Reconstruction of Short Time PET Scans Using Bregman Iterations Jahn M¨ uller, Christoph Brune, Alex Sawatzky, Thomas K¨ osters, Klaus P Sch¨ afers, Martin Burger Abstract—We propose a method for reconstructing data from short time positron emission tomography (PET) scans, i.e data acquired over a short time period. In this case standard recon- struction methods deliver only unsatisfactory and noisy results. We incorporate a priori information directly in the reconstruction process via nonlinear variational methods. A promising approach was the so-called EMTV algorithm, where the negative log- likelihood functional, which is minimized in the expectation maximization (ML-EM) algorithm, was modified by adding a total variation (TV) term. To improve the results and to overcome the issue of the loss of contrast we extend the algorithm by an inverse scale space method using Bregman distances, to which we refer as BREGMAN EMTV algorithm. The methods are tested on short time (5 and 30 seconds) FDG measurements of the thorax. We can show that the EMTV approach can effectively reduce the noise, but still introduces an oversmoothing, which is eliminated by the BREGMAN EMTV method, obtaining a reconstruction of comparable quality to the corresponding long time (20 and 7 minutes) scan. This correction for the loss of contrast is necessary to obtain quantitative PET images. I. I NTRODUCTION I N many applications of positron emission tomography one has to deal with reconstructions from few data. In particular, this is the case when one only a small amount of coincidence events is available, for instance due to tracers with a short radioactive half-life (e.g. radioactive water H 2 15 O), a low tracer dose or a short scan time. The latter one is of particular interest, since for instance in the task of motion correction, the whole scan time is divided into short time frames (so-called gates) , which have to be reconstructed separately (e.g. [1]). Also in dymanic PET imaging one has to deal with short time frames. In the cases above standard reconstruction methods like filtered backprojection or the expectation maximization (ML- EM) algorithm deliver unsatisfactory and noisy results. To improve the reconstruction one would like to make an efficient use of a priori information. Therefore nonlinear variational methods are incorporated into the reconstruction process. A promising approach was the so-called EMTV algorithm [2], [3], [4], where the negative log-likelihood functional, which is This work was partly funded by the Deutsche Forschungsgemeinschaft, SFB 656 MoBil (project B2). J. M¨ uller, A. Sawatzky and M.Burger are with the Institute for Computa- tional and Applied Mathematics, University of M¨ unster, Germany. T. K¨ osters and K. P. Sch¨ afers are with the European Institute of Molecular Imaging, University of M¨ unster, Germany. C. Brune is with the Department of Mathematics, University of California, Los Angeles, USA. Corresponding author: jahn.mueller@uni-muenster.de. minimized in the ML-EM algorithm, was modified by adding a total variation (TV) term. TV regularization, first introduced as the so-called ROF model in [5] for image denoising, is already extensively studied in image processing and it is well known that it suppresses noise effectively while preserving sharp edges. The ROF model for denoising an image g is given by arg min f ∈BV (Ω)  1 2  Ω (f − g) 2 dx + α|f | TV  , (1) where |f | TV denotes the so called total variation seminorm: |f | TV := sup ϕ∈C ∞ 0 (Ω) d ||ϕ||∞≤1  Ω f ∇· ϕ dx. (2) If f is sufficiently smooth, (2) can be formally rewritten as |f | TV :=  Ω |∇f |dx. (3) Unfortunately, TV suffers from a systematic error, namely a loss of contrast in the images. This effect can also be seen in the EMTV algorithm and hence, we propose to extend the algorithm by an inverse scale space method using Bregman distances (cf. [6]) to overcome this issue. This approach has already been successfully applied in optical nanoscopy, where the raw data are also corrupted by Poisson noise [7]. II. METHODS The basic problem arising in PET reconstruction can be written as Kf = g where g are the measurements outside the body, f is the unknown distribution of the tracer inside the body and K is the PET system matrix. The penalized Maximum Likelihood (ML) estimation then leads to the following variational problem: arg min f ≥0  (Kf − g log Kf )    distance measure +αR(f )    regularization , (4) with the so-called Kullback-Leibler divergence as distance measure, a regularization functional R(f ) and a parameter α controlling the strength of regularization. In the absence of regularization (i.e. α =0), the iterative standard reconstruction scheme for solving (4) is the well known ML-EM algorithm (cf. [8]): f k+1 = f k K * K * 1  g Kf k  .