Advanced Graphics Curves

Advanced Computer Graphics I

Yalin Wang

Computer Science and Engineering,Arizona State University

Sep. 01, 2015

Y. Wang (ASU) Shape Analysis Sep. 01, 2015 1 / 34

Outline

1 Motivation

2 Curve Denitions

3 Arc Length Parametrization

4 The Frenet Frame

5 Middle Axis

6 Applications


Motivation

Curve Analysis to Understand Geometry Shapes


Motivation


Quiz

Figure: How can we discriminate these four curves?


Denition

Jordan Curve

A (parametrized plane) curve is a continuous mapping m : I ! R2,where I = [ a;b] is an interval.

The curve is closed if m(a) = m(b).

A curve is a Jordan curve if it is closed and m has noself-intersection: m(x) = m(y) only for x = y or x ;y = a;b.

The curve is piecewise C1 if m has everywhere left and rightderivatives, which coincide except at a nite number of poin ts.


Denition

Regular Curve

A C1 curve m : I ! R2 is a regular curve if _mu 6= 0 for all u 2 I. If mis only piecewise C1, we extend the denition by requiring that allleft and right derivatives are non-vanishing.

For the study of curve, it is essential that the curve is regular.

Singular point: where _m(t) = 0.


Parameterized Curve

Denition

An (innitely) differential map a : I ! R3 of an open interval I = ( a;b) ofreal line R into R3.

a (t) = ( x(t);y(t);z(t)) .

Tangent vector: _a (t) = ( x0(t);y0(t);z0(t)) .

Trace: the image set a (I) R3.


Parameterized Curve

RemarksThe map a needs not to be one-to-one.

a is simple if the map is one-to-one.

Distinct curves can have the same trace:

a (t) = ( cos(t);sin(t)) (1)

b (t) = ( cos(2t);sin(2t)) (2)


Arc Length

Denition

Given t 2 I, the arc length of a regular curve a : I ! R3, from the pointt0, is

s(t) =Z t

t0j _a (t)jdt

where

j _a (t)j =q

(x0(t))2 + ( y0(t))2 + ( z0(t))2

is the length of the vector _a (t).

Since _a (t) 6= 0, s(t) is a differentiable function of t , and dsdt = j _a (t)j.

If the curve is arc length parameterized, then dsdt = 1 = j _a (t)j.

Conversely, if j _a j 1, then s = t t0.


Curves Parameterized by Arc Length

Denition

Let a : I ! R3 be a curve parameterized by arc length s 2 I, thenumber j a j = k is called the curvature of a at s.

t(s) = _a (s)j _a (s) j.

At point where k (s) 6= 0, the normal vector n(s) in the direction of_a (s) is well dened by a (s) = k (s)n(s).

The plane determined by _a (s) and n(s) is called the osculatingplane.

Binormal vector: b(s) = t(s) n(s).


The Frenet Frame

Denition

A curve g : (0;1) ! R3 is called a regular curve, if every component is aC funciton and for all t 2 (0;1); j dgdt j > 0.We have following denitions:

t(s) = _g(s)

n(s) =1k

_t(s)

b(s) = t(s) ^ n(s)

(3)

Therefore, there is an orthonormal frame, the Frenet frame, movingalong the curve:

f g(s) : t(s);n(s);b(s)g:


The Frenet Frame

Illustration

Figure: Frenet frame for a point on a space curve

DemoFrenetFrame-source.nb.


Curvature

Denition

k (t) =jj _a (t) a (t)jj

jj _a (t)jj3

Osculating Circle

Its radius is r = 1=k and its center is

c(t) = a (t) + r (t)n(t)

DemoCircleOfCurvature-source.nb.

OsculatingCircles3D-source.nb.


Osculating Circle

Illustration

Figure: An osculating circle.


Parametric Curve Example: A Straight Line

Straight LineGiven two points, a and b.

The line connecting both points is p = tb + ( 1 t)a, t 2 [0;1].

a

b

pt

1-t

Figure: A simple parametric curve example.


Parametric Curve Example: Cubic B ezier Curves

DenitionCubic B ezier curves

x(t) = B30b0 + B31b1 + b

32b2 + B

33b3

where where Bni (t) =

ni

(1 t)n i t i , with

ni

=

(n!

i !(n i)! if 0 i n0 else:

DemoConstructionOfABezierCurve.cdf


Curvatures for B ezier Curves

B ezier CurvesAt t = 0, we have

k (0) = 2n 1

narea[b0;b1;b2]

jjb1 b0jj3

At t = 1, we have

k (1) = 2n 1

narea[bn 2;bn 1;bn]

jjbn bn 1jj3


Signed Curvature

Signed Curvature

k (t) =det [ _x(t); x (t)]

jj _x(t)jj

Demo: Evolutes of Some Basic Curves-source.nb.


Curvature Tools

max: 0.002

min: -0.038

Figure: A curve with its curvature plot


Curvature Tools

max: 0.000

min: -0.035

Figure: An improved curve with its curvature plot


Middle Axis

DenitionThe medial axis transform associates a skeleton-like structure to ashape.More precisely, represent a shape by an open connected bounded setin the plane,denoted . The skeleton of , denoted () , is the set ofall p such that, open disc B(p; r ) (where p 2 R2 is the center and radiusr > 0), is maximal in for some r > 0, i.e., () is the set of loci of thecenters of maximal discs.


2D Example

Figure: Shapes with horizontal medial axes. The shapes are obtained with2uu(r

2) = 2f , f is in the left column and the curves are in the right one.


2D Example

Figure: Shapes generated from a medial axis with three linear branches.


3D Example

Figure: Medial axis is used to analyze hippocampal morphometry.


Cell Morphometry

segmented lymphocytes0 Video frame grabslymphocyte

T X (t, s) : [0, 2 ) [0, T ] R2extracted curve sequence

A

B CX (t)normalX (t)abnormalFigure: Curve analysis to model dynamic cellular morphology.

X. An, Z. Liu, Y. Shi, N. Li, Y. Wang, S. Joshi. Modeling Dynamic CellularMorphology in Images, MICCAI 2012, Part I. 340-347


Cell Morphometry

A1

A2A3

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A6A7

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A9 A10

A11

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N22

N23N24

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N31N32

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N37N38N39N40

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MDS plot of distances between sequences (DTW)

Dimension 1

Dim

ensi

on 2

A1

A2

A3

A4

A5

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A7 A8

A9

A10

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N35N36

N37N38N39N40

N41N42

MDS plot of distances between sequences (no DTW)

Dimension 1

Dim

ensi

on 2

Figure: Shape space illustration. Projections of the pairwise distancesbetween 42 sequences without dynamic time warping (DTW) (left) and withDTW (right).


Conformal Welding

SketchSuppose = g0;g1; :::;gk are non-intersecting smooth closedcurves on the complex plane.

partitions the plane to a set of connected components 0; 1; :::; s , each segment i is a singly-connected domain.

Construct f k : k ! Dk to map each segment k to a circledomain Dk , 0 k s.

Assume gi 2 = j \ k , then f j (gi ) is a circular boundary in Djwhile f k (gi ) is another circle on Dk .

Let fi = f j f 1k : S

1 ! S1 be the diffeomorphism from the circle toitself. It is called a signature of gi .

ReferenceE. Sharon and D. Mumford. 2D-shape analysis using conformalmapping. Int. J. Comput. Vision, 70:55-75, October 2006


2D Conformal Wedling

Figure: Conformal disparity between the interior and exterior of three planarcurves.


2D Conformal Wedling Application

Figure: Planar signature.


System Overview

Figure: Diffeomorphism signature via uniformization mapping.


3D Shape Signatures

Figure: 3D shape signatures by conformal welding.


Experimental Results

Figure: Distribution of shape signatures (12 control vs 12 Alzheimers diseasepatients).

MotivationCurve DefinitionsArc Length ParametrizationThe Frenet FrameMiddle Axis

Advanced Graphics Curves

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