Magnetic localization and orientation of the capsule endoscope based on a random complex algorithm

Figures & data

Figure 1 Capsule endoscope with magnetic tube.

Abbreviation: LED, light-emitting diode.

Figure 2 Axially magnetized magnetic sets.

Abbreviation: GI, gastrointestinal.

Figure 3 Axial circular magnet (magnetic dipole) model.

Notes: (a, b, c) is the magnet position; (X₁, Y₁, Z₁) is a spatial point; H0 is the normalized vector; R₁ is the distance between the magnet and the spatial point.

Figure 4 The calculation steps for the random complex algorithm (RCA).

Figure 5 Arrangement of sensors.

Figure 6 Position and orientation errors for the error levels of the initial guess of location.

Abbreviation: LM, Levenberg–Marquart.

Figure 7 Proportion of effective points for the different initial errors of position.

Abbreviation: RCA, random complex algorithm.

Table 1 The average and maximum of localization and orientation errors

Number	The average of localization error (mm)		The maximum of localization error (mm)		The average of orientation error (%)		The maximum of orientation error (%)
	LM algorithm	RCA	LM algorithm	RCA	LM algorithm	RCA	LM algorithm	RCA
1	1.68	0.59*	8.06	4.26*	5.36	1.72*	40.15	12.69*
2	44,758	0.81*	175,370	4.76*	99.29	2.25*	354.38	14.58*
3	16,251	0.93*	78,141	9.79*	67.52	2.79*	303.32	30.90*
4	10,239	0.82*	117,680	5.11*	53.62	2.46*	284.96	13.14*
5	19.99	0.84*	8,515.6	4.86*	7.78	2.45*	125.34	12.98*
6	592.59	0.76*	43,437	4.60*	9.76	2.22*	236.93	12.99*
7	2.00	0.91*	7.99	5.47*	6.16	2.51*	35.56	18.90*
8	1.69	0.89*	6.80	5.91*	5.46	2.07*	26.25	12.18*
9	16.89	5.98*	199.58	52.63*	26.18	5.40*	193.56	35.53*
10	16.74	6.60*	414.98	68.71*	27.37	5.99*	779.79	37.86*
11	16.69	5.91*	434.75	42.27*	28.11	5.56*	1,181.80	39.71*
12	131.76	6.28*	48,520	47.88*	30.94	6.22*	829.03	43.97*
13	265.64	6.41*	21,445	81.99*	30.43	6.21*	516.42	35.66*
14	18.68	7.19*	371.99	52.10*	29.94	7.30*	471.32	43.42*
15	19.29	7.52*	462.38	46.17*	31.20	8.76*	408.31	39.52*
16	17.78	6.98*	133.14	42.89*	31.01	9.98*	349.66	41.93*
17	16.72	6.94*	556.32	61.88*	28.62	8.23*	711.56	31.11*
18	15.63	7.37*	100.69	53.37*	26.46	8.26*	613.20	41.71*
19	40.11	6.70*	12,540	43.21*	25.72	8.73*	273.66	44.42*
20	15.44	7.29*	103.39	65.41*	26.79	8.27*	230.89	42.69*
21	15.92	7.87*	129.09	69.82*	27.74	9.10*	231.32	40.39*
22	17.65	7.60*	176.30	45.91*	30.21	10.367*	359.42	41.19*
23	32.04	6.28*	9,413.50	33.67*	24.38	9.34*	184.16	38.89*
24	11.69	4.97*	495.42	27.37*	23.27	5.85*	1,403.70	34.93*
25	52.61	4.82*	20,672	39.15*	20.04	4.64*	183.42	27.41*
26	11.05	4.08*	295.29	30.83*	19.81	4.29*	400.57	30.97*
27	13.45	4.72*	622.95	31.36*	24.14	3.96*	1,146.1	31.14*
28	54.34	5.01*	11,058	47.89*	24.08	6.45*	1,332.7	32.75*
29	12.86	6.08*	113.79	41.50*	25.76	7.98*	600.73	38.21*
30	11.82	4.91*	55.92	34.50*	23.09	7.32*	168.41	33.74*
31	12.49	5.52*	103.99	34.44*	22.94	7.33*	364.48	42.20*
32	11.20	5.22*	80.14	33.60*	20.78	7.22*	154.80	37.07*

Note: The values with asterisks represent the better values of two algorithms.

Abbreviations: RCA, random complex algorithm; LM, Levenberg–Marquart.

Figure 8 Position and orientation errors via noise level; vector Vr1.

Abbreviation: LM, Levenberg–Marquart.

Figure 9 Position and orientation errors via noise level; vector Vr2.

Abbreviation: LM, Levenberg–Marquart.

Figure 10 Comparison for the position and direction error and execution time of real data.

Abbreviations: LMmeanP, the average value of position error of LM algorithm; RCAmeanP, the average value of position error of RCA algorithm; LMmeanO, (the average value of direction error of LM algorithm; RCAmeanO, the average value of direction error of RCA algorithm; LMmeanT, the average value of execution time of LM algorithm; RCAmeanT, the average value of execution time of RCA algorithm; RCA, random complex algorithm.