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

Abstract

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. 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.

Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems

“…Let x ∈ R and t > 0. Then, the unique optimal trajectory for the optimal control problem (6) is given by the function [0, t] s → γ(s; x, t) ∈ R defined in (21). Moreover, the optimal value of the optimal control problem (6) equals V (x, t), as defined in (20).…”

“…Due to the curse of dimensionality, these grid-based methods are infeasible for solving high-dimensional problems, e.g., for dimensions greater than five. Several grid-free methods have been proposed to overcome or mitigate the curse of dimensionality, which include, but are not limited to, optimization methods [16,22,19,21,81,59], max-plus methods [1,2,29,34,37,65,64,66,67], tensor decomposition techniques [27,42,80], sparse grids [11,36,53], polynomial approximation [51,52], model order reduction [4,57], dynamic programming and reinforcement learning [3,10,82], and neural networks [5,6,26,46,39,44,45,58,71,75,76,79,62,18,20,17,69,70,49,…”

SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems

“…In literature, there are some algorithms for solving high-dimensional optimal control problems (or the corresponding Hamilton-Jacobi PDEs), which include optimization methods [18,24,21,23,15,14,87,60,55], max-plus methods [1,2,29,35,39,67,66,68,69], tensor decomposition techniques [28,44,86], sparse grids [10,37,53], polynomial approximation [51,52], model order reduction [4,57], optimistic planning [9], dynamic programming and reinforcement learning [13,11,3,8,89], as well as methods based on neural networks [5,6,27,47,42,45,46,58,73,78,81,84,62,20,…”