FACULTY OF ENGINEERING AND
ARCHITECTURE
Slice Fourier transform: definition, properties
and corresponding convolutions
L. Cnudde
H. De Bie
Ghent University
Dept. of Mathematical Analysis
Clifford Research Group
June 10, 2015, Porto
Contents
1
Introduction
2
Slice Fourier transform
3
Convolutions
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
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Contents
1
Introduction
What defines a Fourier transform?
Classical Fourier transform in a nutshell
Mehler approach
2
Slice Fourier transform
3
Convolutions
L. Cnudde, H. De Bie
Slice Fourier transform
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3 / 30
What defines a Fourier transform?
Its kernel expression?
1
F(f )(y ) = √
2π
Z
e −ixy f (x)dx
R
Its eigenvectors and eigenvalues?
F(ψn ) = (−i)n ψn
Its differential equations?
∂y F(f ) = −iF(xf )
iy F(f ) = F(∂x f )
Its convolution property?
F(f ∗ g ) = F(f )F(g )
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Slice Fourier transform
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Classical Fourier transform in a nutshell
Definition
The classical Fourier transform is defined as
1
F : L2 (R, dx) → L2 (R, dx) : f (x) 7→ F(f )(y ) = √
2π
Z
e −ixy f (x)dx,
R
where
2
L (R, dx) =
Z
f : R → R +∞
∗
(f (x)) f (x) dx < ∞
.
−∞
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Slice Fourier transform
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Classical Fourier transform in a nutshell
Definition
The classical Fourier transform is defined as
1
F : L2 (R, dx) → L2 (R, dx) : f (x) 7→ F(f )(y ) = √
2π
Z
e −ixy f (x)dx,
R
where
2
L (R, dx) =
Z
f : R → R +∞
∗
(f (x)) f (x) dx < ∞
.
−∞
The L2 -space can be endowed with the standard inner product:
1
hf , g i = √
2π
+∞
Z
(f (x))∗ g (x)dx
−∞
.
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Slice Fourier transform
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Classical Fourier transform in a nutshell
Definition (Hermite polynomials)
The Hermite polynomial Hj of order j is defined as
j
d
Hj (x) = 2x −
1.
dx
One might think of 1 as a basis element of ker
d
dx .
Definition (Hermite functions)
The Hermite function ψj of order j is defined as
2
−x
ψj (x) = Hj (x) exp
.
2
With this definition and the above inner product, one has
hψj1 , ψj2 i = hψj1 , ψj1 iδj1 j2 .
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Slice Fourier transform
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Classical Fourier transform in a nutshell
Theorem (Laguerre formulation)
The Hermite polynomials Hj may be expressed as
H2t (x) = (−1)t 22t
−1/2
t! Lt
+1/2
H2t+1 (x) = (−1)t 22t+1 t!xLt
x2
x2 .
where Lkt are the generalised Laguerre polynomials of degree t.
Theorem (scalar differential equation)
One has
d2
x 2 − 2 + 1 ψj (x) = 2(j + 1)ψj (x)
dx
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Mehler approach
Therefore the following formal expression holds:
e−
iπ(H−1)
4
ψj (x) = (−i)j ψj (x)
where the operator H is given by H = x 2 −
L. Cnudde, H. De Bie
Slice Fourier transform
d2
dx 2 .
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Mehler approach
Therefore the following formal expression holds:
e−
iπ(H−1)
4
ψj (x) = (−i)j ψj (x)
where the operator H is given by H = x 2 −
d2
dx 2 .
Theorem (Eigenfunction equation)
F(ψj ) = (−i)j ψj
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Slice Fourier transform
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Mehler approach
Therefore the following formal expression holds:
e−
iπ(H−1)
4
ψj (x) = (−i)j ψj (x)
where the operator H is given by H = x 2 −
d2
dx 2 .
Theorem (Eigenfunction equation)
F(ψj ) = (−i)j ψj
An integral expression can be constructed based on the orthogonality of
the ψj ’s with respect to the inner product. With f ∈ span{ψj } one has
f (x)
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Slice Fourier transform
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Mehler approach
Therefore the following formal expression holds:
e−
iπ(H−1)
4
ψj (x) = (−i)j ψj (x)
where the operator H is given by H = x 2 −
d2
dx 2 .
Theorem (Eigenfunction equation)
F(ψj ) = (−i)j ψj
An integral expression can be constructed based on the orthogonality of
the ψj ’s with respect to the inner product. With f ∈ span{ψj } one has
Z
R
L. Cnudde, H. De Bie
(ψj (x))∗
f (x) dx
hψj , ψj i
Slice Fourier transform
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Mehler approach
Therefore the following formal expression holds:
e−
iπ(H−1)
4
ψj (x) = (−i)j ψj (x)
where the operator H is given by H = x 2 −
d2
dx 2 .
Theorem (Eigenfunction equation)
F(ψj ) = (−i)j ψj
An integral expression can be constructed based on the orthogonality of
the ψj ’s with respect to the inner product. With f ∈ span{ψj } one has
Z
R
L. Cnudde, H. De Bie
(−i)j (ψj (x))∗
f (x) dx
hψj , ψj i
Slice Fourier transform
June 10, 2015, Porto
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Mehler approach
Therefore the following formal expression holds:
e−
iπ(H−1)
4
ψj (x) = (−i)j ψj (x)
where the operator H is given by H = x 2 −
d2
dx 2 .
Theorem (Eigenfunction equation)
F(ψj ) = (−i)j ψj
An integral expression can be constructed based on the orthogonality of
the ψj ’s with respect to the inner product. With f ∈ span{ψj } one has
Z
R
L. Cnudde, H. De Bie
ψj (y )(−i)j (ψj (x))∗
f (x) dx
hψj , ψj i
Slice Fourier transform
June 10, 2015, Porto
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Mehler approach
Therefore the following formal expression holds:
e−
iπ(H−1)
4
ψj (x) = (−i)j ψj (x)
where the operator H is given by H = x 2 −
d2
dx 2 .
Theorem (Eigenfunction equation)
F(ψj ) = (−i)j ψj
An integral expression can be constructed based on the orthogonality of
the ψj ’s with respect to the inner product. With f ∈ span{ψj } one has
Z X
∞
ψj (y )(−i)j (ψj (x))∗
F(f )(y ) =
f (x) dx
hψj , ψj i
R
j=0
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Slice Fourier transform
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Mehler approach
Therefore the following formal expression holds:
e−
iπ(H−1)
4
ψj (x) = (−i)j ψj (x)
where the operator H is given by H = x 2 −
d2
dx 2 .
Theorem (Eigenfunction equation)
F(ψj ) = (−i)j ψj
An integral expression can be constructed based on the orthogonality of
the ψj ’s with respect to the inner product. With f ∈ span{ψj } one has
Z X
∞
ψj (y )(−i)j (ψj (x))∗
F(f )(y ) =
f (x) dx
hψj , ψj i
R j=0
Z −ixy
e
√
=
f (x) dx
2π
R
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Slice Fourier transform
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Contents
1
Introduction
2
Slice Fourier transform
Slice setting
Eigenfunctions and eigenvalues
Mehler construction
Properties
3
Convolutions
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Slice Fourier transform
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Slice setting
Original framework
Real Clifford algebra Rm with basis {e
ei , i = 1, . . . , m}:
e
ei e
ej + e
ej e
ei = −2δij ,
i, j = 1, . . . , m.
x and
Paravector e
x = x0 + r ω
e ∈ R0m ⊕ R1m with r the norm of e
ω
e=e
x /r the unit vector along e
x.
Associated Slice Cauchy-Riemann operator
∂
e CR = ∂ + ω
D
e
0
∂x0
∂r
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Slice Fourier transform
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Slice setting
Original framework left-multiplied with e0
Real Clifford algebra Rm with basis {e
ei , i = 1, . . . , m}:
e
ei e
ej + e
ej e
ei = −2δij ,
i, j = 1, . . . , m.
x and
Paravector e
x = x0 + r ω
e ∈ R0m ⊕ R1m with r the norm of e
ω
e=e
x /r the unit vector along e
x.
Associated Slice Cauchy-Riemann operator
∂
e CR = ∂ + ω
D
e
0
∂x0
∂r
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Slice Fourier transform
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Slice setting
Original framework left-multiplied with e0
Real Clifford algebra Rm+1 with basis {ei , i = 0, . . . , m} where
ei = e0 e
ei , i = 1, . . . , m:
ei ej + ej ei = −2δij ,
i, j = 0, . . . , m.
1-vector x = x0 e0 + r ω ∈ R1m+1 with r the norm of x and ω = x/r
the unit vector along x.
Associated slice Dirac operator
D 0 = e0
L. Cnudde, H. De Bie
∂
∂
+ω
∂x0
∂r
Slice Fourier transform
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Slice setting
Theorem (osp(1|2)-superalgebra)
Defining the Euler operator E as
E=
m
X
i=0
xi
∂
,
∂xi
the operators x, D0 and E constitute a Lie superalgebra isomorphic with
osp(1|2), with relations
{x, x} = −2|x|2
{x, D0 } = −2 (E + 1)
[|x|2 , D0 ] = −2x
[∂x20 + ∂r2 , x] = 2D0
[∂x20 + ∂r2 , |x|2 ] = 4 (E + 1)
L. Cnudde, H. De Bie
{D0 , D0 } = −2(∂x20 + ∂r2 )
[E + 1, D0 ] = −D0
[E + 1, x] = x
[E + 1, ∂x20 + ∂r2 ] = −2(∂x20 + ∂r2 )
[E + 1, |x|2 ] = 2|x|2 .
Slice Fourier transform
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Eigenfunctions and eigenvalues
The space of k-homogeneous polynomials (k ∈ N) in the kernel of D0 is
one-dimensional and a basis is given by
mk (x) = (e0 − 1) (x0 + x)k a,
a ∈ Clm+1 .
Definition (Hermite polynomials)
The Hermite polynomials Hj (mk ) for the slice Dirac operator are defined
as
Hj (mk )(x) = (x − cD0 )j mk (x)
with c ∈ R+
0 a strictly positive, real parameter.
Definition
The Clifford-Hermite functions ψj,k are defined as
|x|2
.
ψj,k (x) = Hj (mk )(x) exp −
4c
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Eigenfunctions and eigenvalues
Theorem (Laguerre formulation)
The Hermite polynomials Hj (mk ) may be expressed as
2
|x|
H2t (mk )(x) = (2c)t t! Lkt
mk (x)
2c
2
|x|
H2t+1 (mk )(x) = (2c)t t! x Lk+1
mk (x)
t
2c
where Lkt are the generalised Laguerre polynomials of degree t.
Theorem (scalar differential equation)
The Clifford-Hermite functions ψj,k are eigenfunctions of the scalar
differential equation
|x|2
cD02 +
ψj,k (x) = (j + k + 1)ψj,k (x).
4c
L. Cnudde, H. De Bie, and G. Ren.
Algebraic approach to slice monogenic functions.
Complex Analysis and Operator Theory 9 (2015), 1065–1087.
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Mehler construction
Because of the scalar differential equation, its formal expression is
π
e −i 2 H ψj,k (x) = (−i)(j+k+1) ψj,k (x)
Construct slice Fourier transform such that Clifford-Hermite
functions ψj,k are eigenfunctions with corresponding eigenvalues.
For an integral expression an inner product is needed such that the
ψj,k ’s are orthogonal.
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Mehler construction
Because of the scalar differential equation, its formal expression is
π
e −i 2 H ψj,k (x) = (−i)(j+k+1) ψj,k (x)
Construct slice Fourier transform such that Clifford-Hermite
functions ψj,k are eigenfunctions with corresponding eigenvalues.
For an integral expression an inner product is needed such that the
ψj,k ’s are orthogonal.
Definition (Inner product)
Z
hf , g i =
f g r
Rm+1
1−m
Z
dx =
0
Rm+1
f g dx0 dr dσx
where
0
n
hR
i
o
1−m
<
+∞
.
f , g ∈ L2 = f : Rm+1 → Clm+1
f
(x)f
(x)
r
dx
m+1
R
0
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Mehler construction
Proposition
On a dense subset of L2 , the inner product shows the relations
hD0 f , g i = hf , D0 g i,
hxf , g i = −hf , xg i.
(
Because
x
2
L. Cnudde, H. De Bie
ψj,k
+ cD0 ψj,k
= 2x − cD0 ψj−1,k
, we get
= c C (j, k)ψj−1,k
Slice Fourier transform
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Mehler construction
Proposition
On a dense subset of L2 , the inner product shows the relations
hD0 f , g i = hf , D0 g i,
hxf , g i = −hf , xg i.
(
Because
x
2
ψj,k
+ cD0 ψj,k
= 2x − cD0 ψj−1,k
, we get
= c C (j, k)ψj−1,k
Theorem (Orthogonality theorem)
The inner product of two Clifford-Hermite functions ψj1 ,k1 and ψj2 ,k2 reads
hψj1 ,k1 , ψj2 ,k2 i = A(j1 , k1 )δj1 j2 δk1 k2
where A(j1 , k1 ) ∈ R+
0.
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Mehler construction
The formal expression for the kernel function is given by
KM (x, y) =
+∞
X
ψj,k (y)(−i)j+k+1 ψj,k (x)
.
hψj,k , ψj,k i
j,k=0
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Mehler construction
The formal expression for the kernel function is given by
KM (x, y) =
+∞
X
ψj,k (y)(−i)j+k+1 ψj,k (x)
.
hψj,k , ψj,k i
j,k=0
Using the definitions and properties of the Clifford-Hermite functions ψj,k
this expression can be simplified by
substituting the known expressions for ψj,k and hψj,k , ψj,k i
using the Hille-Hardy formula
using trigonometric identities
writing a Bessel series as an exponential
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Slice Fourier transform
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Mehler construction
The Mehler kernel function KM thus becomes
|x||y|
KM (x, y) ∼ 2J0 −
2c
+(1 − i)
+(1 + i)e0
+∞
X
!
|x||y|
−
2c
(|x||y|)k
k=1
!#
+∞
X (y0 + y )k (x0 − x)k − (y0 − y )k (x0 + x)k
|x||y|
k
(−i)
J
−
.
k
2c
(|x||y|)k
k=1
L. Cnudde, H. De Bie
(y0 + y )k (x0 − x)k + (y0 − y )k (x0 + x)k
Slice Fourier transform
(−i)k Jk
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Mehler construction
The Mehler kernel function KM thus becomes
|x||y|
KM (x, y) ∼ 2J0 −
2c
+(1 − i)
+(1 + i)e0
+∞
X
!
|x||y|
−
2c
(|x||y|)k
k=1
!#
+∞
X (y0 + y )k (x0 − x)k − (y0 − y )k (x0 + x)k
|x||y|
k
(−i)
J
−
.
k
2c
(|x||y|)k
k=1
(y0 + y )k (x0 − x)k + (y0 − y )k (x0 + x)k
(−i)k Jk
Writing
k


 (y0 + ig ) 
 = cos(kφ) + i sin(kφ)
q
y02 + g 2
and
k
 (x0 + ir ) 
q
 = cos(kχ) + i sin(kχ)
x02 + r 2
we get
"
M
K (x, y) ∼ J0 +
+∞
X
!
k
[cos(k(φ + χ)) + cos(k(φ − χ))] (−i) Jk
k=1
−
+∞
X
!
k
[cos(k(φ − χ)) − cos(k(φ + χ))] (−i) Jk
#
ηω .
k=1
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Slice Fourier transform
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Mehler construction
Finally, using the identity
e iz cos(φ) = J0 (z) + 2
+∞
X
i n Jn (z) cos(nφ),
n=1
the Mehler kernel KM is given by
KM (x, y) =
i
−i
−i
−iΓ(m/2) h
2c (x0 y0 −rg ) + (1 − ηω)e 2c (x0 y0 +rg ) .
(1
+
ηω)e
8cπ m/2+1
Definition (slice Fourier transform)
The slice Fourier transform FS is defined as
Z
FS (f )(y) =
KM (x, y) f (x) dx0 dr dσx .
Rm+1
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Slice Fourier transform
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Properties
For f ∈ span{ψj,k }, one has
Denoting a translation in the e0 -direction over a distance a as
ta f (x0 , r , ω) = f (x0 − a, r , ω), one has
FS (ta f )(y) = e −
iay0
2c
FS (f )(y).
Denoting a reflection with respect to the origin as
sf (x0 , r , ω) = f (−x0 , r , −ω), one has
FS (sf )(y) = sFS (f )(y)
Denoting the complex conjugate of f as f ∗ , one has
∗
FS (f ∗ ) (y) = − FS (f )(−y0 , g , −η)
With respect to e0 , one has FS (e0 f )(y) = e0 FS (f )(y)
The twofold slice Fourier transform gives FS (FS (f )) (y) = −f (−y)
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Properties
KM (x, y) =
+∞
X
ψj,k (y)(−i)j+k+1 ψj,k (x)
,
hψj,k , ψj,k i
j,k=0
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Properties
The following system of differential equations holds
(
M
−i
M
K (x, y)D0x
2c yK (x, y) =
.
D0y KM (x, y) = 2ci KM (x, y)x
KM (x, y) =
+∞
X
ψj,k (y)(−i)j+k+1 ψj,k (x)
,
hψj,k , ψj,k i
j,k=0
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Properties
The following system of differential equations holds
(
M
−i
M
K (x, y)D0x
2c yK (x, y) =
.
D0y KM (x, y) = 2ci KM (x, y)x
M
is given by
The kernel for the inverse transform K−1
i
i
i
iΓ(m/2) h
M
2c (x0 y0 −rg ) + (1 − ωη)e 2c (x0 y0 +rg )
K−1
(y, x) =
(1
+
ωη)e
8cπ m/2+1
∗
= KM (y, x) .
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Properties
The following system of differential equations holds
(
M
−i
M
K (x, y)D0x
2c yK (x, y) =
.
D0y KM (x, y) = 2ci KM (x, y)x
M
is given by
The kernel for the inverse transform K−1
i
i
i
iΓ(m/2) h
M
2c (x0 y0 −rg ) + (1 − ωη)e 2c (x0 y0 +rg )
K−1
(y, x) =
(1
+
ωη)e
8cπ m/2+1
∗
= KM (y, x) .
The computational load of the slice Fourier transform equals that of
two classical Fourier transforms.
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Slice Fourier transform
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Properties (computational load)
Indeed, given that
f ∈ span{ψj,k } so
f (x) = f0 (x0 , r ) + e0 f1 (x0 , r ) + ωf2 (x0 , r ) + e0 ωf3 (x0 , r ),
the classical two dimensional Fourier transform is proportional to
Z
Z
cos(xy ) cos(rg )f (x, r )dxdr −
sin(xy ) sin(rg )f (x, r )dxdr
R2
R2
Z
Z
−i
sin(xy ) cos(rg )f (x, r )dxdr − i
cos(xy ) sin(rg )f (x, r )dxdr ,
R2
R2
a general function f (x, r ) can be written as
f (x, r ) = f ++ (x, r ) + f +− (x, r ) + f −+ (x, r ) + f −− (x, r ),
the slice Fourier transform of f ∈ span{ψj,k } equals
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Properties (computational load)
FS (f )(y ) =

− 2i
4πc
+∞ Z
+∞
Z


0 y0
2c
0 y0
sin
rg 2c
2c
0 y0
cos
rg 2c
2c
f0 (x0 , r )dx0 dr − ie0
+∞
+∞ Z
Z
−iη
cos
x
0 y0
2c
0
L. Cnudde, H. De Bie
sin
rg 2c
0 y0
cos
rg 2c
f1 (x0 , r )dx0 dr
0
0 y0
sin
sin
rg 2c
2c
f3 (x0 , r )dx0 dr
0
+∞ Z
+∞
Z
x
sin
0 y0
cos
rg 2c
−∞
0
x
2c
−∞
sin
−∞
cos
+∞ Z
+∞
Z
x
f2 (x0 , r )dx0 dr − e0 η
0
+∞ Z
+∞
Z
x
−∞
+∞ Z
+∞
Z
f0 (x0 , r )dx0 dr + e0
−∞
sin
−i
rg 0
+∞ Z
+∞
Z
x
−∞
cos
2c
−∞
−η
x
cos
2c
f1 (x0 , r )dx0 dr
0
+∞
+∞ Z
Z
f2 (x0 , r )dx0 dr − ie0 η
−∞
Slice Fourier transform
cos

x
0 y0
2c
sin
rg 2c

f3 (x0 , r )dx0 dr 
0
June 10, 2015, Porto
22 / 30
Properties (computational load)
FS (f )(y ) =

− 2i
4πc
+∞ Z
+∞
Z


0 y0
2c
0 y0
sin
rg 2c
2c
0 y0
cos
rg 2c
2c
−
f (x0 , r )dx0 dr − ie0
0
−iη
cos
x
0 y0
2c
0
L. Cnudde, H. De Bie
sin
rg 2c
0 y0
cos
2c
0 y0
sin
rg 2c
+
f (x0 , r )dx0 dr
1
2c
−
f (x0 , r )dx0 dr
3
0
+∞ Z
+∞
Z
x
sin
0 y0
cos
rg 2c
+
f (x0 , r )dx0 dr − ie0 η
2
rg 0
sin
−∞
0
+∞
+∞ Z
Z
x
2c
−∞
sin
−∞
cos
+∞ Z
+∞
Z
x
−
f (x0 , r )dx0 dr − e0 η
2
0
+∞ Z
+∞
Z
x
−∞
+∞ Z
+∞
Z
+
f (x0 , r )dx0 dr + e0
0
−∞
sin
−i
rg 0
+∞ Z
+∞
Z
x
−∞
cos
2c
−∞
−η
x
cos
2c
−
f (x0 , r )dx0 dr
1
0
+∞
+∞ Z
Z
−∞
Slice Fourier transform
cos

x
0 y0
2c
sin
rg 2c

+
f (x0 , r )dx0 dr 
3
0
June 10, 2015, Porto
22 / 30
Properties (computational load)
FS (f )(y ) =

− 2i
4πc
+∞ Z
+∞
Z


cos
0 y0
2c
0 y0
sin
rg 2c
cos
2c
0 y0
cos
rg 2c
2c
−iη
cos
x
0 y0
2c
0
L. Cnudde, H. De Bie
sin
rg 2c
cos
2c
0 y0
sin
rg 2c
++
f
(x0 , r )dx0 dr
1
2c
−−
f
(x0 , r )dx0 dr
3
0
+∞ Z
+∞
Z
x
sin
0 y0
cos
rg 2c
+−
f
(x0 , r )dx0 dr − ie0 η
2
rg 0
sin
−∞
0
+∞
+∞ Z
Z
0 y0
+∞ Z
+∞
Z
x
−−
f
(x0 , r )dx0 dr − e0 η
2
−+
f
(x0 , r )dx0 dr − ie0
0
x
2c
−∞
sin
−∞
+∞ Z
+∞
Z
++
(x0 , r )dx0 dr + e0
0
0
+∞ Z
+∞
Z
x
−∞
f
−∞
sin
−i
rg 0
+∞ Z
+∞
Z
x
−∞
cos
2c
−∞
−η
x
2c
−+
f
(x0 , r )dx0 dr
1
0
+∞
+∞ Z
Z
−∞
Slice Fourier transform
cos

x
0 y0
2c
sin
rg 2c

+−
f
(x0 , r )dx0 dr 
3
0
June 10, 2015, Porto
22 / 30
Properties (computational load)
FS (f )(y ) =

−i
+∞ Z
+∞
Z


4πc
cos
x
0 y0
cos
rg 2c
2c
−∞ −∞
sin
x
0 y0
sin
rg 2c
2c
x
0 y0
cos
rg 2c
+∞ Z
+∞
Z
−−
f
(x0 , r )dx0 dr − e0 η
2
−∞ −∞
2c
++
f
(x0 , r )dx0 dr
1
sin
x
0 y0
sin
rg 2c
2c
−−
f
(x0 , r )dx0 dr
3
−∞ −∞
+∞ Z
+∞
Z
−i
cos
−∞ −∞
+∞ Z
+∞
Z
−η
+∞ Z
+∞
Z
++
f
(x0 , r )dx0 dr + e0
0
sin
x
0 y0
cos
rg 2c
2c
−+
f
(x0 , r )dx0 dr − ie0
0
−∞ −∞
sin
x
0 y0
cos
rg 2c
2c
−+
f
(x0 , r )dx0 dr
1
−∞ −∞
+∞
+∞ Z
Z
−iη
+∞ Z
+∞
Z
cos
x
0 y0
2c
−∞ −∞
sin
rg 2c
+−
f
(x0 , r )dx0 dr − ie0 η
2

+∞
+∞ Z
Z
cos
x
0 y0
2c
sin
rg 2c

+−
f
(x0 , r )dx0 dr 
3
−∞ −∞
and therefore
FS (f ) =
L. Cnudde, H. De Bie
−i
2DFT (f0+ + ηf2− ) + e0 2DFT (f1+ + ηf3− )
2c
Slice Fourier transform
June 10, 2015, Porto
22 / 30
Contents
1
Introduction
2
Slice Fourier transform
3
Convolutions
Mustard convolutions
Generalised translation
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
23 / 30
Convolutions
There are multiple ways to generalise the classical convolution defined as
Z +∞
Z +∞
f ?g =
f (x − t)g (t)dt =
τt f (x)g (t)dt.
−∞
−∞
Again referring to the classical Fourier transform, one can choose to...
R. Bujack, H. De Bie, N. De Schepper and G. Scheuermann,
Convolution products for hypercomplex Fourier transforms,
J. Math Imaging Vis, volume 48 (2014).
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
24 / 30
Convolutions
There are multiple ways to generalise the classical convolution defined as
Z +∞
Z +∞
f ?g =
f (x − t)g (t)dt =
τt f (x)g (t)dt.
−∞
−∞
Again referring to the classical Fourier transform, one can choose to...
focus on the well-known property
F(f ? g ) = F(f )F(g )
focus on the expression for the translation τt for which:
F(τt f )(y ) = e −ity F(f )(y )
so
τt f (x) = F −1 e −ity F(f ) )(x)
R. Bujack, H. De Bie, N. De Schepper and G. Scheuermann,
Convolution products for hypercomplex Fourier transforms,
J. Math Imaging Vis, volume 48 (2014).
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
24 / 30
Mustard convolutions
Defined as
f ?LM g = FS−1 (FS (f )FS (g )) ,
the expression for the left Mustard convolution can be calculated as
(f ?LM g )(x) = i
!3 Z Z Z h
m
2
m/2+1
8cπ
Γ
z
u
i
i
(1 + ωη)e − 2c (x0 y0 −rg ) + (1 − ωη)e − 2c (x0 y0 +rg )
i
y
i
i
i
(1 + ηζ)e − 2c (z0 y0 −ng ) + (1 − ηζ)e − 2c (z0 y0 +ng ) f (z)
h
i
i
i
(1 + ηχ)e − 2c (u0 y0 −mg ) + (1 − ηχ)e − 2c (u0 y0 +mg ) g (u)
h
dz0 dn dσz du0 dm dσu dy dg dσy
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
25 / 30
Mustard convolutions
Defined as
f ?LM g = FS−1 (FS (f )FS (g )) ,
the expression for the left Mustard convolution can be calculated as
(f ?LM g )(x) = i
!3 Z Z Z h
m
2
m/2+1
8cπ
Γ
z
u
i
i
(1 + ωη)e − 2c (x0 y0 −rg ) + (1 − ωη)e − 2c (x0 y0 +rg )
i
y
i
i
i
(1 + ηζ)e − 2c (z0 y0 −ng ) + (1 − ηζ)e − 2c (z0 y0 +ng ) f (z)
h
i
i
i
(1 + ηχ)e − 2c (u0 y0 −mg ) + (1 − ηχ)e − 2c (u0 y0 +mg ) g (u)
h
dz0 dn dσz du0 dm dσu dy dg dσy
Theorem
With a ∈ Clm , one has
Z
ω a ω dω =
Sm−1
L. Cnudde, H. De Bie
m
2π m/2 X
(−1)k (2k − m)a(k) .
m
mΓ 2 k=0
Slice Fourier transform
June 10, 2015, Porto
25 / 30
Mustard convolutions
Writing f (z) = f1 (z0 , n) + ζf2 (z0 , n), f ∈ span{ψj,k }, we get
F
−1
−i
+∞
Z
(F (f )F (g )) (x) =
4πc
−∞
n
Zr
"
f1 (z0 , n)g1 (x0 − z0 , r − n) +
m
X
(−1)
k
2k
m
k=0
0
+∞"
Z
f1 (z0 , n)g1 (x0 − z0 , n − r ) −
+
m
X
(−1)
k
2k
+∞"
Z
f (z0 , n)g1 (x0 − z0 , r + n) −
+
m
X
(−1)
k
2k
"
n Zr
m
X
0
k=0
f2 (z0 , n)g1 (x0 − z0 , r − n) −
k
(−1)
+∞"
Z
f2 (z0 , n)g1 (x0 − z0 , n − r ) +
+
+∞"
Z
−
f2 (z0 , n)g1 (x0 − z0 , r + n) +
0
2k
m
X
k
(−1)
−1
#
k
(−1)
2k
m
dn
#
(k)
(z0 , n)g2 (x0 − z0 , r + n)
f
2
−1
2k
dn
(k)
f
(z0 , n)g2 (x0 − z0 , n − r )
2
−1
−1
dn
o
#
(k)
f
(z0 , n)g2 (x0 − z0 , r − n)
1
dn
#
(k)
f
(z0 , n)g2 (x0 − z0 , n − r )
1
(k)
f
(z0 , n)g2 (x0 − z0 , r + n)
1
m
m
X
k=0
m
k=0
r
−1
m
k=0
0
#
(k)
f
(z0 , n)g2 (x0 − z0 , r − n)
2
m
k=0
r
+ω
−1
dn
#
dn
o
dz0 .
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
26 / 30
Mustard convolutions
Writing
(
f even (x0 , r ) =
(
f
odd
(x0 , r ) =
f (x0 , r )
f (x0 , −r )
f (x0 , r )
−f (x0 , −r )
r >0
r <0
,
r >0
r <0
we finally get
−i
F −1 (F(f )F(g ))(x) =
×
4πc
m
X
2k
odd,(k)
f1even ? g1even (x0 , r ) +
− 1 f2
? g2odd (x0 , r )
(−1)k
m
k=0
m
X
2k
even,(k)
(−1)k
− 1 f1
? g2odd (x0 , r )
+ω f2odd ? g1even (x0 , r ) −
m
k=0
The expression for the right Mustard convolution, defined as
f ?RM g = FS−1 (FS (g )FS (f )), is obtained by interchanging f ↔ g .
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
27 / 30
Generalised translation
In classical Fourier theory, the Fourier transform of a function translated
over t is given by:
+∞
Z
F(τt f )(y ) =
−∞
e −ixy
√
τt f (x)dx = e −ity
2π
+∞
Z
−∞
e −ixy
√
f (x)dx
2π
+∞
Z
√
= 2π K (t, y )
(K (x, y ))∗ f (x)dx
−∞
and therefore
+∞ Z
+∞
Z
√
τt f (x) = 2π
K (y , x) (K (t, y ))∗ (K (u, y ))∗ f (u) dy du.
−∞ −∞
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
28 / 30
Generalised translation
In classical Fourier theory, the Fourier transform of a function translated
over t is given by:
+∞
Z
F(τt f )(y ) =
−∞
e −ixy
√
τt f (x)dx = e −ity
2π
+∞
Z
−∞
e −ixy
√
f (x)dx
2π
+∞
Z
√
= 2π K (t, y )
(K (x, y ))∗ f (x)dx
−∞
and therefore
+∞ Z
+∞
Z
√
τt f (x) = 2π
K (y , x) (K (t, y ))∗ (K (u, y ))∗ f (u) dy du.
−∞ −∞
We therefore define the generalised translation τy for the slice Fourier
transform as
Z
∗
∗
τy f (x) = K (z, x) (K (y, z)) (K (u, z)) f (u) dz du.
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
28 / 30
Generalised translation
After lengthy calculations, the expression for the generalised translation
reads
Ty f (x) =
m
2
8cπ m/2+1
−iΓ
"
×
m
X
#
2k
odd,(k)
− 1 f2
(x0 − y0 , r − g )η
m
k=0
"
#
m
X
2k
odd,(k)
+ f1even (x0 − y0 , r + g ) +
(−1)k
− 1 f2
(x0 − y0 , r + g )η
m
k=0
"
#
m
X
2k
even,(k)
(−1)k
− 1 f1
(x0 − y0 , r − g )η
+ω f2odd (x0 − y0 , r − g ) +
m
k=0
"
#
m
X
2k
even,(k)
odd
k
+ω f2 (x0 − y0 , r + g ) −
(−1)
− 1 f1
(x0 − y0 , r + g )η
m
k=0
f1even (x0 − y0 , r − g ) −
L. Cnudde, H. De Bie
(−1)k
Slice Fourier transform
June 10, 2015, Porto
29 / 30
Generalised translation
After lengthy calculations, the expression for the generalised translation
reads
Ty f (x) =
m
2
8cπ m/2+1
−iΓ
"
×
m
X
#
2k
odd,(k)
− 1 f2
(x0 − y0 , r − g )η
m
k=0
"
#
m
X
2k
odd,(k)
+ f1even (x0 − y0 , r + g ) +
(−1)k
− 1 f2
(x0 − y0 , r + g )η
m
k=0
"
#
m
X
2k
even,(k)
(−1)k
− 1 f1
(x0 − y0 , r − g )η
+ω f2odd (x0 − y0 , r − g ) +
m
k=0
"
#
m
X
2k
even,(k)
odd
k
+ω f2 (x0 − y0 , r + g ) −
(−1)
− 1 f1
(x0 − y0 , r + g )η
m
k=0
f1even (x0 − y0 , r − g ) −
(−1)k
By definition the corresponding convolution equals the expression for the
left Mustard convolution:
Z
Ty f (x)g (y)dy = FS−1 (FS (f )FS (g )) (x)
y
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
29 / 30
Generalised translation
After lengthy calculations, the expression for the generalised translation
reads
Ty f (x) =
m
2
8cπ m/2+1
−iΓ
"
×
m
X
#
2k
odd,(k)
− 1 f2
(x0 − y0 , r − g )η
m
k=0
"
#
m
X
2k
odd,(k)
+ f1even (x0 − y0 , r + g ) +
(−1)k
− 1 f2
(x0 − y0 , r + g )η
m
k=0
"
#
m
X
2k
even,(k)
(−1)k
− 1 f1
(x0 − y0 , r − g )η
+ω f2odd (x0 − y0 , r − g ) +
m
k=0
"
#
m
X
2k
even,(k)
odd
k
+ω f2 (x0 − y0 , r + g ) −
(−1)
− 1 f1
(x0 − y0 , r + g )η
m
k=0
f1even (x0 − y0 , r − g ) −
(−1)k
By definition the corresponding convolution equals the expression for the
left Mustard convolution:
Z
Ty f (x)g (y)dy = FS−1 (FS (f )FS (g )) (x)
y
though still 6= FS−1 (FS (g )FS (f )) (x)
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
29 / 30
Conclusions and further research
Conclusions
Clifford-Hermite functions
Closed form for kernel function
Clifford convolutions
Future research
Cauchy-Kovalevskaya extension
Segal-Bargmann transform
L. Cnudde, H. De Bie
Slice Fourier transform
June 10, 2015, Porto
30 / 30