  This work deals with the reconstruction of dynamic images that incorporate
characteristic dynamics in certain subregions, as arising for the kinetics of
many tracers in emission tomography (SPECT, PET). We make use of a basis
function approach for the unknown tracer concentration by assuming that the
region of interest can be divided into subregions with spatially constant
concentration curves. Applying a regularized variational framework reminiscent
of the Chan-Vese model for image segmentation we simultaneously reconstruct
both the labelling functions of the subregions as well as the subconcentrations
within each region. Our particular focus is on applications in SPECT with
Poisson noise model, resulting in a Kullback-Leibler data fidelity in the
variational approach.
  We present a detailed analysis of the proposed variational model and prove
existence of minimizers as well as error estimates. The latter apply to a more
general class of problems and generalize existing results in literature since
we deal with a nonlinear forward operator and a nonquadratic data fidelity. A
computational algorithm based on alternating minimization and splitting
techniques is developed for the solution of the problem and tested on
appropriately designed synthetic data sets. For those we compare the results to
those of standard EM reconstructions and investigate the effects of Poisson
noise in the data.
