Nonlinear parameter identification in a corneal geometry model

Figures & data

Table 1. Number of iterations needed for the convergence with different starting points $a_0$ for interior and exterior surfaces of the cornea.

	Starting points $a_0$
	0.1	1	2	3	4	5	6	7	8	9	10
Interior	4	3	2	3	3	3	4	4	4	4	3
Exterior	5	3	4	4	5	7	6	5	4	6	8

Table 2. Results of the numerical experiment for finding true values of parameters $a$ and $b$. The model was constructed from (5151 $$\begin{array}{c}\hfill f(a;x):=\frac{1}{a{I}_0(\sqrt{a})}\left(I_0,(\sqrt{a}x),-,1\right),\end{array}$$51 ) with additive noise on level $\mathit{\delta }$ and $a=1$, $b=2$.

	Noise level $\mathit{\delta }$ (mm)
	0	0.01	0.05	0.1	0.5	1.0	1.5
$a$	0.9936	0.9968	0.9627	1.0532	1.1681	0.8481	1.2239
$b$	1.9979	2.0020	1.9877	2.0187	2.0550	1.9357	2.1082

Figure 1. Contur plots of the absolute errors in fitting $E(x,y)$ defined in (5353 $$\begin{array}{c}\hfill E(x,y):=\left|h,(x,y),-,F,(a_{n_{\ast }}^{†}),(x,y)\right|,\end{array}$$53 ). On the top: exterior surface, on the bottom: interior surface of cornea. Horizontal and vertical axes, shown in millimetres, depict $x$ and $y$ coordinates of the point on the corneal surface. The light colour outside the cornea represents the reference surface.