AGACSE – Leipzig, Germany August 17-19 2008 The Cylindrical Fourier Transform Fred Brackx Nele De Schepper Frank Sommen Clifford Research Group - Department

AGACSE – Leipzig, GermanyAugust 17-19 2008

The Cylindrical Fourier Transform

Fred Brackx Nele De Schepper Frank Sommen

Clifford Research Group-

Department of Mathematical Analysis-

Ghent University


Clifford toolbox …

meee ,...,, 21mIR

: orthonormal basis of

Basis of Clifford algebra mIR ,0 or mC

miiiAeeeehiiiA h

,...,2,1,...,,...2121

with miii h ...1 21

:

CLIFFORD ALGEBRA

Identity element: 1ø e

Non-commutative multiplication: mkjeeee jkjkkj ,...,2,1,;2

W.K. Clifford(1845 – 1879)


. . . Clifford toolbox . . .

Conjugation : anti-involution for which mjee jj ,...,2,1,

Hermitean conjugation : Cee AA

AcA

AAA

Hermitean inner product : 0,

2

0

2,

AAAssociated norm:



m

jjjm

xexxx1

1,...,

m

jjj

yxyxyx1

,

m

i

m

ijijjiji yxyxeeyx

1 1

•

and

yxyxyx •

with

In particular:22 , xxxx

m

m IRIR,0

or mC



mxxF ,...,1

m

jxjx j

e1

0 Fx

mx 2

CLIFFORD ANALYSIS

is left monogenic in

in

with

: Dirac operator


. . . Clifford toolbox

0 xPkx xPtxtP k

kk

and

kMPk

: left solid inner spherical monogenic of order k

Orthonormal basis for L2( IRm ) :

kMjINksx

xPxHx j

kks

ks

m

jks dim,,;2

exp222

2

,2/1

,

4/

,,

with ksH , the generalized Clifford-Hermite polynomials defined by

xP

xxxPxH

k

s

xkks 2exp

2exp

22

,


Clifford-Fourier Transform . . .

Classical tensorial Fourier transform:

xdVxfxifFmIR

m

,exp2

12/

0 2!

1

2exp

k

kHk

ik

HiF

J.-B. Joseph Fourier (1768-1830)

operator exponential form:

with mxH m 2

2

1 : scalar-valued


. . . Clifford-Fourier Transform . . .

HHH2 HH

xx

with

: angular Dirac operator

HiFH 2

exp

: Clifford-Fourier transform

Fi

2exp

HHHH

FFFFF 2

Clifford-Fourier transform = refinement classical Fourier transform

monogenicity = refinement harmonicity


. . . Clifford-Fourier Transform

2

exp2

1IR

HxdVxfxfF

xfFixfxFH

m

H

Two - dimensional Clifford-Fourier transform:

multiplication rule:

differentiation rule:

xfFixfFH

m

xH


Cylindrical Fourier transform – Definition . . .

DEFINITION

xdVxfxfF mIRmcyl

exp2

12/

0 !exp

r r

xx

r

fFfFcylH

with

Remark: for 2m

Expression integral kernel:

xcxxx sincosexp

with x

xxc

sin:sin


. . . Cylindrical Fourier transform – Why “cylindrical”? . . .

WHY “CYLINDRICAL”?

2222222222

,sin,cos1, xxxxxxx

For fixed, the “phase” is constant is constant

for fixed, the “phase” of the cylindrical Fourier kernel is constant on

co-axial cylinders w.r.t.

x

x

,sin xx

or


. . . Cylindrical Fourier transform – Why “cylindrical”? . . .

For fixed, the “phase” is constant

for fixed, the level surfaces of the traditional Fourier kernel are planes perpendicular to

,cos xx

,cos, xxx

Comparison with classical Fourier transform:

,x

or

is constant


. . . Cylindrical Fourier transform – Properties . . .

PROPERTIES

mmm

cylIRCIRLIRLF

01:

1

2/

1

22: ffFIRLf

m

cylm

mIRmcylxcyl

xdVxfxcm

xfFxfF

sin2

22/

For

Differentiation rule:

Multiplication rule:

mIR

mcylcylxdVxfxxc

mxfFxfxF

sin

2

22/


. . . Cylindrical Fourier transform – spectrum L2- basis . . .

Aim: calculate the cylindrical Fourier spectrum of the L2- basis:

kMjINksx

xPxHx jkks

ks

m

jks dim,,;2

exp222

2

,2/1,

4/

,,

kS1 mS

Calculation method is based on Funk - Hecke theorem in space:

: spherical harmonic of degree k; fixed

Notation: for t ,cos, 1 mS

mIR

kmk

mm

mkSdttPttfdrrrgAxdVStfrg

1

1,

2/32

0

1

11



A) The cylindrical Fourier spectrum of

1) k even

2;

2

1,

2

2;

2

1,

2

22

02

2

22

2exp2

2

22

2

,2

kmkkmumkF

p

u

uu

pP

p

kmp

xxPxH

cylF

k

kkp

0 !;,;,

22

z

dc

bazdcbaF

1...1

with

: generalized hypergeometric series

: Pochhammer’s symbol

psjkp

2,,2



2;

2

21;

21

02

1

2

22

1

2exp

2

22

2exp2

2

11

2

22

,2

ukmF

p

u

u

u

k

u

mku

km

u

p

Pp

kmpxxPxH

cylF

kkkp

with

0 !;;

11

z

b

azbaF : Kummer’s function



Special case: p=0

2exp

2;

2

1;2

1

2exp

22

11

2

k

k

Pkm

F

xxP

cylF

2

exp2

x

cylF for m= 3, 4, 5, 6



2) k odd

Special case: p=0

2exp

2;

2

2;2

12

exp

22

11

2

kkP

kmF

xxP

cylF



B) The cylindrical Fourier spectrum of 1,,1

sjk

2exp

2;

2

3;2

11

1

2exp

22

11

2

k

k

Pkm

Fk

mk

xxPx

cylF

1) k even

xx

cylF

2exp

2

for m= 3, 4, 5, 6


2) k odd

2exp

2;

2

2;2

12

exp

22

11

2

kkP

kmF

xxPx

cylF



. . . Cylindrical Fourier transform – example . . .

Characteristic function of a geodesic triangle on S2 :

2,,,1 Sxrrxgeod

rxf

2,exp

2/32

121

S

ebeadSgeod

fcyl

F

Computation by means of spherical co-ordinates:

ddbafcyl

F 2/

0

2/

0sinexp

2/32

1,

with 321

cossinsincossin eee


. . . Cylindrical transform – example . . .

Results simulation (Maple):

real part of e1 e2 – component of bafcyl

F , bafcyl

F ,


. . . Cylindrical transform – example

Results simulation (Maple):

e2 e3 – component of bafcyl

F , bafcyl

F ,e1 e3 – component of


References

1. F. Brackx, N. De Schepper and F. Sommen, The Clifford-Fourier Transform, J. Fourier Anal. Appl. 11(6) (2005), 669 – 681.

2. F. Brackx, N. De Schepper and F. Sommen, The Two-Dimensional Clifford- Fourier Transform, J. Math. Imaging Vision 26(1-2) (2006), 5 – 18.

3. F. Brackx, N. De Schepper and F. Sommen, The Fourier Transform in Clifford Analysis, to appear in Advances in Imaging & Electron Physics.

Documents

AGACSE – Leipzig, Germany August 17-19 2008 The Cylindrical Fourier Transform Fred Brackx Nele De Schepper Frank Sommen Clifford Research Group - Department