ECE468_17 - ECE 468: Digital Image Processing Lecture 17...

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Unformatted text preview: ECE 468: Digital Image Processing Lecture 17 Prof. Sinisa Todorovic [email protected] Outline • • • Homework 5 due Fourier-Slice Theorem (Textbook 5.11.4) Homework 6 Radon Transform ￿ ∞ g (ρ, θ) = −∞ ￿ ∞ −∞ f (x, y )δ (x cos θ + y sin θ − ρ)dxdy continuous space coordinates Radon Transform ￿ ∞ g (ρ, θ) = −∞ ￿ ∞ −∞ f (x, y )δ (x cos θ + y sin θ − ρ)dxdy continuous space coordinates g (ρ, θ) = M −1 N −1 ￿￿ x=0 y =0 f (x, y )δ (x cos θ + y sin θ − ρ) discrete space coordinates Properties of the Radon Transform g (ρ, θ + 180◦ ) = = ￿ ∞ ￿ −∞ ∞ ￿ ∞ −∞ ￿ −∞ ∞ f (x, y )δ (x cos(θ + 180◦ ) + y sin(θ + 180◦ ) − ρ) dx dy f (x, y )δ (−x cos θ − y sin θ − ρ) dx dy −∞ = g (−ρ, θ) 1D Fourier Transform of the Projection G(ω , θ) = ￿ ∞ g (ρ, θ)e −j 2πωρ dρ −∞ 1D FT of the Projection -- Properties G(ω , θ + 180 ) =? ◦ 1D FT of the Projection -- Properties G(ω , θ + 180 ) =? ◦ G(ω , θ + 180 ) = ◦ ￿ ￿ ∞ g (ρ, θ + 180 )e ◦ −j 2πωρ dρ −∞ ∞ = −∞ g (−ρ, θ)e −j 2πωρ dρ 1D FT of the Projection -- Properties G(ω , θ + 180 ) =? ◦ G(ω , θ + 180 ) = ◦ ￿ ￿ ∞ g (ρ, θ + 180 )e ◦ −j 2πωρ dρ −∞ ∞ = −∞ g (−ρ, θ)e −j 2πωρ dρ dρ dρ =− = ￿ ￿ −∞ g (ρ, θ)e j 2πωρ ∞ ∞ g (ρ, θ)e −j 2π (−ω )ρ −∞ = G(−ω , θ) Fourier Slice Theorem G(ω , θ) = ￿ ∞ g (ρ, θ)e−j 2πωρ dρ −∞ Fourier Slice Theorem G(ω , θ) = ￿ ∞ g (ρ, θ)e−j 2πωρ dρ −∞ by definition G(ω , θ) = ￿ ∞ −∞ ￿ ∞ −∞ ￿ ∞ −∞ f (x, y )δ (x cos θ + y sin θ − ρ)e−j 2πωρ dx dy dρ Fourier Slice Theorem G(ω , θ) = ￿ ∞ g (ρ, θ)e−j 2πωρ dρ −∞ by definition G(ω , θ) = ￿ ∞ −∞ ￿ ∞ −∞ ￿ ∞ −∞ f (x, y )δ (x cos θ + y sin θ − ρ)e−j 2πωρ dx dy dρ = ￿ ∞ −∞ ￿ ∞ f (x, y )e−j 2πω(x cos θ+y sin θ) dx dy −∞ Fourier Slice Theorem G(ω , θ) = ￿ ∞ g (ρ, θ)e−j 2πωρ dρ −∞ by definition G(ω , θ) = ￿ ∞ −∞ ￿ ∞ −∞ ￿ ∞ −∞ f (x, y )δ (x cos θ + y sin θ − ρ)e−j 2πωρ dx dy dρ = ￿ ∞ −∞ ￿ ∞ f (x, y )e−j 2πω(x cos θ+y sin θ) dx dy −∞ = F (ω cos θ, ω sin θ) Fourier Slice Theorem relates 1D Fourier Transform of the projection with 2D Fourier Transform of the original image Fourier Slice Theorem 1D FT = a slice of 2D FT Reconstruction Using Backprojections Given g (ρ, θ), that is G(ω , θ) find f (x, y ) Example: Backprojecting a 1D signal Backprojection from the Radon Transform • • Given point g (ρ, θ ) Backprojection = Copy the value of g (ρ, θ ) on the entire line ∀ρ, fθ (x, y ) = g (x cos θ + y sin θ, θ) ⇒ f (x, y ) = ￿ π g (x cos θ + y sin θ, θ)dθ 0 Reconstruction Using Backprojections by definition f (x, y ) = ￿ ∞ −∞ ￿ ∞ F (u, v )ej 2π(ux+vy) du dv −∞ Reconstruction Using Backprojections by definition f (x, y ) = ￿ ∞ −∞ ￿ ∞ F (u, v )ej 2π(ux+vy) du dv −∞ polar coordinates in the frequency domain u = ω cos θ, v = ω sin θ, ⇒ dudv = ω dω dθ f (x, y ) = ￿ 2π 0 ￿ ∞ F (ω cos θ, ω sin θ)ej 2πω(x cos θ+y sin θ) ω dω dθ 0 Reconstruction Using Backprojections f (x, y ) = ￿ 2π 0 ￿ ∞ F (ω cos θ, ω sin θ)ej 2πω(x cos θ+y sin θ) ω dω dθ 0 Reconstruction Using Backprojections f (x, y ) = ￿ 2π 0 ￿ ∞ F (ω cos θ, ω sin θ)ej 2πω(x cos θ+y sin θ) ω dω dθ 0 by Fourier Slice Theorem ￿ 2π f (x, y ) = 0 ￿ ∞ G(ω , θ)ej 2πω(x cos θ+y sin θ) ω dω dθ 0 Reconstruction Using Backprojections ￿ ￿ f (x, y ) = 2π ∞ G(ω , θ)e j 2πω (x cos θ +y sin θ ) ω d ω dθ 0 0 Reconstruction Using Backprojections ￿ ￿ f (x, y ) = 2π ∞ G(ω , θ)e j 2πω (x cos θ +y sin θ ) ω d ω dθ 0 0 G(ω , θ + 180 ) = G(−ω , θ) ◦ f (x, y ) = ￿ π 0 ￿ ∞ 0 |ω |G(ω , θ)e j 2πω (x cos θ +y sin θ ) dω dθ Reconstruction Using Backprojections Reconstruction Using Backprojections f (x, y ) = ￿ π 0 ￿￿ ∞ 0 |ω |G(ω , θ)ej 2πωρ dω ￿ dθ ρ=x cos θ +y sin θ Reconstruction Using Backprojections f (x, y ) = ￿ π 0 ￿￿ ∞ 0 |ω |G(ω , θ)ej 2πωρ dω ￿ dθ ρ=x cos θ +y sin θ 1D filtering Box + Ramp Filter Algorithm for Filtered Backprojection 1. Given projections g(ρ,θ) obtained at each fixed angle θ 2. Compute G(ω,θ) = 1D Fourier Transform of each projection g(ρ,θ) 3. Multiply G(ω,θ) by the filter function |ω| modified by Hamming window 4. Compute the inverse of the results from 3. 5. Integrate (sum) over θ all results from 4. Examples naive backprojection zoom ramp filter windowed ramp filter ramp filter windowed ramp filter Next Class • • Multiresolution image processing (Textbook 7.1) Image Pyramids (Textbook 7.1.1) ...
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This note was uploaded on 12/05/2010 for the course ECE 468 taught by Professor Lucchese during the Fall '08 term at Oregon State.

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