Bayesian Estimation Applied to Stochastic Localization with Constraints due to Interfaces and Boundaries

Table 3

Statistical values about the estimation results of 3D cone beam geometry for cases (A)–(E) with 𝑀= 10,000 sample points 𝐱 each are presented for the three 3D estimators. The radial estimation error is presented at first and the coordinate estimation errors of (𝑥₁,𝑥₂,𝑥₃) are shown in brackets. The same formulas described as in the caption of Table 2 results are utilized.

Case	Method	Mean	st. deviation	abs. maximum	Radial RMSE	[%]
	ML	2.52 (−0.0,−0.0,−0.0)	1.25 (1.8,1.8,1.1)	8.89 (7.7,7.1,4.7)	2.81 (1.8,1.8,1.1)	—
A	MAP	2.20 (0.8,0.8,0.3)	1.03 (1.4,1.4,1.0)	7.86 (6.1,6.9,4.1)	2.43 (1.6,1.6,1.0)	+14
	MMSE	1.77 (0.0,0.0,0.0)	0.78 (1.2,1.2,0.9)	5.99 (4.8,5.2,4.1)	1.93 (1.2,1.2,0.9)	+31
	ML	3.36 (0.1,0.1,0.0)	1.50 (2.3,2.3,1.7)	10.55 (9.5,9.1,7.2)	3.68 (2.3,2.3,1.7)	—
B	MAP	2.79 (0.9,0.9,0.6)	1.21 (1.6,1.6,1.4)	8.72 (7.9,8.3,5.5)	3.04 (1.9,1.9,1.5)	+17
	MMSE	2.27 (0.0,0.0,0.1)	1.00 (1.5,1.5,1.3)	7.80 (7.1,7.4,5.2)	2.48 (1.5,1.5,1.3)	+33
	ML	1.89 (0.0,0.0,0.0)	0.98 (1.4,1.4,0.8)	7.57 (5.5,6.9,2.9)	2.13 (1.4,1.4,0.8)	—
C	MAP	1.74 (0.6,0.6,0.2)	0.88 (1.1,1.1,0.7)	7.04 (4.8,5.5,3.0)	1.95 (1.3,1.3,0.8)	+8
	MMSE	1.42 (0.0,0.0,0.0)	0.68 (1.0,1.0,0.7)	5.18 (4.3,3.9,3.0)	1.57 (1.0,1.0,0.7)	+36
	ML	2.56 (0.0,0.0,0.0)	1.28 (1.9,1.9,1.1)	10.99 (7.4,8.7,4.7)	2.86 (1.9,1.9,1.1)	—
D	MAP	1.86 (0.9,0.9,0.5)	0.77 (0.9,0.9,0.8)	5.73 (5.7,5.3,3.6)	2.02 (1.3,1.3,0.9)	+29
	MMSE	1.31 (0.1,0.1,0.0)	0.59 (0.9,0.9,0.8)	5.00 (4.7,4.0,3.2)	1.43 (0.9,0.9,0.8)	+50
	ML	1.27 (−0.0,−0.0,−0.0)	0.63 (0.9,0.9,0.5)	4.78 (3.9,3.8,2.2)	1.41 (0.9,0.9,0.5)	—
E	MAP	1.21 (0.3,0.3,0.1)	0.60 (0.8,0.8,0.5)	4.55 (3.8,3.5,2.3)	1.35 (0.9,0.9,0.5)	+4
	MMSE	1.07 (0.0,0.0,0.0)	0.51 (0.8,0.8,0.5)	3.93 (3.0,2.8,2.0)	1.19 (0.8,0.8,0.5)	+16