Localization of immersed obstacles from boundary measurements

Figures & data

Figure 1. Reconstruction results from two boundary velocities satisfying (right plot) and not satisfying (left plot) the orthogonal condition (1919 $$\begin{array}{c}\hfill x_{f_1}=\frac{{\mathcal{R}}_{\mathrm{\Gamma }}\left(-,\frac{1}{{\int }_{\mathit{\omega }}xdA},{\mathbf{v}}_1,,,0\right)}{{\mathcal{R}}_{\mathrm{\Gamma }}\left(-,{|\mathit{\omega }|}^{-1},{\overrightarrow{\mathbf{e}}}_2,,,0\right)}\overline{x}\text{and}{y}_{f_2}=\frac{{\mathcal{R}}_{\mathrm{\Gamma }}\left(-,\frac{1}{{\int }_{\mathit{\omega }}ydA},{\mathbf{v}}_2,,,0\right)}{{\mathcal{R}}_{\mathrm{\Gamma }}\left(-,{|\mathit{\omega }|}^{-1},{\overrightarrow{\mathbf{e}}}_1,,,0\right)}\overline{y}\end{array}$$19 ).

Figure 2. The same as the previous figure when considered a different size for the kite.

Table 1. Reconstruction of a small circular obstacle $\mathit{\omega }=B_{0.03}(4.2,0)$ immersed in $\mathrm{\Omega }=B_5(0,0)$.

Observations	without noise	$10\%$ noise	$20\%$ noise
10	$(3.02,-0.15)$	$(3.15,-0.06)$	$(5.33,0.17)$
30	$(4.18,-0.02)$	$(5.12,0.03)$	$(4.32,0.91)$
60	$(4.20007,0.00014)$	$(4.33,0.024)$	$(5.83,0.45)$

Table 2. The same as the previous Table but considering the domain $\mathrm{\Omega }=B_{30}(0,0)$.

Observations	without noise	$10\%$ noise	$20\%$ noise
10	$(4.19,0.004)$	$(3.97,0.23)$	$(4.74,0.66)$
30	$(4.19,0.0003)$	$(3.54,0.09)$	$(4.36,0.14)$
60	$(4.20,0.0003)$	$(4.26,0.19)$	$(4.201,0.06)$

Figure 3. Reconstruction results considering 10 boundary observations and several obstacle shape and location.

Figure 4. Two different schemes for the observation points.

Figure 5. Localization of a shark. On the left, a plot of the obstacle, observation points and detected location (red bold dot). On the right, a plot of the obstacle and the retrieved location, $(-5.1,-14.4)$.

Figure 6. Localization of a smaller shark. The retrieved location was $(0.15,9.33)$.

Figure 7. Reconstruction from partial data. The obtained points are $(3.03,4.07)$ on the right, and $(8.1,2.2)$ on the left.