In this paper, we analyze the minimization of seminorms ∥L · ∥ on R^n under
the constraint of a bounded I-divergence D(b,H·) for rather general linear
operators H and L. The I-divergence is also known as Kullback–Leibler
divergence and appears in many models in imaging science, in particular when
dealing with Poisson data but also in the case of multiplicative Gamma noise.
Often H represents, e.g., a linear blur operator and L is some discrete derivative
or frame analysis operator. A central part of this paper consists in proving
relations between the parameters of I-divergence constrained and penalized
problems. To solve the I-divergence constrained problem, we consider various
first-order primal–dual algorithms which reduce the problem to the solution of
certain proximal minimization problems in each iteration step. One of these
proximation problems is an I-divergence constrained least-squares problem
which can be solved based on Morozov’s discrepancy principle by a Newton
method. We prove that these algorithms produce not only a sequence of
vectors which converges to a minimizer of the constrained problem but also
a sequence of parameters which converges to a regularization parameter so
that the corresponding penalized problem has the same solution. Furthermore,
we derive a rule for automatically setting the constraint parameter for data
corrupted by multiplicative Gamma noise. The performance of the various
algorithms is finally demonstrated for different image restoration tasks both for
images corrupted by Poisson noise and multiplicative Gamma noise.