Telescope Reviews: Newbie Corner- 30 second exposier = 30 1 sec stack

Its not possible to correctly answer this question without a lessson on noise as you originally requested, the process of stacking is ALL ABOUT NOISE. Lets simplify the problem, and, use nice round numbers to make the math clear. Assume a camera that reads out in the range of 0 to 100. Read noise is constant, random in the range of 0 to 10. Simplify again, dark current is constant, and 0. Further simplify, for each photon that hits the sensor, the readout number bumps up by 1. Now, point this camera at a target that emits one photon per second on a small portion of the sensor.

If we take a one second exposure, and look at the results. Home in on a single pixel that's recieving photons from the target, it will have a value of 1 for signal, and noise somewhere between 0 and 10. Only the pixels reading out a full value of 11 allow us to be 100% confident that there is signal in the pixel, but if it reads out 9, it could be all noise, or, it could be signal and noise, we cannot tell with any degree of confidence the difference.

Same conditions, but, take a 30 second exposure. Pixels that are recieving signal will read out 30+noise which means in the range of 30 to 40, while pixels not recieving signal will still read out in the range of 0 to 10. It will now be clear which are recieving signal, and which are not.

So, now, lets look at the stack of 30 1 second exposures, again, looking at a single pixel recieving signal, simple addative stack. Each frame has a readout of 0 to 11, but, we know the average of the noise will be 5, the midpoint of the noise on the readout. If we read that pixel 30 times, the total noise will sum to approximately 150 (30x5), and the signal will sum to 30. The entire background of the image will be pixels in the area of 150, while the signal portion will be pixels in the area of 180, after adding all 30 frames together. There will be some variance, and this is the noise characteristic of the camera.

Now lets compare the values on the 30 second image, and those on the 30 second stack. Totals on the single frame for a pixel with signal will be in the 30 to 40 range, with noise in the 0 to 10 range. The ratio of signal to noise is now 40 to 10, so the SNR = 4. Good solid signal to noise ratio, and well defined signal. On the stack, looking at that same pixel after adding them all, noise is 150, signal is 180, so the signal to noise ratio is 180 to 150, SNR = 1.2. Note that the absolute value of the difference is similar, 40-10=30 while 180-150=30.

So, for this setup, with appropriate post processing, we _can_ get similar results in the final image for a star the emits 1 photon per second into our image rig by shooting one 30 second exposure, vs 30 1 second exposures. Now, re-run the math for a star that emits 1 photon every 3 seconds.

In the single 30 second exposure, that pixel will read out 10, plus a random noise component between 0 and 10. It may or may not stand out from the read noise initially, but mathematically we can say with certainty it's there, it has read out a value between 11 and 20. The math will show us, that pixel has _some_ signal with 100% certainty. Same pixel in the 1 second exposures, after 30 reads we will have a noise total of 150. On 10 of those reads we will get a signal bump and 20 of them will have no signal at all, so that particular star will total 160 in the stack. 160/150 = 1.06 so our signal to noise ratio on that particular star is 1.06, and once we do the math, we will discover that with only 30 frames, we cannot statistically verify it even exists, we'll likely need to take another 30 to 50 frames before we can statistically validate the existance of that one signal point, and confirm it's not just a random bump in the noise readouts.

Repeat the math one more time for a star emitting 1 photon every 5 seconds, and you will find, the 30 second exposure still shows it, while in the stack of 1 second exposures, it cannot be differentiated from the background noise at all. Depending on how well we have characterized the noise readout on the camera, a stack of 120 one second exposures _may_ be able to pick that star out of the background, but it's a statistically challenging problem. If we were accounting for dark currents, vignetting, and a few more details, the problem becomes statistically insurmountable. And, that's all stacking really is, a practical application of statistics.

Now, another very interesting detail comes out when doing this kind of stacking, and, that's going to DIRECTLY answer your initial question. If we do the addative stack of 120 shots, look at what kind of readout values we are working with. Consider that your dslr reads out in the range of - to 4096, and we stack 100 of them, then we end up with pixels reading out 0 to 409,600, but to display them on the computer screen, they get scaled into the range 0 to 255 in the primary color channels on your screen. All that effort to capture the detail, and, when displaying on the computer screen, everything in the range 0 to 1600 gets scaled to 0 on the display, and everything from 1601 to 3200 crunches to 1 on the display. Until you apply non linear stretching (levels and curves), you wont see any of that detail you worked so hard to capture.

Another detail one _must_ keep in mind. Stacking is nothing more than a practical application of statistical principles. Statistics = Lies, Lies, and more *BLEEP* lies.....