MAT vs. CAT

Maximal disc
        Any circle along with its interior points that is contained in the shape 
        and touches the boundary at two or more points

MAT
        Centers and radii of all maximal discs contained in the shape

Problems with MAT (Pages 5-6)
        Indirect inversion
        Nonlocal transform
                Boundary feature may induce skeletal feature far away
                Boundary feature may be exaggerated or underplayed in skeleton
        Only info about distances of shape features, none about girths or sizes
                Hard to evaluate the significance of a feature
                Difficult to prune MAT to form "basic" skeleton
        MAT does not exist for discrete shape

Requirements of skeleton for 2D planar shape:
        (i) connected 1D subset of shape
        (ii) maximally axial to shape (local axis of symmetry)
        (iii) Feature of skeleton should be nearby corresponding shape feature
        (iv) invertible
        (v) multiresolution
        (vi) applicable to discrete representations of shapes,
             coarser discretization should lead to coarse skeleton

Maximal chord of tangency
        Chord of the bounding circle of a maximal disc

CAT
        Midpoints and half-lengths of all maximal choords of tangency,
        and the centers and radii of a maximal disc with three maximal chords
        of tangency that form an acute angled triangle

Differences between MAT and CAT
        MAT is connected
        CAT is disconnected protoskeleton
                Contiguous arcs are local axes of symmetry
                Isolated points are branch points
                -> Features of skeleton correspond to features of shape
                -> Size of shape feature corresponds to size of skeletal feature
                -> Protskeleton can be connected

        MAT is nonlocal
                Each pair/triple of boundary points can lead to 
                infinite number of skeletal points (Figure 2e)
        CAT is local    
                Each pair of boundary points leads to only one skeletal point
                and it is nearby and maximally axial

	Shape of CAT degrades gracefully with coarser discrete sampling
	MAT does not 
		CAT stays along local symmetry axes
		MAT can go outside shape (Figure 3f)

        MAT is invertible
        CAT is strongly invertible
                Boundary can be recovered directly from CAT
                Each maximal chord is perpendicular to CAT axis
                Allows partial reconstruction

CAT for discrete shapes
        Connect midpoints of delauney edges
        Prune based on d/|AB|, where d is distance from furthest point to edge AB

Morphological congruency
        CAT provides attributed graph
        Grammar for shape description (J = T +2g -2)
        J = joints, S = sleeves, T = terminals