Fig 1.
Overview of objects fusion by the proposed method.
The objects f1 and f2 in the left side of the figures are transformed using Nabla operator, then they are fused, and finally the inverse transform reconstructs the fused object.
Fig 2.
The discrete 1D heat equation.
Manually initializing the blocks in the first row, the heat equation converges U(i, m) to the U(i).
Fig 3.
Obtaining U(i,j) from the Jacobi’s iterative method.
The problem has N = n2 equations and N unknowns. U(i,j) is calculated from points that are connected to it with red line segments. The boundary values are 0.
Fig 4.
Experiment 1: Results of fusing CT and MR images of brain with acute stroke.
(a) CT image; (b) MR-T2 image; (c) DWT; (d) NSCT; (e) NSCT-SR; (f) m-PCNN; (g) PCNN-NSCT; (h) SCM-F; (i) SCM-M; (j) Laplacian Pyramid; (k) Morphological Difference Pyramid; (l) PCA; (m) the proposed method with PCA, (n) max, and (o) weighted averaging model, respectively.
Fig 5.
Experiment 2: CT and MR-T2 images of brain with acute stroke, and the fusion results.
(a) CT image; (b)MR-T2 image; (c) DWT; (d) NSCT; (e) NSCT-SR; (f) m-PCNN; (g) PCNN-NSCT; (h) SCM-F; (i) SCM-M; (j) Laplacian Pyramid; (k) Morphological Difference Pyramid; (l) PCA; (m) the proposed method with PCA, (n) max, and (o) weighted averaging models, respectively.
Fig 6.
Experiment 3: Axial MR-T1 and MR-T2 images of the normal brain.
(a) MR-T1 image; (b)MR-T2 image; (c) DWT; (d) NSCT; (e) NSCT-SR; (f) m-PCNN; (g) PCNN-NSCT; (h) SCM-F; (i) SCM-M; (j) Laplacian Pyramid; (k) Morphological Difference Pyramid; (l) PCA; (m) the proposed method with PCA, (n) max, and (o) weighted averaging models, respectively.
Fig 7.
Experiment 4: MR-T1 and MR-T2 images of the brain of patients with vascular dementia, and the fusion results.
(a) MR-T1 image; (b)MR-T2 image; (c) DWT; (d) NSCT; (e) NSCT-SR; (f) m-PCNN; (g) PCNN-NSCT; (h) SCM-F; (i) SCM-M; (j) Laplacian Pyramid; (k) Morphological Difference Pyramid; (l) PCA; (m) the proposed method with PCA, (n) max, and (o) weighted averaging models, respectively.
Fig 8.
Experiment 5 [53]: Results of fusing CT and MR images of a brain.
(a) CT image; (b) MR image; (c) NNSST [93]; (d) FMSAP [95]; (e) CST [97]; (f) ST-NSST [96]; (g) MFDF-NSST [53]; (h) the proposed method with PCA, (i) max, and (j) weighted averaging model, respectively.
Fig 9.
Experiment 6 [53]: Results of fusing CT and MR images of a brain.
(a) CT image; (b) MR image; (c) NNSST [93]; (d) FMSAP [95]; (e) CST [97]; (f) ST-NSST [96]; (g) MFDF-NSST [53]; (h) the proposed method with PCA, (i) max, and (j) weighted averaging model, respectively.
Fig 10.
Experiment 7 [53]: Results of fusing CT and MR images of a brain.
(a) CT image; (b) MR image; (c) NNSST [93]; (d) FMSAP [95]; (e) CST [106]; (f) ST-NSST [96]; (g) MFDF-NSST [53]; (h) the proposed method with PCA, (i) max, and (j) weighted averaging model, respectively.
Fig 11.
Experiment 8 [53]: Results of fusing CT and MR images of a brain.
(a) CT image; (b) MR image; (c) NNSST [93]; (d) FMSAP [95]; (e) CST [97]; (f) ST-NSST [96]; (g) MFDF-NSST [53]; (h) the proposed method with PCA, (i) max, and (j) weighted averaging model, respectively.
Fig 12.
Experiment 9–12: Fusion of Multimodal medical images of patients with vascular dementia.
(a, b) Experiment 5: CT and MR-T2 images. (c, d) Experiment 6: MR-PD and MR-T2 images. (e, f) Experiment 7: CT and MR-GAD images. (g, h) Experiment 8: CT and MR-T1 images.
Chart 1.
The average results of metrics for known fusion methods and proposed ones on Experiments 1–12.
Chart 2.
The average results of metrics for recent hybrid fusion methods and proposed ones on Experiments 1–4.
Chart 3.
The average results of metrics for recent hybrid fusion methods and proposed ones on Experiments 5–8.
Table 1.
The result of fusion methods’ average ranks using Friedman test in Experiments 1–4.
Fig 13.
The differences between neighboring fusion methods’ average ranks in Table 1.
Fig 14.
Comparison of fusion methods when no method is singled out.
The results of Kruskal-Wallis in wider gray lines and Bonferroni-Dunn in thinner black lines.
Fig 15.
Result of Holm and Hochberg methods.
In the Holm row, all the connected fusion methods are significantly different from Nabla-max method. On the other hand, the Hochberg method cannot show significant difference between Nabla-max to Nabla-PCA, but reject all the remaining hypotheses.
Table 2.
The details of implementing Holm and Hochberg method to compare fusion methods.
Fig 16.
Result of Holm and Hochberg post hoc tests for Experiments 5–8.
Fig 17.
Result of Holm and Hochberg post hoc tests for all above 12 Experiments.
Table 3.
The result of fusion methods’ average ranks using Friedman test in Experiments 5–8.
Table 4.
The result of fusion methods’ average ranks using Friedman test in all above 12 Experiments.
Fig 18.
The result of post-hoc tests’ significance differences between the proposed fusion methods and: (a) some recently hybrid methods applying on experiments 1–4; (b) some other recent hybrid methods applying on experiments 5–8; (c) some known methods applying on experiments 1–12.