Study Of Algorithm In Image Processing Based On Multiscale Geometric Analysis And Energy Functional

In mathematical imaging and visual of computer, multi-scale geometric analysis and minimization energy functional are the most representative two paradigms at current. It has attracted attention of researchers widely. A lot of new productions are presented about the two paradigms. On the other hand, the productions proposed still exist some problems which are worthy of researching further. In this dissertation, based on the two paradigms, the following problems are discussed:1.To represent curve singularity of image and deal with redundancy of Candes's monoscale ridgelets, a new ridgelets frame, monoscale orthonormal ridgelets frame (MORF), is presented, which use the localization principle and the orthonormal ridgelet constructed by Donoho. The MORF not only remains directionality, but also bears orthonormality. We discuss the nonlinear approximation of the MORF for image which is smooth away from discontinuities across curves. The experiments demonstrate that the new frame have preferable application for image compression, image reconstruction and image denoising.2.A new minimization energy functional based on the basis pursuit is presented to improve on Starck's decomposition model. In this new functional, the second generation curvelets and wave atoms are used to represent structure and texture respectively, and the total variational semi-norm is added for restricting structure parts. In addition, the generalized homogeneous Besov norm proposed by Meyer is used to constrain noisy components. Finally, the basis pursuit denoisiing algorithm is used to solve the new model. Experiments show that the new model is very robust for noise, and that can keep edges and textures stably.3. To reduce the"aliasing effects"resulted from using multi-scale geometric analysis for image denoising, a minimization total variational energy functional with constraint condition on transform domain is presented. Firstly, the nonlinear thresholding strategy associated with certain multi-scale geometric analysis is applied to the transform coefficients of noisy image. And then, the feasible domain of the proposed model is determined by the coefficients remained. Finally, the projected gradient algorithm is used to solve the proposed model. Experiments show that the presented model can remove noisy and remain edges, while the"aliasing effects"such as psudo-Gibbs effects,"wrap around"effects and the"curvelet like"aliased curves are suppressed efficiently, when the finite ridgelet transform and the curvelet transform are applied respectively.4.In the classical minimization energy functional, the balance parameter often depends on the priori information of the oscillatory component. To reduce this dependence, Firstly, a more general decomposition model is discussed, while the existence and uniqueness of solution about this model are proved. And then, the balance parameter is taken as scale. Instead of L2 space, we discuss the oscillatory component in distribution space W ?1 ,∞. To this end, a multiscale hierarchical decomposition model is established, which still obtains a multiscale representation of image, at the same time, its convergence is analyzed. Finally, a new algorithm is proposed by applying BV to approximate W 1,1. The numerical experiments show that the multiscale hierarchical decomposition model has well application in many image processing.5.In the two different viewpoints, we investigate staircasing effect and losing of small scale texture which are caused by TV regularization in the classical minimization energy functional, and present two solutions as follows:(1) A new minimization energy functional with adaptive regularization is proposed, which decomposes image into structure component and oscillatory component, where the regularization for the structure component is obtained by interpolation of TV regularization and isotropic smoothing, namely, take adaptive regularization in terms of the local feature of image; the oscillatory component is investigated in div(BMO) space. In addition, we proof the existence and uniqueness of the presented model. Finally, the corresponding Euler-Lagrange equations are derived to numerically implement. Experimental results and comparisons demonstrate that the proposed model has an advantage of improving visual effect of image decomposition, namely, both edge and texture are well remained, while the staircasing effect is avoided efficiently.(2) Using oscillating patterns theory in image processing proposed by Meyer, a total variation image denoising method is presented, which based on smoothing flow field. Firstly, through applying Hilbert-Sobolev norm to measure fidelity term, a total variation filter is used to smooth the normal vectors of the level curves in noise image. And then, a model is constructed to find a surface which fit smoothed normal vectors. Finally, finite difference schemes are used to solve the Euler-Lagrange functions derived from above models. The experiments show that the approach not only can remove noisy efficiently, but also can retain edges and texture and the staircasing effect also is avoided.