Some Applications Of Geometric Calculus Of Variation In Digital Image Processing

The main idea of this paper is to use geometric analysis methods, including calculus of variation and PDE methods, to study the problems, e.g., video/image sequences stabilization and moving objects tracking, image segmentation, chromaticity denoising, colorization and segmentation of color image based on chromaticity-brightness decomposition, spatially varying noise removing. 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; then we use the steepest method to derive the heat flow equation, or use intermediate method to separate the proposed energy minimization problem into some manageable sub-problems, and then solve them by fast algorithm developed recently; Finally, we get the numerical solution and experimental results.The main research results are as follows:1. Shaky video analysis based on calculus of variationGenerally, there exists shake phenomenon in real-world videos. Former methods pre-process the shaky video by video stabilization techniques and then analysis it, such as moving object tracking. We propose a uniform variational framework by modifying the KDA model. We add an affine transform in the energy functional so as to stabilize the shaky video, track the moving objects, and restore the background at the same time. We firstly prove the existence of minimizer of the energy in constraint space, and then show the equivalence between this minimizer and the minimizer in unconstraint space. When solving the minimization problem, we utilize an alternate iterative method by regard the energy minimization problem as three sub-problems, and solve them by the fast dual method of total variation norm. The efficiency of the above-mentioned study of video images is verified by the numerical experiments.2. Improvement of the Chan-Vese segmentation modelThe classical Chan-Vese model cannot provide good results in textured image segmentation. In this paper, we use the KuUback-Leibler divergence in information theory to improve the Chan-Vese model. Firstly, we propose a segmentation model for gray-level image, then we extend it to color image segmentation. Moreover, we prove the existence of minimizer of the proposed energy functional, derive the Euler-Lagrange equations, and solve the minimization problem by fast global minimization method. In experimental part, we compare our method with some state of arts.3. Color image processing using chromaticity-brightness decomposition The color image processing methods based on chromaticity-brightness decomposition model provide more satisfactory results than the RGB model. However, such methods have to deal with the unit sphere constraint of chromaticity information. Based on the study of existed methods, we propose a new method to handle the unit sphere constraint by using an intermediate variable. Then we show the applications in chromaticity denoising and image colorization. In the rest of this part, we propose a novel color image segmentation model by using chromaticity-brightness decomposition, the chromaticity term of the proposed functional follows the data term of the color Chan-Vese model with constraint on unit sphere, and the brightness term is formulated by the Wasserstein distance. Then we prove the existence of minimizer of the proposed energy functional, derive the Euler-Lagrange equations, and solve the minimization problem by fast global minimization method.4. Removal of spatially varying noiseThe image obtained from scatter media such as frog or human body usually contains inhomogeneous noise, so called spatially varying noise. In this paper, we discuss this kind of noise, and propose a variational model to remove spatially varying noise and meanwhile keep the image details (texture). Then we give a proof of the existence of the minimizer of the energy functional, and solve the minimization problem by steepest descent method. The comparisons between our results with ROF model and non-local means are shown in the experimental part.