Image Reconstruction from Projections • Computed Tomography (CT) is an important medical imaging technique – Puts to use two important image transforms (Radon and Fourier transforms) that were mathematically formulated many years before its inventionWhat do X-rays measure? • Detector strip measures absorption profile. – Energy of any given beam is absorbed depending on what it passes on its way from the source to the detector. We will see that this can be represented as an integral. Signal absorption Position ?What do X-rays measure? • Detector strip measures absorption profile. – Energy of any given beam is absorbed depending on what it passes on its way from the source to the detector. We will see that this can be represented as an integral. Signal absorption PositionBackprojection • Single projection does not carry enough information to reconstruct an image, but it is our starting point • Project the 1D signal back along the direction from which the beam came. This is called backprojection • What can we do next? © 1992–2008 R. C. Gonzalez & R. E. WoodsMultiple projections • Sum the backprojections from multiple directions – This starts to give us an idea about the image we are reconstructing © 1992–2008 R. C. Gonzalez & R. E. Woods© 1992–2008 R. C. Gonzalez & R. E. Woods© 1992–2008 R. C. Gonzalez & R. E. WoodsPrinciples of Computed Tomography (CT) • G1: Pencil beam & single detector • G2: Fan beam & multiple detectors – Fewer translations required • G3: Wide beam & detector bank – No translation necessary • G4: Circular beam of detectors – Only the source rotates. Fast but costly. © 1992–2008 R. C. Gonzalez & R. E. WoodsRaysum along a line gρj,θk( )= f (x , y)δx cosθk+ ysinθk−ρj( )−∞∞∫−∞∞∫dxdy© 1992–2008 R. C. Gonzalez & R. E. Woods x cosθ+ ysinθ=ρRadon transform • Arbitrary line – Any θ, any ρ#• Notice Radon transform is a function of two variables; hence, it can be displayed as an image. This is called a sinogram. • Discrete version: gρ,θ( )= f (x , y)δx cosθ+ ysinθ−ρ( )−∞∞∫−∞∞∫dxdygρ,θ( )= f (x , y)δx cosθ+ ysinθ−ρ( )y=0N −1∑x = 0M −1∑Sinograms © 1992–2008 R. C. Gonzalez & R. E. WoodsReconstruction by backprojection • Backproject each projection – How can we back project a single point g(ρj,θk) ? – Copy value of g(ρj,θk) on to the line x cos θk + y sin θk = ρj – Repeat and add for all values of ρj for fixed θk fθkx, y( )= g(ρ,θk)= g x cosθk+ ysinθk,θk( )Reconstruction by backprojection • 0o and 180o are mirror images so the summation is carried out in the range [0,180] • Reconstructed image is called a laminogram – It is only an approximation to the true image – Blurring problem f (x, y) = fθx, y( )dθ0π∫f (x, y) = fθk(x, y)k∑© 1992–2008 R. C. Gonzalez & R. E. WoodsFourier-Slice Theorem • 1D Fourier Transform of g(ρ,θ) with respect to ρ is a slice through the 2D Fourier Transform of f(x,y) G(ω,θ)=F(ω cos θ , ω sinθ ) © 1992–2008 R. C. Gonzalez & R. E. WoodsFiltered Backprojection f (x, y) =ωGω,θ( )ej 2πρdω−∞∞∫⎡⎣⎢⎤⎦⎥0π∫ρ= x cosθ+ y sinθdθ© 1992–2008 R. C. Gonzalez & R. E. WoodsFiltered backprojection example © 1992–2008 R. C. Gonzalez & R. E. Woods 1. Compute the 1D FT of each projection 2. Multiply by |w| and Hamming window 3. Compute the 1D inverse FT of the filtered transform 4. Integrate over all 1D inverse FTs to get f(x,y)Filtered backprojection example © 1992–2008 R. C. Gonzalez & R. E. WoodsNotes • In practice, the data are discrete. Computations are carried out with the 1D DFT • Sampling rates important – Number of rays per angle – Number of angles • Undersampling results in