Efficient computation of Hessian-based enhancement filters for tubular structures in 3D images

This work presents guidelines for a computationally efficient implementation of multiscale image filters based on eigenanalysis of the Hessian matrix, for the enhancement of tubular structures. Our focus is the application to 3D medical images of blood vessels. The method uses matrix trace, determinant and sign to discard voxels unlikely to belong to vessels, prior to the calculation of the Hessian eigenvalues. As example of time savings, we provide results obtained in four computed tomography datasets (300 × 300 × 300 voxels) containing coronary and pulmonary arteries. The test based on the Hessian trace avoided the computation of the eigenvalues in half of the voxels on average, while the test combining the Hessian determinant and sign eliminated up to 10% additional voxels. The actual time savings depend on the algorithm used to compute the eigenvalues for the remaining voxels. With a very fast algorithm using a closed-form solution, the computational time was reduced from 20.5 to 12.5 seconds per scale, but the time gained thanks to the more complex of the two tests was negligible. However, this fast algorithm is prone to numerical instabilities. Accurate computation of the eigenvalues requires the use of iterative or hybrid algorithms. In this case, both tests produce time savings and the computational time can be reduced by several minutes per scale.