Clifford Fourier Transforms in Colour Image Analysis Hardy Spaces and Paley-Wiener Spaces
David Franklin University of Newcastle
September 5, 2015
Introduction to Image Analysis
Introduction to Image Analysis
R : 20 G : 30 B:0
Introduction to Image Analysis
R : 20 G : 30 B:0 Colour images f : R2 → R3 .
The Classical Fourier Transform
The Fourier Transform acts on f : R → C by Z F {f (x)}(y ) = e −ixy f (x)dx.
The Classical Fourier Transform
The Fourier Transform acts on f : R → C by Z F {f (x)}(y ) = e −ixy f (x)dx. On f : Rn → C by Z F {f (x)}(y ) =
e −i f (x)dx.
Properties of the Classical FT For any f , g : Rn → C, λ, µ ∈ R and x0 ∈ Rn Linearity Translation Differentiation Scaling Plancherel Convolution
F {λf + µg } F {f (x − x0 )} F { dxd j f } F {f (λx)} kF {f }k2 F {f }F {g }
=λF {f } + µF {g } =e −ix0 y F {f } =iyj F {f } = |λ|1 n F {f }( λy ) =kf k2 R = F { f (x − y )g (y )dy } = F {f ∗ g }
Problems
x2
x1
Problems
x2
x1
Problems
F F F
x2
x1
Intro to Clifford Algebras
Create new units e1 , ..., en such that ej2 = −1 and ej ek = −ek ej . A number is a = a0 +
n X j=1
aj e j +
X j 0. Further |f (x, t0 )|2 dx ≤ kf k2 .
Classical Paley-Wiener Theorem
x
y
If a function f : Rn → R has a Fourier Transform supported on a interval of length R,
Classical Paley-Wiener Theorem
x
−R
support of Ff (x) y R
If a function f : Rn → R has a Fourier Transform supported on a interval of length R,
Classical Paley-Wiener Theorem
t
x
−R
R
y
If a function f : Rn → R has a Fourier Transform supported on a interval of length R, then it has an analytic extension f (x, t),
Classical Paley-Wiener Theorem
t t0
x
−R
R
y
If a function f : Rn → R has a Fourier Transform supported on a interval R of length R, then it has an analytic extension f (x, t), such that |f (x, t0 )|2 dx ≤ e 2|t|R kf k2 .
Clifford Paley-Wiener Theorem
y1
x1 x2
y2
Theorem (Franklin, 2015) If a function f : Rn → R(n) has a Clifford Fourier Transform supported on the ball of radius R,
Clifford Paley-Wiener Theorem
x1 x2
y2
R
support yof Ff (x) 1 R
Theorem (Franklin, 2015) If a function f : Rn → R(n) has a Clifford Fourier Transform supported on the ball of radius R,
Clifford Paley-Wiener Theorem
t
x1 x2
y2
R
R
y1
Theorem (Franklin, 2015) If a function f : Rn → R(n) has a Clifford Fourier Transform supported on the ball of radius R, then it has a monogenic extension f (x, t),
Clifford Paley-Wiener Theorem
t t0 x1 x2
y2
R
R
y1
Theorem (Franklin, 2015) If a function f : Rn → R(n) has a Clifford Fourier Transform supported on the ball of radius monogenic R R, then 2it has a 2|t|R extension f (x, t), such that |f (x, t0 )| dx ≤ e kf k2 .
Conclusion and Questions
Thanks to Jeff Hogan and Kieran Larkin for their valuable support and advice. Thanks for listening.