I am new to wavelets and for several times I stumbled over these 2D wavelet transformation pictures, but I do not understand how they are created.

Can anybody try to explain in his own words how this wavelet transformation in 2D works and how or what we can see it on the picture?

enter image description here

I took this image from http://graphicon.ru/oldgr/courses/cg_el01/lect_wav/wav_cg/fig1.gif


You can notice that:

  • the top left looks like the original image, albeit smaller
  • the top right shows mainly horizontal features
  • the bottom left shows vertical trends
  • the bottom right is much less clear

What you show "is not wavelets" per se. It is another image, normally with the size of the original one, with four quadrants. However, it illustrates some wavelet features. The top left is a coarse approximation of the image, resulting from filtering and downsampling, obtained from a scaling function. The three others are details, with at least one high pass component. In the classical version, this image combines low- and high-pass on rows, and the same on columns, each followed by downsampling on both directions, which explains the above items. If you iterate the process on the low-pass image, under some conditions, you get a separable two-dimensional wavelet transformation. The key point is that in your image, all the original information remains.

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