On the Laplacian operation with applications in magnetic resonance electrical impedance imaging

Figures & data

Figure 1. The approximation to α* by above iterative algorithm.

Figure 2. The relative errors RE of the regularizing solution u^α,δ for different noise level δ when α = α_H, α_G and α_M, respectively (left), the RE for α = α_G and different δ when q = 1, 0.6, 0.2 and 0, respectively (right).

Figure 3. Exact solution u = Δf and the regularizing solutions u^α,δ for δ = 0.005, 0.01 and 0.05 when α = α_H.

Table 1. The values of α*, α_H and the iteration number it for solving the generalized discrepancy principle.

δ	0.001	0.005	0.01	0.05	0.1
α*	1.659 × 10^−3	3.139 × 10^−3	4.495 × 10^−3	1.106 × 10^−2	1.692 × 10^−2
αH	1.659 × 10^−3	3.139 × 10^−3	4.495 × 10^−3	1.106 × 10^−2	1.692 × 10^−2
it	5	6	6	8	10

Table 2. The values of α_H, α_M, α_G and the relative errors RE of the corresponding regularizing solution u^α,δ.

δ	0.001	0.005	0.01	0.05	0.1
αH	1.674 × 10^−3	3.130 × 10^−3	4.513 × 10^−3	1.109 × 10^−2	1.667 × 10^−2
RE	2.792%	7.259%	10.57%	23.21%	31.41%
αG	7.814 × 10^−3	1.501 × 10^−2	2.223 × 10^−2	6.104 × 10^−2	9.883 × 10^−2
RE	13.12%	22.75%	30.44%	54.71%	66.13%
αM	1.583 × 10^−3	2.427 × 10^−3	3.300 × 10^−3	7.248 × 10^−3	1.034 × 10^−2
RE	2.670%	7.471%	11.17%	25.64%	35.50%

Figure 4. The functions f and u = Δf.

Figure 5. The regularizing solutions u^α,δ for different noise level δ.

Figure 6. The picture with grey level f.

Figure 7. The exact magnetic flux and .

Figure 8. The reconstructive results of σⁿ when noise level δ = 3%.

Table 3. The relative error E(n) of the iterative sequence σⁿ for different noise level δ and iteration number n.

	n = 0	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7
δ = 0%	0.4603	0.1469	0.0688	0.0547	0.0520	0.0514	0.0513	0.0513
δ = 1%	0.4603	0.1566	0.0991	0.0924	0.0916	0.0915	0.0914	0.0914
δ = 3%	0.4603	0.1801	0.1383	0.1334	0.1328	0.1327	0.1327	0.1327
δ = 5%	0.4603	0.1943	0.1621	0.1574	0.1566	0.1564	0.1564	0.1564