Multiresolution representations using the auto-correlation functions of compactly supported wavelets, (with G.Beylkin), Research Note, Schlumberger-Doll Research, Aug. 1991.
In this paper we propose a "hybrid" shift-invariant multiresolution
representation which utilizes dilations and translations
of the auto-correlation functions of compactly supported wavelets.
In this representation, the exact filters for the decomposition
are the auto-correlation of the quadrature mirror
filter coefficients of the compactly supported wavelets. The decomposition
filters are, therefore, exactly symmetric. Moreover, the auto-correlation
functions of the compactly supported wavelets may be viewed as
pseudo-differential operators of the even order and behave, essentially, as
the derivative operators of the same order. This allows us to relate the
zero-crossings in this representation to the locations of edges at different
scales in the signal. The recursive definition of the compactly supported
wavelets and, therefore, their auto-correlation functions, allows us to
construct fast recursive algorithms to generate the multiresolution
representations. Though it is not an orthogonal representation, there is
a simple relation with the wavelet-based orthogonal representations on each
scale. We describe a simple reconstruction algorithm to recover functions
from such expansions.
A remarkable feature of the representation using the
auto-correlation function of compactly supported wavelets
is a natural interpolation algorithm associated with it.
This interpolation algorithm, the so-called symmetric iterative
interpolation, is due to Dubuc  and Deslauriers and Dubuc .
The coefficients of the interpolation scheme of  and  generated from the
Lagrange polynomials are the auto-correlation coefficients of
quadrature mirror filters associated with the compactly supported wavelets of
Finally, we consider the reconstruction of signals from zero-crossings
(and slopes at zero-crossings) of their multiresolution representations.
Our approach permits a non-iterative reconstruction from zero-crossings
(and slopes at zero-crossings).
Using the interpolation algorithm mentioned above, we locate the
zero-crossings and compute slopes at these points within the
prescribed numerical accuracy.
We then set up a linear system, where the entries of the matrix
are computed from the values of the auto-correlation function and its
derivative at the integer translates of zero-crossings.
The original signal is reconstructed within the prescribed accuracy by solving
this linear system.
Get the full report: gzipped PS file or PDF file.
me if you have any comments or questions!
back to Naoki's Publication Page