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

Table 2

Statistical values about the estimation results of 2D parallel beam geometry for cases (A)–(E) with 𝑀= 10,000 sample points 𝐱 each are presented for the six 2D estimators. The radial estimation error is presented at first and the coordinate estimation errors of (𝑥₁,𝑥₂) are shown in brackets. In the following the formulas for the statistical values are presented for the radial error: the mean value =‖𝜹(𝐮)‖−‖𝐱‖=1/𝑀∑^𝑀_𝑗=1‖𝜹(𝐮_𝑗)‖−‖𝐱_𝑗‖, the standard deviation =(1/(𝑀−1)∑^𝑀_𝑗=1(‖𝜹(𝐮_𝑗)‖−‖𝜹(𝐮)‖−‖𝐱_𝑗‖+‖𝐱‖)²)^1/2, the absolute maximum =max_{𝑗=1⋯𝑀}|‖𝜹(𝐮_𝑗)‖−‖𝐱_𝑗‖|, the Root Mean Square Error (RMSE) =(1/𝑀∑^𝑀_𝑗=1(‖𝜹(𝐮_𝑗)‖−‖𝐱_𝑗‖²)^1/2 and thegain of method 𝑋 relative to ML =(RMSE_ML−RMSE_𝑋)/RMSE_ML.

Case	Method	Mean	st. deviation	abs. maximum	Radial RMSE	[%]
	2×2	3.77 (−0.0,0.0)	1.94 (3.0,3.0)	11.90 (11.1,10.9)	4.24 (3.0,3.0)	−39
	ML	2.67 (−0.0,0.0)	1.48 (2.2,2.2)	9.75 (9.2,8.3)	3.05 (2.2,2.2)	—
A	MAP^np	2.35 (−0.3,−0.3)	1.35 (1.9,1.9)	9.75 (7.5,8.3)	2.71 (1.9,1.9)	+11
	MMSE^np	2.31 (−0.9,−0.9)	1.32 (1.6,1.7)	9.65 (7.5,8.2)	2.67 (1.9,1.9)	+13
	MAP	2.05 (0.6,0.6)	1.11 (1.5,1.5)	8.38 (6.5,6.3)	2.33 (1.7,1.7)	+24
	MMSE	1.79 (0.0,0.0)	0.96 (1.4,1.4)	7.93 (6.1,5.8)	2.03 (1.4,1.4)	+33
	ML	3.79 (−0.0,−0.0)	1.99 (3.0,3.0)	13.43 (13.2,11.7)	4.28 (3.0,3.0)	—
	MAP^np	3.29 (−0.4,−0.4)	1.83 (2.7,2.6)	12.77 (12.6,11.5)	3.76 (2.7,2.7)	+12
B	MMSE^np	3.10 (−1.2,−1.2)	1.68 (2.2,2.2)	10.99 (10.3,10.9)	3.52 (2.5,2.5)	+18
	MAP	2.56 (0.7,0.7)	1.34 (1.9,1.9)	8.95 (8.8,8.3)	2.89 (2.0,2.1)	+33
	MMSE	2.28 (0.0,0.0)	1.20 (1.8,1.8)	8.84 (8.6,8.1)	2.58 (1.8,1.8)	+40
	ML	1.99 (−0.0,−0.0)	1.13 (1.6,1.6)	8.40 (7.0,7.2)	2.29 (1.6,1.6)	—
	MAP^np	1.81 (−0.2,−0.2)	1.05 (1.5,1.5)	7.00 (5.8,6.2)	2.09 (1.5,1.5)	+9
C	MMSE^np	1.76 (−0.6,−0.6)	1.02 (1.3,1.3)	7.00 (5.8,6.2)	2.03 (1.4,1.5)	+11
	MAP	1.67 (0.4,0.4)	0.94 (1.3,1.3)	7.18 (5.5,5.7)	1.92 (1.4,1.4)	+16
	MMSE	1.48 (0.0,0.0)	0.82 (1.2,1.2)	6.95 (5.0,5.6)	1.70 (1.2,1.2)	+26
	ML	2.68 (0.0,0.0)	1.48 (2.2,2.2)	9.47 (7.8,9.0)	3.06 (2.2,2.2)	—
	MAP^np	2.23 (−0.4,−0.4)	1.32 (1.8,1.8)	9.47 (7.4,7.9)	2.59 (1.8,1.8)	+15
D	MMSE^np	2.35 (−1.2,−1.2)	1.32 (1.5,1.5)	9.45 (7.5,7.9)	2.69 (1.9,1.9)	+12
	MAP	1.38 (0.4,0.4)	0.76 (1.1,1.1)	5.78 (5.0,4.4)	1.57 (1.1,1.1)	+49
	MMSE	1.29 (0.0,0.0)	0.67 (1.0,1.0)	5.69 (4.6,4.2)	1.45 (1.0,1.0)	+53
	2×2	1.88 (0.0,0.0)	0.99 (1.5,1.5)	6.48 (6.4,6.2)	2.13 (1.5,1.5)	−40
	ML	1.33 (0.0,−0.0)	0.74 (1.1,1.1)	4.93 (3.8,4.2)	1.52 (1.1,1.1)	—
E	MAP^np	1.24 (−0.1,−0.1)	0.70 (1.0,1.0)	4.93 (3.8,4.2)	1.42 (1.0,1.0)	+7
	MMSE^np	1.20 (−0.3,−0.3)	0.68 (0.9,0.9)	4.93 (3.8,4.2)	1.38 (1.0,1.0)	+9
	MAP	1.22 (0.2,0.2)	0.67 (1.0,1.0)	4.74 (3.5,4.6)	1.39 (1.0,1.0)	+9
	MMSE	1.11 (0.0,0.0)	0.61 (0.9,0.9)	4.72 (3.4,4.6)	1.27 (0.9,0.9)	+16