Differential Geometry

Computer Vision #8

Differential Geometry

1. Curvature of curve

2. Curvature of surface

3. Application of curvature

Parameterization of curve

1. curve -- s arc lengtha(s) = ( x(s), y(s) )

2. tangent of a curvea’(s) = ( x’(s), y’(s) )

3. curvature of a curvea”(s) = ( x”(s), y”(s) )

|a”(s)| -- curvature

s

a(s)

ssa )('

)( ssa )(sa

)(' ssa )(' sa

)(' sassa )("

)(' ssa

Example (circle)

1. Arc length, s

2. coordinates

3. tangent

4. curvature

rs

rsrryr

srrx sinsincoscos

rsrr

srsysxsa sin,cos,

rs

rssa cos,sin'

sarrrsrrssa 21sin,cos"

rsa 1"

xrs

y

sa' sa

S

Definition of curvature

The normal direction (n) toward the empty side.

snsksa "

Corner model and its signatures

ba

cd

s=0

a b c

ab c d a

arc length

d

ab

c

s=0

s=0

Gaussian filter and scale space

a

a

a

bc d

e

b

c

d

e

f

g

h

i j

k

+

+

+

Curvature of surfaces

normal section non-normal section

normal curvature

Principal directions and principal curvatures

Principal curvaturesplane: all directions

sphere: all directions

cylinder:

ellipsoid:

hyperboloid:

021 KK

021 KK

00 21 KK

00 21 KK

00 21 KK

Gaussian curvature and mean curvatureKKK 21 curvatureGaussian

HKK

2 curvatureMean 21

0

0

2

1

K

K

0

0

2

1

K

K

0

0

2

1

K

K

0

0

2

1

K

K

0

0

2

1

K

K

0

0

H

K

0

0

H

K

0

0

H

K

0

0

H

K

?0

H

K

Parabolic points0K Parabolic point

0K elliptic point

0K hyperbolic point

F.Klein used the parabolic curves for a peculiar investigation. To test his hypothesis that the artistic beauty of a face was based on certain mathematical relation, he has all theparabolic curves marked out on the Apollo Belvidere. But the curves did not possessa particularly simpler form, nor did they followany general law that could be discerned.

Lines of curvature

Principal directions, whichgives the maximum and the minimalnormal curvature.

Principal direction

curves along principal directions

PDPD

PD

Lines of curvature

Curvature primal sketches along lines of curvature

Important formula1. Surface

2. surface normal

3. the first fundamental form

4. the second fundamental form

yxzyx

vuzvuyvuxvuX

,,,

,,,,,,

vu

vu

XX

XXvuN

,

vvvuuu XXGXXFXXE

vvvv

vuuvuv

uuuu

XNXNg

XNXNXNf

XNXNe

uX

vX

N

2

2

2

2

0

22

2

2

1 curvatureMean

curvatureGaussian

area

2length arc

FEG

gEfFeGFEG

feg

dudvFEG

dvGFdudvduES

Y

Z

X

cos,sinsin,cossin

,,,,,,

rrr

ZYXX

0,cossin,sinsin

sin,sincos,coscos

rrX

rrrX

22

222222

22

22222

22222222

sin

cossinsinsin

0

cossincossincossincossin

sincos

sinsincoscoscos

r

rrXXG

rrXXF

rrr

rrrXXE

XX

XXN

kjir

rr

rrr

kji

XX

cossinsincossinsin

0cossinsinsin

sinsincoscoscos

2

cos,sinsin,cossinN

0,sinsin,cossin

0,coscos,sincos

cos,sinsin,cossin

rrX

rrX

rrrX

22222

22222

sinsinsincossin

0cossincossincossincossin

cossinsincossin

rrrXNg

rrXNf

rrrrXNe

rrrr

r

FEG

gEfFeG

rrrr

r

FEG

feg

drrdddr

ddFEG

drdr

dGdFddE

11

2

11

sin2

sin21

2

21 MC

111

sin

sin GC

sinsin

area

sin

2lenght arc

24

23

2

224

22

2

2

2

2

22222

22

rd

dr sin

r

1

Summary

1. curvature of curve

2. curvature of surface– Gaussian curvature– mean curvature

Surface Description #2(Extended Gaussian Image)

Topics

1.Gauss map

2.Extended Gaussian Image

3.Application of EGI

Gauss map

Let S R3 be a surface with an orientation N.⊂The map N: S→R3 takes its values in the unit sphere

The map N: S→S3 is called the Gauss map.

1Dgauss map

gauss map 2D

zyxNzyxRzyxS ,,,1;,, 22232

Characteristics of EGI

EGI is the necessary and the sufficient condition for t

he congruence of two convex polyhedra.

Ratio between the area on the Gaussian sphere and th

e area on the object is equal to Gaussian curvature.

EGI mass on the sphere is the inverse of Gaussian cur

vature.

Mass center of EGI is at the origin of the sphere

An object rotates, then EGI of the object also rotates.

However, both rotations are same.

Relationship between EGI and Gaussian curvature

object Gaussian sphere

large small

small

large

(K: small)

(K: large)

O

SK

O

0lim

O

S

small

O

S

large

O

O

S

S

Gaussian curvature and EGI maps

Since and exist on the tangential plane at , we can represent them by a linear combination of and

dudvXX

dvXduXO

vu

vu

dudvNN

dvNduNS

vu

vu

vuv

vuu

XaXaNXaXaN

2212

2111

vuvu XXaaaaNN

)( 21122211

2212

2111detaaaa

XX

NN

O

S

vu

vu

uN

vN

X

uX

vX

dvNN v

duNN u

N

v

u

duX u

dvX v

dudvduNu

dvNv

GFFE

aaaa

gffe

2212

2111

FaEaeXXaXXaXN vvuuuu

2111

2111

FaEafXXaXXaXN uvvuuv

2212

2212

GaFafXXaXXaXN vvvuvu

2111

2111

GaFagXXaXXaXN vvvuvv

2212

2212

kFEG

fegaaaa

O

S

2

2

2212

2111det

k

1 map

mass

EGI

EGIS

O

Implementation of EGI

Tessellation of the unit sphere•all cells should have the same area

have the same shape occur in a regular pattern

•geodesic dome based on a regular polyhedron

semi-regular geodesic dome

Example of EGI

Cylinder Ellipsoid

side view

top view

Determination of attitude using EGI

viewing direction

EGI table

10 20 0

0

08

85

5

The complex EGI(CEGI)

Normal distance and area of a 3-D object are encoded as a complex weight. Pnk associated with the surface normal nk such that:

k

kn

kAn

kdk

kk

jdeAP nn 0

(note: The weight is shown only for normal n1 for clearly.)

The complex EGI(CEGI)

1n

6n3n

2n5n

4n

2n5n

4n

1n

6n

3n

1d

1A1

1jdeA

Gauss mapping

Origin

(a) Cube (b) CEGI of cube

Bin picking system based on EGIPhotometric stereo

segmentation

Region selection

Photometric stereo

EGI generation

EGI matching

Grasp planning

Needle map

isolated regions

target region

precise needle map

EGI

object attitude

Lookup table for photometric stereo

Calibration

Hand-eye calibration

Photometric　 Stereo　 Set-up

Bin-Picking　 System

Summary

1. Gauss map2. Extended Gaussian Image3. Characteristics of EGI

congruence of two convex polyhedraEGI mass is the inverse of Gaussian curvaturemass center of EGI is at the origin of the sphere

4. Implementation of EGITessellation of the unit sphereRecognition using EGI

5. Complex EGI6. Bin-picking system based on EGI7. Read Horn pp.365-39 pp.423-451