Clifford Fourier Transforms in Colour Image Analysis

Clifford Fourier Transforms in Colour Image Analysis Hardy Spaces and Paley-Wiener Spaces David Franklin University of Newcastle September 5, 2015 ...
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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.