Variational Models For Image Enhancement And Numerical Simulations

Digital images are usually degraded(such as low contrast,low resolution,contaminated by noise and haze,blur and so on)by the improper operations in the processes of acquirement and transportation.Therefore,image enhancement is an important and fundamental task in the digital image processing field.In general,the methods,which meets the specific needs to make images clearer and visually better perceived,can all be regarded as the image enhancement methods.In recent,image enhancements in the variational framework have been widely used and researched.Their basic idea is to propose diverse energy functionals according to the different reasons that makes the images degrade,and then solve the minimization problems of the energy functionals by optimization methods or the calculations of variation.This thesis focuses on the variational models for image enhancement and their corresponding numerical algorithms.The main work involves the following four parts:1.The uneven illumination correction of imageDue to the uneven illumination,some areas of the captured image are often too dark or too bright and so difficult to observe.Thus,a lot of methods for correcting the uneven illumination and enhancing the images have been proposed in the literature.However,these methods usually couldn't control the degree of enhancement and also may enhance the noises in images.We propose a variational image enhancement model for the correction of uneven illumination and develop an algorithm to solve it.The proposed method and algorithm can correct the uneven illumination effectively,while suppressing the noises to some extent.2.Image dehazingThrough the formula describing the hazy image,we can know that the scene depth plays a very important role in the dehazing process.Thus,estimating the scene depth is the key step for image dehazing.However,traditional image enhancement methods don't utilize the scene depth information,so they usually have no positive effect on the removal of haze from images.We propose a variational model to estimate the scene depth,by which further dehaze images by combing the image forming theory and the formula describing the hazy image.Experiments show that our method is very effective for removing the haze.3.A fast and effective algorithm for solving a well-known variational Poisson denoising model.This Poisson denoising model was originally solved by gradient descent and explicit difference scheme.However,due to the limitation of the CFL condition,the time step must be chosen small enough to keep the iteration stable,which may result in slow convergence.In this study,we propose a semi-implicit difference scheme to discretize the gradient descent flow equation.Our scheme can ensure that the restored solution is strictly positive in image domain,while it is proved to be stable and convergent under mild conditions.4.Beltrami regularization based model for image denoisingThe Beltrami framework regards a graylevel image as a surface of the high-dimension space,while the Beltrami functional indicates the superficial area of the surface.Minimizing the Beltrami functional is equivalent to minimizing the area of the surface,thus the Beltrami functional can be used as a regularization term for image denoising.In this study,we propose a Beltrami regularization based model for image denoising.We derive the prime-dual formula of the proposed model by using the Legendre-Fenchel transform,which is solved by the alternating method.The proposed model can effectively remove different types of noise,depending upon the various selections of fidelity terms.