ECE468_16 - ECE 468: Digital Image Processing Lecture 16...

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Unformatted text preview: ECE 468: Digital Image Processing Lecture 16 Prof. Sinisa Todorovic sinisa@eecs.oregonstate.edu Outline • • Image reconstruction from projections (Textbook 5.11) Radon Transform (Textbook 5.11.3) Computed Tomography Image Projections X-ray computed tomography: X-raying an object from different directions 3D object representation Computed Tomography Two Equivalent Definitions of the Line y = ax + b x cos θ + y sin θ = ρ Radon Transform A point in the projection g (ρj , θk ) is the ray-sum along x cos θk + y sin θk = ρj 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 Example: Radon Transform f (x, y ) = ￿ A 0 , x +y ≤r , o.w 2 2 2 Sinogram = Image of Radon Transform Properties of Objects from Sinogram • • Sinogram symmetric = Object symmetric Sinogram symmetric about image center = Object symmetric and parallel to x and y axes • Sinogram smooth = Object has uniform intensity Computed Tomography (CT) • Key objective: Reconstruct f (x, y ) from its projections g (ρ, θ ) • How: Backproject all projections and sum them all in one image Example: Backprojecting a 1D signal ...As We Increase the Number of Backprojections halo effect Example: Backprojecting a 1D signal Backprojection from the Radon Transform • • Given point g (ρj , θk ) Backprojection = copy the value of g (ρj , θk ) on the entire line ∀ρ ⇒ fθk (x, y ) = g (x cos θk + y sin θk , θk ) ⇒ f (x, y ) = ￿ π fθ (x, y )dθ 0 Backprojection from the Radon Transform • • Given point g (ρj , θk ) Backprojection = copy the value of g (ρj , θk ) on the entire line ∀ρ ⇒ fθk (x, y ) = g (x cos θk + y sin θk , θk ) ⇒ f (x, y ) = ￿ π fθ (x, y )dθ 0 Laminogram Obtained from Sinogram • Backprojection for a specific angle fθk (x, y ) = g (x cos θk + y sin θk , θk ) • Summation over all theta f (x, y ) = π ￿ θ =0 fθ (x, y ) Laminogram Obtained from Sinogram • Backprojection for a specific angle fθk (x, y ) = g (x cos θk + y sin θk , θk ) • Summation over all theta f (x, y ) = π ￿ θ =0 fθ (x, y ) Example Laminograms Significant improvements can be obtained by reformulating backprojections! Next Class • • • Homework 5 due Fourier-Slice Theorem (Textbook 5.11.4) Homework 6 ...
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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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