# Reconstruction From Dyadic

Wavelet Modulus Maxima

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.

## Outline

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.

## Implementation

A signal is to be reconstructed from the values
and locations u_{j,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.

An example in
PDF format (32 Kb) is available. Here is a
preview of it:

Images,
Edge Dectection and Reconstruction