We can deconstruct images as the sum of periodic variations of brightness and represent the frequencies we get by sine and cosine functions. This process is a Fourier transform. In fact, you can represent images by transforming them from brightness at any pixel position to a frequency with a particular amplitude.
Wavelets get around this restriction because these functions have shapes that limit their oscillations. You can create a new image in a wavelet domain instead of the strict frequency domain by using these shapes.
Using a particular wavelet function, I’ve broken an image of the Lagoon Nebula (M8) into small-scale features around four pixels in size, large-scale features 32 pixels in size, and a residual image that is everything else (Image #2). Adding these “layers” back together gives me my original image. But now I can modify any layer to enhance or diminish the information there.
You can set the layers up as a doubling scheme (a scale of one pixel for the first, two for the second, etc.). Turning off (removing) the first layer and combining the remaining ones will remove features on the order of one pixel in size. This example shows how you might remove noise in your image.
Another popular program that uses wavelets for planetary processing is RegiStax. The layer adjustments are similar with the sliders being like the “Bias” setting in PixInsight. In fact, here’s a secret many of the top planetary imagers know: They modify the wavelet (filter kernel) and increase the weight of the center of the matrix (Image #4). Changing the shape of the wavelet in this way (making it more “peaked”) better probes small features when you adjust the scale size.
In my next column, I’ll show you how I used wavelet layers in combination with high dynamic range processing to handle the M8 image.