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
Recommended
