Signal and noise and their typical measurement as a signal-to-noise ratio are central to the assessment of the quality of digital images.
The signal is the thing of interest in the digital image, such as the light from a deep space object, the ever-changing magnitude of a variable star, or the intensity of nebulous gases glowing with a certain wavelength. Signal is often measured as analog-to-digital units (ADUs) or the pixel values detected by a CCD camera.
Noise is the uncertainty in the measurement of signal caused by anything which might give rise to spurious data which interferes with the direct observation of the signal. Some common causes of noise are diffraction limits of telescope designs, astronomical seeing, dark current and readout noise in CCD cameras, and the quantum efficiency of the CCD detector or individual pixels. Noise is typically quantified as some dispersion of the signal, such as standard deviation (sigma), variance (sigma-squared), full-width at half-maximum (FWHM), or half flux diameter (HFD), depending upon the specific purpose of the noise measure.
More signal and less noise are always desirable. The relative relationship between signal and noise is often expressed as a signal-to-noise ratio (signal divided by noise), where noise would be measured as the standard deviation (sigma) of the signal.
In the case of focus and collimation the signal of interest is a definitive measurement of the quality and effects of focus or collimation. A purpose-designed diffraction mask is the most direct way to measure the signal generated by focus or collimation. A diffraction mask, such as the GoldFocus High-Precision or Plus Collimation masks, creates diffraction fringes (spikes) at very specific locations relative to the focus or collimation.
To achieve the goals of a high signal-to-noise ratio in the measurement of focus or collimation requires both the measurement of a higher signal and the signal being in the presence of a lower noise.
The greatest shortcoming of using solely the minimum of any of full-width at half-maximum (FWHM), half flux diameter (HFD), or sigma as a measure of focus (see Focus Techniques) is that each of these is a measurement of noise, not signal. Using these techniques amounts to excluding signal and signal-to-noise ratio altogether and relying solely upon minimum noise. It is clearly to the advantage of the astro-imager to utilize signal-to-noise rather than simply minimum noise.
The Diffraction Spike Mask or Modified Hartmann Mask (see Focus Techniques) rely upon two or more diffraction fringes being made coincident or overlapping, which by their very nature blurs (adds noise) to the distinction of when they are coincident. Although far less obvious, a Bahtinov Mask also introduces a very large amount of poor signal and increased noise due to its very design. In some ways the designs of all these masks are self-defeating with respect to signal-to-noise, a fact made clear by a detailed diffraction analysis of the masks at varying amounts of defocus.
For example, some portions of the diffraction fringes (spikes) generated by a Bahtinov mask move very little with poorer focus while other portions of the diffraction fringes (spikes) move in a more or less intended way. The differential movement of various portions of the diffraction fringes (spikes) has the net effect of adding a great deal of noise to the movement of the spikes, effectively reducing the average signal by a factor of 2 and increasing the noise several fold, both to the detriment of the signal-to-noise ratio in a Bahtinov mask measuring focus.
By comparison, the signal and noise generated by a GoldFocus High-Precision Mask has a minimum of 10 times higher signal-to-noise ratio compared to a Bahtinov mask and the GoldFocus Plus Mask has a minimum of 3 times higher signal-to-noise ratio compared to a Bahtinov mask. These large and very significant differences are due to the GoldFocus High-Precision and GoldFocus Plus Masks being specifically designed to optimize signal-to-noise.
Even when a Bahtinov mask is used with 3rd party software, the GoldFocus Masks used with the GoldFocus Analysis Software will still have 10 times or 3 times higher signal-to-noise, respectively, because these signal-to-noise advantages are inherent in the masks themselves.