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We consider image denoising problems formulated as variational problems. Itis known that Hamilton-Jacobi PDEs govern the solution of such optimizationproblems when the noise model is additive. In this work, we address certainnon-additive noise models and show that they are also related toHamilton-Jacobi PDEs. These findings allow us to establish new connectionsbetween additive and non-additive noise imaging models. Specifically, we studyhow the solutions to these optimization problems depend on the parameters andthe observed images. We show that the optimal values are ruled by someHamilton-Jacobi PDEs, while the optimizers are characterized by the spatialgradient of the solution to the Hamilton-Jacobi PDEs. Moreover, we use theserelations to investigate the asymptotic behavior of the variational model asthe parameter goes to infinity, that is, when the influence of the noisevanishes. With these connections, some non-convex models for non-additive noisecan be solved by applying convex optimization algorithms to the equivalentconvex models for additive noise. Several numerical results are provided fordenoising problems with Poisson noise or multiplicative noise.

On Hamilton-Jacobi PDEs and image denoising models with certain non-additive noise