Description and Quantification

In the previous sections, the enhancement of the local structures based on the eigenvalues of the Hessian matrix was discussed. In this section, we further combine the gradient vector with the Hessian matrix to perform explicit detection, localization, and description of the local structures. Especially, we focus on the line and sheet structures. The methods are formulated as a 3-D extension of

2-D line description presented in [36]. The 3-D line model consists of the medial axes of lines and the cross-sectional shape associated with each point on these axes, while the 3-D sheet model consists of the medial surfaces of sheets and the width associated with each point on these surfaces. The medial axes and medial surfaces are detected and localized by fully utilizing formal analyses of

3-D second-order local intensity structures based on the gradient vector and the Hessian matrix.

The following is an overview of the method:

Step 1: Existing filtering techniques for line and sheet enhancement are used to extract the initial regions, which should include all potential medial axes and surfaces [7,11]. These are then used as initial values for the subsequent subvoxel edge localization. The candidate regions, which should include all potential line and sheet regions, are also extracted.

Step 2: The medial axes and surfaces are extracted using local second-order approximation given by the gradient vector and Hessian matrix. The eigenvectors of the Hessian matrix define the moving frames on medial axes/surfaces. After this, the moving frames are embedded in a 3-D image such that each point within the candidate regions is directly related to its corresponding moving frame.

Step 3: Subvoxel edge localization of the region boundaries is carried out using adaptive 3-D directional derivatives, whose directions are adap-tively changed depending on the moving frame, to accomplish accurate segmentation, model recovery, and quantification.

In the following, we begin with a description of Step 2 of the method since Step 1 has been already described in the previous sections.

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