Consider a 2-D object consisting of two triangle compartments, as shown in Figure P9.4. Suppose a solution containing a 511 KeV gamma ray emitting radionuclide with concentration f = 0.5….

## Give the mean and the variance of the reconstructed image, mean[ˆ f(x, y)] and var[ˆ f(x, y)].

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

where c˜() = {||W()} and W() is a rectangular windowing filter that cuts off at = _{0}. Suppose we use M projections (θ_{1}, θ_{2}, …, θ_{M}) uniformly spaced over the range [0, π), and N + 1 (odd) ray paths per projection. Assume the spacing between detectors is T, and g_{ij} = g_{θj} (iT). A discrete approximation to the reconstruction of ˆ f(x, y) can be written as

(a) The observation g_{ij} is proportional to the number of photons hitting the detector i at angle θ_{j}, N_{ij}, that is, g_{ij} = kN_{ij}. Assume N_{ij} is a Poisson random variable with mean N_{ij} 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

can be approximated as .

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

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

(d) Define SNR = . Assume that we double the photon counts, that is, N = 2N. Before doubling, the SNR is SNR_{1}; and after doubling, the SNR is SNR_{2}. What is the ratio ?