8 Multiscale entropy and optimal compressibility

Subsequently we looked for the relation between the multiscale entropy and the optimal compression rate of an image which we can obtain by multiresolution techniques (Starck et al. 1998a). By optimal compression rate we mean a compression rate which allows all the sources to be preserved, and which does not degrade the astrometry (object positions) and photometry (object intensities). Louys et al. (Louys et al. 1999) and Couvidat (Couvidat 1999) have estimated this optimal compression rate using the compression program of the MR/1 package (MR/1 1999). We note that photometry and astrometry (and visual quality) were used in this work, given the crucial importance in astronomy of flux conservation and completeness. The cited references may be referred to for further details.

Figure 12 shows the relation obtained between the multiscale entropy and the optimal compression rate for all the images used in our previous tests, both digitized plate and CCD images.

Figure 12: Multiscale entropy of astronomical images versus the optimal compression ratio. Images which contain a high number of sources have a small ratio and a high multiscale entropy value. With logarithmic numbers of sources, the relation is almost linear

The power law relation is obvious thus allowing us to conclude that:

• The compression rate depends strongly on the astronomical content of the image. We can then say that compressibility is also an estimator of the content of the image;
• The multiscale entropy allows us to predict the optimal compression rate of the image.