Study On Partial Differential Equations Based Image Denoising And Enhancement

Image processing has been an active research field resulting from the development ofcomputer technology and increasing requirements of multimedia data processing. Sinceimage denoising and enhancement are often applied to improve the image quality and tomake them fulfill the specific application requirements, both of them are importantresearch subjects in image processing. In this paper, more attention is paid to the partialdifferential equation based methods, especially to the theoretical and applicationaldifficulties in these methods. With the topics of image denoising and enhancement, wehave reviewed and studied the following aspects in this thesis: the axiomal PDEs based onscale space theory, variational PDEs and geometrical PDEs based on curve or surfaceevolution. The difficulties in them are analyzed and solutions to them are proposed.Some novel models and algorithms have been proposed as following:(1) Anisotropic diffusion based image denoising couplying adaptive data fidelity termis proposed in this paper. At first, three requirements of image denoising are proposed,based on which the defects of using the fidelity term in classical image denoising modelsare investigated. After that, an adaptive data fidelity term based on local image structuralinformation is brought up to preserve the objects in the images. The new adaptive datafidelity term will make nonlinear PDE based denoising methods capable of preservingsufficiently the geometric structures such as edges and comers besides its effectiveness inimage denoising. To the numerical schemes, a more stable and reliable implementation ofproposed denoising model is introduced.(2) Image denoising with second order non-linear diffusion PDEs often leads to anundesirable staircase effect, namely, the transformation of smooth regions into piecewiseconstant regions. In this paper, the gradient fidelity term is introduced which describles thesimilarity between the restored images and noise ones in gray variation in order to alleviatethe staircase effect. Anisotropic diffusion models are improved by adding the Eulerequation derived from the gradient fidelity term. After coupling the new restriction derivedfrom the gradient fidelity term, the classical second order PDE-based denoising modelswill produce piecewise smooth results, while preserving sharp jump discontinuities inimages. In addition, the gradient fidelity term is integrabel in bounded variation functionspace which makes our model outperform fourth order nonlinear PDE based denoisingmethods suffering from leakage problems and sensitivity to texture components. Experimental results show that our new model alleviates the staircase effect to some extentand preserves the image features, such as textures and edges.(3) The research on the geometrical image modeling has been made and we haveproposed to alleviate staircase effect with the uniform distribution restriction of imagelevel sets. In order to inhibit the staircase effect, a new third order PDE model is derived toimplement the uniform distribution restriction of image level sets. Coupling new modelinto classical PDE-based denoising models will make them produce piecewise smoothresults. Moreover, we prove that the proposed third order PDE model does not change theposition of edges in the noisy images and preserves the image edges well. On the otherhand, the surface evolution equation driven by normal curvature is also proposed in thispaper after having reviewed the existing surface evolution equation based image denoisingmethods. To normal curvature driven diffusion PDE model, we find that it will preservetexture components in images better than other geometrical diffusion model because thenormal curvature along the texture direction is smaller.(4) A pseudo-linear diffusion filter was proposed to enhance fingerprint imagesaccording to the directional infomation of them. In this part, the defects of coherentdiffusion in divergence form are discussed which produce artificial structure and a newnonlinear directional diffusion PDE is proposed in order to improve its inability. The newalgorithm outperforms the coherent diffusion based image enhancement in denoisingability and structure preservation. At last, a pseudo-linear diffusion equation derived fromnew method is proposed in order to decreasing the computational burden. Thepseudo-linear method is more suitable to real-time fingerprint recognition system becauseof its efficiency.(5) The contrast enhancement models to images with uneven illumination distributionare reviewed and an efficient image enhancement method based on the PDE is proposed toadjust the uneven distributed illumination in the image. This model improves the imagedetails in shadows by adjusting the distribution of image gradient firstly and reconstructsthe result image from new gradient field in least square sense finally. With the introductionof Lab color space, the new method is extended to color image applications. The efficientnumerical scheme for Poisson equation is very important in real application and a moreefficient scheme is proposed according to the features of Laplacian operator.