sparse image representation via combined transforms - Convex ...

B.1. TRANSFORMS FOR 2-D CONTINUOUS FUNCTIONS 151
R + ,θ is fixed}. Thenwehave
ˆf (s)
θ
(ρ) = ˆf(ρ cos θ, ρ sin θ).
Let f (p)
θ
(ρ ′ ) be the projection of f onto the line having angle θ ′ in the original domain:
′
∫
f (p)
θ
(ρ ′ )dρ ′ =
f(x, y)dxdy.
′
x cos θ ′ +y sin θ ′ =ρ ′
(B.2)
Later we see that this is actually the continuous Radon transform. Substituting them into
equation (B.1), we have
ˆf (s)
θ
(ρ) = ˆf(ρ cos θ, ρ sin θ)
∫ ∫
= f(x, y)e −2πi(xρ cos θ+yρ sin θ) dxdy
=
x,y
∫
f(x, y)dxdye
∫ρ ′ x cos θ+y sin θ=ρ
∫
′
= f (p)
θ
(ρ ′ )e −2πiρρ′ dρ ′ .
ρ ′
Note the last term is exactly the 1-D Fourier transform of the projection function f (p)
θ
(ρ ′ ).
This gives the Fourier slice theorem.
B.1.2
Continuous Radon Transform
The Radon transform is essentially a projection. Suppose f is a 2-D continuous function.
Let (x, y) be the Cartesian coordinates, and let (ρ, θ) be the Polar coordinates, where ρ is
the radial parameter and θ is the angular parameter. The Radon transform of function f
is defined as
∫
R(f,ρ,θ) =
f(x, y).
ρ=x cos θ+y sin θ
Recalling function f (p)
θ
(·) in (B.2) is defined as a projection function, we have
R(f,ρ,θ) =f (p)
θ
(ρ).