Stable Convergent Learnable Optimization For Low-level Vision

The optimization method is one of the most important strategies to solve the low-level vi-sion.It usually uses the empirical knowledge to establish an optimization objective function,and then adopts a theoretically guaranteed numerical iterative algorithm to solve it.However,it is difficult to solve when the objective function is enough complex.Some works intuitively embed deep network into the optimization model to avoid complex solutions.Unfortunately,the simple and rudely combination results in the stability or convergence of the original opti-mization algorithm will be destroyed.To overcome this issue,we propose a series of learnable optimization with stability or convergence analysis by fine design.Meanwhile,we also explain that deep architectures can learn a better descent direction and verify the effectiveness of the proposed methods in the low-level vision problems.The main contributions are as follows:1.Stable feedback control system for blind image deconvolution.For the unstableness of the image deconvolution algorithms,we propose a control system to simulate the gradient de-scent in image propagation processing based on the observation that the famous gradient descent method can be regarded as a discrete feedback control system.Specifically,we build a learnable controller and feedback to simulate the propagation of the latent image,and provide the sta-bility analysis by adjusting parameters.Combining the sparse priors and data-driven network,we design the adaptive cross-layer guidance,learnable convolutional filter and smooth sparsity measure to improve the restored results.We verify the effectiveness of the proposed feedback control system in non-blind deconvolution.More experiments illustrate that our method achieve predominant deblurring results in various of sceneries.2.A global convergent general framework for model optimization and deep network.Due to these naive combination strategies for the combine of deep network and optimal model,it is still challenging to provide strict convergence analysis on their resulted deep models.Inspired by the inexact proximal gradient methods,a bridging framework for model optimization and deep propagation is proposed by introducing a checking mechanism.We combine the traditional model and network architecture by unrolling the mathematical modeling and embedding the learnable network.To avoid the blank of the theorem,we introduce an optimality error checking condition together with a proximal feedback mechanism,which can judge whether the network has a positive influence for the image propagation in current stage.Moreover,we prove in theory that the propagation sequence generated by this manner is globally convergent to the critical point of the given optimization model.This provides new ideas for the interpretability and convergence analysis of deep networks.The experimental results on super-resolution and deconvolution verify that the proposed method can converge to a critical point of the original energy function and achieve superior effects.3.Deep operator splitting method based on multi-variable nonconvex optimization.In or-der to solve the multi-variable problem in the visual tasks,a learnable multi-variable non-convex model is proposed and the operator splitting method is used to simplify the problem.Choosing the solution strategy of the specific vision task as the launch point,we first introduce the Breg-man distance penalization when solving each sub-problem.Then we split the traditional model and deep network into three operators,including network architecture operator by data-driven,gradient operator by fidelity-driven,and proximal operator by prior-driven.The above special design leads to our method can not only adapt plenty sceneries,but also has theoretical conver-gence guarantee.The experiments on image deconvolution illustrate that the proposed method can converge in a few iterations.The quantitative and qualitative results on image inpainting show the effectiveness of the proposed method.