Research On Image Restoration Models Based On Partial Differential Equation Regularization

Digital image,as an important component of information space,is widely used in media communication,biomedicine,public security and other fields.Image restoration covers hot issues such as image denoising,deblurring,super-resolution reconstruction,etc,aiming at inferring high quality images of real scenes from degraded observations,and providing reliable data sources for high level computer vision tasks.Image restoration is a typical ill-posed problem.Usually,multiple reasonable restoration results can be inferred from a known observed image.Regularization technology is an important means to deal with this kind of ill-posed problem.By using domain knowledge and prior information to establish reasonable regularization terms,the obtained model can have good mathematical properties,and when embedded in the deep network method the problems such as generating phantom details can be weakened.This thesis focuses on establishing more effective adaptive regular variational denoising models and fast numerical solution algorithms,and building a deep super resolution(SR)network fused with anisotropic diffusion regularization.The main contents are as follows:Firstly,a globally convex adaptive total variation(TV)model is proposed for speckle removal of images heavily polluted by multiplicative gamma noise.Since multiplicative noise causes different degradation degrees of image at different gray level,the gray prior of image can be used to describe the contamination level of noise at the corresponding pixel.Here,a gray indicator framework is constructed,which is used as the weight function of the regularization term and enables the high gray level area to be fully denoised while the structure of the low gray level area is protected.Combined with the convex fidelity term obtained from the statistics of the inverse of the noise,the final adaptive regularized multiplicative denoising model is obtained.The existence,uniqueness and comparison principle of the solution of functional minimization problem and the maximum principle of the solution of corresponding evolution equation are proved by the basic principle of variational calculus.In addition,the explicit finite difference method with scaling technique and the primal dual algorithm with adaptive step size are adopted to quickly solve the model.Compared with the classical multiplicative noise removal model,the new model can better preserve the low gray level information and maintain the image contrast.Secondly,to reduce the staircase effect caused by low-order TV regularization,a high-order multiplicative denoising model using adaptive Euler’s elastica as the regularization is proposed.The core is to treat the recovery of a surface as dealing with a series of level set curves simultaneously.Image denoising can be achieved by minimizing the total perimeter and curvature of the level set curves.Moreover,two e cient numerical algorithms are developed.Aiming at the evolution equation obtained by gradient descent method,a semi-implicit and semi-explicit iterative scheme is designed and the additive operator splitting algorithm is used to speed up the calculation.Also,the restricted proximal augmented Lagrangian algorithm is proposed.The auxiliary variables are introduced to transform the unconstrained minimization problem into a constrained minimization problem,then the alternating direction multipliers method is used to update each variable and the Lagrangian multipliers in turn.Notably,proximal operator and truncation technique are introduced in the subproblem to enhance the numerical stability of the algorithm and to ensure the consistency between the algorithm and the model.Numerical experiments show that the new model has significant advantages in mitigating staircase effect and recovering small geometric structures.Finally,a deep network method that incorporates edge guidance and diffusion equation constraint is proposed for single image SR with a large scale factor.Since it has been proved in traditional methods that the more accurate the prior information is,the better the restoration result will be.Therefore,we plan to introduce the manually designed prior into the PSNR-oriented network and propose a branch network to learn and yield more accurate prior information.Specifically,we first select a proper edge detector to obtain the low resolution edge map,second construct an edge SR branch to predict high-quality high resolution edge map,and third embed the edge SR branch into the image SR branch to provide deep edge priors.Finally,we use the traditional diffusion mechanism to construct a loss function,propose the Perona-Malik type SR diffusion equation constraint loss,which is combined with the content loss to optimize the network parameters.In addition,we use ablation studies to verify the effectiveness of edge SR branch,fusion module,and diffusion constraint.Comparative experiments indicate that the proposed method yields high-resolution reconstruction results that are faithful to the structure.