Queen's University Belfast and Strasbourg Observatory

CEA, DAPNIA, Saclay

A new measure of information is defined which is based on
noise modeling and incorporates
resolution scale. Properties of this measure are discussed.
Use of such an entropy measure for
filtering and deconvolution is exemplified. Implications
for structuring of, and access to,
image archives are noted.

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 *h*_{s},
and *h*_{n}. The expression
*h*_{s} is called the signal information and represents the part
of *h* which is definitely not contaminated by the noise. The
expression *h*_{n} is
called the noise information and represents the part
of *h* which * may* be contaminated by the noise. We have *h* = *h*_{s}+ *h*_{n}.
Following this notation, the corrected (i.e., filtered) wavelet
or multiscale coefficient should
minimize:

i.e. there is a minimum of information in the residual () due to the significant signal, and a minimum of information which can be due to the noise in the solution .

In order to verify a number of properties, the following functions are
proposed for *h*_{s} and *h*_{n} 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, *D*_{m}, for the data we can use

The regularized entropy-based filtering algorithm is as follows.

- Estimate the noise in the data (see Olsen 1993; Starck & Murtagh 1998a).
- Wavelet transform of the data.
- Calculate from the noise standard deviation at
each scale
*j*. - Set , .
- For each scale
*j*do- Set
- For each wavelet coefficient
*w*_{j,k}of scale*j*, find by minimizing or - Calculate the standard deviation of the residual:

- If then the regularization is too strong, and we set to , otherwise we set to .

- If then goto 5.
- Multiply all by the constant (default: 1).
- For each scale
*j*and for each wavelet coefficient*w*find by minimizing or . - Reconstruct the filtered image from by the inverse wavelet transform.

The minimization of *j*_{m} or *j*_{ms} (step 5.2) can be done by any method.
For instance,
a simple dichotomy can be used in order to find such that

(1) |

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*

- Noise separation, for compression and quicklook
- Variable resolution over time, for progressive transmission, and variable resolution over space leading to multifoveated images.
- Usable approaches to information characterization.

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

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