Estimation method for inverse problems with linear forward operator and its application to magnetization estimation from magnetic force microscopy images using deep learning

Figures & data

Figure 1. MFM.

Figure 2. Cell discretization.

Figure 3. Numbering cells.

Figure 4. Magnetic moments and charges.

Figure 5. Complexity

Table 1. ‘Practical’ rank of G (observation points: $20\times 20\times 1$).

	Charges
Length $\parallel u_n\parallel$	$20\times 20\times 10$	$20\times 20\times 6$	$20\times 20\times 1$
$\parallel u_n\parallel \ge 0.9$	400	21	9
$0.1\le \parallel u_n\parallel <0.9$	0	71	19
$0.01\le \parallel u_n\parallel <0.1$	0	94	29

Table 2. ‘Practical’ rank of G (Observation points: $20\times 20\times 4$).

	Charges
Length $\parallel u_n\parallel$	$20\times 20\times 10$	$20\times 20\times 6$	$20\times 20\times 1$
$\parallel u_n\parallel \ge 0.9$	399	22	8
$0.1\le \parallel u_n\parallel <0.9$	66	76	22
$0.01\le \parallel u_n\parallel <0.1$	214	99	29

Table 3. ‘Practical’ rank of G (observation points: $40\times 40\times 1$).

	Charges
Length $\parallel u_n\parallel$	$20\times 20\times 10$	$20\times 20\times 6$	$20\times 20\times 1$
$\parallel u_n\parallel \ge 0.9$	355	22	9
$0.1\le \parallel u_n\parallel <0.9$	45	79	22
$0.01\le \parallel u_n\parallel <0.1$	370	93	27

Table 4. ‘Practical’ rank of G (Observation points: $40\times 40\times 2$).

	Charges
Length $\parallel u_n\parallel$	$20\times 20\times 10$	$20\times 20\times 6$	$20\times 20\times 1$
$\parallel u_n\parallel \ge 0.9$	352	22	8
$0.1\le \parallel u_n\parallel <0.9$	65	78	24
$0.01\le \parallel u_n\parallel <0.1$	359	103	27

Figure 6. Distributions of $r^{\left(\alpha \text{β}\right)}$

Table 5. Minimum and average of $\{1000\cdot r_{\left[\ell \right]\downarrow }^{\left(\alpha \text{β}\right)}\}$ $(n_x=n_y=20)$.

Piece ℓ	1	2	3	4	5	6	7	8	9	10
Minimum	4832	271	59.6	18.0	7.12	3.57	1.78	0.943	0.459	0.256
Average	5715	347	82.6	30.4	13.9	7.27	4.12	2.46	1.48	0.827

Table 6. Minimum and average of $\{1000\cdot r_{\left[\ell \right]\downarrow }^{\left(\alpha \text{β}\right)}\}$ $(n_x=n_y=32)$.

Piece ℓ	1	2	3	4	5	6	7	8	9	10
Minimum	5076	303	67.9	22.8	9.97	4.58	2.52	1.43	0.835	0.437
Average	5715	348	83.7	30.9	14.2	7.44	4.27	2.58	1.56	0.864

Table 7. Minimum and maximum of $\{1000\cdot \parallel GR{\mu }_{\left[\ell \right]}{\parallel }_O\}$ $(n_x=n_y=20).$

Piece ℓ	1	2	3	4	5	6	7	8	9	10
Minimum	117,000	7550	1860	695	304	150	79.9	48.2	32.3	22.5
Maximum	124,000	10,100	3220	1440	799	495	337	239	175	131

Table 8. Minimum and maximum of $\{1000\cdot \parallel GR{\mu }_{\left[\ell \right]}{\parallel }_O\}$ $(n_x=n_y=32).$

Piece ℓ	1	2	3	4	5	6	7	8	9	10
Minimum	189,000	12,900	3370	1320	620	333	183	109	70.6	49.0
Maximum	197,000	15,400	4830	2120	1140	702	460	314	222	170

Table 9. Minimum and average of $\{\parallel GR{\mu }_{\left[\ell \right]}{\parallel }_O/\parallel GR{\mu }_{[\ell +1]}{\parallel }_O:{\mu }_{\left[\ell \right]}={\mu }_{[\ell +1]}\}$ $(n_x=n_y=20)$.

Pieces pair $(\ell ,\ell +1)$	(1,2)	(2,3)	(3,4)	(4,5)	(5,6)	(6,7)	(7,8)	(8,9)	(9,10)
Minimum	11.7	3.00	2.02	1.70	1.50	1.39	1.32	1.27	1.23
Average	13.8	3.57	2.42	1.99	1.76	1.62	1.52	1.45	1.40

Table 10. Minimum and average of $\{\parallel GR{\mu }_{\left[\ell \right]}{\parallel }_O/\parallel GR{\mu }_{[\ell +1]}{\parallel }_O:{\mu }_{\left[\ell \right]}={\mu }_{[\ell +1]}\}$ $(n_x=n_y=32)$.

Pieces pair $(\ell ,\ell +1)$	(1,2)	(2,3)	(3,4)	(4,5)	(5,6)	(6,7)	(7,8)	(8,9)	(9,10)
Minimum	12.4	3.12	2.13	1.77	1.52	1.42	1.36	1.31	1.26
Average	13.7	3.54	2.40	1.97	1.75	1.61	1.51	1.44	1.39

Table 11. Execution environment.

OS	Windows 10 Home
CPU	Core i7 4.20 GHz
GPU	NVIDIA GeForceGTX 1080
Memory	16.0 GB
Language	Python 3.6.0
AI Library	TensorFlow 1.4.0, Keras 2.0.9

Figure 7. Convolutional auto encoder.

Table 12. Reversal rates.

r (rate of data)	0.1	0.2	0.3	0.4	0.5
$r^{\ast }$	0.1	0.3	0.3	0.5	0.5

Table 13. Estimation performances of each rate.

Rates		Piece
r	s	1	2	3	4	5	6	7	8	9	10
0.1	${0.1}^{\ast }$	1.000	1.000	1.000	1.000	0.996	0.964	0.866	0.773	0.731	0.772
0.1	0.3	1.000	1.000	1.000	0.997	0.971	0.831	0.654	0.614	0.612	0.654
0.1	0.5	1.000	0.978	0.861	0.652	0.511	0.487	0.524	0.505	0.541	0.487
0.2	0.1	1.000	1.000	1.000	0.991	0.923	0.752	0.683	0.673	0.640	0.704
0.2	${0.3}^{\ast }$	1.000	1.000	1.000	0.999	0.978	0.826	0.621	0.583	0.578	0.601
0.2	0.5	1.000	0.997	0.978	0.868	0.636	0.514	0.518	0.505	0.530	0.492
0.3	0.1	1.000	1.000	0.993	0.922	0.715	0.612	0.609	0.595	0.592	0.627
0.3	${0.3}^{\ast }$	1.000	1.000	1.000	0.999	0.975	0.788	0.580	0.554	0.553	0.563
0.3	0.5	1.000	1.000	0.998	0.979	0.834	0.569	0.514	0.505	0.517	0.494
0.4	0.1	1.000	0.995	0.942	0.721	0.571	0.538	0.536	0.533	0.545	0.558
0.4	0.3	1.000	1.000	1.000	0.991	0.909	0.644	0.537	0.528	0.529	0.532
0.4	${0.5}^{\ast }$	1.000	1.000	1.000	0.997	0.938	0.663	0.515	0.505	0.508	0.497
0.5	0.1	1.000	0.972	0.787	0.537	0.501	0.501	0.500	0.500	0.500	0.500
0.5	0.3	1.000	0.999	0.991	0.927	0.706	0.518	0.502	0.501	0.500	0.500
0.5	${0.5}^{\ast }$	1.000	1.000	1.000	0.999	0.968	0.721	0.517	0.503	0.501	0.501

Table 14. Estimation performances of each $d(O,M)$.

	Piece
$d(O,M)$	1	2	3	4	5	6	7	8	9	10
0.1	0.500	0.500	0.500	0.500	0.499	0.500	0.500	0.499	0.500	0.500
0.4	0.500	0.502	0.500	0.500	0.500	0.499	0.500	0.500	0.500	0.499
0.5	1.000	1.000	1.000	0.999	0.994	0.880	0.570	0.506	0.501	0.500
0.6	1.000	0.999	0.999	0.999	0.990	0.844	0.548	0.506	0.501	0.501
0.9	1.000	0.999	0.999	0.999	0.975	0.755	0.522	0.503	0.501	0.501
1.0	1.000	1.000	1.000	0.999	0.968	0.721	0.517	0.503	0.501	0.501

Table 15. Estimation performance for the size of the problem and the number of filters.

	Piece
size & $N_f$	1	2	3	4	5	6	7	8	9	10
$20\times 20,$ 256	1.000	1.000	1.000	0.998	0.961	0.710	0.517	0.503	0.501	0.500
$20\times 20,$ 512	1.000	1.000	1.000	0.998	0.964	0.727	0.524	0.505	0.503	0.501
$32\times 32,$ 256	1.000	1.000	1.000	0.999	0.968	0.721	0.517	0.503	0.501	0.501

Table 16. Estimation performance when no error occurs in the upper pieces.

	Piece
Rate	1	2	3	4	5	6	7	8	9	10
0.1	1.000	1.000	1.000	1.000	0.998	0.986	0.952	0.921	0.907	0.906
0.3	1.000	1.000	1.000	0.999	0.986	0.903	0.829	0.781	0.754	0.749
0.5	1.000	1.000	1.000	0.999	0.981	0.882	0.795	0.738	0.704	0.693

Table 17. Estimation performance for cases with noise.

	Piece
Noise	1	2	3	4	5
0	1.000	1.000	1.000	0.999	0.968
0.001	1.000	1.000	0.993	0.545	0.504
0.01	1.000	1.000	0.618	0.506	0.500

Table 18. Estimation performance of the iteration method ($n_x=n_y=20$).

	Piece
Rate	1	2	3	4	5	6	7	8	9	10
0.1	1.000	0.990	0.912	0.795	0.810	0.808	0.801	0.782	0.732	0.664
0.3	1.000	0.999	0.894	0.645	0.625	0.578	0.541	0.505	0.512	0.514
0.5	1.000	0.999	0.938	0.608	0.499	0.500	0.502	0.505	0.501	0.495

Table 19. Estimation performance of the iteration method ($n_x=n_y=32$).

	Piece
Rate	1	2	3	4	5	6	7	8	9	10
0.1	1.000	0.924	0.728	0.603	0.595	0.645	0.716	0.759	0.781	0.780
0.3	1.000	0.993	0.809	0.559	0.577	0.614	0.636	0.629	0.532	0.559
0.5	1.000	0.999	0.908	0.583	0.499	0.504	0.508	0.495	0.507	0.506

Data availability

All the data sets used in this study were randomly generated.