A taxonomy of boundary descriptions

Transcription

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

2 Polnomials Functions of the form: 1 2 f a a a a n a a1 a2 an n n Degree of Name Number of coefficients polnomial (descriptors) 1 Linear 2 2 Quadratic 3 3 Cubic 4 4 Quartic 5 5 Quintic 6 etc. etc. etc.

3 Effect of adding terms to a polnomial: = 3 1 = = = = =

4 Fitting polnomials to a set of boundar points:.7.25 = = = = = =

5 When fitting polnomials to scatters of data: Regular polnomials: As new terms are added to model, eisting coefficients change. Orthogonal polnomials: As new terms are added to model, eisting coefficients i remain the same.

6 E: Cifelli (1996): upper dentitions of phllostomid bats. Upper dentitions Tooth cusps Interpretation? PCA of coefficients

7 E: Rødven et al. (26): dental wear in reindeer. Reindeer teeth Regression of goodness-of-fit against wear 2 nd degree polnomials

8 Interpolation with polnomials and cubic splines:

9 Fitting polnomials to forms: (1) Eactl fit an n th -degree polnomial lthrough h n+1 points. Provides n+1 characters ( a, a1, an ). (2) Fit a smaller-degree polnomial through the same number of points b least-squares regression. (3) Polnomial splines: (a) Linear splines: Connect consecutive points (knots) with straight-line segments (polgons). Continuous but not differentiable. (b) Quadratic splines: Join knots b constraining tangents (1 st derivatives). (c) Cubic splines: Join knots b constraining:» Identical tangents (1 st derivatives).» Identical curvature (2 nd derivatives). Constrain 2 nd derivatives to vanish at open endpoints.

10 Can reparameterize polnomials as functions of distance along the boundar: f t, f t Permits curves to be free of orientation. Relativel eas to produce 2D splines. Etends to 3D. t 9 t 8 t 7 t 6 t 5 t 4 t t 3 t 1 t 2

11 Most common polnomial splines: Quadratic (Oberhauser): 33 parameters for 11 segments Smooth bulbous transitions between points Cubic: Tighter curves 44 parameters

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

13 Circular functions = Periodic functions, polar functions. Used with polar coordinates (r, θ) rather than rectangular (cartesian) coordinates: r cos r sin

14 Circular functions: Involve trigonometric functions as a function of the angle (in radians) about a central point: Sines, cosines, tangents, etc. sin( ) cos( )

15 Produce wonderful figures. Difficult to fit to real forms. (D Arc Thompson, 1917)

16 Supershapes Gielis (23) etended periodic functions to include supertrigonometric functions: Circles and squares, ellipses and rectangles, spirals. Asmmetrical forms. r Generalized 6-parameter model: 1 f 1 m 1 m cos cos a 4 b 4 n n 1 n Can be statisticall fitted to actual forms. Represents a 6-dimensional morphospace.

17 Supershapes:

18 Supershape model developed for stud of plant forms. Developmental rules. Biophsical optimization. Can be etended to stud other radial forms: Echinoderms: Snowflakes:

19 Single forms: Modeling sets of forms Description is arbitrar: conic splines, polnomial splines, circular functions, etc. Sets of forms: All components must be strictl comparable (i.e., homologous) among forms, in terms of: Number and placement of points. Kinds of mathematical objects. E.g., parabolic arc, 2 nd degree polnomial, etc. Bottom line: For parameters to be treated as characters, must correspond eactl in kind and number. Aim: to get efficient description, as complete as possible, with ihfewest practical number of characters.