Wavelets

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Wavelets are a relatively recent invention, a hybrid between the spatial methods of operators and kernels, and the frequency-based methods of the Fourier Transform. In many ways wavelet image processing offers the best of both worlds. In this chapter, we'll explore one method that uses wavelet analysis to enhance detail and two methods that use wavelet analysis to reduce image noise.

So—what is a wavelet? The formal definition—a function with positive and negative values locally, zero everywhere else, and an integral of zero—does little to illuminate the matter. However, a graph of a wavelet is more revealing:

this wavelet has a positive region closely flanked by negative regions, but elsewhere its value is zero. It resembles a ripple on a smooth pond, which originally suggested the name.

This particular wavelet is the "Mexican Hat function," so called because of its sombrero appearance. Integrating from +00 to -00 yields zero. Suppose that you applied a Mexican Hat function as a convolution kernel—what would happen? The answer, not surprisingly, is that it would act as a bandpass filter. The skinny positive core acts as a low-frequency high-pass filter, while the wide negative wings act as a high-frequency low-pass filter. Between them, the two filters isolate a narrow band of spatial frequencies.

Wavelet analysis uses functions like the Mexican Hat to split an image into different wavelet scales or levels, each corresponding to a specific band of spatial frequencies. As does convolution, wavelets operate locally on an image; and as the Fourier transform does, wavelets allow you to process images in the frequency domain. In this chapter, we shall see how this hybrid between the spatial domain and frequency space produces an powerful image processing tool for astronomy.

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