A taxonomy of boundary descriptions

Transcription

1 A taxonomy of boundary descriptions (1) Model the best fit of a simple mathematical object Conics, conic splines, log-spirals, and other geometric objects Polynomials Circular functions (2) Redescribe the boundary in a different form Radial functions Tangent-angle functions Medial axis transforms (3) Decompose the boundary into independent components Classical Fourier functions Elliptical l Fourier functions Wavelets

2 Radius functions Closed boundaries are difficult to deal with mathematically. Radius functions of closed outlines: Provide method for converting boundaries into mathematical functions. Boundaries then more easily analyzed and compared. Redescription: no loss of information.

3 Boundary points, connected by linear segments Radius function sampled at points Radius function resampled by equal angular arcs Ra adius Ra adius Theta Theta Ra adius Ra adius Distance along boundary Distance along boundary

4 Radius function: (1) Choose a starting point on the boundary. For single form, choice is arbitrary. For set of forms, the starting points must be comparable (homologous) by some criterion. (2) Choose a center point (focus) within the form. For single form, choice is arbitrary. For set of forms, the center points must be comparable (homologous) by some criterion. i Might be chosen geometrically, e.g.:» center of area (centroid).» center of mass (weighted centroid). Better: might be provided by biological forms:» e.g., growth centers of bones and scales.

5 Radius function: (3) Radius function generated by either: (a) Varying the central angle counterclockwise: Begin at boundary point. Measure the radius at each angle. (b) Varying the distance along the perimeter: Begin at boundary point. Measure the radius at each distance. r r p θ

6 Radius functions for skulls of fossil temnospondyl amphibians (Sengupta et al., 2005):

7 Note: a radius function can be calculated with respect to a line as well as a point. E.g., pennate diatoms:

8 Uses of radius functions: (1) Can be standardized and compared Radius Theta

9 Uses of radius functions: (2) Can be used to average and reconstruct forms. Mean form Reconstructed average form Radius Mean radius Theta Theta

10 Superimposed midsagittal contours of chimpanzee skulls, and mean contour (Jacobshagen 1997):

11 Uses of radius functions: (3) Can be sampled at regular intervals for multivariate analysis Theta (4) Can be decomposed into orthogonal components using classical Fourier functions or other mathematical tools. (5) Can be extended to 3D. dius Rad Problems with radius functions: Highly dependent on positions of center and starting points. Depends on projection for 2D analysis of 3D objects.

12 A taxonomy of boundary descriptions (1) Model the best fit of a simple mathematical object Conics, conic splines, log-spirals, and other geometric objects Polynomials Circular functions (2) Redescribe the boundary in a different form Radial functions Tangent-angle functions Medial axis transforms (3) Decompose the boundary into independent components Classical Fourier functions Elliptical l Fourier functions Wavelets

13 Tangent-angle functions Tangent angle: the counterclockwise angle between a tangent and a standard reference line. Q P θ Q θ P Reference line

14 Tangent-angle functions of closed outlines: (1) Choose a starting point on the boundary. For a single form, choice of the point is arbitrary. For sets of forms, the points must tbe comparable (homologous) by some criterion. (2) Find the tangent line to the boundary at that point. Calculate angle (=tangent angle) between the tangent line and the reference line. (3) Move counterclockwise along the boundary. At each point, find tangent-line and calculate angle. (4)E Express tangent tangle as a function of fdistance along the boundary.

15 Tangent-angle g function of a circle: 6 5 le Tangent-ang Distance along boundary

16 Tangent-angle function of closed forms: 5 Tangent-angle e Distance along boundary 5 4 le Tangent-ang Distance along boundary

17 Uses of tangent-angle g functions: (1) Can be standardized and compared. (2) Can be averaged and reconstructed. (3) Can be sampled at regular intervals for multivariate analysis. (4) Can be decomposed into orthogonal components using tangential Fourier functions or other mathematical tools. (5) Can be extended to 3D. (6) Serve as the basis for eigenshape analysis.

18 Comparison of radius functions and tangent-angle functions: Capture basically the same information. Little loss of information. Forms can be reconstructed from both kinds of functions. Tangent-angle functions require smooth boundaries. No corners or cusps. Radius functions can describe any kind of boundary. Tangent-angle angle functions don t require an interior center point. Radius functions depend on choice of center.

19 A taxonomy of boundary descriptions (1) Model the best fit of a simple mathematical object Conics, conic splines, log-spirals, and other geometric objects Polynomials Circular functions (2) Redescribe the boundary in a different form Radial functions Tangent-angle functions Medial axis transforms (3) Decompose the boundary into independent components Classical Fourier functions Elliptical l Fourier functions Wavelets

20 Medial axis transform = Medial axis, = symmetric axis (Blum & Nagel 1978). Used to define the middle of a form of arbitrarily complex shape. Decomposes form into two independent (orthogonal) components: (1) Line skeleton, a geometric object. (2) Set of corresponding radius functions. Therefore is a kind of orthogonal decomposition. The two components can be recombined to reconstruct the form. No (or little) loss of information.

21 Medial axis defined in two equivalent ways: (1) Quenching function: Grass-fire metaphor. Propagation pg from boundary inward at a constant rate. Axis defined by points where propagations from opposite directions meet. (2) Locus (collection) of centers of maximal discs inside the form.

22 Decomposes form into two components: (1) Line skeleton: Set of curves passing along middle of form in precisely defined manner. Provides a graph of the skeleton of the form. Triple-points: points: Points at which the line skeleton branches. Can be used:» To subdivide form into regions.» For quantification: lengths of line segments between triple-points; angles between corresponding line segments around triple-points.» As internal pseudolandmarks provided by the shape of the boundary.

23 Decomposes form into two components: (2) Radius function(s): Each segment of the line skeleton has a corresponding radius functions. For each point on the line skeleton, records the distance to the nearest point on the boundary. = Radius of corresponding maximal disc.

24 Medial axis

25 Medial axes

26 Biological medial axes Human mandibles (Bookstein) Baculus shape in bats (Straney) Mapping onto phylogenetic tree Growth series

27 Medial axes Ge & Stelts : central path of the human colon. Created a movie that follows this path through the colon.

28 Medial axes Used extensively in pattern recognition and computer graphics to recognize, characterize, and compare complex shapes.

29 Medial Surface Medial axes are extensible to 3D objects: Hassouna and Farag