Estimating Curvature on Triangular Meshes

8/2/2019 Estimating Curvature on Triangular Meshes

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TRIA LAR ME HE

. .

Department of Computer Science and Engineering.

P R E S E N T E D B Y :P A R D E E P K U M A R

C I V I L A N D E N V I R O N M E N T A L E N G I N E E R I N G D E P A R T M E N T

,


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Curvature


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Parametric Representation of Surfaces

Curved sur ace described b a ol nomial in terms

of two parameters u and w,

=

Jacobian of curved surface

, , , , , ,

( )

( )

( )

( )

( )

( )

, , ,, ,

, , ,x y z

y z z x x yJ J J

u w u w u w

( ),y z

y z u uwhere

=,

w w


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Assum tion: Three Jacobian are not all zeros at

same time2 2 2

0J J J+ +

This assumption makes sure that curved surface: Do not degenerate to a point or a curve, and

Do not contain any singular points such a spikes.


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u-curve = P u w

w-curve = P(u0,w)

( ) ( ) ( )0 0 0, , ,x u w y u w z u w

( ) ( ) ( )0 0 0

, ,

, , ,

ou uu u u

x u w y u w z u w

=

u

Tangent Vectors

at pointP(u0,w0) , ,ow w

w w w=

w


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Parametric Representation of Surfaces( ) ( ) ( ) ( ) ( ) ( ), , , , , ,y u w z u w x u w z u w x u w y u w

( ) ( ) ( ) ( ) ( ) ( )

, ,

, , , , , ,o

o

u uw w

y u w z u w x u w z u w x u w y u ww w w w w w =

=

=

u wP P

( )

( )

( )

( )

( )

( )

, , ,, , , ,

, , , oooo

u ux y zw wu u

w w

y z z x x yJ J J

u w u w u w===

=

= =

u wP P

or non-zeroJacobian conditionto be true we must

have cross productabove to be nonzero.


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Curve in arbitrary direction on curved surface can be

d du dw= +

u w

PP P

expressed in terms of parameter t as P(u(t),w(t)). Tangent atP(u0 ,w0)

0 0 0t t t t t t = = =


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u w u wP P P P

( )2 2 2 2

x y zJ J J+ +u wP P


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Equation of Tangent Plane R be position vector on tangent plane

( )( ) ( )( ) ( )0 0 0 0, . 0, , . 0u w u w= =u wR - P e R - P P P


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Unit Tangent Vector

) ) ) ),Define t u t w t and curved surface t= u P u

[ ]Tangent vector u w u w = + = u

u w

w

PP P P uA

P

( ) ( ) ( )22

,T

agnitude s of tangent vector s t t t = =

2

P P P

[ ] [ ] [ ]2

2

s u w u ww w

= =

=

u u u w

u w 2

w w u w

P PP P P P

T =

( )1

, .

,t

Unit Tangent Vector = =

P uAt

2T uF u


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Curvature of a Surface

: ' . .d d dt

Lets now use length s as parameter = =

P P PP

, . ' 1.

'

s t s s

Length of tangent s

Thus Unit Tan ent Vector

= =

=

2P P

t P

( ) ( )2 0 02

0

' 'lim .

s

s s sd

ds s

+

P PPP" =

( )0 0

1lim lim

0s s

s

ss s

= = =

P"

( )0s

s + s

=

=

P" n = t'

P = t t2s + st t'

2s + s = P t n


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Curvature of a Surface

( ) ( ) ( ) ( )( ),Lets find fo r curve t u t w t on curved surface t=

P u P u

2 2

,u w

d du u w w u uw u w wu w

dt dt

+

+ + + = + + + + +

u w

u w

u w uu uw u w w w u w

=

P PP = P P P P P P P P

,Taking inner product of unit normal with both the expressions we get

=

e

e. P

2 2s + s s =

0

e. t e. n n.e

( ) ( ) ( )

( ) ( ) ( )2 2u uw u w wu w= + + + + +

uu uw u w w w u w

0 0

e. P e.P e.P e.P e.P e.P e.P

[ ]

2

T

T

u ww

Equating two we get s

= =

=

uu uw

w u w w

. .uG u

e.P e.P

n.e uG u

Second Fundamental Matrix of Curved S=G urfaces


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Normal Curvature

Space curve u on curved surface P(u,w). urve : n ersec on o curve sur ace w p ane con a n ng

tangent vector and unit normal at P. Curvature of C is called normal curvature relative to (du/dt)A

. Normal curvature is projected length of curvature vector of

curve u to e .

2

,n

T

Lets be normal curvature

2

n

T T

n T = =

u G u u G u

s u u


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Normal Curvature

is calledPr i n c ip l e D ir e ct io n o f N o r m a l Cu r v a t u r e .

, , , ,Let u w L M N = = = = = = uu uw w u w we.P e.P e.P e.P

2 2 2 2

, . , .T

n T

T T

E F G Then can be written as= = = =

2 2

u u w w

uG uP P P P

uF u

n n


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Normal Curvature

0 0,n nTo find extreme values set and to get

= =

( ) ( ) ( ) ( )0, 0:

n n n nL E M F M F N G

Elimin ating and from th ese equations gives

+ = + =

( ) ( ) ( )2 2 2

max min

2 0

, &

n n

n n

EG F EN GL FM LN M

Roots of this quadratic equations are always real an

+ + =

d

.

are maximum and minimum values of normal curvature called

the principle curvatures

max min

max min

, n n

n n

Total Curvature or Gaussian Curvature K

= =

+

F

,2

=


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Objective

urvature

EstimationMethods forTriangular Meshes

curvature

curvaturedirections

Set of Test Cases


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Curvature Estimation Methods

Curvature Calculation Methods for Triangular

Discrete Estimation

Fitting Methods

CurvatureDirections

Curvature tensor


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Goal of Test Cases

MeshRegularity

Noise in Mesh

Accuracy

and Stabilityof CurvatureEstimation


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Introduction

Set o test cases to model mesh variations was

presented. Accuracy of curvature calculation methods for

triangular meshes was assessed based on noise in

the data, mesh resolution, regularity and valence. tat st ca ana ys s was nc u e to a ress

different aspects of curvature estimation error.


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Curvature Estimation for Triangular Meshes


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Basic Approaches

Sur ace ittin usin anal tic unction that its the

mesh locally. Discrete estimation of curvature and curvature

directions.

Estimation of Curvature tensor from whichcurvature an curvature rect ons can ecalculated.


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Fitting Methods

Fittin methods var de endin on t e o anal tic

function chosen for the fitting. Function can be parametric or implicit.

F t unct on separate y at eac vertex.

Picking a local coordinate frame is useful.

.

Minimum number of vertices are picked depending onnumber of coefficients of the function being fit.


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Fitting Methods

Local Coordinates Origin at target vertex i.e. vertex where the curvature is being

calculated. Normal at target vertex is one of the axis of the system.

Normal can be calculates directly from the analytic function used.

Average of unit normals of triangular faces that surround target

vertex.

Parameterization Re resent sur ace as unction o two arameters u v).

F(u,v) = [x(u,v), y(u,v), z(u,v)] Simplest representation-Height function

F(u v) = [u v (u v)]


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Fitting Methods

Re resentin curved sur ace: F(u,v = [u, v, (u,v)]

Parametric coordinates can be found by projectingthe vertices onto the tangent plane.

Quadric Fitting

Cubic fitting with normal

( ) 2 2, ,i i i i i i i i iz f u v Au Bu v Cv Du Ev F= = + + + + +

Main focus is to calculate curvature directions.

Implicit conic functions

0i i i i i i i i i i i i

ax by cz dx y ey z fz x gx hy jz k + + + + + + + + + =


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Quadratic Fitting Methods

Representing curved surface: F(u,v) = [u, v, f(u,v)]

If origin is located at target vertex constant term can be( )

2 2

, ,i i i i i i i i iz f u v Au Bu v Cv Du Ev F= = + + + + +

roppe .

Dropping the linear term forces the normal to line up with

- .

( )

)

0, 01

, 0, 0

u

u vv

fF F

Unit normal vector at origin e f

= =

( ) ( ) ( )( ) ( ) ( )

2

22 2

0, 0 0, 0 1 1

0, 0 0, 0 0, 0,

u v u v

uu vv uv

F F f f

f f fGaussian Curvature K

+ +

= =

G

, ,u v+ +


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Quadratic Fitting Methods

Quadratic surface passing through the origin

Requirement that it passes through 5 points can be expressed bylinear system:

2 2z Au Buv Cv Du Ev= + + + +

( ), , , 1, 2, ..5i i iU V Z i =

1

2 2

1 1 1 1 1 1 2

3... ... ... ... ...

ZA

U U V V U V Z B

ZC

=

2 2

5 5 5 5 5 5 4

5

U U V V U V Z D

ZE

t s system as un que so ut on, t e un t norma an curvature atorigin are given by2

1D

AC BE and

( )2 2 2 21 11D E D E+ + + +


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Discrete Methods

Avoid computational cost associated with fittingalgorithms.

o no nvo ve so v ng e eas square pro ems.

Only provide subset of gaussian, mean and.


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Discrete Methods

Spherical Image Method

Angle Excess Method An le De icit Method

Integral of Absolute Mean Curvature Method


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Spherical Image Method

Non planar polygon P .

Spherical image isapproximated by spherical

polygon ni.

Curvature at O,( )1i o i

i

Ar n n n

+

=

1i i

iAr P OP +


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Angle Deficit Method

Non planar polygon Pi.

Spherical image isapproximated by spherical

i,i+1.

Join points on spherical

ima e with arcs. Area of spherical polygon is

angle deficit of polygon,

Area related to target point O, 1i i +

, 13

i i

i

S +


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Angle Deficit Method

, 12

i i +

Gaussian Curvature at O,, 1

1

i

i i

KS +

= i


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Discrete Methods

Integral Formulation

Per Face Tensor Calculation


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Evaluations


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Evaluations

Test cases to hi hli ht both detailed behavior o

curvature estimation methods and statistical erroranalysis.

Detailed behavior test case defines meshparameters to distinguish between noise and

.

Statistical analysis test case creates meshes,

regular and irregular mesh regions.


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Test Cases

2 2 2: 4Sphere x y z+ + = 2 2: 2Elliptical Paraboloid z x y= +

2 2 2

:

: 13 2 4

y n er x z

x y zEllipsoid

+ =

+ + =

2 2: 0.4 -Hyperboloid z x y=


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Test Cases

2 2

3 2

: 2Elliptical Paraboloid z x y= +

( )3 2 2

. -

: 0.15 2 - 2

on ey a e z x xy

Cubic Polynomial z x x y xy y

=

= + +

[ ]0.1 cos( ) cos( )

Trignometric Function

z x y = +

22 -

0.1x y y

Exponential Function

z e+=


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Detailed Mesh Parameters

To assess local curvature at a oint on sur ace.

Project the planar mesh onto surface of study. Center the mesh at tar et oint.

Mesh is regular N-ring neighborhood, N = {1,2,3}


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Detailed Mesh Parameters

1. Number of vertices in first ring, n.

2.The scale,

.

3. Target vertex displacement, dRT.

4. Adjacent vertex displacement, dRA.

5. Target vertex displacement, d

T,owar s a acen ver ex.

6. Adjacent vertex displacement, dA,.

7. Adjacent vertex displacement, d,towards a neighboring adjacent vertex.


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Statistical Analysis Case

Create a mesh1. 72 interior vertices (112 total),

2. Valence ranging from three to ten, and

3. onta n ng ot o tuse an non-o tuse tr ang es.


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Experimental Results


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Curvature Estimation Based on Fitting


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Curvature Estimation Using Discrete Methods


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Curvature Estimation Using Curvature Tensor


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Statistical Analysis Results

Considering both Gaussian and Mean Curvature,most accurate methods:1. The cubic fit with exact normal

-. ,

3. Two-ring conic fit,

4. Two-ring quadric natural fit, and

5. u c t w t ca cu ate norma .

-

curvature magnitude, where as two ring conic fitpredictions are larger.


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Discussion


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Discussion

Accuracy of conic fitting methods is very dependent ontype of surface being fit.

Two ring fitting methods are superior than one ringme o s.

One ring methods are highly noise sensitive.

,using weighted face average in quite sensitive.

Princi al curvature directions are more stable and less

sensitive to mesh regularity than curvature magnitude. Fitting methods based on two ring neighborhood is

recommen e t an t ree r ng, to avo ncrease cost.


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Estimating Curvature on Triangular Meshes