By Wim Sweldens and Peter Schroder

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7 Multiresolution Analysis In this section, we will go into some more mathematical detail about multiresolution analysis as originally conceived by Mallat and Meyer 20, 19 . In the previous section we de ned the notion of a scaling function 'x and saw how all scaling functions are simply translates and dilates of one xed function: 'j;l x = '2j x , l: In this section we will use these functions to build a multiresolution analysis. Assume we start a subdivision scheme of the previous section on level j from a sequence fsj;lg.

From left to right, top to bottom the coarsest level used in the transform is V9 , V7 , V5 , and V3 . We hasten to point out that this is is a very simple and naive smoothing technique. Depending on the application and knowledge of the underlying processes much more powerful smoothing operators can be constructed 15, 14 . This example merely serves to suggest that such operations can also be performed over irregular samples. 3 Weighted Inner Products When we discussed the construction of scaling functions and wavelets we pointed out how a weight function in the inner product can be incorporated in the transform.

After performing this step the usual interpolating subdivision would follow. Depending on the application one of these schemes may be preferable. Next we took some random data over a random set of 16 sample locations and applied linear N = 2 and cubic N = 4 interpolating subdivision to them. 10. These functions can be thought of as a linear superposition of the kinds of scaling functions we constructed above for the example j = 3. Note how sample points which are very close to each other can introduce sharp features in the resulting function.

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Building Your Own Wavelets at Home by Wim Sweldens and Peter Schroder

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