Dyadic wavelets are wavelets which satisfy an additional scaling property.
This property allows the implementation of a Fast Dyadic Wavelet Transform with filter banks.
The definition of dyadic wavelets comes from the definition of multiresolution approximations.
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This idea of consecutive approximations at finer and finer resolutions is formalized by the concept of multiresolution approximation (or multiresolution analysis).
A sequence V_{j} , j in Z, of subspaces of L^{2}(R) is a multiresolution approximation if the six following conditions are satisfied:
Condition |
Interpretation |
V_{j+1} is obtained from V_{j} by a factor 2 rescaling |
There exists an underlying dyadic sequence of time grids, e.g., the intervals satisfy a geometric progression with reason 2. |
For any j, V_{j+1} is a subspace of V_{j} |
Any low resolution signal is also a high resolution signal. |
V_{j} is 2^{j} translation invariant. |
There is an underlying time grid with step 2^{j}. Condition 1 shows that this grid is obtained from the case j=0 by a 2^{j} rescaling. |
The intersection of the V_{j} is 0 in L^{2}. |
A a zero resolution, the only finite energy signal is 0. |
The union of the V_{j} is dense in L^{2}. |
At the infinite resolution, all finite energy signals are perfectly reproduced. |
There is a function such that the integer translations of q make a Riesz basis of V_{0}. |
The resolution V_{j} is generated by a basis which is obtained by 2^{j }translations of a 2^{j} rescaled q. A Riesz basis is a frame of linearly independent vectors. |
A less literary definition is available.
The rescaling of q does not modify the area of its Heisenberg box, but it changes the proportions of the box, like for non dyadic wavelets.
The wavelets are used to build a basis in which are represented the details that are gained between a resolution and the next finer one.