Digital Signal Processing Reference
In-Depth Information
Figure 1.7. (top row) Four images used and (bottom row) each with added Gaussian noise of
standard deviation 20.
In Murtagh and Starck (2008), we do not seek to discriminate as such between
particles of varying sizes and granularities, but rather, to directly classify mixtures.
Our work shows the extent to which we can successfully address this more practical
and operational problem. As a “virtual sieve,” this classification of mixtures is far
more powerful than physical sieving, which can only handle individual components
in the mixtures.
1.4.4.2 Assessments of Higher-Order Wavelet and Curvelet Moments
We took four images with a good quantity of curved, edgelike structure for two
reasons: first, owing to a similar mix of smooth but noisy in appearance and edgelike
regions in our construction images, and second, to test the curvelet as well as the
wavelet transforms. To each image we added three realizations of Gaussian noise of
standard deviation 10 and three realizations of Gaussian noise of standard deviation
20. Thus, for each of our four images, we had seven realizations of the image. In all,
we used these 28 images.
Examples are shown in Fig. 1.7. The images used were all of dimensions
512
512. The images were the widely used test images Lena and a landscape, a
mammogram, and a satellite view of the city of Derry and the river Foyle in north-
ern Ireland. We expect the effect of the added noise to make the image increasingly
smooth at the more low (i.e., smooth) levels in the multiresolution transform.
Each of the 28 images is characterized by the following:
×
For each of five wavelet scales resulting from the starlet transform, we deter-
mined the second-, third-, and fourth-order moments at each scale (hence vari-
ance, skewness, and kurtosis). So each image had 15 features.
For each of 19 bands resulting from the curvelet transform, we again determined
the second-, third-, and fourth-order moments at each band (hence variance,
skewness, and kurtosis). So each image had 57 features.
