Features Extraction Extremal Points and Lines

To extract reliable curves on a surface, most approaches try to generalize the notion of "edges" to smooth surfaces to find the most salient features on the surface: ridges. Prior to the late 1980s and early 1990s, the interest in ridges was mostly theoretical, in areas of mathematics related to catastrophe theory [1,20,24,32,33] Crest lines were then defined as the cuspidal edges of a caustic surface, and the link between caustics and curvatures on a surface was established.

Practical applications were then discovered by researchers in computer vision, graphics, and medical imaging together with the specification of algorithms to extract ridges. In the following, we focus on the crest and extremal line as introduced in [25,26] and developed in [37,38]. Basically, these curves are (subsets of) the loci of the surface where one of the principal curvatures reaches a locally optimum in the associated principal direction. In these works, the crest lines are extracted using third order derivatives of the image intensities. An alternative approach was presented in [16] with the use of a B-spline approximation of the surface.

Adifferentnotionofridgesisgivenin [10,40]: They are defined as the salient structures of the intensity surface defined by I = f (x, y, z). Here, the ridges are surfaces and are more like results of the medial axis transform than the intuitive notion of salient lines on a surface. Cutting et al. [6] also developed a method using a template of ridges to assist in their extraction from image data. This method was extended by Dean etal. [7]. A good review of the many definitions of ridges can be found in [4].

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