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Ignoring the inverse square law and attenuation, an approximate reconstruction for SPECT imaging is given by

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where c˜(Best Essay Writing Services | EssayBureau.com) = Best Essay Writing Services | EssayBureau.com {|Best Essay Writing Services | EssayBureau.com|W(Best Essay Writing Services | EssayBureau.com)} and W(Best Essay Writing Services | EssayBureau.com) is a rectangular windowing filter that cuts off atBest Essay Writing Services | EssayBureau.com = Best Essay Writing Services | EssayBureau.com0. Suppose we use M projections (θ1, θ2, …, θM) uniformly spaced over the range [0, π), and N + 1 (odd) ray paths per projection. Assume the spacing between detectors is T, and gij = gθj (iT). A discrete approximation to the reconstruction of ˆ f(x, y) can be written as

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(a) The observation gij is proportional to the number of photons hitting the detector i at angle θj, Nij, that is, gij = kNij. Assume Nij is a Poisson random variable with mean Nij and is independent for different i and j. Give the mean and the variance of the reconstructed image, mean[ˆ f(x, y)] and var[ˆ f(x, y)].

(b) Show that

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can be approximated as Best Essay Writing Services | EssayBureau.com.

Now, like in CT, we assume that Nij ≈ N.

(c) Find var[ˆ f(x, y)] using the result in (b).

(d) Define SNR = Best Essay Writing Services | EssayBureau.com . Assume that we double the photon counts, that is, N = 2N. Before doubling, the SNR is SNR1; and after doubling, the SNR is SNR2. What is the ratio Best Essay Writing Services | EssayBureau.com ?

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