  We consider image denoising problems formulated as variational problems. It
is known that Hamilton-Jacobi PDEs govern the solution of such optimization
problems when the noise model is additive. In this work, we address certain
non-additive noise models and show that they are also related to
Hamilton-Jacobi PDEs. These findings allow us to establish new connections
between additive and non-additive noise imaging models. Specifically, we study
how the solutions to these optimization problems depend on the parameters and
the observed images. We show that the optimal values are ruled by some
Hamilton-Jacobi PDEs, while the optimizers are characterized by the spatial
gradient of the solution to the Hamilton-Jacobi PDEs. Moreover, we use these
relations to investigate the asymptotic behavior of the variational model as
the parameter goes to infinity, that is, when the influence of the noise
vanishes. With these connections, some non-convex models for non-additive noise
can be solved by applying convex optimization algorithms to the equivalent
convex models for additive noise. Several numerical results are provided for
denoising problems with Poisson noise or multiplicative noise.
