9 Conclusion

We have seen that information must be measured from the transformed data, and not from the data itself. This is so that a priori knowledge of physical aspects of the data can be taken into account. We could have used the Shannon entropy, perhaps generalized (Sporring & Weickert 1999), to measure the information at a given scale, and derive the bins of the histogram from the standard deviation of the noise, but for several reasons we thought it better to directly introduce noise probability into our information measure. Firstly, we have seen that this leads, for Gaussian noise, to a very physically meaningful relation between the information and the wavelet coefficients: information is proportional to the energy of the wavelet coefficients normalized by the standard deviation of the noise. Secondly, this can be generalized to many other kinds of noise, including such cases as multiplicative noise, non-stationary noise, or images with few photons/events. We have seen that the equations are easy to manipulate. Finally, experiments have confirmed that this approach gives good results.

For filtering, the multiscale entropy has the following advantages:

• It provides a good trade-off between hard and soft thresholding;
• No a priori model on the signal itself is needed as with other wavelet-based Bayesian methods (Chipman et al. 1997; Crouse et al. 1998; Vidakovic 1998; Timmermann & Nowak 1999);
• It can be generalized to many different noise distributions;
• The regularization parameter $\alpha$ can be easily fixed automatically. Cross-validation (Nason 1996) could be an alternative, but with the limitation to Gaussian noise.

Replacing the standard entropy measurements by the Multiscale Entropy avoids the main problems in the MEM deconvolution method.

We have seen also how our new information measure allows us to analyze image background fluctuation. In the example discussed, we showed how signal which was below the noise level could be demonstrated to be present. Our SNR was 0.25. This innovative analysis leads to our being able to affirm that signal is present, without being able to say where it is.

To study the semantics of a large number of digital and digitized photographic images, we took already prepared - external - results, and we also used two other processing pipelines for detecting astronomical objects within these images. Therefore we had three sets of interpretations of these images. We then used Multiscale Entropy to tell us something about these three sets of results. We found that Multiscale Entropy provided interesting insight into the performances of these different analysis procedures. Based on strength of correlation between Multiscale Entropy and analysis result, we argued that this provided circumstantial evidence of one analysis result being superior to the others.

We finally used Multiscale Entropy to provide a measure of optimal image compressibility. Using previous studies of ours, we had already available to us a set of images with the compression rates which were consistent with the best recoverability of astronomical properties. These astronomical properties were based on positional and intensity information, - astrometry and photometry. Papers cited contain details of these studies. Therefore we had optimal compression ratios, and for the corresponding images, we proceeded to measure the Multiscale Entropy. We found a very good correlation. We conclude that Multiscale Entropy provides a good measure of image or signal compressibility.

The breadth and depth of our applications lend credence to the claim that Multiscale Entropy is a good measure of image or signal content. Compared to previous work, we have built certain aspects of the semantics of such data into our analysis procedures. As we have shown, the outcome is a better ability to understand our data.

Acknowledgements

Work described in Sect. 7 was contributed to by S. Couvidat. We wish to thank F. Bouchet and R. Teyssier for providing us with the Planck simulated dataset. This work was partially supported by USA National Science Foundation grant DMS 98-72890 (KDI).