Conclusion

We have revealed that there exists an upper bound of the sampling interval of the scanning parameter in the Hough Transform by the analysis of the general lines which have 2-dimensional parameters. We have also derived and proposed the method for finding the upper bound.

We have derived the upper bounds of the sampling intervals for two famous parameterizations (a-b and $\rho$-$\theta$) which are often used for line detection. The sampling interval for a-b parameterization agrees with the one proposed by Svalbe. The condition for the sampling interval in $\rho$-$\theta$ parameterization also agrees with the condition which Van Veen et al. showed. We have made it clear that these conditions proposed previously also guarantee the precision of the Hough Transform.