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Do you see that brighter vertical band just right of center? It's not actually there. Look again and insist otherwise, but it remains a figment of your visual processing system.
It took two frustrating days of rehashing my code (for creating seamless time-slices) and a final painstaking inspection of brightness values pixel by pixel, for me to conclude the bright stripe was an optical illusion. After a couple of unhelpful Google searches, I was still pretty sure this couldn't be a new discovery. So I contacted Stephen James O'Meara, the expert in visual observations of the sky, figuring that he would have seen this and know the name of the phenomenon. In less than a day he answered me with "Mach bands." [Link to Wikipedia article].
Our visual system performs edge-detection on the fly, and highlights this to us as Mach bands; they appear whenever there is an abrupt change in shading, in my case: a linear ramp up in brightness ending at a flat plateau. Mathematically this is called a discontinuity in the slope.
A classic time-slice
A classic vertical time-slice copies a strip from the left side of the first image, then appends consecutive strips from the next images, in this case at 5 minute intervals.
Linear gradient mask for blending. Note the "whiter than white" band at the join. There is also a "blacker than black" band on the left side of the gradient except that those are tougher to see.
I'm not a fan of the hard-edged strips, so I figured out how to use a gradient mask that softens them by blending an overlap region. Although this led to a nicer result, I was not satisfied and so pondered on how to make a smooth background then overlay the classic time-slice image using the lighten function, which led me to the following:
It's not a perfectly smooth background across the left half, but it's pretty decent. The Mach band near center remains noticeable because the gradient suddenly flattens out.
Google is a lot more helpful when your query contains good keywords. It turns out that computer graphics people are very familiar with the issue, dating back to 1971. They are an "artefact" of the Gouraud polygon shading technique. I put artefact in quotes, because unlike sharpening and noise suppression tools that do leave artificial features, this all happens inside our heads. Mach bands commonly appear to radiologists and dental technicians, wherever there are sudden changes in brightness in X-ray images. On the astronomy side, those trying to tease out fine detail on the surface of Mars or striations in the cloud tops of Venus, as did Percival Lowell and Schiaparelli, will be subject to the phenomenon.
Knowing it is an illusion doesn't make it go away, which left me annoyed and wondering how I might apply a filter to mitigate the effect. I fiddled for a half an hour manually "rounding the corner" at the top of the ramp, which seemed to reduce the effect, but then left it for the next morning and a refreshed mind to make a reasonable flattened S-curve. With a good dose of sleep supporting me I conjectured that surely someone has already figured this out. With the search terms "reduce Mach bands in computer graphics," Google's first hit was the paper "Methods of Reducing the Visibility of Mach bands during Gouraud Shading."
Figure 3 from Sean Bangay's paper. His experiments led to him to the two most important parameters: relative width of the transition to the viewing window, and the curvature at the edges.
Not only is our vision sensitive to a change in the slope, but also how quickly this slope changes (in math terms, the second derivative). Bangay noted that it goes just past the next level, but with diminishing impact. There is a dependence on colour, which I presume is related to predator-prey detection (yellow is when it is most obvious).
The S-shape, termed a sigmoid, is commonly used in image processing software as an image-wide contrast-enhancing tool, and comes with adjustable parameters. Although I quickly found settings that mitigated the Mach band, the extra contrast created in each vertical time-slice was distracting. I realized I needed to keep the linear change through the left strips and have a single half-S gradient blend for the final transition. Since there were no significant transitions to follow on the right, I could take the liberty to spread the top of the S across a strip that was four times as wide. The concept was to blend a linear change in the first 0.5 to an arctangent in the remaining 3.75:
Linear function
But of course the resulting stitch did not produce the hoped-for result. The slope discontinuities at 0.5 and at 4 (where it goes flat) still triggered Mach bands. Realizing that a Bezier curve would eliminate the discontinuities, I made one to fit, only to find that the software I was using, ImageMagick, did not accept polynomials of that form, only the classic type. Using an online calculator to interpolate chosen points to a cubic polynomial, I arrived at a solution:
The cubic polynomial did not quite have a slope of 1 at 0.5, but it was an easy stretch in the horizontal to make that happen. Where it leveled off, I crop it then it would join nicely with the final vertical slice on the far right. The Mach band disappeared nicely, but another one showed up at the join, where the eye picks up the subtle discontinuity to a flat intensity. Artificially adding a very shallow linear gradient to the right side solved it!
Result: a de-Mached smooth background (admittedly not 100%). Using an offset of a couple of minutes on either side, I made two more smooth backgrounds and averaged them. Combining them and using lighten with the classic hard-edged time-slice yields the final seamless time-slice:
The beauty of this method is that for the next event, it will take less than 20 minutes of wall-clock time to create this type of composite!
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