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Cake Wavelets

Here the cake wavelets can be generated and are visualized. The first collection of images shows the cake wavelets in the frequency domain. In this domain it becomes clear why they are called cake wavelets: they look like pieces of a cake. The last kernel in this collection is the direct sum of all the cake wavelets in the frequency domain, and should look like a complete cake except with its center removed. By taking the inverse discrete Fourier transform of the cake wavelets in the frequency domain one gets the complex-valued cake wavelets in the spatial domain. The second collection shows the real part of the cake wavelets in the spatial domain. The real parts are line detectors. The last kernel in this collection is again the direct sum of the kernels and should look like a radial $$\operatorname{sinc}$$. The third collection shows the imaginary part of cake wavelets in the spatial domain. The imaginary parts are edge detectors. In every image the color black indicates the lowest value, and white indicates the highest value.

The drawing illustrates how the settings influence the cake wavelets in the frequency domain. The $$i_b$$ and $$i_e$$ labels in the figure correspond to the inner taper begin and inner taper end settings. The $$o_b$$ and $$o_e$$ correspond to outer taper begin and outer taper end. The $$a$$ label indicates the angle taper size.

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