## 3 Curvelet Identification of MRE and MRE with MWCNTs
### Curvelet Transform  
Curvelet transform is a mapping principle expressed as a transform \(f\rightarrow\Gamma_{f}(a,b,\theta)\) of a function \(f(x_{1},x_{2})\) on \(R^{2}\) into a transform domain regarding a specific scale \(a>0\), location \(b\in R^{2}\), and orientation \(\theta\in[0,2\pi)\). The function assumed herein is on the two-dimension plane \(R^{2}\) with spatial variable \(x\), frequency-domain variable \(\xi\) denoted with \(r\) and \(\omega\) polar  
Figure 4: The saturated magnetic induction: MRE and MRE with MWCNTs is 1.296 T and 1.316 T respectively.  
Figure 5: SEM Image at 1000\(\times\) magnification.  
coordinates. Radial and angular windows are respectively expressed as \(W(r)\) and \(V(r)\). They satisfy admissibility conditions:  
\[\int_{0}^{\infty}W(ar)^{2}\frac{da}{a}=1,\ \ \ \ \ \forall r>0 \tag{6}\]  
\[\int_{-1}^{1}V(u)^{2}du=1 \tag{7}\]  
A family of analyzing elements can be constructed using these windows. At scale \(a\), the family is generated on the basic element \(\gamma_{a00}\) with different levels of transition and rotation.  
\[\gamma_{aab}(x)=\gamma_{a00}(R_{\theta}(x-b)) \tag{8}\]  
Accordingly, the Polar Fourier coordinates \((r,\omega)\) at the scale \(a\) are defined as:  
\[\hat{\gamma}_{a00}(r,\omega)=W(ar)V(\omega/\sqrt{a})a^{3/4} \tag{9}\]  
Thereby, the support of \(\gamma_{aab}\) can be expressed as combination of polar \"wedge\" shapes with scale-dependent radial and angular windows in different directions.  
Continuous curvelet transform \(I_{f}\) can be defined with different scales, locations and directions in space.  
\[\Gamma_{f}(a,b,\theta)=\gamma_{aab},f\ \ \ \ \ a<a_{0},b\in R^{2},\theta\in[0,2\pi) \tag{10}\]  
Assuming \(V\) and \(W\) satisfy admissibility conditions, the _Calderon_-like reproducing formula for high-frequency functions is expressed as:  
\[f(x)=\int\Gamma_{f}(a,b,\theta)\gamma_{aab}\mu(dadbd\theta) \tag{11}\]  
where a Parseval formula for high-frequency functions is:  
\[\left\|f\right\|_{L^{2}}^{2}=\int\left|\Gamma_{f}(a,b,\theta)\right|^{2}\mu( dadbd\theta) \tag{12}\]  
Assuming \(\Phi\) is a band-limited radial function in \(L^{2}\)  
\[\Phi_{a_{0},b}(x)=\Phi(x-b),\ \text{thus} \tag{13}\]  
\[f(x)=\int\langle\Phi_{a_{0},b,t}f\rangle\Phi_{a_{0},b}(x)db+\int_{0}^{a_{0}} \iint\langle f,\gamma_{aab}\rangle\gamma_{aab}\mu(dadbd\theta)\,\ \text{and} \tag{14}\]  
\[\left\|f\right\|_{L^{2}}^{2}=\int\langle\Phi_{a_{0},b,t}f^{2}\rangle db+\int_ {0}^{a_{0}}\iint\langle f,\gamma_{aab}\rangle^{2}\mu(dadbd\theta) \tag{15}\]  
Therefore, the \"full\" curvelet transform consists of the fine-scale directional elements and coarse-scale isotropic father wavelets.  
Conventional Fourier transform can map a function to a combination of a series of trigonometric bases with limited frequency domain information. Wavelet transform can identify horizontal, vertical and diagonal details with poor directional selection ability but may lose the feature of edges with changing orientations and angles. Thus, it cannot effectively deal with line singularity. The biggest advantage of curvelet transform is that it can extract spatial and directional feature information with optional-scale multiresolution analysis. Curvelet has a good property of representing border lines with highly directional anisotropy and sensitivity regarding various locations, orientations and scales of curved morphological structures at the same time, rather than just simple thresholding.  
As shown in Figure 6, the spatial Cartesian grid is related to a specific scale and orientation, while in the Fourier space, curvelet is supported near a \"parabolic\" wedge indicated with a shaded area in the frequency plane.