by Barry Van Veen
A great deal of signal processing is concerned with linear systems. The mathematics for analyzing linear systems are very powerful. However, in some applications nonlinear systems are a very useful signal processing tool. One such application is in astrophotography, where a nonlinear system is used to make faint details more visible in images of the night sky.
Why Use Nonlinear Systems in Astrophotography?
Astrophotography can produce stunning images that are filled with detail that cannot be seen with the human eye. The human eye does not "see" color for very faint objects because color sensitive cones in the eye require much higher light levels than the much more prevalent light-dark sensitive rods. Furthermore, the opening or aperture of the eye is relatively small, limiting the amount of light received, and the length of time over which the brain forms a conscious image is also limited. A camera can integrate received light over a much longer time - hours if one compensates for the earth's rotation, use a large aperture, and is not limited to rods and cones.
Fine details in a faint object like a galaxy or nebula are made visible in an image by increasing the differences of the perceived brightness of such features. This is typically accomplished in post processing using a nonlinear system to "stretch" the image. Stretching increases the difference in apparent brightness of faint features, but decreases the difference in apparent brightness of the bright objects like stars. Decreasing the apparent difference in bright objects is a worthwhile tradeoff for seeing the details in fainter objects.
This must be accomplished in an environment with very large dynamic range. For example, Venus, one of the brightest objects in the night sky, is 
brighter than Pluto in terms of light intensity. Some of the faintest objects imaged by the Hubble Space Telescope produce many orders of magnitude less light intensity than Pluto.
Examples of Nonlinear Systems in Astrophotography
Figure 1 depicts an image of a section of the Milky Way in the vicinity of the constellation Cygnus. This image was produced by stretching the image captured by the camera (see Figure 2) with the nonlinear system characteristic shown in Figure 3. The brightness each pixel in Figure 2 is the input, and the curve shown in Figure 3 is used to determine the brightness of the corresponding output pixels in Figure 1.
The nonlinear system in Figure 3 maps the lowest 25% of input intensities (horizontal axis) into the lowest 45% of output intensities (vertical axis). If we normalize the maximum input and output value to one, then this stretches input intensities on the interval [0, 0.25] to output intensities on the larger interval [0, 0.45]. On the other hand, differences in bright pixels are reduced by this nonlinear system. The brightest 50% of input pixels are mapped into the brightest 25% of output pixels, as we see the input interval [0.5, 1] is mapped into the output interval [0.75, 1].
The definition of a linear system states that if the input is multiplied by a constant, then the output must be multiplied by the same constant. The system in Figure 3 is clearly nonlinear, since doubling the input intensity, say from 0.5 to 1, does not double the output intensity. The output only increases from 0.75 to 1, which is much less than a factor of 2.
The faint features in Figure 1 are more visible than those in Figure 2 due to this stretch. For example, the branches of the trees are much easier to distinguish from the sky background, and the detail of the Milky Way is significantly increased. More faint stars are now visible. A bit of pink color associated with several emission nebula is also evident, even though the infrared filter built into this camera (Canon T2i) blocks most of the light from hydrogen-alpha emission nebulae.
Figure 4 illustrates an even stronger stretch of the low intensity pixels in the image. Note that here the input interval [0, 0.25] maps to an output interval of [0, 0.62]. The result of applying this nonlinear system to the image in Figure 2 is shown in Figure 5. It is is not clear that Figure 5 is preferable to Figure 1. While more of the detail in the Milky Way is now visible, the stronger stretch significantly brightens the sky background associated with light pollution.
Limiting Consequences
Several factors limit the degree to which the image can be stretched in practice. One is the presence of light pollution. Light pollution creates a bright sky background which limits the faintest objects that can be seen. Light pollution is evident as a brown cast that gets brighter with increased stretching of low intensities. One way to eliminate light pollution is to capture images shot from dark sky locations. In this case stretching is limited by the presence of noise. Noise results from thermal and other effects in camera electronics. Significant amplification of small intensities can result in the noise becoming visible and degrading image quality.
Ultimately the quantization step size used to sample the photodetector outputs limits the degree of stretching that can be applied to an image even in the absence of light pollution and noise. A digital camera encodes light intensity into a discrete set of intensity levels. For example, the camera used to capture Figure 2 has a 14-bit analog-to-digital converter, which means there are 
possible levels for the light intensity at each pixel. Typically the differences between distinct levels are sufficiently small that we perceive continuous variation in intensity within the image. Very aggressive stretching can make the differences between distinct intensity levels very visible, an effect known as posterization. Posterization is generally very undesirable in astrophotography.
Nonlinear systems are a powerful tool for producing visually pleasing images of faint objects in the night sky.
