# A Coupled Variational System for Image Decomposition along with Edges Detection
## 4 Experimental Results and Discussion
### Image Decomposition Results Discussion  
Figure 3 displays the decomposition results for Synthetic image. The parameters for the proposed model are \(\mu=0.1,\ \beta=0.04,\ \tau=1,\ \lambda=0.2,\ \theta=0.5\). The parameters in [8] are taken \(\tau=0.002,\ \mu=0.001,\ \beta_{1}=1,\ \beta_{2}=0.1,\ \beta_{3}=0.001\). The parameters in [15] are \(\gamma=1,\ \alpha=0.1,\ \mu_{1}=1,\ \mu_{2}=0.1,\ \mu_{3}=0.01\). From Figure 3, we can see that the new model gains much better decomposition results and some edges are kept in the cartoon image. The other methods contain cartoon contours (edges) which were wrongly assigned to the texture component. For the pseudo edge function, we only give the results of [40] and ours, since other methods without it. From Figure 3d,e, we can see that the pseudo edge imageappears many false edges in Figure 3d which are not consistent with the decomposed cartoon image (see some circular areas in Figure 3d), however the proposed method obtains the better results. In short, the proposed method is more effective than other four methods. Figure 4 takes Fabric as the tested image, and the experimental parameters are taken \(\mu=0.04,\ \beta=0.01,\ \tau=1,\ \lambda=0.4,\ \theta=0.5,\) respectively. Those parameters in [8] are manual mediated to achieve good results, which are selected \(\tau=0.001,\ \mu=0.001,\ \beta_{1}=1,\ \beta_{2}=0.001,\ \beta_{3}=0.01.\) The parameters in [39] are \(K=5,\ \lambda=0.2,\ \Delta t=0.1\). For the parameters in [40], \(\lambda=0.2,\ \mu=0.1,\ \theta=5,\ \Delta t=0.02\) are chosen. From Figure 4a-d, we can see that the texture images in Figure 4b,c contain more contours which should be retained in the cartoon image. However, these characteristics are not obvious in Figure 4a,b. In addition, from the perspective of vision, the separated cartoon by the proposed method is better than [8]. For the pseudo edge image in Figure 4, we can observe that the obtained pseudo edge using [40] doesn't match the cartoon image. In Figure 5, we take Barbara image to test the different methods. The experimental parameters are \(\mu=0.05,\ \beta=0.04,\ \tau=2,\ \lambda=0.4,\ \theta=0.5\). The parameters in [15; 40] are \(\gamma=2,\ \alpha=0.1,\ \mu_{1}=0.1,\ \mu_{2}=0.01,\ \mu_{3}=1\) and \(\lambda=0.2,\ \mu=0.2,\ \theta=2,\ \Delta t=0.1,\) respectively. From the decomposed cartoon images, we can observe that the cartoon image in Figure 5a contains some textures which are not extracted in the upper right area; the cartoon image in Figure 5d is smoother than other cartoon images. For the decomposed texture images, we can observe that the texture in Figure 5b is enhanced, and the texture in Figure 5d contains more contour structure which should be in the cartoon. Relatively speaking, the proposed method achieves better results. For the pseudo edge images in Figure 5d,e, we can see that the pseudo edge image in Figure 5d is not consistent with the cartoon image in Figure 5d, which can be seen from the neckline area of the [PERSON] image, while the proposed method better consistent with its cartoon image which can be seen from the Figure 5e. Figure 2: The test images. (**a**) Synthesis image; (**b**) Fabric image; (**c**) [PERSON] image; (**d**) Table image; (**e**) [PERSON] image; (**f**) Boy image. Figure 3: The decomposition results using different algorithms. From left to right: (**a**) Cartoon and texture obtained by [4]. (**b**) Cartoon and texture obtained by [8]. (**c**) Cartoon and texture obtained by [18]. (**d**) Cartoon, the pseudo edge and texture obtained by [40]. (**e**) Cartoon, the pseudo edge \(|\omega|\) and texture obtained by the proposed method. Figure 4: The decomposition results using different algorithms. From left to right: (**a**) Cartoon, texture and the module of gradient for the cartoon obtained by [8]. (**b**) Cartoon, texture and the pseudo edge obtained by [39]. (**c**) Cartoon, texture and the pseudo edge obtained by [40]. (**d**) Cartoon, texture and the pseudo edge \(|\omega|\) obtained by the proposed method.