Concept of second-order Gaussian derivative for edge estimation and ellipsoidal concept for direction estimation. Left: Second-order Gaussian derivative. Right: Three-dimensional ellipsoid.
where G (x, a) is the three-dimensional Gaussian kernel defined as:
The main concept behind the eigenvalue of the Hessian is to extract the principal directions in which the local second-order structure of the image can be decomposed. Because this directly gives the direction of the smallest curvature (along the direction of the vessel), application of several filters in multiple orientations is avoided. This latter approach is computationally more expensive and requires a discretization of the orientation space. If Xa,k is the eigenvalue corresponding to the kth normalized eigenvector ua,k of the Hessian H0,a computed at scale a, then from the definition of the eigenvalues:
5Known as the Lindeberg Constant (LC).
The above equation has the following geometric interpretation. The eigenvalue decomposition extracts three orthonormal directions that are invariant up to a scaling factor when mapped by the Hessian matrix. In particular, a spherical neighborhood centered at x0 having a radius of unity will be mapped by H0 onto an ellipsoid whose axes are along the directions given by the eigenvectors of the Hessian, and the corresponding axis semi-lengths are the magnitudes of the respective eigenvalues. This ellipsoid locally describes the second-order structure of the image (see Figure 4.30, right). Thus, the problem comes down to an estimation of the eigenvalues and eigenvectors at each voxel location in the three-dimensional volume. The algorithm for filtering is framed in the next section.
4.4.2 Algorithmic Steps for BBA Ellipsoidal Filtering
The algorithm we used consisted of the following steps and is in the spirit of Frangi et al.'s approach . The diagram showing the algorithm pipeline is seen in Figure 4.31, left, and discussed below as:
1. Preprocessing of the MRA data sets. This consists of changing the anisotropic voxels to isotropic voxels. We used trilinear interpolation
Scale-Space Ellipsoidal Filtering
Raw Angiographic Volume
Sine Interpolation and Down-Sampling
3-D separable Gaussian Convolution in X, Y and Z
Cj3jD Ellipsoidal Direction Estimation
Nonvascular Separatjon^I> .
Performance Evaluation^—H Means, SNR/CNRs
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