Research And Application Of Image Decomposition Based On Variational PDE

Stochastic modeling,wavelet analysis and partial differential equation are the main mathematical research methods used in the field of modern image processing. Image decomposition based on variational partial differential equation is proposed in the 90s of the 20th century and it forms at the beginning of this century.Its basic idea is to construct a variational model which is suitable for image decomposition,then gain the corresponding variational partial differential equations according to the variational principle,and finally use the numerical methods to optimize the objective function. This idea based on mathematical methods can describe image decomposition as an mathematical inverse problem:a given image f can be decomposed into two components u which is the main component and v which is the texture and noise component.At present,image decomposition based on partial differential equations is a new field which is important and challenging in image processing technology.This paper focuses on studying three different image decomposition models and their applications. The main work contains:1.Analyse the development of partial differential equations in the image processing and the application in image decomposition,and summary the idea of modeling based on three different theories including multi-scale theory,variational theory, and curves and surfaces theory,then generalize the advantages of image decomposition based on variational partial differential equation.2.Study two main mathematical tools of variational partial differential equation: Variational Methods and Partial Differential Equation.Variational methods is the filed of the functional analysis,and the use of variational methods in the field of functional analysis provide a powerful theoretical support for the mathematical formulation of image decomposition.Moreover,using the framework of variational partial differential equation can show image decomposition problem as an well-posed problem which can ensure the solutions of equations existence, uniqueness and regularity.3.Compare and apply three image decomposition models based on variational partial differential equation:TV-L~2 model,TV-G model and MS-G model.From the view of subjective and objective,compare the difference of these models to select MS-G model applying to the identification.According to the differential characters of the cartoon and texture,choose Hu invariant moment(regional moment) which have the invariance of the scale,translation rotation and fractal dimension which have the invariance of multi-resolution multi-scale variance respectively to describe the cartoon and texture.A new identification of image object based on decomposition was proposed,and it is proved feasible through the experiments.