Research Of Variational Calculus And Nonconvex Regularization For Image Processing

In recent years, some mathematical tools such as variational calculus (VC) andpartial differential equation (PDE) are widely used in the fields of image processingand computer vision. These tools not only provide very theoretial supports for aproposed model, but also facilitate the discussion on the performance of the model. Inthis thesis, we do some research on some basic but important problems in imageprocessing, including denoising, segmentation, and registration. Some novel models aswell as their corresponding efficient algorithms are proposed. The main work issummarized as follows.1. A new additive noise removal model and a new multiplicative noise removalmodel are proposed. In the additive noise removal model, the nonlocal structure tensorof images is defined by using the nonlocal spatial gradients. The eigenvectors of thenonlocal structure tensor consist in a characteristic space for the image, based on whichthe nonlocal diffusion tensor is constructed. Using the nonlocal diffusion tensor, wepropose the nonlocal anisotropic diffusion model for image denoising. This model isdifferent from the local anisotropic diffusion in that, not only neighboring pixels butalso pixels faraway with similar intensities are concerned in our model. The mainadvantage of taking those pixels faraway but with similar intensities into considerationis that that model protects edges and textures much better than the local model. In themultiplicative noise removal model a nonconvex regularization term is used tomeasure the smoothness of the restored image. Different from the classical totalvariation (TV) regularization term, the proposed nonconvex regularization term has amuch sparser property, and it can preserve the geometrical structure of the image better.In order to obtain a more efficient algorithm to solve the model, we first take use of thesplitting technology to convert the original model into an equivalent one. Then, thesplit Bregman algorithm is used to solve the equivalent model. To overcome thenonconvexity of our model, an iteratively reweighting process is incorporated into thesplit Bregman algorithm. From the well-designed algorithm, we can obtain the restoredimage as well as the edge indicator of the image. Comprehensive experiments areconducted to measure the performance of the proposed denoising methods in terms ofvisual evaluation and a variety of quantitative indices.2. Image segmentation is introduced into the process of color transfer, and a new image-segmentation and color-theme-based color transfer method for textile images isproposed. The method contains three phases. The first phase is to segment the inputtextile image into several regions. As the introduction of a bias field function, thisphase can not only partition images with nonhomogeneous illumination, but alsooutput color means of different regions of the image. The combination of these colormeans is considered as the color theme of the input image. The second phase is toretrieve the relevant color themes from a database of color themes. The third phase isto reconstruct new images with different appearances from the input image by usingthe retrieved color themes. In the three phases mentioned above, the most importantphase is the image segmentation phase which is the main innovation of the proposedcolor transfer method. Numerical results indicate that the proposed color transfermethod can provide a powerful tool for designers to generate textile patterns.3. To improve the segmentation performance of the so called multiphase fuzzyregion competition (MFRC) model, we propose two new image segmentation models,one for gray-scale images, and the other one for color-scale images. In both of the twomodels, we introduce a nonconvex regularization term on the fuzzy membershipfunctions. This regularization term performs better than the usual convex TV used inthe MFRC model in that it can protect edges of the fuzzy membership functions fromover-smoothing. In addition, in the gray-scale image segmentation model, the intensitydistribution of each homogeneous region is not assumed to be the Gaussian distributionas in the MFRC model, but estimated from the kernal estimation method. For thisreason, our model can be used to partition more complicated images. In the color-scaleimage segmentation model, to overcome the difficulty that the classical MFRC modelcannot be simply used to partition natural (color-texture) images, we introduce thePCA descriptors so that the natural image segmentation problem can be discussed inthe framework of region competition. Numerical results demonstrate that the proposedtwo models achieve better segmentation results compared with some other well-knownmodels, such as the level-set (LS) model and the MFRC model, while the algorithm ofour natural image segmentation model is much more efficient than the state-of-the-artgraph cut-based algorithm.4. Two variational calculus-based nongird registration models are proposed, one ofwhich is the one-modality model, and the other is the multi-modality model. In theone-modality model, the weighted L2norm is used as the regularization term, whichbrings out two advantages. Firstly, it avoids the imbalance problem of the convergingspeed in different regions. Secondly, it preserves the important geometric structures of an image while restrains the staircase effect. In the multi-modality model, imagesobtained from different modalities are converted into the one-modality ones. Then themethods which are used to handle the one-modality problems can be used to deal withthe multi-modality problems. By exploiting the techniques of the operator splitting andthe alternative iteration minimization, we solve the models by shrinking and additiveoperator splitting (AOS). Numerical results demonstrate that the proposed registrationmethods perform well for noisy images and images with large deformation.5. To overcome the problem that local deformation are not aligned well by mutualinformation, we propose a new registration model in which local joint entropy is usedto measure the similarity between the moving image and the template image. In themodel, the weighted Horn-type regularizer is used to protect the displacement fieldfrom over-smoothing. Numerical results demonstrate that the proposed model has theadvantage of aligning local edges of the images better than classical models.