth noting that the master frames have already been flatted, so either my system is dust-free, or the flatting took care of any dust donuts. What I'm about to do is apply a simple correction that's limited to the central area of the image.

At the very center the brightness doesn't seem diminished and no correction is needed. As distance from the center increases the distortion progressively darkens to form a ring, then quickly brightens again. It's this ring that must be brightened. Because I'm correct the master linear frames I'll use simple multiplication of the existing pixel values.

Eyeballing this led me to a polynomial model of the needed correction. Here it is in an Excel graph.

Sixth-order polynomial fit to estimated donut darkness

The horizontal axis is scaled distance from the donut's center, with the value 1 corresponding to radius of the ring's darkest values at about 700 pixels from center. The vertical axis is a dimensionless value for the darkness of the donut relative the image's true background.

This is a complex shape that is fit well by a sixth-order polynomial 

So, how to make use of that polynomial? Let's look at the PixelMath script!

Script for hole correction. Click for larger version

This is a simple script. Basically it looks at the distance a pixel is from the center of the hole, calculates the factor by which to increase the pixel's brightness, and multiplies the pixel's existing brightness by that factor. Pixels outside the donut (R > 1.38)  are left unchanged. 

The constants are essentially input parameters that can be played with to refine the correction. The main adjustment is Amplitude; too large a value and the donut becomes a light ring, too small and there's not enough improvement.

The multiplier is 

        1.0 + (polynomial value times Amplitude). 

I found the best result used Amplitude = 0.04, so at most the correction is a 4% increase in linear value.

So what did this do? Here is the Sh2-264 Green Master before and after the script is applied:

Uncorrected

Corrected

Note that these are given the "Boosted" PixInsight stretch to emphasize the hole; it won't be this evident in the final image.  The script did a fair job, I think. 

 

My post-WBPP workflow for this image uses ABE to subtract first order light pollution from each master light frame; the script is then applied to each channel. After that I return to my conventional workflow: CombineChannels, BlurXTerminator to reduce chromatic aberration prior to color calibration, and so on. 

The result:

Finished Image, corrected (1/4 scale)

North is up; The Angelfish swims nicely westward! I think the best way to validate any image is to compare it with clearly superior one by an accomplished imager, so I used Adam Block's superb image for this one. I think my processing does a very decent job of reproducing his in the central area of the nebula, so I'm happy!

This completes the rehabbing of the Iowa Star Party images.  Next I'll be revisiting some of my images from 2022. Also, Masters of PixInsight are doing a workshop on mosaics in December, so maybe I'll finally process that Veil Nebula mosaic data I've been sitting on.