Variation Based Image Restoration Models And Density Estimation Method With Regional Constraint

In recent years,some mathematical tools such as variational methods,optimization algorithms and statistical methods have been widely used in the fields of image processing.With the development of the study,the research on the additive denoising is becoming more and more mature,while some challenges still exist in the problems of multiplicative denoising and image deblurring.In this thesis,we do some research on some basic but important problems in image processing and its applications,including multiplicative denoising,image restoration,and crime density estimation with regional restriction.Some new models as well as their corresponding fast optimization algorithms are proposed.The main contributions lie in the following aspects:1.A cartoon-texture decomposition based multiplicative noise removal method is proposed.For the multiplicative noise existing in aerial image,remote sensing image,ultrasonic image or synthetic aperture radar imaging,it is usually modeled as Gamma or Rayleigh distribution,rather than Gaussian distribution.Based on the convex denoising method,we first analyze the multi-scale property of the TV-L1 model,then incorporate the idea of cartoontexture decomposition,we design a new denoising model.The convexity of the new model makes it easy to apply the optimization algorithm to solve the problem,so we introduce operator splitting method to solve the model.Experimental results confirm that the new denosing method can not only effectively remove the noise,but also simultaneously preserve the edge and details.2.Based on the statistical characteristics of the Gamma noise,a root transformation based multiplicative denoising model is proposed.It is known that the most common distribution of the multiplicative noise is the Gamma distribution.Therefore,under the hypothesis of Gamma noise,we implement root transformation on the degradation model,analyze the statistical properties of the mth root,and obtain its asymptotic distribution.Based on the conclusion,we formulate a new denoising model,which can not only well model the statistical property of noise,but also be effectively solved by some optimization algorithm on account of its convexity.Finally,a modified alternating direction multiplier algorithm is introduce to solve the proposed model.The experiments show that the new model performs better than the existing ones.3.Based on the sparsity of gradient,we propose an image restoration model including gradient prior information.One challenge in image restoration is the damage of the edge and the loss of the detail information.Therefore,we first analyze the characteristics of the gradient histogram and the best representation of gradient sparsity.Based on the conclusion,we design a new regularization method to reflect the gradient sparsity,and propose new image restoration model.The new model can not only retain the details of the image,but also achieve a good balance between removing the blur and noise and retaining the edges and details.Moreover,we design a new optimization algorithm to solve the model.Experimental results show that the new algorithm is efficient and convergent,and the new model can effectively remove the blur and noise,and simultaneously preserve the edges and textures.4.We propose a crime density estimation algorithm based on regional constraints.In order to improve the geographical accuracy of the estimation algorithm,the aerial image or remote sensing image of the target area is analyzed and processed to determine the support area of the density.Based on this,a new estimation model is designed to avoid positive density generated in invalid regions,such as lakes,rivers,forests or parks.In the numerical algorithm,we improve the algorithm by introducing plug-and-Play scheme,which makes the new estimation algorithm more applicable and efficient.By means of theoretical analysis and experimental comparison,the new algorithm is confirmed to be more extensive and stable.