Figure 430

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:

G (x, a) = d e ^ (4.23)

  • 2va2) d where D = 3, due to three-dimensional processing. The parameter 7 was introduced by Lindeberg5 (see [83-89]). This was used to define a family of normalized derivatives. This normalization was particularly important for a fair comparison of the response of differential operators at multiple scales. With no scales used, LC = 1.0. The second-order information (called Hessian) has an intuitive justification in the context of tubular structure detection. The second derivative of a Gaussian kernel at scale a generates a probe kernel that measures the contrast between the regions inside and outside the range (—a, a). This can be seen in Figure 4.30 (left). The third term in Equation 4.22 gives the second-order directional derivative:
  • I d \ ( d

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 [73]. 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

Preprocessed Volume

3-D separable Gaussian Convolution in X, Y and Z

Convolved Data

Cj3jD Ellipsoidal Direction Estimation

Direction

Nonvascular Separatjon^I> .

Filtered Volume

Performance Evaluation^—H Means, SNR/CNRs

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