## 4 Experimental Results
### ISAR Imaging  
We utilize the CS ISAR imaging model as discussed in [43], and assume that the translational motion compensation has been already achieved and thus only the rotational motion is considered. The relation between the received signal and the complex reflection coefficients can be written as Equation (10) in [43], which is equivalent to Equation (2) in this paper. We use the quasi real data of an airplane (B-727) provided by the U.S. Naval Research Laboratory to test the feasibility and performance of the proposed methods, which is available on the website [[http://airborne.nrl.navy.mil/](http://airborne.nrl.navy.mil/)]([http://airborne.nrl.navy.mil/](http://airborne.nrl.navy.mil/))\(\sim\)vchen/tftsa.html. The stepped frequency radar operates at 9 GHz with the equivalent pulse repetition interval (PRI) of 3.2 ms and has a bandwidth of 150 MHz. For each pulse train, 64 complex range samples were saved, and the file contains 200 successive pulse trains within the long coherent processing interval (CPI). An additive noise is added to the original B-727s data, and the SNR is set to 10 dB. Herein we choose 40 range cell numbers and 20 cross range cell numbers for short CPI test, and the reconstruction of target spatial domain is discretized with 64 range bins and 32 cross range bins, besides, we set \(H=2,L=S=60\) and \(Q=5\). Since in radar imaging field, the position estimates are very important when off-grid problem exists. Figure 8: MSE of imaging results _versus_ SNR. MSE of position _vs_. SNR (**a**) and amplitude _vs_. SNR (**b**). Figure 9 are the position recovery results by MP, SBL, FOCUSS, BP, Matrix-Pencil, ESPRIT, H\(l_{1}\)-DCD and the proposed methods (HR-DCD, Hlog-DCD and H\(l_{p}\)-DCD) shown as the red circles, and we use the conventional FFT-based ISAR image as the background for fair comparison. As stated above, the imaging performances of the classical methods (MP, BP, FOCUSS, SBL) are not good because of the off-grid errors. Matrix Pencil and ESPRIT can direct solve the positions of the plane, but their performances suffer from low SNR and few snapshots, therefore the imaging results are not good enough. In contrast, our methods can deal with the off-grid target, and achieve satisfying imaging performances which have better visual effect for the airplane shape. Moreover, it is obvious that the imaging results of our methods are better than H\(l_{1}\)-DCD with the same parameters, especially on the details of airplane wings and tails as illustrated in zoomed-in regions of Figure 9h-j, which demonstrate the advantage of homotopy DCD method with non-convex regularization compared with convex regularization under the same measurements. Figure 9: ISAR imaging results of B-727 by MP (**a**), SBL(**b**), FOCUSS (**c**), BP (**d**), Matrix-Pencil (**e**), ESPRIT (**f**), H\(l_{1}\)-DCD (**g**), HR-DCD (**h**), Hlog-DCD (**i**), H\(l_{p}\)-DCD (**j**). The red circles denote the corresponding position recovery result, and background represents the FFT-based reconstruction. In order to quantitatively evaluate the amplitude estimation performances of the obtained ISAR images via different methods, we use the correlation value [44] to evaluate the similarity between the recovered images and the reference image, and the image entropy [44] to measure the focusing quality of the recovered images. They are defined as follows:  
\[Cor\big{(}\mathbf{x}_{r},\mathbf{x}_{o}\big{)}=\frac{\left|\big{\langle}\mathbf{x }_{r},\mathbf{x}_{o}\big{\rangle}\right|}{\left|\mathbf{x}_{r},\right|\left| \mathbf{x}_{o}\right|_{2}} \tag{22}\]  
\[Ent\big{(}\mathbf{x}_{o}\big{)}=-\sum_{i=1}^{N}p_{i}\log p_{i} \tag{23}\]  
Figure 9: _Cont._  
where \(\mathbf{x}_{r}\) and \(\mathbf{x}_{o}\) denote the reference image vector using full measurements and the recovered image vector by different methods, and \(p_{i}\) is the histogram of the recovered gray level (0-255) image.