Mosaic overlap strategy

Interesting considerations! I like the idea of tackling the problem from a theoretical standpoint.

I definitely agree with the points you raise in an ideal case. My guess is that when put into practice there are a number of factors that come into play, for instance:

• The edges of images usually have vignetting from the scope : flats allow to correct that very well, but I don't know what the impact might be on SNR.
• Dust tends to collect more on sensor/scope edges, so defects are more concentrated there
• Small errors in tracking/guiding means that the edges of the image do not have exactly N subs

So having better concentration of signal into the edges/corners area might be a way to compensate the unwanted errors. Added to that, the size of the object(s) being photographed, combined with the size of your field might push you to want a bit more or fewer panels.

HOWEVER: this comes from my experience where a number of factors mostly due to my incompetence or lack of resources make SNR considerations less of a concern (there's a pile of other issues with my images that I should fix before I think about that).

Of course you are correct that, optically speaking, edges are usually not as good as image centers.   

With a single image that is often OK because the subject is in the center.   

With a mosaic, we take our image edges and put them into the subject area.  That's potentially a problem, regardless of what overlap scheme you use.   

As you point out, there are fixes for the edge problems, which remove the vignetting.   Many people crop off the ragged edge of the panels where guiding has made overlap imperfect.

But I view these as different problems that are largely unrelated to SNR, because SNR does not fix them.  If the edges of your frame have stars that are not round due to coma, no amount of stacking more copies of bad edge stars on top is going to help.

In minimum overlap (i.e. 10% overlap) mosaics, you are stacking N copies of the extreme right edge of one frame with N copies of the extreme left edge of the next frame.   The outermost 10% of the frames is going to be the worst.  While coma can be directional, it's not usually the case that two wrongs make a right.  So the places where edges are stacked are going to look bad.   The overall mosaic will have regions that are good (center-stacked-on-center) and strips which are bad (edges-stacked on edges).

In maximum overlap mosaics - which is the approach I am exploring - where you overlap by 50% - that tends to average together the edges and the centers.   So if your edges are bad, this will effect the whole image, but I believe it would be more uniform.

I think that the lesson here is: don't have bad edges!  If you have poor optical quality at the edge of frame, your mosaic will suffer.   So you need to either get a different optical system, or stop the lens down, or crop off the bad portion off of the frames. 

Makes a lot of sense! And I wonder whether it would improve things to actually crop the edges of your integrated panels and THEN combined them together. (Or even better, change weigh the averaging by how far you are from the center of the image when integrating things). In APP the blending options already allow to do that (to some extent), so with maximum overlap you'd probably be getting the best of both world (you increase the weight of the center pixel on image A while decreasing the corner of image B).

I did some tests of cropping lens versus upgrading.   That, along with basic considerations of absolute aperture size, has lead to use longer lenses for wide field panoramas.

Recall that for stars the f-stop does not matter - the light gathering power depends on the area of the aperture.  So a 50 mm f/2 lens and a 100mm f/2 lens both are f/2, but for stars the 100mm lens has 4X the light gathering power.

Plus, it is harder to have good edges with wide angle designs.  I could not find a wide angle lens that had good stars at the edge wide open.  When you start stopping down the lens you strangle the light gathering.

A 24mm f/1.4 lens has terrible edges at f/1.4.  Most of the need to go to f/4, at which point its aperture is 

Cropping the 24mm images does work - but it is not competitive with a decent 50mm lens stopped down until 

Meanwhile a 135mm Samyang/Rokinon f/2   (or Zeiss for more money) is good to the edges wide open at f/2.   

The 135mm at f/2 has 126X the light gathering power of the 24mm at f/4.   So you need to take more images to cover the same field but you get a better result.

Very interesting discussion! I think, although I'm not a maths genius, your points seem valid @nathanm. It's something I've been thinking about as well, more to see if it would indeed reduce coma overall in my images, by having more overlap of the center across the mosaic. Big downside is time though, I would need way more frames to be able to cover the subject with 50% overlap. But.. I think I'll still would like to give that a try.

Actually, this is the best part - the number of frames is SMALLER than the other approach.  The reason is that you take fewer subs per panel.

This is counterintuitive so here is an explanation.

Here is a conventional 3 x 3 mosaic with 10% linear overlap - meaning that the horizontal frame overlap is 10% of the horizontal frame length, and vertical overlap is 10% of the vertical length.   Of course our frames don't really look this when projected on the sky due to spherical trig, but it is a reasonable approximation for long focal length frames.

I am working on figuring out cases for wide field mosaics which then have substantial spherical impacts.  More on that when I am done with it.

The x and y axes here refer to the panel size which is taken to be full frame (i.e. 36 x 24 units).  Ultimately that turns into some angular coverage on the sky but we don't need to calculate that here.

There are  9 panels, and suppose we use 30 subs/panel.  Then that is 270 subs.

The area covered is reduced a bit because of the overlap - it is (1+2*(1-overlap))^2 or 7.84 panels, treating a panel as having area = 1.

Here is an example of the new approach

This is 2 grids of 3 x 3 panels, each with zero overlap, but the two are shifted relative to each other, here by 10%.

Each panel now gets 15 subs - half of the conventional number.

The total panels is 18, so the total subs are 18*15 = 270 subs.  The same as above!    Yet each point within the "good" area of our picture (where the two grids overlap) still has 30 subs.

The area of the good area is slightly larger than the conventional case, it is (3 - overlap)^2 = 8.41.

In order to compare the relative efficiency, we need a metric

A reasonable metric is  (total subs count)/(subs coverage *good area).  Of course one could invent a better metric, but this corrects for the fact that the good areas are slightly different.

The conventional case is 270/(30*7.84) = 1.15, while the new case is 270/(30*8.41) = 1.07.    

Roughly speaking what this tell us is the "sub efficiency" of a mosaic plan.  I should expect to waste roughly 15% of the subs due to mosaic overlap in the conventional scheme with 10% linear overlap.   I only waste 7% in the new scheme.

That difference is pretty small, but it turns out that the advantage of the new method grows with the size of the mosaic

This is a simple analysis that only counts N x N mosaics, where N is the x-axis in the plot.   I have plotted the conventional case for overlap = 10%, 20%, 30%

In the new approach there is little reason to vary the shift between the 2 grids so I have left it at 10%.  

The metric shows us that as N becomes large the metric tends to 1 for the new scheme.  At N=10, it is 1.02.  That is because the only wastage in this approach are the narrow strips where the grids do not overlap.

Meanwhile for N=10, the conventional approach with 10% overlap is 1.42.  

10 frames overlapped by 10%, creates a grid that is 8.4 frame units long.  The overlap compounds with N.

If we were doing 100 panel mosaics (which I am gearing up to do), then the conventional method with 10% overlap would require 4260 subs, while the new method would take 3060, so it would save roughly 28% of the time to do the new method.

One could say that the new method requires too much accuracy because I am assuming that one can have the frames touch each other.  I think that is actually quite reasonable if you do a plate solve for each position - the accuracy can be within a very small fraction of the frame size (10s of pixels).   

Thank you for this elaborate explanation! I had to read through it very slowly, but I do get it. Very very interesting indeed. So indeed, because of the overlap the FOV shrinks, while in your scheme it doesn't basically and therefore requires less frames in total. Never seen this approach before, but as the overlap remains similar with the old case I don't see why this won't work.

Yes, the shrinking of the area is one reason, the other is that in the old method the area was covered at N, 2N, 3N.   In the new method the area is entirely covered at N.

Here is an example of a wide angle mosaic (74 degrees width) of milky way core region

The shot is set up along alt-az lines - that works best near the horizon.

Here is how the mosaic panels are laid out, for a 200 mm focal length, full frame lens

As with the examples above, there are two grids - blue and red.  Each one has very low overlap within the grid, but complete overlap between the two of them.

Each panel has edges which are great circles, so they are not rectangles.  In the chart I approximate the great circles as polygons, but the calculations to line them up are correct.   Altogether there are 84 panels in this mosaic.

This mosaic is for end of next week, when the bottom edge of the shot will be just above the horizon.  The imaging quality is not great there but this is meant as the sky portion of a landscape astro shot, so I actually want the horizon effect.

Each panel has an associated orientation angle, so a field rotator will keep it in the proper orientation.  

However the horizon imposes a problem - some of the subject will set below the horizon during the shot.   This can be seen in the first chart above, where the horizon at various times is shown as the thin blue lines, which are spaced an hour apart. 

To cope with the setting subject, the frames have to be shot in a priority order.  That is shown by the Cyan colored lines, and the numbers. 

This is truly amazing, thank you for sharing your strategy here! I'm really curious how this is going to work out.

Forest fires have made the air terrible where I live, but later this week I will travel to the desert and try these shots.  I will be using APP to process them.