Nonlinear Analysis On Riemannian Manifolds And Its Applications In Digital Image Processing

The thesis consists of three parts:The first part deals with optimal Sobolev inequalities on compact Riemannian manifolds;the second part is concerned with some variational models (energy minimization methods) in image processing,and we propose variational models as well as experimental results for detection,denoising and mosaicking;the last part deals with inverse differential geometry in 3-D non-local image restoration.Partâ… :(Chapter 1)In Chapter 1,we study some optimal Sobolev inequalities(I_p,opt^θ) on compact Riemannian manifolds.We end a program which was started by Druet and Hebey in the last 90's.Surprisingly, we discover that some optimal inequalities are locally valid without being globally valid. This is the first time such a Sobolev inequality is found to behave like this.Usually,the proof of the global validity relies on a local to global argument which fails here.Partâ…¡:(Chapters 2-5)In Chapter 2,we propose a modification of the Chan-Vese model to extract object(s) of interest(OOI) in a specified region.This modification permits to deal with objects only in the specified zone.This is achieved by adding to the Chan-Vese functional a penalization term taking care of the chosen region.And the principal component analysis and interval estimation are used to extract the statistical information of OOI.Existence of a solution is proved through a gradient flow technique.The chapter is completed with examples of the effect of this model on some images.Chapter 3 deals with variational methods for denoising.The first and second sections are devoted to speckle removal of Synthetic Aperture Radar(SAR),respectively single-polarized and multi-polarized.We define an energy functional which consists of a regularization term and two constraints.The regularization term is the integral of the norm of image gradient,taking into account the fact that the noise should have some statistical distribution.The speckle reduction result,which is the minimizer of the functional,is obtained by solving the corresponding Euler-Lagrange equation.To deal with multi-polarization SAR,we present both the results of the channel by channel model and of the coupled model.The latter performs better.Chapter 4 is concerned with registration and seamless mosaicking of cloud-contaminated image.In order to register two images,a natural way is to define an energy functional which is the integral of square of the difference between corresponding points.However,since some points are contaminated by clouds,this functional is no longer small.Thus,we modified it by eliminating cloud points from the functional,using the cloud detection method previously introduced.After registration,two images are mosaicked by using a weighted sum of two corresponding pixels.The weight is determined by solving the Laplace equation with Dirichlet boundary condition.This approach is very simple but outperforms other methods in the literature.In the second section,we propose a forward image mapping method using image inpainting.Forward mapping will cause overlap and gap phenomena.We use anisotropic diffusion to fill the gaps.Image inpainting can be regarded as an alternative of the interpolation method,and its superiority is obvious in the case that interpolation method fails.In Chapter 5,we propose a new variational method for reconstruction in point clouds by modifying the weight in the usual energy functional.Point cloud reconstruction is a challenging work since it needs to retrieve information from unorganized point cloud data.Inspired by tensor voting technique,we use both distance filed and saliency field as weights in our intrinsic model.Partâ…¢:(Chapters 6 and 7).In Chapter 6,we present a new way to denoise and detect crest lines on raw 3D point sets. Our direct method can denoise effectively point clouds and preserve the geometric feature of object by using modified bilateral filter.And,we developed a Crest Lines Flow Algorithm (CLF) by iterating projection operator.No organized structure for this point set is assumed,so the process can work with any scanned shape.In Chapter 7,we deal with C²-curves by a hierarchy of local projection smoothing methods. We present the projection algorithms and explicit estimates of the curvatures of a surface or a curve.This theorem illustrates the denoising of the adapted bilateral algorithm in regular facets. It also explains why the obtained crest lines in the previous chapter undergo a slight curvature correction.