<title>Geometric transform for shape feature extraction</title>

Abstract

A novel and efficient invertible transform for shape segmentation is defined that serves to localize and extract shape characteristics. This transform-the chordal axis transform (CAT}-remedies the deficiencies of the well-known medial axis transform (MAT). The CAT is applicable to shapes with discretized boundaries without restriction on the sparsity or regularity of the discretization. Using Delaunay triangulations of shape interiors, the CAT induces structural segmentation of shapes into limb and torso chain complexes of triangles. This enables the localization, extraction, and characterization of the morphological features of shapes. It also yields a pruning scheme for excising morphologically insignificant features and simplifying shape boundaries and descriptions. Furthermore, it enables the explicit characterization and exhaustive enumeration of primary, semantically salient, shape features. Finally, a process to characterize and represent a shape in terms of its morphological features is presented. This results in the migration of a shape from its affine description to an invariant, and semantically salient feature-based representation in the form of attributed planar graphs. The research described here is part of a larger effort aimed at automating image understanding and computer vision , 6

Rectification of the chordal axis transform skeleton and criteria for shape decomposition

“…8). Henceforth, we will restrict ourselves to the structure of the CAT skeleton in the rest of the paper and direct the interested reader to [11] and [17] for other details and implications of the CAT.…”

“…Both these methods, however, require uniform or well-sampled representations of the shape or shape boundary to yield satisfactory skeletons. In [9][10][11] we proposed the chordal axis transform (CAT) as a more stable definition of the skeleton of a 2D shape that is robustly extendable to sparse and uneven discrete samplings of shape boundary, and is easy to compute. Since then it has gained currency among some researchers in the area of 2D shape analysis and 3D shape modeling [12][13][14][15].…”

<title>Geometric morphology of cellular solids</title>

“…5, we transform through the CDT the spectral pixel information of our given micrograph into an affine geometric description. Our previous 4,5 and ongoing 11 work on 2-D shape analysis has amply demonstrated the value of Delaunay triangulations in obtaining structurally meaningful decomposition of shapes into simpler components, much along the lines the human visual system parses complex shapes. 2-D shapes can be decomposed into limbs and torsos 4,5 , which are generic shape components.…”

“…A "limb" is a chain complex of pairwise adjacent triangles, which begins with a junction triangle and ends with a termination triangle. A "torso" is a chain complex of pairwise adjacent triangles, which begins and ends with a junction triangle 4,5 . In Fig.…”

Practically oriented parallel Delaunay triangulation in E2 for computers with shared memory

“…Due to these good properties, Delaunay triangulation is used in many areas such as terrain modeling (GIS) [5], scientific data visualization [6][7][8][9] and interpolation [10], robotics, pattern recognition [11,12], meshing for finite element methods (FEM) [13][14][15], natural sciences [16,17], computer graphics and multimedia [18,19], etc.…”