Study On Geometry-driven Variation Calculus And Partial Differential Equations In Image Processing

Image processing is a developing rapidly interdisciplinary field in information science and engineering,which plays a very important role in information society. A main challenge of image processing is to preserve and enhance multi-level image features(such as edges,details and textures),and to avoid false artifacts and over smoothing to images during accomplishing the given tasks effectively.In recent years, image processing methods based on variation calculus and partial differentiation equations infuse new life into this field.In this paper,we study the applications and theories of variation calculus and partial differentiation equations in image processing.As for the applications,we have studied the modeling,model analysis and numerical implementation with high precision of the geometry-driven fractional total variation and bidirectional shock-diffusion equations,and have applied them to image denoising,edge sharpening,image resolution enhancement,image inpainting and image measurement. As for the theories,we have studied the proposed algorithmic models and their computing schemes,intrinsic relations among different image processing methods,such as mathematical morphology,Gibbs random field/ Bayesian statistical inference,fuzzy mathematics,fractional total variation and bidirectional shock-diffusion equation.Finally, after having given a summary of several characteristics and algorithmic mechanisms of image processing,methods based on nonlinear evolving equations,we elaborate its advantage and theoretical principle.In this paper,we have made innovations in the following aspects:(1)Geometry-driven bidirectional shock-diffusion equations.Having researched deeply on nonlinear evolving equations in image processing,we propose several advanced adapting algorithms:a kind of geometry-driven bidirectional shock-diffusion equation adapted to different imaging modes and image characteristics,and unified them into a bidirectional diffusion frame including two kind equations of anisotropic diffusion and shock filter.This frame enhances and sharpens important features of image by reducing edge width of image,and fuses image smoothing and sharpening into a nonlinear evolving equation.Finally,we apply the model to image denoising,edge sharpening,image resolution enhancement,image inpainting and image measurement, and obtain better corresponding image results.(2)Fractional total variation.We generalize the total variation(p=1) into a fractional total variation model(0＜p＜1),which improves the adaptability of total variation model. Choosing adaptively parameters according to image features,we fuse the forward and backward diffusion into a bidirectional diffusion model.Based on image processing strategies and the mathematical morphology,we apply it to image resolution enhancement and image inpainting.Finally,by highly efficient numerical computing we obtain better corresponding image results.(3)Fast and highly efficient numerical schemes with high precision.By studying their behavior,we reveal the essence and characteristic of image enhancement of inverse diffusion equation,shock filter equation,and so on.Fusing ideas in computed fluid dynamics into image processing,we construct fast and highly efficient numerical schemes with high precision by examining different effects between the shock and diffusion terms with feature-adaptive discontinuous coefficients,where the bidirectional diffusion is split into a type of coupling equations to remove the cancelling effect of the forward and backward diffusions.Finally,we analyze systematically the well-posed property(existence,uniqueness and stability) of the proposed finite difference scheme, maximum principle,TVD(Total Variation Diminishing) and the behavior of the solution of the model equation in theory.(4)Intrinsic relations among different image processing methods.Image processing needs increasingly the introduction and promotion of modern mathematics,such as the applied harmonic analysis centered by the wavelet analysis,the variation calculus integrating various geometric regularities,the linear and nonlinear partial differentiation equations,the stochastic modeling and analysis based on the Gibbs/Markov random fields and Bayesian statistical inference,the computational intelligence methods(including fuzzy mathematics,artificial neural networks and genetic algorithm).Although these methods treat image processing problems from a different point of view,they share some common ideas and methods.We reveal intrinsic relations among different image processing methods such as mathematical morphology,Gibbs random field/ Bayesian statistical inference,fuzzy mathematics,fractional total Variation and bidirectional shock-diffusion equation.This helps for revealing algorithmic mechanisms of these methods and developing new image processing methods by borrowing ideas from each other.(5)Characteristics,strategies and algorithmic mechanisms of the method based on partial differentiation equation in image processing.We dissect characteristics of image processing methods based on nonlinear evolving equations,such as"locality", "iteration"and"feature dependence",and we expatiate on its advantages,theoretical principles and basis in image processing.Then,we present three strategies of the method:"To divide into steps"—Image processing is divided into two steps:image feature detection,and different processing according to different features;"To divide into regions"—According to different image features(such as edge,detail,texture and flat region),an image is divided into some regions,where a region adaptive image algorithm is developed.We carry out a feature detection robust against image noise and blur using anisotropic diffusion of the data field of the structure tensor of image,and controlled image enhancement of features with different scales of nonlinear shock-diffusion equation using the generalized fuzzy technique,and designed model parameters using local differential geometries of image(such as the gradient,the curvature and eigenvalues of the structure tensor),to preserve important image features during the evolution of the equation;"To divide into soft and hard parts"—We employ the bidirectional shock-diffusion equation as the"hard"image processing frame;at the same time,we employ other modern image processing tools(such as wavelet analysis,stochastic analysis,fuzzy mathematics,and other computational intelligence methods) to design"soft(adaptive)"model parameters and coefficients of the frame.Above methods and strategies improve the validity and adaptability of the method based on partial differentiation equation in image processing.The key techniques proposed in this paper have wide promising applications in fields such as medical image processing,image measurement,vision surveillance,digital television,long-distance meeting television system and image magnifying software. As an across subject related to information science and mathematics,our research will enrich the application of partial differentiation equation in image processing,and has important theoretic value and wide applied prospect.