Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Clif

ford

Con

volu

tion

and

Fou

rier

Tra

nsfo

rm o

n V

ecto

r Fi

elds

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Uni

vers

ity o

f Lei

pzig

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Flow

Fea

ture

Det

ectio

n U

sing

Imag

e Pr

oces

sing

?

Pat

tern

mat

chin

g in

tuiti

ve

Con

volu

tion

robu

st in

term

s of n

oise

Ana

lysi

s of f

ilter

beh

avio

ur

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Con

volu

tion

and

Four

ier

Tra

nsfo

rm o

n V

ecto

r Fi

elds

Con

volu

tion

of e

ach

coor

dina

te se

para

tely

[Gra

nlun

d, K

nuts

son

1995

]

Sca

lar p

rodu

ct in

con

volu

tion

[Hei

berg

200

1]

Clif

ford

Con

volu

tion

[Ebl

ing,

Sch

euer

man

n 20

03]

Clif

ford

Fou

rier T

rans

form

[Ebl

ing,

Sch

euer

man

n 2

004]

Julia

Ebl

ing,

Ger

ik S

cheu

erm

annO

verv

iew

Clif

ford

Alg

ebra

Clif

ford

Con

volu

tion

Clif

ford

Fou

rier t

rans

form

Gab

or F

ilter

Futu

re w

ork

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Clif

ford

Alg

ebra

For

Eucl

idea

n 3D

-spa

ce,

we

use

a 8-

dim

ensi

onal

real

alg

ebra

G3

with

the

vec

tor

basi

s {1

, e 1, e

2, e 3,

e 1e 2, e3e 1, e

2e 3, e1e 2e 3}.

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Clif

ford

Alg

ebra

1ej

=e j

j=1,

2,3

e je j

=1

j=1,

2,3

e je k

=­e ke j

j,k=

1,2,

3,j

≠k

Mul

tiplic

atio

n is

bili

near

and

ass

ocia

tive

with

the

rule

s:

Mul

tiplic

atio

n is

not

com

mut

ativ

e!

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Vec

tors

are

des

crib

ed a

s

and

an a

rbitr

ary

mul

tivec

tor c

an b

e de

scrib

ed a

s

with

i=e 1e 2e 3.

v=xe

1ye

2ze

3∈E

3 ⊂G

3

A=

aib

∈G

3,

∈ℝ,a,b

∈E

3

Clif

ford

Alg

ebra

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

In C

liffo

rd a

lgeb

ra, t

he m

ultip

licat

ion

of v

ecto

rs

desc

ribes

the

com

plet

e ge

omet

ric re

latio

n be

twee

n tw

o ve

ctor

s:

Her

e,

is th

e sc

alar

(inn

er) p

rodu

ct b

etw

een

two

vect

ors a

nd

is t

he o

uter

pro

duct

:

ab=a⋅ba∧b

a⋅b

a∧b a⋅b

=∣ a

∣∣ b∣ cos

a;b

∣ a

∧b∣

=∣ a

∣∣ b∣ sin

a;b

Clif

ford

Alg

ebra

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Let

be

a m

ultiv

ecto

r fie

ld. A

s dire

ctio

nal

deriv

ativ

e, w

e de

fine

and

as to

tal d

eriv

ativ

e

As i

nteg

ral,

we

defin

e

A br

=lim

0

1 [A

r

b­A

r],

∈ℝ

∂A

r=

∑k=

1

3e kA e

kr

∫ E3Adx

=lim

i∞∑

iAx i

xi

A:E

3G

3Clif

ford

Alg

ebra

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Cur

l and

div

erge

nce

of a

vec

tor v

alue

d fu

nctio

n

ar

e de

fined

as:

divergence

f=

⟨∇,f

⟩=∂f

f∂

2

Clif

ford

Alg

ebra

f

curl

f=

∇∧f=

∂f­

f∂

2

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Clif

ford

Con

volu

tion

With

the

int

egra

l, th

e C

liffo

rd c

onvo

lutio

n is

def

ined

as A

com

paris

on w

ith [

Hei

berg

200

1] s

how

s th

at w

e ge

t fo

r a v

ecto

r filt

er h

is s

cala

r con

volu

tion

plus

a b

ivec

tor

part

desc

ribin

g th

e re

lativ

e po

sitio

n in

spac

e.

F∗V

x

=∫ E

3F

yV

x­ydy

F∗V

j,k,l

=

∑s,t,u=

­r

rF

s,t,u

Vj

­s,k­t,l­u

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Inte

rpre

tatio

n

Inst

ead

of th

e co

nvol

utio

n, w

e m

ay lo

ok a

t the

spa

tial

cohe

renc

e:

At e

ach

posi

tion

x, w

e co

mpu

te th

e co

here

nce

betw

een

the

mas

k ce

nter

ed a

t its

orig

in a

nd t

he v

ecto

r fie

ld!

By

the

Clif

ford

pro

duct

, we

get t

he r

elat

ive

geom

etric

po

sitio

n be

twee

n m

ask

and

vect

or fi

eld.

F×P

x

=∫ E

3F

yP

xydy

F×P

j,k,l

=

∑s,t,u=

­r

rF

s,t,u

Pj

s,kt,lu

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Vec

tor

Filte

r

As

vect

or

valu

ed

mas

ks fo

r pa

ttern

m

atch

ing

we

can

use

typi

cal

patte

rn su

ch

as ro

tatio

n or

co

nver

genc

e.

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Vec

tor

Filte

r

Her

e ar

e so

me

vect

or v

alue

d fil

ter i

n 3D

:

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Patt

ern

mat

chin

gC

liffo

rd C

onvo

lutio

n:A

ppro

xim

atio

n of

rota

tion

betw

een

loca

l st

ruct

ure

in fi

eld

and

mas

kR

otat

e m

ask

to a

lign

field

and

mas

kC

ompu

te sc

alar

con

volu

tion

for s

imila

rity

➔ R

otat

ion

inva

rian

t pat

tern

mat

chin

g

App

roxi

mat

ion

not g

ood

enou

gh:

Use

3 (2

D) /

6 (3

D) m

ask

dire

ctio

ns

to c

ompu

te th

e lo

cal d

irect

ion

2D

3D

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Patt

ern

mat

chin

g

2D3D

Com

puta

tion

of lo

cal d

irec

tion:

2D: a

ppro

xim

atio

n w

ith sm

alle

st a

ngle

3D: w

eigh

ted

aver

agin

g of

app

roxi

mat

ions

with

posi

tive

scal

ar

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Gas

furn

ace

cham

ber

Gas

furn

ace

cham

ber.

Patte

rn

mat

chin

g w

ith d

iffer

ent m

ask

size

s: 3◊3◊3

(red

), 5◊

5◊5

(yel

low

), 8◊

8◊8

(gre

en)

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Flat

tene

d su

rfac

e of

del

ta w

ing

Del

ta w

ing

at 0

.2 m

ach

vel

ocity

, a

ngle

of a

ttack

: 25∞

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Clif

ford

Fou

rier

tran

sfor

m

{1,I 3=e

1e 2e 3 }

isom

orph

to c

ompl

ex n

umbe

rs

I 3 co

mm

utes

with

eve

ry m

ultiv

ecto

r

Use

I 3 in

stea

d of

i in

Fou

rier K

erne

l

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Clif

ford

Fou

rier

Tra

nsfo

rm in

3D

:

F{f

}u

=∫fx

e­2I 3

⟨x,u

⟩ dx

F{f

}u

=∫fx

e2I 3

⟨x,u

⟩ dx

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Clif

ford

Fou

rier

Tra

nsfo

rm in

3D

:

Shift

The

orem

:

Con

volu

tion:

Der

ivat

ion:

F{x

­x'

}u

=F

{f}u

e­

2I 3

⟨x',u⟩

F{h

∗f}

u=F

{h}u

F{f

}u

F{∇f}

u=

2I 3uF

{f}u

F{f }

u=

­4

2 u2F

{f}u

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Clif

ford

Fou

rier

Tra

nsfo

rm in

3D

:

Bas

icly

4 c

ompl

ex F

ourie

r tra

nsfo

rms o

f:

1 -

e1e 2e 3

e 1 -

e2e 3

e 2

- e 3e 1

e 3

- e 1e 2,

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

{1,I 2=e

1e 2 }

isom

orph

to c

ompl

ex n

umbe

rs

I 2 co

mm

utes

with

eve

ry sp

inor

I 2antic

omm

utes

with

vec

tor

U

se I 2

inst

ead

of i

in F

ourie

r Ker

nel

Th

eore

ms a

bit

mor

e co

mpl

icat

ed

Clif

ford

Fou

rier

Tra

nsfo

rm in

2D

:

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Clif

ford

Fou

rier

Tra

nsfo

rm in

2D

:

Bas

icly

2 c

ompl

ex F

ourie

r tra

nsfo

rms o

f:

1 -

e1e 2

e 1 -

e2

as

a 1e 1+ a 2e 2 =

e1 ( a

11 +

a2e 1e 2 )

can

be

unde

rsto

od a

s a c

ompl

ex n

umbe

r

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Pars

eval

s the

orem

Sam

plin

g th

eore

m

Dis

cret

izat

ion

Fast

tran

sfor

m

Clif

ford

Fou

rier

Tra

nsfo

rm:

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Turb

ulen

t sw

irlin

g je

t en

teri

ng fl

uid

at r

est

Res

olut

ion:

256

*128

Col

or c

odin

g of

the

abso

lute

val

ues o

f the

(mul

ti) v

ecto

rs

Top:

or

igin

al v

ecto

r fie

ld

Bot

tom

: fa

st C

FT, 2

D v

ecto

rs a

re

conv

erte

d to

vec

tors

(3D

vec

tors

con

vert

to

vect

or +

biv

ecto

r)

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Vec

tor

Filte

r in

Fre

quen

cy D

omai

n

Rot

atio

n,

Div

erge

nce,

Sa

ddle

Poi

nts:

Diff

er o

nly

in

phas

e, n

ot in

am

plitu

de!

Julia

Ebl

ing,

Ger

ik S

cheu

erm

annGab

or F

ilter

Shor

t tim

e or

win

dow

ed F

ourie

r tra

nsfo

rm

Opt

imal

ly lo

caliz

ed in

bot

h sp

atia

l and

Fou

rier

dom

ain

Gab

or e

xpan

sion

, wav

elet

s, fil

ter b

anks

Julia

Ebl

ing,

Ger

ik S

cheu

erm

annGab

or F

ilter

scal

ar G

abor

filte

r in

spat

ial d

omai

n:

➔ m

ultiv

ecto

r val

ued

Gab

or fi

lter i

n sp

atia

l dom

ain:

hx

=gx

∗e­

2i⟨x,U

⟩

hx

=gx

∗e­

2I k

⟨x,U

⟩

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Con

clus

ion:

Pro

/Con

tra

Clif

ford

Con

volu

tion:

+ un

ifyin

g no

tatio

n fo

r sca

lar /

vec

tor f

ield

s+

vect

or fi

elds

: sim

ilarit

y an

d ge

omet

ric p

ositi

on

Patte

rn m

atch

ing:

+

robu

st in

term

s of n

oise

+ ro

tatio

n in

varia

nt+

appl

icab

le to

irre

gula

r grid

s- s

low

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Con

clus

ion:

Pro

/Con

tra

Four

ier t

rans

form

:+

conv

olut

ion

thor

em, .

..+

mat

hem

atic

al b

asis

for a

naly

sis o

f filt

er+

acce

lera

tion

of c

onvo

lutio

n vi

a FF

Ts-

irreg

ular

grid

s

Gab

or F

ilter

:+

inhe

rent

mul

tisca

le a

ppro

ach

- di

rect

app

roac

h ga

ve n

o bi

g ad

vant

ages

for

m

atch

ing

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Futu

re W

ork

Irre

gula

r Grid

s

Ana

lysi

s of i

nter

pola

tion,

smoo

thin

g, sa

mpl

ing,

de

rivat

ion

and

the

indu

ced

erro

rs

➔Fi

lter D

esig

n

Scal

e sp

ace

cons

ider

atio

n / h

iera

rchi

cal f

eatu

res

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Ack

now

ledg

men

ts

FAnT

oM so

ftwar

e de

velo

pmen

t tea

m(T

om B

obac

h, C

hris

toph

Gar

th, D

avid

Gru

ys, K

ai H

erge

nrˆt

her,

Nik

olai

Ivle

v, M

ax L

angb

ein,

Mar

tin ÷

hler

, Mic

hael

Sch

lem

mer

, X

avie

r Tric

oche

, Tho

mas

Wis

chgo

ll)

Com

pute

rgra

phic

s Gro

up a

t TU

Kai

sers

laut

ern

and

Uni

vers

ity o

f Lei

pzig

Dat

a Se

ts:

Wol

fgan

g K

ollm

ann,

MA

E D

epar

tmen

t, U

C D

avis

Mar

kus R

¸tte

n, D

LR G

ˆttin

gen

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Four

ier

Tra

nsfo

rm o

f Sec

ond-

Ord

er T

enso

r Fi

elds

F{f

}u

=∫fx

e­2iI

⟨x,u

⟩ dx

f∗v

x=

∫ E3fyvx­ydy

Con

volu

tion

usin

g m

atrix

mul

tiplic

atio

n

Four

ier t

rans

form

➔ C

onvo

lutio

n th

eore

m:

F{h

∗f }

u=F

{h}u

F{f

}u

Julia

Ebl

ing,

Ger

ik S

cheu

erm

ann

Four

ier

Tra

nsfo

rm o

f Arb

itrar

y O

rder

Ten

sor

Fiel

ds

F{f

}u

=∫fx

e­2i⟨x,u⟩dx

f∗v

x=

∫ E3fyvx­ydy

Con

volu

tion

usin

g te

nsor

pro

duct

Four

ier t

rans

form

➔ C

onvo

lutio

n th

eore

m:

F{h

∗f}

u=F

{h}u

F{f

}u