Homologous point matching simulations

This section illustrates the use of MS_reg_error to analyze homologous point matching registration. The simulation was performed using Matlab and proceeded through the following steps.

1. Landmark points in imaging space were randomly chosen in a cubic region of 400 mm on a side, centered at the origin.
2. A rotation matrix and a translation vector were chosen randomly.
3. Homologous points in surgical space were generated by transforming imaging space points with and according to the equation:

   

   Note that the scale was kept unchanged between the two spaces.

4. Normally distributed random noise with standard deviation was added to the surgical space points.
5. The homologous point matching algorithm was applied to the two sets of points to find the transformation , and the squared modulus of the registration error for 10 points from 0 to 200 mm along the x-axis was computed. The choice of the x-axis is arbitrary and of no consequence on the final result since the average result is independent of the direction of the vectors in imaging space. The procedure was repeated from step 1.
6. After a number of repetitions, the means and standard deviations of the squared registration errors were computed for several equally spaced points on the x-axis.
7. A parabola was fitted to the mean squared registration error in order to find the parameters and . The square root of these numbers is shown in reg for different numbers of homologous points and noise levels (STD). In addition, some graphs of the RMS registration error as a function of the distance from the origin are presented in simul_4_2simul_50_3. The solid line represents the parabola MS_reg_error fit to the data and the error bar length is twice the square root of the standard deviation of the squared registration error. The dashed line represents the fit of VS_reg_error.

Examination of reg and the graphs clearly shows the reduction of the RMS registration error with the number of homologous pairs observed in Neelin et al. [42]. Furthermore, it can also be observed that the variability of registrations is reduced by increasing the number of homologous pairs used.

In order to verify that and are representative measurements of the registration method, plots of the distribution of the rotational and translational errors (for 10 homologous pairs and noise standard deviation of 2 mm), are presented in hist_rothist_trans. These graphs show clearly that the mean values of the translational and rotational error have the highest probability of being observed.