## 3 Curvelet Identification of MRE and MRE with MWCNTs
### Curvelet Identification  
Due to various and complex morphological patterns of features of images, sparse representation in the curvelet domain is required so that one type of feature can be identified via one or several related decomposed terms. The maximum number of decomposed scales is \(\log_{2}N-3\) for an \(N\times N\) image. Three steps are implemented for feature extraction. First, an image is decomposed via curvelet transform into multiple scales with different orientations and angles. Second, different regular main features of decomposed components in the transform domain are grouped and clustered at large scales while irregular and isolated dots or noise denoting other features are obtained and classified into other groups. Finally, inverse curvelet transform is applied for reconstruction of features based on different scales. The SEM image of MRE and nanocomposite has the size of 128 pixel \(\times\) 128 pixel. The maximum number of decomposition scale via curvelet is \(\log_{2}128-3=4\). Figure 7 shows the absolute values of coefficients of curvelet transform, where coefficients in different scales are distinguished by interval gray bands indicating no curvelet coefficients. Areas in lighter color represent larger value of coefficients, which means more energy is distributed.  
Illustration of reconstructed image at each scale is shown in Figure 8. For both MRE, and MRE with MWCNTs, the first scale reveals the coarse and overall characteristics of the composites, the second and third scale reflects more concrete composition of the composites  
Figure 6: Curvelet Transform Domain.  
Figure 7: Absolute Value of Coefficients via Curvelet Transform.  
in the curvelet domain, and the forth scale shows the finest part and a few noise effects on the image.  
Areas in strip-shape light color are denoted as iron particles, dark colors are silicon rubber matrix, and cotton-like material with small clusters is the additives of MWCNTs. Specifically, scale four of MR nanocomposite reveals MWCNTs closely wrap the scattered iron particles, showing a distinct bonding between matrix and particles.  
It can be seen from Figure 9 that the normalized energy distribution of curvelet coefficients at each scale is further studied by calculating the sum of squares of the amplitude of curvelet coefficients, denoted as: \(E^{(a)}=\sum_{\theta,\theta}\lvert c(a,b,\theta)\rvert^{2}\).  
Figure 8: Curvelet Decomposition by Scales 1–4 (**a–d**) for MRE, Scale 1–4 (**e–h**) for MRE with MWCNTs.  
Figure 9: Normalized Energy Distribution.  
Relative significance of each constituent of MRE or MRE with MWCNTs is measured at each scale. Scales 1 to 4 have an increasing trend of energy distribution. MRE with MWCNTs has relatively larger relative energy than MRE for almost all the scales. It indicates the importance of MWCNTs for function enhancement of MRE, since they change the energy distribution of curvelet coefficients.  
From Figure 10, it can be seen that, as the scale increases from one to four, the variance first increases and then gradually decreases. Large variance indicates the discrimination degree of energy concentration is relatively obvious along specific directions, angles or a few fractions of areas. Isotropic or stochastic characteristics of features are implied with smaller variance. This phenomenon is clearly reflected at scale four, where the energy variance is 0.62 for MRE with MWCNTs compared to 0.55 for MRE. The addition of MWCNTs renders the particles aligned in parallel and the composite more structured.