Some Applications Of Geometric Analysis Methods In Image Processing

The main idea of this paper is to use geometric analysis methods, including variational methods and PDE methods, to study the problems, e.g., image denoising, image decomposition and image segmentation. Firstly, we transform the discussed problem to an energy minimization problem, establish the variational model and discuss the existence and uniqueness of the minimizer; Secondly, the Euler-Lagrange equation of the problem is derived; Thirdly, we use steepest method to derive the heat flow equation and study the existence, uniqueness, stability and approximation behavior of the heat flow; Finally, we get the numerical solution and experimental results.The main research results are as follows:1. The unified study of the six classical variational denoising modelsIn the same framework, we study six variational denoising models which includes additive noise models: ROF model, TV-L¹ model, and multiplicative noise models: RLO model, LCA model, AA model, RLO-L¹ model. In the first, we use regularization method to derive the existence of minimizer of the energy and discuss under what conditions uniqueness holds; In the second, using the theory of subdifferential and BV function, we derive the Euler-Lagrange equation; In the end, we discuss the properties of the heat flow equation derived by steepest method. Here we use two methods. According to the idea of numerical solution, One method is discussing the discreted heat flow first in order to get some estimations, and then using these estimations to prove the solution of the continuous heat flow exists. According to the theory of quasilinear equation, the other method is discussing the heat flow directly.2. Improvements of ROF modelROF model is successful in image denoising but it leads to "staircase effect". In order to overcome this shortage, by analyzing the formulation of this phenomenon, we design three algorithms. First, we introduce variable exponent Sobolev space as the base space; Second, we combine the Gauss diffusion model and ROF model, then we exploit to the fully one's favourable conditions and avoid unfavurable ones; Third, we introduce 4-th order equation and using the smoothness of high order equation to control the "staircase effect". We fully discussed the first improved method in the base space-variable exponent Sobolev space.ROF model can also be seem as a decomposition model. In this view, ROF model decompose an image into two components: cartoon and noise. Considering the diversity and complexity of texture image, we propose a new model to decompose a texture imageinto three components: cartoon, H^-1 type texture and L² type texture. The model can detect different textures in some extent.3. Improvements of active contour modelsBy analyzing the function of each terms in GAC model and CV model, we propose two new improved models. One is improved GAC model, the other is improved active contour model. The improved models are simple in formulation and can segment some images which can neighter be segmented by GAC model nor by CV model. The improved models become standard parabolic equations, and then we get the properties of the solutions by standard arguments. However, it is very difficult to establish the theories for GAC model and CV model which involving the complex viscosity solution theory.We propose a relative simple algorithm for texture image segmentation. According to the characteristic of texture image, we use diffusion tensor to extract the texture features, and then segment the image based on the features. Furthermore, we can construct a texture detection function using these features.4. Denoising and segmentationIn some cases, denoising and segmentation can be achieved in the same time. In other cases, we can view denoising as a pre-processing method before segmentation in order to remove the influence of noise.Based on Ginzburg-Landau equation and weighted ROF model, we propose a new variational model for image denoising and segmentation. Based on diffusion, it has advantages over active contour model in image segmentation. For example, this model is able to detect non-closed curves and quadruple junctions.SAR images are always contaminated by large speckle noise. It is difficult to segment this kind of image directly by active contour models. In this article, we use the multiplicative noise model- RLO-L¹ model to denoise a SAR image first, and then use fast CV model to segment the image. The segmentation algorithm is efficient, correct and stable.