Are all isotropic bandpass filter kernels, wavelets?
Are there any properties that the radial frequency profile has to meet to be considered a wavelet?
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Certainly not. ,,Isotropic'' means that the impulse response is independent of direction, that is, that the filter is rotationally symmetric. Wavelets are orthogonal basis functions (filters, maybe) with certain properties. Not every orthogonal basis function will be isotropic, and not every isotropic filter is orthogonal. It is even hard to see how a isotropic filter may be a wavelet.
Look up ,,wavelet conditions'' to learn more about how this orthogonality has to be understood to build an invertible transform. Banded Matrices with Banded Inverses might help the algebraically inclined.
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$\begingroup$ A simple example: and isotropic filter with rectangular function for the (radial) passband. This could be shifted to cover all the spectrum, would that not meet your conditions? Now if there is some random function for the passband would that be sufficient? Or would the sum of the scaled passbands have to be constant? $\endgroup$ Commented Jan 27, 2014 at 12:22