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In signal processing, a transform is a mathematical technique to convert data in one domain to another. The most common example is using the Fourier Transform to convert data from the time domain to the frequency domain.
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Critically Sampled Laplacian Pyramid from 2nd Generation Wavelet
Your description sounds very similar to the "Red Black" Wavelet Transform.
It is a useful transform, and is used for instance in "Edge Avoiding Wavelets". …
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Critically Sampled Laplacian Pyramid from 2nd Generation Wavelet
The usual case for the wavelet transform in 2d is that the sample count quarters with each level. … If you applied this transform recursively, you'd end up with a series of detail bands at half octave intervals. …