An Analysis of Vessel Enhancement Filters Based on the Hessian Matrix for Intracranial MRA

Introduction Vessel enhancement filters applied to 3D MRA data sets prior to rendering as a 2D image may improve visualization of vessel detail. We previously compared several line enhancement filters for intracranial 3D MR angiography images[1]. We examined filters based on discrete lines (e.g., Du and Parker's filter [2]) and on the Hessian matrix (Frangi [3]). We found the Du and Parker filter to provide the best discrimination between vessel and background tissue. However, the continuous directional properties available from using the Hessian matrix are important features not available from discrete kernels. As such we have made further investigations into the Hessian matrix as a basis for vessel enhancement filtering. In this abstract we describe a detailed analysis of Frangi's filter applied to intracranial TOF MRA images. We propose an alternative filter, based on differences in the eigenvalues of the Hessian matrix (HessDiff). Methods The Frangi filter provides a measure of “vesselness” ( V) based on the eigenvalues of the Hessian matrix. Let l1l2 and l3 be the eigenvalues of the Hessian matrix sorted by increasing magnitude (|l1|, |l2| and |l3|). For l2 or l3 greater than zero V is set to zero otherwise V=(1-exp(-Ra2/ 2a2))exp(-Rb2/2b2)(1-exp(-S2/2c2), where Ra is the ratio |l2| / |l3|, Rb is the ratio of |l1| to the geometric mean of |l2| and |l3|, S is the summation of|l1|, |l2| and |l3|, and a, b, and c are normalizing constants taken to be 0.5, 0.5 and 0.5max(S) respectively. The HessDiff filter is a generalization of Du and Parker's filter that uses the continuous directionality properties of the Hessian matrix. As with Frangi's filter, we solve for the eignevalues of the Hessian matrix and sort them according to magnitude. For the filter response to be nonzero we again require that both l2 and l3 be less than zero (for bright blood) otherwise we take the filter response (V) to be simply V=|l3|-|l1|. This is a natural extension toDu and Parker's filter. For both Frangi's filter and the HessDiff filter an identical multiscale implementation was used. Successfully larger Gaussian functions for the filter kernels were used. Scales were grown linearly. The maximum response over all the scales was selected at each voxel.