Image Processing Based On Variational Problems And Partial Differential Equations

Image processing is an interdisciplinary topic, which is connected to photology, electronics, mathematics, imaging and computer techniques. There are lots of important applications about image processing in many scientific areas and engineering fields. In the current stage, image processing approaches can be divided mainly into three classes, i. e., stochastic modelling, wavelets theory, and partial differential equations (PDEs). This thesis focuses on some key issues of PDEs, in the topics of image enhancement, image fusion, impulsive noise removal and structure-texture decomposition. The main work and innovations are listed as follows:According to the observation that local variation is the important issue in an image, a simple method for image enhancement is designed, which linearly magnifying the gradient; considering the serious noise in infrared images, we design, to the gradient, a new magnification factor, which is adaptive to the gradient itself, such a factor makes the large gradient almost unchanged, which can avoid the halo from large dynamic range, meanwhile, an extra term, TV norm, is involved into the reconstruction to further compress the noise; according to the high demands of brightness preservation of image enhancement in consumer electronics, we interpret the histogram equalization as the maximum entropy, and using the variational perspective, we find the close-form solution for the optimal histogram, and a histogram transform is employed to enhance the image with brightness preservation; furthermore, about the determination of the target histogram, we select a more intuitive way to measure the flatness, and the convex optimization can help us determine such a solution, based on the characteristics of the solution, we design a simplified algorithm for the target histogram, and also an exact histogram specification algorithm is employed to better our enhancement result.We extend the variational image fusion from 2D to 3D, define the contrast of the multi-channel 3D image, and a variational method help us effectively fuse the multichannel 3D medical images; with the concept of Just-Noticeable-Differences in the Weber's law, we introduce the perceptual contrast into the variational PDE image fusion, and derive the perceptual contrast based variational image fusion method; according to the different importance of each channel, we set different weights to each pixel in different channels, and the statistics are extracted to make the result preserve the salience well; since local variation is very important in image fusion, we design a measure to evaluate the quality of the fusion results, which employs the similarity of the gradient amplitude and direction between the source and the result, such a method has some desired property in differentiating the inverse direction.About the two kinds of impulse noise removal, we employ a two-steps approach, i.e., detection and restoration, among which, the restoration is achieved by the edge preserving TV image inpainting technique. As for the salt-and-pepper noise, since its value has some distinct characteristics, an adaptive median filter is employed to identify them, and the result is better than the state-of-the-art methods; for the random valued impulse noise, we design an adaptive neighborhood noise map to identify the corrupted pixels, with a TV inpainting followed, it can overcome some weakness of existing methods and the performance is better.According to the description ability of the existing model for characterizing structure and texture, we combine the Mumford-Shah model in modelling structure and G-space in modelling texture, and advance a structure-texture decomposition method, namely Mumford-Shah-G method, it can make the resultant structure component piece-wise smooth except some simple edges without the staircase effect from the TV-type model, while the texture can be well separated from the source image; when we consider to compress the structure and texture component of an image respectively, we propose an improved method, which adds the error from structure compression to the texture component to reduce the error source, such a scheme has a theoretical boost in performance, comparing to the existing method; for a structure image, edge is important, we can restore the whole image using only edge together with its neighborhood by structure inpainting, upon which we propose a compression method to such edge-like information, it combines edge tracking, running length encoding and vector quantization technique, and effectively compress the structure image, the result has a perceptually good performance.