A signal is representated by its low pass approximation and the modulus maxima of its dyadic wavelet transform.
This representation allows an almost perfect reconstruction of a signal.
The continuous wavelet transform detects isolated singularities with their order of singularity. The regular part of the signal is coded in its coarsest approximation. It is sensible to try to reconstruct a signal from this coarse resolution and from its wavelet modulus maxima.
In practice, only the dyadic wavelet transform is considered to take advantage of the fast algorithme à trous which implemented by filter banks.
From a theoretical point of view, Meyer and Berman have proved that the representation by dyadic maxima is not complete because several signals may exhibit the same wavelet maxima.
In practice, numerical experiments have shown that it is possible to reconstruct usual signals with a relative mean sqaure error smaller than 10-2. On images, the difference is not visible.
A signal is to be reconstructed from the values and locations uj,p of its wavelet modulus maxima, j being the scale and p the time localization. This difficult problem is replaced in practice by a simpler one which consists in finding a minimum norm signal among those which have the assigned wavelet coefficients at the maxima locations. Solving this problem tends to create signal with modulus maxima at the right locations with the correct values.
Since this problem actually bears on discrete signals, this simplified probleme is an inverse frame problem, which can be solved using a conjugate gradient algorithm. To this reconstruction a previously stored low frequency component defined by the sample averages is added.
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Images, Edge Dectection and Reconstruction