Augmented Lagrangian methods for penalized likelihood reconstruction in emission tomography

In emission tomography, the Poisson statistics of the observations make penalized-likelihood reconstruction with an ℓ₁ penalty more difficult than in the case where the observed data is Gaussian. Previously proposed methods for enforcing sparsity of the reconstructed image with respect to some transform use approximations of the ℓ₁ norm. These approximations facilitate the derivation of monotonie algorithms using optimization transfer methods. Recently, augmented Lagrangian methods have been applied to restoration of images corrupted by Poisson noise without requiring approximations to the ℓ₁ norm. This work extends previously derived augmented Lagrangian-based algorithms to penalized likelihood reconstruction for emission tomography with an exact ℓ₁ penalty. We compare the proposed algorithm to an incremental optimization transfer algorithm that performs penalized-likelihood reconstruction with a hyperbolic approximation to the ℓ₁ penalty. The results show that the proposed algorithm reduces the cost function nearly as quickly as the incremental optimization transfer algorithm. The potential advantage of the proposed method is that it solves the ℓ₁ penalized-likelihood problem exactly.