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Immediate applications include image filtering and deconvolution. Other objectives include object finding and definition, and feature characterization. More globally, we also note that such vision modeling will be necessary in future generation image databases.
Background on these methods can be found in Starck, Murtagh, & Bijaoui (1998). Information on a multiresolution image and vision software package, MR/1, can be found at http://visitweb.com/multires.
The multiscale entropy method consists of measuring
the information h
relative to wavelet coefficients, and of separating this into two parts hs,
and hn. The expression
hs is called the signal information and represents the part
of h which is definitely not contaminated by the noise. The
expression hn is
called the noise information and represents the part
of h which may be contaminated by the noise. We have h = hs+ hn.
Following this notation, the corrected (i.e., filtered) wavelet
or multiscale coefficient should
minimize:
In order to verify a number of properties, the following functions are
proposed for hs and hn in the case of Gaussian noise:
The regularization parameter, , can be determined by using the
fact that we expect a residual with a given standard deviation at each
scale j equal to the noise standard deviation
at that
scale. Hence we have an
per scale.
If we have a model, Dm, for the data we can use
The regularized entropy-based filtering algorithm is as follows.
The minimization of jm or jms (step 5.2) can be done by any method.
For instance,
a simple dichotomy can be used in order to find such that
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The idea to treat the wavelet coefficients such that the residual respects some constraint has also been used in Nason (1996) and Amato & Vuza (1998) using cross-validation.
Figure 1 shows a difficult case of smooth and sharp transitions. The filtering method described here allows an excellent noise filtering of it to be carried out. Figure 2 shows a spectrum, and an effective noise filtered version.
We note that large image repositories require
Further reading is available in Starck, Murtagh, & Gastaud (1998) and Starck & Murtagh (1998b). The methods described here are available in the multiresolution analysis software package, MR/1, Version 2.0. Details of the MR/1 software package can be found at http://visitweb.com/multires.
Amato, U. & Vuza, D. T. 1998, Rev. Roumaine Math. Pures Appl., in press
Nason, G. P. 1996, J. Roy. Stat. Soc. B, 58, 463
Olsen, S. I. 1993, Comp. Vis. Graph. Image Proc., 55, 319
Starck, J. L. & Murtagh, F. 1998a, PASP, 110, 193
, 1998b,
Signal Proc., in press
Starck, J. L., Murtagh, F., & Bijaoui, A. 1998, Image and Data Analysis: The Multiscale Approach, (Cambridge: Cambridge Univ. Press)
Starck, J. L., Murtagh, F. & Gastaud, R. 1998, IEEE Trans. CAS II, 45, 1118