Resolution and noise properties of MAP reconstruction for fully 3-D PET

We derive approximate analytical expressions for the local impulse response and covariance of images reconstructed from fully three-dimensional (3-D) positron emission tomography (PET) data using maximum a posteriori (MAP) es timation. These expressions explicitly account for the spatially variant de tector response and sensitivity of a 3-D tomograph, The resulting spatially variant impulse response and covariance are computed using 3-D Fourier tra nsforms. A truncated Gaussian distribution is used to account for the effec t on the variance of the nonnegativity constraint used in MAP reconstructio n. Using Monte Carlo simulations and phantom data from the microPET small a nimal scanner, we show that the approximations provide reasonably accurate estimates of contrast recovery and covariance of MAP reconstruction for pri ors with quadratic energy functions. We also describe how these analytical results can be used to achieve near-uniform contrast recovery throughout th e reconstructed volume.