radon_2 - 2.2 Filtered Back projection Recall where 1 f (x)...

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2.2 Filtered Back projection Recall f ( x )= 1 2 π ° π 0 A ( Rf )( x · ω,ω ) dω, where A ( Rf )( t,ω )= 1 2 π ° −∞ ± Rf ( r, ω ) e irt | r | dr = i H t ( Rf )( t,ω ) . Here H is the Hilbert transform. We note that A ( Rf )( t,ω ) is a one-dimensional shift invari- ant Flter which does not depend on ω . This Flter can be built into the scanner once the data { Rf ( t,ω k ) } are collected for each ω k , giving A ( Rf )( t,ω k ), and then f can be approximated with the formula f ( x n ) θ 2 π M ² k =0 A ( Rf )(( x n · ω k ) k ) , (5) which requires O ( M ) operations for each n =1 , ··· K . In the end the total complexity is O ( MK ). Discretely, suppose A ( Rf )( t j k ) is known, then in computing (5), interpolation is need to compute A ( Rf )(( x n · ω k ) k ). There is one other draw back to computing A ( Rf )( t,ω ): Suppose that there is noise measuring Rf , that is we measure ± Rf v ,where v is additive noise. In computing A ( Rf ), the noise is ampliFed by ˆ v ( r ) | r | . To overcoming this problem, we need to apply a smoothing Flter to A ( Rf ). 3 Ram-Lak Filters and other approaches In the Flter A , we would like to replace | r | with a A ( r ) | r | where A ( r ) 0as | r |→∞ .L e t φ be the function such that ˆ φ ( r )= A ( r ) | r | . DeFne A φ ( Rf )( t,ω )= 1 2 π ° −∞ ± Rf ( r, ω ) ˆ φ ( r ) e irt dr. Then we can rewrite A φ (( Rf )( t,ω )= ° −∞ Rf ( r, ω ) φ ( t r ) dr. Then f is approximated by A φ ( Rf )( t,ω )as f ( x ) f φ ( x )= 1 2 π ° π 0 A φ ( Rf )(( x · ω ) ) dθ. Suppose ω are sampled at M = ³ ω k = ω ( k θ ): k =1 , ··· ,M, θ = π M +1 ´
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and t are sampled at T N = ° t j = j t : j = N, ··· ,N, t = L N ± .
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radon_2 - 2.2 Filtered Back projection Recall where 1 f (x)...

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