Study On Image Denoising And Segmentation Algorithm Based On Fractional-Order Variational

The inverse problem is a very hot research topic, which exists widely in the fields of science and engineering. This paper chooses denoising and segmentation inverse problems in image processing as research object, uses variational method, fractional calculus theory, duality theory, and saddle point theory as mathematical tool, carries out intensive study for the modeling methods and the numerical methods in denoising and segmentation inverse problems. The main work can be summarized as follows:1. A primal-dual model for image denoising is proposed based on duality principle. We theoretically analyze its equivalency with the ROF denoising model, and its structural similarity with the saddle-point optimization model. A primal-dual algorithm for solving the saddle-point problem is used for solving the model, and the convergence condition is deduced. In terms of model’s parameter selection, the regularization parameter is updated adaptively based on the Morozov’s discrepancy principle which can guarantee the denoised image in the feasible set, and protect more image feature. The experiment results show that the primal-dual algorithm can convergent rapidly. The proposed regularization parameter selection strategy is effective in improving the denoising effect.2. The denoised image exhibits staircase effect when the integer order variational method is used. To solve this problem, combining fractional calculus theory and duality theory, a fractional-order variational denoising model is proposed, and its saddle point structure is deduced. On this basis, the primal-dual algorithm based on resolvent is used for solving the proposed model. The adaptive variable step size iterative optimization strategy is used, which can improve the optimizing efficiency, and remedy the step size limitation of the traditional numerical algorithms. The convergence condition is deduced. Using the proposed regularization parameter optimization strategy, the edge preservation ability and the fidelity are balanced properly. The experiment results show that the proposed fractional-order variational algorithm is effective in avoiding the staircase effect, preserving texture and detail information, and having faster convergence speed.3. As for multiplicative Gamma noise removal, we analyze the characteristic and correlation of several classical variation models. On this basis, combining the frequency characteristic of fractional differential, the classical I-divergence variation model is generalized, a fractional-order convex variation model is proposed. Based on duality theory and saddle-point theory, a fractional-order primal-dual algorithm for solving the model is proposed. And the convergence of the algorithm is analyzed. To balance the edge preservation ability and the fidelity, an adaptive parameter regularization strategy based on the balancing principle is proposed, which does not require priori knowledge of the noise. On frequency-domain aspects, the experiments analyze and verify that the proposed fractional-order variation model is effective in relaxing the staircase effect and preserving some medium frequency texture and high frequency edges information compared with the classical first-order variation model. And the proposed fractional-order primal-dual algorithm can convergent effectively, and has faster convergence speed.4. In the traditional edge detection differential operators, the first-order derivative masks are easy to loss image details information, and the second-order derivative masks are more sensitive to noise. As for these problems, combining the frequency characteristic and the memorability of fractional differential, the classical first-order Sobel operator and the second-order Laplacian operator are generalized to fractional-order mode, the fractional-order differential masks are constructed for extracting the edge feature of medical images. The experiment results show that compared with the integer order differential, the fractional-order differential can detect more edge details feature of the medical images, and is more robust to noise.5. In order to further depict the edges and "texture of the important visual geometric structure in the image, combining the modeling idea of fractional-order variation, a fractional-order level set image segmentation model is proposed, which improves the edge extraction ability of the depressions area in the target object. The corresponding fractional Euler-Lagrange equation is given, and the gradient descent method is used to achieve the solution of the model. The experiment results show that the proposed fractional-order level set model can extract more depressions area contour, and has good performance for segmenting image details.